,
bIMULATION OF A MULTI STATION SYSTEM OF QUEUES
WITH DELAYED EXPOm:NTIAL SERVICE
by
Frank D. Mason
Institute of Statistics
Mimeograph Series No.
•
1961
541
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
• vi
1.
INTRODUCTION
2.
THE GENERAL PROBLEM •
2.1
2.2
2.3
1
The Arrival Distribution •
The Para~le1-Series Network
The Service Time Distribution
3.
THE SPECIFIC PROBLEM
4.
REVIEW OF THE LITERATURE
4.1
4.2
e
t
• •
4
6
6
•
• 10
•
• •
MIMl1 •
MIDll •
M/Dfc
MIMIc.
MIGll •
MIGlc •
M/Er/l
DIMll •
Ef/M/l
G IGll
GIGlc •
..
-;0
4.4
5.
M/M/l
MIMIc
+
+
8
•
•
•
•
·.
•
•
•
•
•
•
4.3 Queues in Series •
4.3.1
4.3.2
4
•
General Queueing Theory
Some Analytic Results
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.2.11
·.
•
. . •• 1013
•
•
•
•
•
•
• •
13
13
14
15
15
16
16
17
• 17
• 18
• 18
• 19
MIl.
Mlc +
... Mlc
• 19
• 20
•
Other Queueing Studies •
• 20
DEVELOPMENT OF EXPERIMENTAL PROCEDURES
• 23
5.1
5.2
Introduction •
Origin of the Example
5.2.1
•
•
• 23
•
Restrictions and Assumptions
5.2.1.1
5.2.1.2
5.2.1.3
5.2.2
• • 23
•
Parameters
• 25
Stationary State •
•
Arrival Time Distribution
Service Time Distribution
•
·.
• 26
• 26
• 27
•
• 29
v
TABLE OF CONTENTS (continued)
Page
5.3 Single Channel Queue • • • • • • • • • • •
• 30
5.4 Parallel Channels • • • • • • • • • • •
• • • 33
5.5 Selection of Regression Analysis Models ••• · • • • • • 35
·· .. .. .. . · .· .. · . ·· .. •• 4136
SIMULATION DESIGN • · . . . . . .
· . · . . · . . • • 43
6.1 Experimental Design Requirements
• · • • 44
• · ·
6.2 Design for the Distribution Variables · ·
• • · · • • · · • • 45
6.3 Design for the Operational Variables
· •• 4746
·
·
·
6.4 Selection of Design Parameter Values ·
· · · · • · • 50
6.5 Selection of Design Parameter Values - k ·
• •
6.6 Selection of Interaction Terms •
· · · · · · · · · · · • .53
EXPERIMENTAL PROCEDURE
• • • · · · • • 55
• · · · · ·
RESULTS . . . . . .
···
· • • · · · • · • · 59
8.1 The First Rank Queue
·• ·• · · ·· ·· ·• ·· ·· ·· • · · · · · 5962
8.2 The Second Rank Queue ·
8.3 Implications
• 66
· ······
·
· · · . Cases
8.4 Analysis of Combined
· · · • · • • · 67
CONCLUSIONS AND RECOMMENDATIONS
· · · • · · · • · · · · 70
·
9.1 Conclusions • • • • • • •
• •
· . · . . • •• 7170
9.2 Recommendations for Further Research • • •
5.5.1
5.5.2
6.
First Rank Model • • •
Second Rank Model • • •
- p
7.
8.
e
•
9.
9.2.1 Variations of the Queueing Systems • • • •
• 72
9.2.2 Exploration of Systems Characteristics • • • • • 73
9.2.3 Extensions of the Method • • • • • • • • • • • • 73
10.
11.
•
· . . · . · . . · . . . · . . • • 81
APPENDICES
· . . . . . . . . . • • 85
·.
11.1 Formulae for Average Queue Lengths for the MIMIc Queueing
System •
· . · · ·· ·· ·· ·· • · · ·· · · ·• · · · · ·• ·· 8587
· ·Designs
11.2 The Simulation
89
11.3 Parametric Values Used
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
11.4 Interaction Analysis
• 97
• ·
·• · · · · · · · · · .105
11.5 The Distribution of Queue Length.
11.6 Simulation Results--Queue Lengths
• .108
·
·
·
·
.113
11. 7 Regression Analysis of First Queue
• · ·
·
·
·
11.8 Regression Analysis of Second Queue
• · · • • · .124
·
·
11.9 Combined Regression Analysis
· · · · · · • · • · · · · .149
LIST OF REFERENCES
vi
LIST OF TABLES
Page
11.1.
Factorial arrangement used for distribution variables • • • • 87
11.2.
Factorial arrangement for case c l ' c2 • 1, 1 with arrangement for operational variables superimposed
•
•
11.3.
11.4.
11.5.
11.6.
11.7.
11.8.
11.9.
11.10.
e
•
11.11.
11.12.
11.13.
11.14.
11.15.
· · · · · · 88
Parameter values for c 1 ' c2 • 1, 1 •
· · • · • • · • • • · • 89
Parameter values for c ' c2 • 1, 3
1
· · · · • · · · · · · 90
Parameter values for c l ' c2 • 3, 1
· · · · · · · · · 91
Parameter values for c ' c • 3, 3 •
l
2
· • · · • • · • • • • • 92
Parameter values for c ' c2 • 3, 5
l
· · · • · · · · · · · · • 93
Parameter values for cl' c 2 • 4, 4 •
· · · • · · · • • · • · 94
Parameter values for c ' c 2 • 5, .3 •
1
· · • · • · • · • • · • 95
Parameter values for c1' c 2 • 5, 5
· · · • · · · 96
List of interaction tables . . . .
• • · · • · · · · 97
Example of observations used in interaction tables •
· • · • 99
Examples of interactions for L (1, 1) • • • • • • • • • • .100
q2
Summary of L
(1, 1) interaction calculations • • • • • • .102
q2
List of simple algebraic forms and associated interaction
tables
• .103
............
·...·...
11.16.
Sampled queue length distribution •
11.17.
11.18.
Selected simulation results for case c l ' c2 • 1, 1 for
first rank queue length, L
•••••••
ql
Average length, first rank queue
11.19.
Average length, second rank queue •
11.20.
First regression analysis of L ; order of distribution
ql
variable entry into solution and F values to remove • • •• 113
11.21.
Second regression analysis of L ; order of operational
ql
variable entry into solution and F values to remove • • • • 115
•
.104
·.
·.
·.
· . .108
.109
• •• 111
vii
LIST OF TABLES (continued)
Page
11.22.
Third regression analysis of L ; order of operational
ql
variable entry into solution and F values to remove • • • • 117
11.23.
Third regression analysis of L • coefficients and their
ql'
standard errors • • • •
• ••• 119
·..
, coefficients
Third regression analysis of L .
· . • · . • .122
ql
Compilation of residuals for L , case 1, 1
·.
· .123
ql
Analysis of variance for factorial arrangement L q for
2
case 1, 1 . . . . . . . . • . . . . . . .
· •
· .124
00
11.24.
11.25.
11.26.
••
00000
••
0
11.27.
Analysis of variance of factorial arrangement L for all
q2
8 cases, list of mean squares
•
•
....·. .
0
11.28.
e
11.29.
.
11.30•
11.31.
11.32.
11.33.
••••••••
.127
q2·
Order of significance of variables from analysis of variance
of factorial arrangement, by mean square • • • • •
• • .129
0
•
•
0
•
•
•
0
0
•
•
0
•
•
•
11. 35.
,
11. 37.
At
•
•
•
.130
Order of variables from first regression for L
.131
q2
Order of operational variables from second regression
analysis of L
• • • • • • • • • • • • • • • • • • • ••• 132
q2
Order of operational variables from third regression
analysis of L
• • • • • • • • • • • • •
o
•
• .133
q2
Step by step summary of fourth regression analysis of L
•• 134
q2
Summary of final coefficients from fourth regression
analyses for L
•••
137
q2
Coefficients from fourth regression analysis for L
arranged by number of service channels • • • • ~2. ~ ••• 138
0
11. 36.
•
Analysis of variance for factorial arrangement, L for
case 1, 1, significance tests • • • • • • • • • q2
0
11.34.
· .125
Order of mean square in factorial analysis of variance for
L
•
0
•••
0
•
0
•
•••••••••••
Increments of R2 from fourth regression ana1ysis for Lq
arranged by number of service channels • • • • • • • 2• • • 139
viii
LIST OF TABLES
Page
11.38.
Step by step summary of fifth set of regression analyses
for Lq • • • • • • • • • •
.... .........
2
11.39.
Comparison of order of significance of terms from
second, third, fourth and fifth sets of regression
analyses for L ,using operational variables • • • •
q2
F values to remove variables, fourth set of regression
analyses for L
q2
F values to remove variables, fifth set of regression
analyses for L
q2
Distribution of residuals of L
and L
from regression
ql
q2 .
.
analysis of all eight cases combined • • • • • •
• •
•
11.40.
•
11.41.
•
•
Cl
....
..
0
•
11.42.
11.43.
11.44.
11.45.
0
•
0
•
0
•
•
0
0
•
...
146
147
148
149
Step by step sUDlllaryof regression analysis of L ,all
ql
eight cases combined • • • • • • • • • • • • • • • •
150
Step by step summary of regression analysis of Lq , all
eight cases combined • • • • • • •
• • • •2 •
151
Comparison of regression coefficients for first and second
rank queues
0
•
•
•
0
140
0
•
0
•
0
0
0
0
0
0
0
0
•
0
0
154
l.
INTRODUCTION
A queue, or waiting line, occurs when a sequence of items or
customers arrive at a server or servers for service of some sort.
The service takes time to accomplish.
Any customers arriving while the
service channel is occupied must wait, and as succeeding customers
arrive a waiting line or queue is formed.
The queue may be of zero
length, it may be of a scheduled or a conceptual rather than a physical
positional character, and so not exist in a geometric sense, but is a
queue nonetheless.
The queueing system includes the customers in the
service channel as well as those waiting in the queue itself.
It is obvious that the manner of the timing with which
~he
custom-
ers arrive will be an important factor determining the nature of the
..
queue.
The arrival,process will ordinarily be described in probability
terms.
The two simplest sorts of arrival schemes are the perfectly regular
rate, where the inter-arrival times are constant, and the Poisson or
so called random or perfectly random arrival rate.
This is taken to
mean that the arrivals occur independently at some average rate 1 and
that the number arriving in any given period of time is distributed as
the Poisson density function.
This can be shown to imply that the inter-
arrival times are distributed as the negative exponential with parameter
1.
See, for example, Feller (l950) •
Except in one or two special cases the Poisson arrivals are easier
,
to handle mathematically than any other arrival rate distribution.
.e
I
Poisson process being at the same time a pure birth process, a
The
2
stationary process, and a Markoff process has many nice and tractable
mathematical properties.
Markoff processes have the property, useful
in queueing theory, that the future state of the system depends only on
the current state, and not on the previous history.
Variations on the arrival schemes which have been investigated are
few.
Among them are batch arrivals, where the customers arrive in
groups, and so-called Erlangian arrivals, where the distribution is a
chi-square, and can be considered to be a filtered Poisson distribution.
The second principal characteristic of queueing systems is the
queueing discipline.
By this is meant the rules followed by the arrivals
in forming the queue.
served discipline.
Most straightforward is the first-come, first-
Others which have been investigated are: priority
systems, where the arrival may have pre-emptive or non-pre-emptive
priority; last-come first-served; random selection from the queue for
service; etc.
The third primary determinant of the nature of the queueing system
is the distribution function of the service times.
Again, the common
distributions are the constant service time and the negative exponential
distribution with parameter 1I, where 1I is the service rate of the service
channel when fully occupied, and is the reciprocal of the mean service
time.
Many other service distributions have been investigated, notably
the Erlangian (or chi-square) which in this case acts in a manner somewhat intermediate between the constant and the negative exponential.
..
Queueing systems may be joined together in "networks" to form a
large system of queues.
.e
Common arrangements will include series or
tandem queues, where the output of one queue is the input to another,
3
parallel queues, where customers may enter any of two or more service
channels, perhaps forming a single line before the parallel set of
servers or perhaps forming a line before each service channel.
Also
possible are closed loops, bridge arrangements, completely closed
systems with a finite number of customers, and other intricate systems.
The series case has been investigated for Poisson arrivals and service,
and some special conditions.
The parallel case has received considera-
bly more attention.
Queueing theory is as yet in its infancy.
In the past decade it
has undergone tremendous growth, but as yet no general theory has
developed capable of handling any really large class of queues •
..
4
2.
THE GENERAL PROBLEM
We have here undertaken to explore some of the characteristics of
a multiple station (parallel and series) system of queues having delayed
exponential service.
2.1
The Arrival Distribution
Customers arrive at the first queue with inter-arrival intervals
having a negative exponential distribution with mean l/A (the arrivals
therefore being Poisson distributed).
The queue discipline is strictly
"first come, first served," and as each customer eventually works his
way to the head of the queue, he is served by the first of the c
l
servers to become unoccupied, or by any of the servers (chosen randomly
.
with equal probabilities) who may be or become unoccupied at that time •
Arrival intervals are almost always assumed distributed as the
negative exponential, rarely constant in applications and in development of queueing theory.
This is variously justified on the grounds
that many (and by implication most) actual queueing situations have indeed
Poi~son
arrival rates, that many others have normal or Gaussian
distributions which can be approximated by a Poisson distribution with
large mean, and that those with obviously non-Poisson rates may be
approximated by the Poisson over short segments of time.
Approximation
methods are available for estimating the effect of variations in the
mean arrival rate, for peak load periods, for example.
.
An important difficulty which may underlie the lack of theoretical
development of various types of arrival rate distributions is the lack
of any obvious simple, believable, and moderately general
~del
for
5
producing such processes.
produce
It is easy to conceive of models which would
regular arrivals - some sort of machine with a constant
periodic output, or scheduled output.
It is possible to consider the
Poisson arrival process as the result of a "perfectly random" mechanism
of some sort.
Though this is a little more vague it at least can be
visualized as a sort of limiting distribution, in that any other distribution may be thought of as "less random."
A clue to another type of
arrival process mechanism lies in the Erlangian distribution.
The time
necessary to observe r arrivals has a density function of the form
1
('t-l)! (rp)
r
y
r-l
e
-rpy
•
Though this was developed for other reasons, it has since been interpreted as representing the output of a filtered Poisson arrival process, where only each r th customer is permitted to arrive.
This leads
us to the possibility of using a queue system as a mechanism to produce
an arrival process.
For this reason we investigate a two-rank series
- parallel system of queues.
The first rank of service channels serves
as a subject of investigation in its own right, and also modifies the
flow of customers which arrive at the second rank of service channels,
changing the arrival distribution from the original Poisson to some
distribution with a generally lesser variance.
It is easily seen that in general, the distribution of a flow of
customers through a service facility will be altered by the service
time distribution of the server.
•
Therefore t in a series system each
server in a series of servers will be faced with an arrival rate having
a different arrival distribution.
6
2.2
The Parallel-Series Network
Practically no work at all has been done on networks of queues
involving both parallel and series elements.
Numerous practical examples exist of this sort of network, of
all shades of complexity and importance.
Illustrations are customers
queueing up to a parallel set of shoe shine boys in series with a
parallel set of barbers; customers queueing before ticket sales windows,
then before ticket takers; aircraft queueing at a holding pattern before
landing, then before the discharge gates before letting off passengers,
customers in a restaurant waiting for seats, then waiting for meals.
These are all 2-level examples, any job shop, any warehouse, and a
great many factories have an intricate multi-level queue system.
De-
livery systems such as the Railway Express and the Post Office have
complex queue systems
2.3
0
The Service Time Distribution
There are large numbers of practical situations, many of great
importance, which have a service time distribution which is in a sense
intermediate between the negative exponential and the constant distributions.
Numerous instances can be cited where service always requires
a minimum finite time plus a variable amount of time which varies in
accordance with some distribution law.
A cashier always requires at least a certain time to ring up a
sale, sometimes more but never less.
An airplane always requires a
minimum time to land, perhaps more but never less.
Machines in a
repair shop may all require some given minimum time to check in, inspect,
7
disassemble, reassemble, inspect and checkout, and in addition a variable time to perform the actual repairso
It is quite possible that this situation is more common than the
usually used negative exponential distributiono
Many published
examples of service time distributions given to demonstrate the validity
of the negative exponential distribution have at least a short finite
delay, or an initial dip in the curve which suggests that a further
refinement in the data would reveal a finite minimum service time
0
The data presented by Erlang himself which he uses to justify the
negative exponential character of telephone call durations in his
famous pioneering papers on queueing theory applied to telephone exchange design is of this sort o (Erlang, 1920).
The rationale for a finite minimum service time is rather well
established.
The distribution of the variable amunt of time in ex-
cess of the minimum as being negative exponential follows from the same
arguments and much of the data used to justify the "pure" negative
exponential.
There exists, then a wide variety of practical cases of queueing
systems where one or more service channels in parallel feed into a
succeeding rank of one or more service channels in parallel, and in
which the service time distributions have a finite delay followed by
a variable delay which may be reasonably taken to be a negative exponential.
No existing theory can handle these situations
0
I,
8
3.
THE SPECIFIC PROBLEM
The specific queueing system investigated has a Poisson distribution of customer arrivals from an infinite source.
The arrivals form
a single queue with the queue discipline "first come, first served,"
before the first rank of service channels.
Each arrival, when it reaches
the head of the queue, enters the first channel to empty, if all are
occupied.
If two or more are simultaneously unoccupied, the arriving
customer enters one of the two or more service channels selected on an
equal probability basis.
The service for all customers has the same
distribution, which consists of a "delayed negative exponential."
That is, there is always a finite constant delay, or service time.
When this constant delay has passed the customer remains in service for
an additional interval of time which is distributed as the negative
exponential.
(Thus, the term "delayed negative exponential. ")
There
may be from one to five service channels in parallel in the first rank.
All customers departing after service in the first rank of
parallel service channels form a single queue in front of the second
rank of parallel service channels.
Again, the queue discipline is
"first come, first served tl and that any customer reaching the head of
the queue will enter with equal probability any service channel if
more than one is available.
empty is entered.
If none are available the first one to
From one to five service channels may be in the sec-
ond rank and the distribution of service times is again a delayed
negative exponential, although the parameter values are not necessarily
the same as in the first rank.
After customers finish service in the
9
second rank, they leave the system.
The cases of interest are those
in the "stationary state, It or statistical equilibrium situation wherein
the queue runs for a very long time at a constant average rate of input
arrivals, thus damping out any transient or short term effects.
Measures of interest are the average queue lengths of each queue
and the average waiting times in each queue.
For a given queue of
these types the aver.age queue length or number waiting in the queue
(not including those in service) is the product of the average waiting
time (not including time spent being served) and the average arrival
rate.
Thus only the average queue length will be considered, as
average waiting time is easily calculated from average queue length.
The objective will be to derive approximate formulae by simulation
methods for the average queue len&tDs, which will be functions of the
first rank service distribution parameters, the arrival rates and
bers of parallel servers for the first queue.
n~
For the second queue
they will be functions of the second rank service distribution parameters, the numbers of parallel servers in the second rank and the
arrival rates.
They will also be functions of the distribution of
arrivals from the first rank of servers, and this distribution is
generally not known.
It is presumed to be a function of the first rank
service time distribution parameter values, the arrival rates and
traffic density (thus also the first queue length) and the number of
parallel servers in the first rank.
Any exact formulation for the average queue lengths under these
conditions would be extremely complex, judging from available queueing
theory results.
We will be satisfied with expressions using power
series approximations to the experimental
simulati~
results.
10
·e
4.
REVIEW OF THE LITERATURE
4.1
General Queueing Theory
Queueing theory as an element of the scientific literature seems
to stem from two papers by Fr. Johannsen (1907, 1908) "Waiting Times
and Number of Calls," and "Busy. II
These were approximations by use
of probability theory to estimates of delays of incoming calls to manual
switchboards and to busy periods of subscribers with one or more telephone lines.
Johannsen, Managing Director of the Copenhagen Telephone
Company, suggested to A. K. Erlang, a mathematician, who had been appointed head of the telephone company's newly formed physico-technical
laboratory for scientific research, that he should take these problems
up for mathematical treatment.
This Erlang did and soon published the
first of a lifelong series of papers (Erlang, 1909) on queueing theory.
This first paper proves that the number of arrivals during a given time
interval, assuming random origination of the arrivals, follows the
Poisson law, then deals with the simplest case of the waiting time for
a single service channel with constant service time.
His most important
publication (Erlang, 1917) handles waiting times for parallel service
channels with constant service time distributions and also with negative
exponential service time distributions.
An interesting account of these
early beginnings is given by Brockmeyer I I aI., 1948.
Development of queueing theory was largely confined to those involved in telephone exchange design until after World War II.
•
Publica-
tion of Feller's book (Feller, 1950) and development of the field of
Operatio~s
Research were important in the rapid expansion of this branch
11
of mathematics following World War II, and its application in widely
diverse areas.
The first published bibliography on the theory of queues (Doig,
1957) contained over 700 references.
Saaty, in his book (Saaty, 1961)
gives 910 references, over half of them less than six years old at the
time.
Kendall, in his review of queueing theory (Kendall, 1964), notes
that according to the "Mathematical Review," papers on queueing theory
were then appearing at a rate of one per week, and he alone cites 63
papers, 31 of them from 1960 or later.
There are now a number of
abstracting and reviewing lists available, and for practical queueing
theory "International Abstracts in Operations Research" is useful.
Volume 6, for 1966, for example, abstracts 56 articles, of worldwide
origin.
"A Comprehensive Bibliography on Operations Research," John
Wi ley and Sons, New York, 1958, gives extensive references on queueing
and classifies them by type.
The literature on queueing theory is
widely scattered and fast growing but is well cataloged.
Kendall (1953) proposed a classification scheme for queueing
systems which has gained general recognition and will be used here.
Three successive symbols separated by two virgules describe a queueing
system.
M
The symbols are:
negative exponential distribution (of inter-arrival or of
service times)
•
D
constant distribution
K
chi square with even degrees of freedom
E
Erlangian distribution with mean independent of r
GI
general independent
n
r
.e
(~.A.,
input intervals may be independently
and identically distributed by a general distripution)
12
G
general, not requiring independence
c
c channe18 •
We shall add
DE
delayed negative exponential.
Thus, MIMic indicates the c-channel case with Poisson input (negative exponential distribution of inter-arrival times) and negative
exponential distribution of service times.
D/M/I indicates input
regular (constant spacing) and negative exponential service with a
single channel.
M/DE/c.
The systems we investigate are, in the first rank,
In the second rank the input process is the output of the
first rank, which we shall indicate as does Kendall (1964) by use of
an arrow.
Thus:
M/DE/c+DE/c.
Other symbols to be used herein are as listed:
Essential Symbols:
a
st'andard deviation; standard deviation of the service time
p
mean service rate for a fully occupied server
A(t)
cumulative inter-arrival interval distribution
a(t)
corresponding density function
B(t)
cumulative service time distribution
b(t)
corresponding density function
A
mean arrival rate
steady state probability of having n units in the system
number of parallel service channels in the i th rank
•
.e
traffic intensity or
u~ilization
factor; the ratio of the
arrival rate per channel to the service rate
13
L
q
W
average waiting time in the queue
L
average number in the queue and in service
W
average waiting time in the queue and in service.
q
."
average number in the queue
4.2
Some Analytic Results
Both the M/Ml1 and the M/D/l systems were first, analyzed by
Er1ang (1917 and 1909).
The expressions for average queue length are
given below.
4.2.1
M/M/l
L
•
e
• ...JL
1-P
(4.1)
2
Lq •
4.2.2
...L-
(4.2)
1-p
M/D/1
L
•
L
q •
(2-p)
2
P
(l-p)
p2
2(l-p)
(4.3)
(4.4)
Er1ang (Er1ang, 1920) gave results for M/D/2 and MlD/3 and illustrates
an awkward method of finding a general solution to M/D/c.
(1932, 1934) and Pol1aczek (1930) improved on his methods •
.e
CroDlll8l1n
14
4.2.3 M/D/c
I»
L - A
tt'
re
i-1
ipc
(ipc)j _ 1 I
(iPc)j] + pc
r
j-ic+1
j!
j!
j-ic
I»
[
(4.5)
P
(ipC)j
j!
I
(iPc)j].
p j-ic+1
j!
_!
(4.6)
Mo1ina (1927) offers an approximation to this rather complicated function:
_
L
p
•
c
(1_p c - 1)
(1-p)
(c+1) - - - - - (P > 0) + pc
(1_pc)
(4.7)
where
(P
>
c(cp)c e- cp I c!
0) _
1 - e-cp
L
q
- --IP~­
c(1-p)
I»
~
L
n-c
c
(c+l)
n
(cp ) - +
nr--
)c -cp /
~..J"c;;"Jp:.:.-~e~---,:.....;;:;c....
!
(l-p c-1.)
-- - c
(4.8)
(l-P)
- (P > 0) •
(4.9)
(l-p )
Both Cromme1in and Edie (1954) find the Molina approximation to give
delays much too high.
For large c, ! ..!.., c > 20, Po11aczek (1930) gives as approximations:
pc e(1-p)c
(P > 0) - ~---­
(4.10)
L __AP__C~e_(1_-_P_)c--,._ [ 1 + 0(1) ] •
c
q
c /2wc (1_p)2
(4.11)
(1-p) ~1rC
and
.e
15
Everett (1953) has developed an iterative computational scheme for this
case, which, however, has an apparent error in it.
•
The complete proof
is not given in his paper and attempts to reach the author for clarification have failed.
Erlang also solved the MIMIc system.
4.2.4
MIMIc
~(,p~c"-)c_+_l
L
c
I
(c-l) I
(pc)
_
n
(4.12)
n-O
p(cp)c P
_ _ _ _ _=-0
L
q
(4.13)
c l (1_p)2
where
p
1
o - -~-------c-l
I (cp) n + (cp) c
n-O
nl
cl (l-p)
(4.14)
The solution to the problem of the single channel queue with Poisson
input and general or arbitrary service time distribution was given by
Pollaczek (1930a) and Khintchine (1932).
Known as the Pollaczek-
Khintchine formula, the expression for the mean waiting time is:
4.2.5
M/G/l
2
L - P+
.e
P
2 (l-p)
[ 1 + (11<1)2 ]
(4.15)
2
2
2
Lq - 2(l-p)
e
[1 + (11<1)2 ] - 2(~-P) [ 1 + 11 var(t) ] , (4.16)
16
where var(t) is the variance of the service tiue and
sq~are
l.?
of the coefficient of variation of service tiue.
var(t) is the
It is readily
seen that for negative exponential service, a - IIp, and for constant
service, a - 0, these convert to the MIMl1 and MIDl1 cases, respectively.
Many others have worked with this system, for example, Kendall (1951).
4.2.6
M/G/c
The case of Poisson input, senera1 service tiue distribution and
multiple channels has been studied by a number of investigators.
The
most successful of these were Kiefer and Wolfowitz (1955) who give a
general expression for the waiting tiue distribution.
This senera1
result is difficult to compute for any but the simplest of single
server cases •
•
These systems describe some of the limiting situations for the
first rank of servers and queues.
The system with Poisson input and
Er1angian service tiue distribution is an interuediate case, the
Erlangian ranging from the constant to the negative exponential as its
parameter r is varied from infinity to one.
4.2.7
MIE /l
r
The Poisson input, Erlangian service tiue single channel queue
system is an intermediate system, with randoDless in service in between
the MIMl1 and M/D/1.
(The Er1ang distribution was originally suggested
by Erlang and is a chi-square with 2 r degrees of freedom.)
The solu-
tions are given in varying degrees of detail by several authors inc1uding MOrse (1951), Saaty (1961) (incorrectly), and others so that priority
is difficult to assign.
17
L • 2 rp - p2(r_l)
2~ (l-p)
(4.17)
2
L
q
.!:tL.-'L..2r
(l-p)·
(4.18)
The second rank of servers and the second queue will be identical
to the first excepting that the input or arrival distribution will no
longer 'be Poisson. but will be determined by whatever transformation is
made by the first-rank servers and the subsequent combination of their
outputs into the single second rank queue.
Since the input distribu-
tion to the second queue is not .!. priori known, again only limiting
case investigations in the literature can be of any help.
4.2.8 D/M/l
Raymond
~.!.i.
(1956), have some results on the regular input,
negative exponential service time, single channel queue, where
the inter-arrival interval, and
p
o
<5
is
is determined from
-l.IP 15
l-p o • e
wq
4.2.9
o
.l-p
o
(4.19)
(4.20)
liP 0
E/M/l
The Erlang arrival distribution with exponential service times,
single channel, has been studied by Morse (1958),
(4.21)
18
(4.22)
Lq •
where u is a root of a secular equation, less than unity.
4.2.10
GI/G/l
Lindley (1953), using integra-differential equations, studied the
general independent input, general service, single channel case.
He
obtains a Wiener-Ropf type integral equation for the distribution of
the difference of input times and
servle~
time's" in terms of the inter-
arrival and service time densities and distributions.
of a simple character, solution is difficult.
Unless these are
Inspection of the general
result is unedifying as to the nature of the queueing process.
(1961, 1962) works with the situation where
p is
Kingman
nearly 1, the "heavy
•
traffic" condition.
He shows that
t variance (1' - ~)
W:
(4.23)
mean of (1 - 1 )
A
II
and that this is an.. exact upper bound.
4.2.11
G/G/c
Kiefer and Wolfowitz (1955)
established stability criteria and
gave a general expression for the waiting time distribution for which
Lindley's result is the special case of a single channel.
Others have contributed also, for example, Kendall (1955) and
Pollaczek.
The difficulty here is that results tend to be very
general, in part at least due to the generality of the input distribu-
.e
tion a,sumed.
Until specific distributions are specified and results
19
computed, little can be said about the effects of specific distributions
on the queueing process.
4.3
Queues in Series
Most of the work which has been done on tandem or series queues
has concerned the Poisson input, negative exponential service cases.
Burke (1956), by showing the independence of time intervals between
departures for the system MIMIc, has proven the output inter-service
completion time distribution to be the same as the input interarrival time distribution.
Reich (1951) has shown that it is not
generally the case that the output distribution will match the input
distribution, by working with normalized chi-squared distributions,
single channel.
Thus, the Poisson and negative exponential distribu-
tions are unique in this respect.
Finch (1959) showed that exponential
service is in fact a necessary and sufficient condition for the independence of the interdeparture intervals, for the Poisson input single
server queue.
Expected numbers and waiting times in the two servers in
series queue system MIMII
+
MIl were obtained by O'Brien (1954).
A
rather complete and methodical analysis has been performed by R. R. P.
Jackson (1954) of two tandem queues with Poisson input and negative
exponential service.
He determines the expected numbers of customers
at each point in the system, and the totals.
4.3.1 MIMII
+
MIl
.
(4.24)
In his 1956 paper, Jackson generalizes to k queues and service ranks
in series, each i th rank with a single queue before r
i
service channels
20
in parallel.
All channels in a rank have identical negative exponential
service, but each rank may differ.
4.3.2
MIMIc
L
i
+
Mlc
+
... Mlc
• ..-l
l-P
(4.25)
i
p2
L
qi
.i
I-Pi
k
L
•
2
I
Pi
(l-P
k
Pi
i·l
Lq •
(4.26)
i
2
I (l-"Pi )
i·l
(4.i27)
)
.
(4.28)
'fI
4.4 Other Queueing Studies
Hunt (1956) has studied the Poisson input, negative exponential
service time series queueing systems, emphasizing the utilization
aspects and exploring restrictions on queue sizes (other than the first
rank queue).
Sasieni
~
al. (1959) illustrate an elementary but interesting
three rank queue and a partial solution by manual Monte Carlo methods.
Cohen (1956) has solved the queueing problem of a tandem or series
system
~ith
any number of ranks, and also of parallel series system
followed by a common final series system, all Poisson inputs and negative
..
..
.e
exponential service.
The problem is also solvable even if at the final
rank the service time does not have a negative exponential distribution •
21
Finch (1959) also developed a formulation for the output distribution of a single channel queue with Poisson input and any service time
distribution.
Chang (1963) has obtained results for the distribution
of the output intervals for the GI/GI/l system.
An
attempt at the
waiting time 'distributions for two single channels in series is made
by Ghosal (1962).
Reich (1963) shows that, for a large number of single
channel queues in series, Poisson input to the leading queue and negative exponential service at each (but with different parameters allowed
at each) the random variables which represent the waiting times at
each queue are mutually independent.
He gives the form of the waiting
time distribution in the k th queue.
,
-
•
Avi-Itzhak (1964) has obtained some results with series queues
with only finite sizes allowed for any but the leading queue.
The
arrivals are arbitrarily distributed, and all service time distributions
are constant.
Parallel servers are allowed, all with the same constant
service time in the same rank.
He shows that total time in the system
for any customer is independent of the order of the servers and of the
sizes of the queue allowed.
He also proves a theorem on the equivalence
of a series of queues to a single parallel queue, so far as waiting
time of any individual customer is concerned.
Various analytic approaches to the problems of systems of queues
have made little headway except for the simplest of cases.
Some at-
tempts have been recently made to treat the system as a whole rather
.
.e
than as a collection of individual queues.
Syski (1960., 1961) proposes
a "generalized congestion process" form of network theory.
Benes (1962,
1963) considers the topological and algebraic character of networks.
22
Though it is doubtful that these approaches will be the much sought
breakthrough in queueing system analysis', they may very well constitute
"
the next significant step or steps.
Transient behavior of queues has been studied by many of the
previously cited authors.
Volberg (1939), Clarke (1956), Cox and Smith
(1961), Cox (1956), and Sonnenschein (1962) give particularly good
accounts.
Monte Carlo methods have been expounded in a voluminous literature
since the advent of the electronic computer, although the method itself
is as old as applied statistics.
here.
Kalin (1954) reviews
Only a few citations will be made
variance reduction schemes.
A recent
book by Schreider (1964) covers the Monte Carlo method well and devotes
an entire chapter to its uses in queueing simulation.
raphy of nearly 300 references.
J
It gives a bibliog-
Recently, special computer languages
have been adapted or formed for the treatment of queueing problems,
Page (1964).
Two recently developed and promising techniques for variance
reduction in simulation of queueing systems are the "Control Variate"
method ,Page, and the "antithetic Variate" method, Page and Hammersley
and Morton (1956).
.e
See also Page (1965) •
23
5.
DEVELOPMENT OF EXPERIMENTAL PROCEDURES
5.1
'!t\'troduction
In this chapter we will consider a specific example of a parallelseries problem which requires a simulation procedure to attempt a
solution.
In the development of this procedure we need to consider
certain restrictions which affect the simulation process and at the same
time restrictions which affect the experimental design to be used in
investigating the behavior of the queueing process under a predetermined pattern of different parametric combinations.
5.2
Origin of the Example
This problem originated in the summer of 1957 when the development
of intercontinental and intermediate range ballistic missiles was well
under way.
.'
The first missiles were about to leave the factories and
were then to be taken to Cape Canaveral.
At Cape Canaveral, the missiles
were to enter one9f a rapk of checkout and assembly hangars where
certain tests and subsequent modifications or repairs were to be made
as necessary-.
Upon leaving the checkout and assembly hangars, the
missiles were to be installed on one of a rank of launchers.
Again time
consuming and detailed tests and modifications as necessary were to be
accomplished, culminating in a missile launch.
The launch itself was
an uncertain process, previous experience with other types of missiles
having shown that the probability of a completed launch on anyone try
was appreciably less than one.
This resulted in a later try, after
corrective action (unless missile destruction occurred) which again, it
.e
.e
24
was anticipated, would have a probability of completion about the same
as before.
,.
These programs had the highest of national priorities, and the
significance of their completion without undue delay had both national
and international implications of the greatest import.
It was crucial
that there be enough checkout hangars and missile launchers available
at Cape Canaveral to handle the flow of developmental missiles without
undue risk of delay.
The other horn of the dilemma was that costs in
these programs were enormous.
A checkout and assembly
hanga~
and all
the complex, handmade, almost one-of-a-kind equipment needed in it was
a very expensive thing, as was a launcher with its associated huge
gantry-cranes, blockhouses, remote control and inspection equipment,
etc.
But even more important was the requirement for highly skilled
people of a sort in very short supply.
It was essential then to
determine just how many checkout and assembly hangars and how many
launchers would be needed to reach a balance between minimizing potential delays and simultaneously minimizing costs in dollars and scarce
skilled manpower.
This problem was recognized as a queueing problem of a previously
unsolved nature.
Immediate numerical results were imperative.
Analytic'
solution of the queueing system was recognized to be extremely difficult,
and in fact to date all attempts in this direction have been to no
avail.
•
Therefore, the problem was attacked by simulation or "Monte
Carlo" methods.
A program was written for the Univac 1101 electronic
computer which simulated a flow of missiles through a rank of checkout
.e
and assembly hangars, then through a rank of launchers.
The Univac
.e
25
computer, though it has been far outclassed by its current successors,
was among the biggest, fastest and most versatile of its day.
"
5.2.1
Restrictions and Assumptions
It was recognized that the general form of the queueing system was
not unique to the field of ballistic
g~ided
missile development, but
had almost innumerable counterparts in all walks of civil as well as
military life.
The specific values anticipated in the missile develop-
ment problem were rather restrittive.
The attempt was therefore made
to so design the experiment and to so program the simulation that after
the specific numeric results designed for the missile program were
obtained, further statistical analysis could be made.
The dual intent
of this further analysis was first to provide approximate formulae for
the calculation of queue lengths over a rather broad range of parametric
values, and second to shed what light could be shed by simulation methods
upon the queueing process, both as an aid to basic understanding of
queues and as an aid to future simulation.
A principal justification for the use of statistical designs is
economy in experimentation.
Great pains are taken to choose the correct
statistical design and analysis, largely to be able to reach a significant result with the least number of experiments, and least cost.
The [experimental design and analytic] method adopted should
be that for which the desired standard of accuracy can be
attained with the smallest expenditure • • • (Cochran and
Cox, 1950, p. 41.)
..
In this simulation, the basic cost limitation was the number of hours
of computer time allocated for performing the simulation.
.e
Thh com-
pelled compromises to be made between the data outputs to be recorded,
·e
26
the type of computations to be made by the computer, and the length of
time the computer was to run with each set of parameters, corresponding
•
to the number of replications to be made.
It also enforced limitations
on the size of the experimental design and compromises on the symmetry
of the designs permitted.
5.2.1.1
Stationary State.
The first decision was to restrict
the investigation to the steady state or stationary state.
The
transient state is far less well understood and much more difficult
analytically, and far fewer results are available from known queueing
systems to act as a guide to the experimental design.
Furthermore,
the inherent variability is far greater for the transient state and a
great deal more computer time, probably by orders of magnitude, would
be needed to achieve comparably accurate results.
5.2.1.2
Arrival Time Distribution.
The arrival time distribution
for the missiles at Cape Canaveral was investigated by means of the
production predictions.
date established.
Every missile contracted for had a delivery
Particularly in the earlier models, these missiles
were handmade, hardly any two were exactly alike, and engineering changes,
experiments, all sorts of interruptions in part deliveries, tests, rework,
etc. made alterations in the predicted delivery dates a daily occurrence.
As a result, all three of the missile programs considered had predicted
arrival times distributed approximately as the Poisson, Chi-square tests
of goodness of fit being significant in all three cases at the 80 percent level.
In one program, an Erlangian distribution with r - 2 gave a
27
somewhat better fit than the Poisson.
It is safe to say that, as all
these arrival schedules had started as almost completely regular distri-
•
butions, by the time delivery actually
occurred, the distributions
would probably be even more closely Poisson in nature than the Chi-square
tests indicated.
This was a convenient result, for most of the queues
which have been investigated analytically had Poisson inputs.
In two programs, the maximum number of checkout hangars or
launchers to be considered was three.
In the other program at least
three and not over five of each were to be considered.
In one program
at that time, four checkout hangars and four launchers were already
planned.
It was found essential to include this case as an actual
simulation point.
All other combinations could be interpolated or
predicted in any way desirable.
from c
l
This led to a first estimate of using
• 1 to 5 servers in the first rank (representing checkout and
assembly hangars), in each case with c
• 1 to 5 servers in the second
2
rank (representing launchers), for a total of 25 cases.
5.2.1.3
Service Time Distribution.
The service time distributions
for the checkout and assembly hangars were based on standard check
lists of work and tests which had to be performed on every missile.
As
the inevitable discrepancies occurred, further time delays resulted,
in a manner leaving considerable variation.
Information available from
the factory on early models of the missiles was helpful in estimating
the times required for the basic check list of tests and for estimating
•
the variation of service time needed in addition to the minimum time.
These estimates turned out to involve a constant minimum time in each
•
28
case (the no-error, perfect check list time), and a variable time after
that which was approximately distributed as a negative exponential.
Similar results were found for the expected distributions of service
times on the launchers, although this was a little less firmly based
on factory experience due to the nature of some of the tests.
This
confirmed previous experience at Cape Canaveral where it was shown by
the author that, with one exception for every large and complex missile
system ever tested there to that date, the time necessary to prep.re
for launch was a finite minimum based on the check list plus a variable
delay which was distributed as the negative exponential.
Some work,
by its nature, could only be performed in the checkout and assembly
hangars, some likewise could only be done on the launchers.
.
e
A good deal
of the work however could be done at either location, whichever might
prove to be desirable.
Since this was a cheaper and much more flexible
method of altering the queueing process than building either checkout
hangars or launchers, it was highly desirable to simulate the queueing
system over a wide range of the service time parameter values.
Whereas the Poisson arrival rate distribution had only a single
parameter, A, the arrival rate, each service time distribution had two
parameters.
These were designated as P and P , the mean service rates
2
l
for servers on the first and second ranks, respectively, and a l and a2 ,
the standard deviations of the service times for servers in the two ranks.
Thus, the mean service times are l/P
l
and l/P2 and the coefficients of
variation of the service time are k l • Pral and k 2 • P202.
Since the
mean and the standard deviation of the negative exponential distribution
(with no delay) are identical, it is easily shown that the delays in
29
the delayed negative exponential distributions are l/Pl-ol and l/p2-o2'
respectively.
5.2.2
Parameters
The queueing system, once c
and c are established, is completely
2
l
described by the five "distribution" parameters, A, P , P2 ' ° ,° ,
l
2
1
Alternatively the system is completely prescribed by the four "operational" parameters, k , k , P and P ' where P _..la.- and p -...!.-.
l
l
2
2
1
Plcl
2
P2c2
The coefficient of variation of the service time distribution is k. In
e
our case, with the delayed negative exponential distribution,
k -
°t
IIp - po.
The traffic intensity (or utilization factor), P,
is the ratio of average arrival to average service rates per service
.
e
channel or altematively the ratio of mean service time to mean time
between arrivals per channel.
The value of P must always be less than
one or the queue will grow indefinitely.
It should be intuitively clear
that when the arrival rate exceeds the service rate, customers cannot
be served as fast as they arrive and will have to wait.
If the average
ratio of arrival rate to service rate for a service channel is less
than unity, then at some time the instantaneous arrival rate must drop
below the average service rate and then customers will be served faster
than they arrive.
Thus any existing queue will decrease in size.
The specification of the system arrival distribution as Poisson
or negative exponential allows the incoming stream of customers to be
completely described by a single parameter, A.
Departure from this
distribution for the service time introduces a requirement for at least
.e
two parameters, P and
0.
Were we interested only in the statistical
30
properties of the system this is as far as it would be necessary to go
excepting possibly for mathematical convenience.
...
From a practical point
of view, interest centers on the operationally descriptive quantities,
traffic density and the coefficient of variation of service time, rather
than the parameters of the distributions of service time and arrival
time.
The most important of these is the traffic density p.
By defini-
tion, this accounts for two of the three statistical distribution
parameters, A and p.
p.
The remaining parameter a could be used along with
However, the relative variation in service time as expressed by the
coefficient of variation is more operationally meaningful than the
standard deviation alone.
We are encouraged in this selection also by
the presence of k in the Pollaczek formula (4.16) where it appears as
.
e
k2 .
It would be possible to relate A and p in other ways and frequently
(p-A) is a term found in queueing theory work, where it expresses the
differential rate in service and arrivals.
5.3
Single Channel Queue
Inspecting the equation for queue length in the MIMll queueing
system
2
L
q
•
...El-p
where
and
c1 • 1
(5.1)
and comparing it to the MIDll system, where
L
.e
q
p2
2(1-p)
• --,,;.--
(5.2)
it becomes obvious that by changing from completely regular to perfectly
31
random (negative exponential) service time distributions, the queue
length increased by a factor of 2.
L
The delayed negative exponential
distribution is an intermediate case with these two as extremes.
is, when
0 •
That
0, the service time distribution is perfectly regular,
with mean l!p.
When there is no delay, l!p -
which is the usual negative exponential.
0 •
0, so that
0 •
l!p,
For the first rank single
service channel case these two relationships establish bounds on the
expected queue lengths.
However, for this case an exact result is avail-
able for M!G!l,
p
[
(
2
L • 2(1-p) 2 - 1 - k )]
2
L.
.
e
q
Q
2(1-p)
(S .3)
(1 + k 2)
(S.4)
which reduces appropriately to the other two expressions.
These three equations for L q are pleasingly simple.
They show
that for the regular and the perfectly random service distributions the
average queue length is a simple function of p, the traffic intensity,
only.
No matter what the service time distribution, the effect on queue
length is a very simple function of the square of the coefficient of
variation and the traffic intensity.
The queue length is extremely sensitive to the traffic intensity
as p nears unity.
let
•
p
If for example p • 0.8 in M!M!l, then Lq • 3.2, but
increase by 10 percent to 0.88 and L increases from 3.2 to over
q
6.4S or more than a 100 percent increase •
dL
--S •
dp
(2-P)2
(l-p)
which. 2, when p • 0 and + • as p + 1. (S.S)
32
The variance of the instantaneous number of customers in the system, N,
is also quite large, as would be expected from this,
Var (N) • L + L2 •
P
(l-p) 2
which • 0 when P • 0
and ....
CD
(5.6)
as p .... 1
being always greater than the mean, L.
The distribution of customers waiting in the queue or being served
in MIMII is expressed as
and
Po • 1 - p
P
n
• pn (1 - p)
(5.7)
where P and P are the probabilities of zero or n (n • 1,2, ••• ) customo
n
ers in the system. This is the geometric distribution. Since the
..
e
probability of no customers waiting or in service is P • 1 - p, the
o
probability of one in service must be p.
This expresses the fraction
of the time the server is occupied and accounts for the alternative name
for p, the "utilization factor."
This willaIso be the expected number
of customers in service, so that
L
• Lq
+
(5.8)
P •
The total customer time expected to be spent in service and in the
queue, W, is the product of the mean number in service and in the queue
and the mean time between arrivals.
W • L
(5.9)
).
..
And the same relation holds for the queue, W , being the mean waiting
q
time in the queue.
33
(5.10)
The equations for L, W, and W , hold for the multiple channel cases
q.
also.
Of course, all customers do not wait in the queue.
lucky enough to arrive while the server is unoccupied.
Some are
The probability
a customer will be delayed, other than for the actual service time, is
p.
For those customers who are delayed, the mean delay will be W':
W
W' • -S-
(5.11)
A
5.4 Parallel Channels
With parallel channels with identical service time distributions,
the arriving stream of customers is split with equal probability into
each channel, thus the traffic intensity in each channel is the original
traffic intensity divided by the number of channels, for first-comefirst-served queue discipline, with all channels fed from a common queue.
Now the equations for queue length become far more complicated and
devoid of any clear intuitively meaningful structure.
Whereas for
M/M/l we had
2
.--'L-
L
1-P
and
L
q
.~
1-p
(5.12)
now for MIMIc we have
p(cp)c P
L
•
q
wQere
.e
0
c! (1_p)2
(5.13)
34
P
.::.1
o
c-l
~
(cp)
n~O
nl
n
+
_
(5.14)
c
(cP)
cI (I-p)
These equations have been recast in a multitude of forms in an attempt
to gain some clue as to parameters which would shed some light on the
queueing process and on likely models to use, with little success.
For particular values of c, the formula for L reduces to a ratio of
q
polynomials.
See Appendix 11.1.
The other bound on the service time distribution, M/D/c, gives,
if anything, even less edifying results.
Pollaczek (1930, 1931) and
Crommelin (1932, 1934).
•
L - A
q
•
r
e- iPc [r
(iPc)
i-l
j-ic
jI
j .
r
j
- 1
~
P j-ic+1
j
].
(5.15)
This is completely different in form from the exponential service case,
apparently involving series of Poisson probabilities.
An
intermediate situation, M/E Ic, has been worked out by Morse
(1958).
Explicit solutions are given only for the case of r - 2 and
c - 2.
r
Whereas in the M/E
L
-
r
11 system
2
p
[2 - (r- 1 ) P ] and L - ...-..:;..-.2(1-p)
r
q
2(1-P)
P
r
+ 1
r
(5.16)
the similar functions for M/E 2 /2 are
P
4(1-P)
L
_
L
- ~p~,,:""
(64 + 50p - l4p2 - 11p3 - 2p4)
(8 + 14p + 6p 2 + p 3)
(5.17)
(2 + SOp + 29p2 + 6p3)
(8 + 14p + 6p 2 + p 3)
(5.18)
•
2
.e
q
4(l-p)
.e
35
There is some similarity here to the expressions for L and L for the
2
q
MIMIc system, in that the initial term of ~ or ~ is multiplied
by a quotient of polynomials.
The polynomials do not however give by
inspection any clear clue as to the nature of the relationship to the
means or variances, for example, of any of the distributions involved.
The result is, in summary, that we know, for the steady state,
first-come-first-served queue discipline, Poisson arrival systems the
following things:
The variance of the queue length is very large, 81-
ways larger than the queue length itself and increases as p approaches
unity; the formula for queue length of the delayed negative exponential
service for parallel channels is not known and is currently unsolvable;
greater variability in service time gives a greater queue length; an
exact form for the single channel system is known; the only c01lllDOn
element among any of the formulae for queue length is
l~P
2
or
-t:P
multiplied by a quotient of polynomials; and all but a few of the very
simplest of systems give solutions which are extremely complicated.
certain general relations are also available for this type of system,
Lq • L - p
L
W • ...9q
A
(5.19)
This is the extent of relevant prior knowledge of the behavior of the
first rank of servers.
5.5
Selection of Regression Analysis MOdels
Once the resultant data from the simulation of the queueing systems
are available, it will be necessary to analyze them by statistical
.e
methods, since the systems are basically stochastic in nature and, as
·e
36
has been indicated, there will be a great deal of variation.
The ob-
vious way to establish the relationships between queue performance and
the various system characteristics is by means of regression analyses.
Careful attention must be paid to the proper choice of the regression
model, based to the maximum extent feasible upon previous knowledge of
queue behavior.
5.5.1
First Rank Model
Selection of a regression model for the first queue requires first
a dependent variable or variables.
In all the known formulations, the
equations for L contain an added term cpo
This may sometimes be re-
structured. so it is not explicitly visible, but is there.
sents a discontinuity in the growth of L.
This repre-
There is no waiting for
service until c servers are occupied, then succeeding customers wait
until one or more servers are free.
of L begins.
q
This point is where the calculation
Thus L measures a single process while L measures two
q
generally different processes.
This is one reason L is selected as
q
the dependent variable to be measured.
it.
1J
So also can W and Wq •
L can readily be obtained from
The known equations for Wq all contain either
or A, as well as p, thus being a mixture of the "distributional" and
"operational" variables.
Whereas any equation containing operational
variables only may be transformed readily to the distributional form,
not all equations formed only of the distributional variables have
equivalent operational forms.
This is the second reason L is chosen
q
•
as the dependent variable, to allow models to be written exclusively
.e
in terms of the operational variables.
They can then be rewritten in
terms of the distributional variables as desired.
.e
37
The most general exactly known model for L is for the M/G/l system.
q
2
p
For M/G/1:
L •
(5.20)
• (l + k 2 )
q
2 (l-P)
For M/E 11:
r
2
1
e
L •
• (l + -)
r
q
2 (l-P)
(5.21)
For M/E2/2:
2
!2 + 50p + 29p 2 + 6p3)
P
Lq •
2(1-P) • 2 (8 + 14p + 6p2 + p3)
(5.22)
For M/M/2:
2
P
Lq • 2(l-p)
(5.23)
and for M/M/3:
e~
L •
q
2 (l-P)
4p
(l + P)
18p 2
(5.24)
(2 + 4p + 3p2)
etc.
It must be admitted the Po11aczek formula (4.6) for the M/D/c
system does not fit well into this grouping.
In fact it does not
appear to be dimensionally consistent, though opportunity to check
this point thoroughly has not arisen.
Now in each of the above cases
2
the L is partitioned into a factor
p
and into a quotient of two
q
2(l-p)
polynomials, in P for the multiple server cases, in k for the single
channel system.
In the M/G/1 case, if the service distribution is
negative exponential, the coefficient of variation of service time
becomes unity, k 2 becomes 1 and the second factor becomes 1+1, a constant, leaving Lq as a function of P only.
For the case M/E 11, the
r
second factor is also a constant (for given r), 1 being the coefficient
r
of variation squared.
For the multiple channel Er1ang service cases,
the second factor combines the effects of having more than oU4 server
with the statistically descriptive variables such as mean,
v~riance,
·e
38
coefficient of variation, which for Erlang service distributions are
l!p, 1!rp 2, l!~ , respectively.
It is not at all certain, and in fact is highly unlikely, that for
other service time distributions the second factor would reduce to
terms in p only or to constants.
Our model then should be of this form:
2
Lq - 2(i-p) • f (p, p2, p3, ... k 2)
(5.25)
where f is some function of powers of p and of k 2 •
The first factor may
be transferred to the left side.
232
- f (P, P , P , ••• k ) •
(5.26)
However, this adds computational work for the eventual user of the
equations.
Since
p2
2
---:__ ....e..2(1-p)
(1
2
+ P + P + p3 +
2
... )
(5.27)
we may leave it on the right side where it will be combined with the
second factor.
The second factor will be represented by the product
of a power series in p and 1+k2 •
Thus
232
L - (Ao + AlPl + A2Pl + A3Pl + ••• )(1 + ~kl)
ql
L - Ao + AlPl +
ql
~pi
+'
A3P~
+ ••• +
Ao~k~
222
+ Al~Plkl + A2~Plkl + •••
.e
(5.28)
or, collecting coefficients of like powers of P and k,
(5.29)
39
(5.30)
where the subscripts on L , p and k indicate that these refer to the
q
first queue and the first rank of serverso
The second rank of servers
is identical in all respects to the first rank, excepting slight changes
in some of the numerical va1ueso
If the input to the second rank were
Poisson, the same model should be used o However, the output of the first
rank of servers, which is the input to the second rank, is not Poisson.
Consider non-Poisson arrivals.
If a queue is
MIMIc
then the output
inter-departure times are distributed negative exponentially also, so
that a next succeeding queue would have a Poisson input.
When the serv-
ing time distribution is not negative exponential, the queue output is
not in general Poisson.
In our case, the serving time in the first rank
is a delayed negative exponential.
This being more regular than the
exponential, the first rank output has an element of regularity greater
than the Poisson.
servers, increases.
This effect is lesser in degree as c, the number of
It may be seen that, for a single channel, for exam-
ple, the output inter-departure intervals will be of two kinds.
As long
as the server is occupied, the inter-departure intervals (excepting the
first one in each busy period) will be distributed identically as the
service times.
When a busy period finishes, the inter-departure interval
starting with that final customer will be the sum of the remainder of
the simultaneous inter-arrival interval and the next service intervalo
•
The server is a busy fraction P of the time, thus when P1 is large, the
l
output will most closely approach the service time distribution, and
40
when P is small the output will most closely approach the arrival
1
distribution. In our case, a high PI means a greater queue length in
the first queue, and greater regularity in the output.
This should
correspond to decreased queue length in the second rank queue.
As Cox
and Smith (1961) point out with Poisson arrivals, the distribution of
the length of idle periods is negative exponential with mean
l/~.
The
server is idle a fraction 1 - PI of the time, so in a very long time
period T, the server is idle for a time T(l - PI)' or
periods.
T~(l
- PI) idle
Since idle and busy periods alternate there are also
busy periods of total time TP •
I
T~(l
- PI)
Mean busy period length is
•
(5.31)
Similarly, it can be shown that the mean number of customers served per
busy period is
1
(5.32)
The inter-departure intervals are clearly not independent for a
queueing system with a delayed negative exponential service time distribution.
For a single channel queue no two customers may depart at a
lesser interval than the minimum service delay.
For a multiple channel
system it is not impossible for departures to occur at a lesser interval
than the minimum service delay interval, but the probabilities are reduced.
It would be of interest to know the distribution of interdeparture
•
times, to be able to relate this to the causative factors in the first
rank of servers, and to be able to predict the effects on the subsequent
.e
41
queue in the second rank.
undoubtedly functions of A,
The parameters of the output distribution are
~,
and o.
Measuring the shape of the distri-
bution, estimating the mean .and variance, and making a good guess as to
the analytic formulation seem quite possible.
The relation to the
para~
eters of the first rank seems more difficult to assess by simulation
means, but with good luck it might be possible to do so analytically.
The prediction of the effect of the output distribution on the queue of
the second rank of servers would be a more formidable task by simulation
methods.
The overall probability of success here by analytic methods
seems remote, though still theoretically possible.
The solution chosen to this dilemma was to ignore the output distribution from the first queue.
The simulation was then used to relate
the second queue characteristics directly to the first queue parameters.
This method is a practical one entirely in keeping with established applied statistical methods.
What it may lack in developing insight into
the basic nature of the queueing process is more than made up for in the
directness, simplicity and practical utility of the results and by the
considerable reduction in computation necessary.
5.5.2
Second Rank Model
The second rank model should include the exact same parameters
as the first rank model since they are identical except for the input
distributions.
Then in addition it must contain terms which represent
the effect of the non-Poisson input.
These terms can only be known very
poorly at first and must be estimated with the aid of inspection of the
.e
simulation results, and possibly iterative statistical analysis.
They
42
will certainly involve the first rank parameters as well as interaction
effects with both first and second rank parameters included.
As a guide we have the following general rules:
the output of
shorter; when
L should be
q2
long; when Pl
when P is large
l
the first queue is more regular, thus, L should be
q2
k is large the first queue output is more variable, thus,
l
longer; when P is small and k is large L should be
l
1
q2
is large and k is small the effect is not obvious but
l
depends on the relative magnitudes of the two factors.
Furthermore, it
is known A priori that each and every distribution parameter has an
effect on L
and each and every interaction of these parameters also
q2
has an effect, although the relative significance and magnitudes are
unknown.
We postulate then, subject to further modification, that the model
for the queue length of the second rank queue should be
3 2
L - (B + B P + B2P22 + B P + ..• ) . (1 + ~k2)
2
o
2
1
3
q2
22
+ a o + a l Pl + a P2 + •.. + a k 2 + a 4Pl k 2
l + aSPlk l
2 l
3 l
(5.33)
or, again collecting terms of like powers,
_e
22222
+ a 4Pl k i + aSP1k l + ... + g(P , P2' k l , k2 ) ,
I
(S .34)
where g(p l , P2 ' k 2 , k 2) represents the combined interaction effects ;be2
l
tween first rank and second rank parameters.
43
6.
SIMULATION DESIGN
In Chapter 5 we considered in some detail the physical system to
be simulated and some restrictions necessary for delimiting the problem.
We further considered the exact results from previous analytical
studies and formulated a regression model which purported to relate
queue length to a power series involving the parameters used in the
simulation.
We consider in this chapter the actual choice of the
parameter space in which the simulation was done.
The principal procedure of interest here is the experimental
design used.
As mentioned previously it was first intended to run from
one to five servers in parallel in each rank for a total of twenty-five
.
e
cases.
Due to the enormous variance encountered in test runs the
computer time needed to reach acceptably stable outputs was far greater
than expected.
As a result, with a fixed computing budget, it was
found necessary to reduce the number of cases simulated from the anticipated 25 down to 8 cases.
3, 1
Those chosen were:
3, 5
3, 3
4, 4
5, 3
5, 5
This set of cases was selected on the following basis:
c , c • 4, 4 was an essential point for direct simulation. This was
2
l
because one of the real missile systems being simulated was already
.e
44
tentatively committed to a 4 hangar, 4 launch pad system.
Those direct-
ing that project, understandably, desired a direct simulation of the
exact system chosen.
Project officials felt that interpolation or ex-
trapo1ation from nearly alike systems would not give as useful results
from their point of view as direct simulation of the system at issue,
the c ' c 2 • 4, 4 case. Those cases where c differed too much from
1
1
c were of less interest practically speaking. The remaining seven cases
2
available were arranged in two sets of four
c I ' c2 • 3, 3.
~ith
a common corner at
This arrangement enables complete coverage of the range
of one to three hangars and one to three launchers previously estimated
for two of the programs, and the three to five of each type of server
estimated as the requirement for the remaining program.
.
e
the c
l
• c 2 diagonal excepting 2, 2.
The cases cover
It was with considerable regret
that the completely symmetric arrangement with c
l
• 1, 3, 5 and
c 2 • 1, 3, 5 (plus c 1 ' c 2 • 4, 4) was abandoned. It would have meant,
however, a 25 percent increase in computing time, which the budget did
not allow.
6.1
Experimental Design Requirements
The first objective of the regression analysis is to provide a set
of approximating equations which can be used to calculate queue lengths
i~
practical situations using the operational variables
p
and k.
There-
fore p and k must be varied over a range of values of practical interest.
A regression model must be established which will be capable of "explaining" most of the changes in queue length resulting from variation in
e
•
parameter values in the simulation.
The design should provide some
-e
45
simulations of simple queues whose properties are theoretically known,
to act as controls.
The design should be such that the raw results of
the simulation should be capable of being analyzed to provide tests to
determine the relative importance of the various parameters and their
interactions.
To establish the regression model it will be desirable
not only to know which interactions are important but to have some idea
of the algebraic form of these interactions or at least an adequate
approximating algebraic form.
6.2
Design for the Distribution Variables
There are five distribution variables, A, P1' a 1 , P and 02.
2
From
statistical considerations, the best arrangement to use to test for the
existence and relative importance of interactions is the factorial de-
•
e
signs.
"If there is interaction this design is indispensable, 11 Anderson
and Bancroft (1952, p. 268).
The most economical way to establish a
factorial design with five variables is to use two levels of each variable, for a total of 2 5 or 32 different combinations. An analysis of
5
variance of a 2 factorial arrangement will give inferential statistical
evidence of the existence and relative importance of all possible interactions of the five distribution variables.
This will enable the impor-
tant interaction terms of the regression model for the average queue
length to be established.
Since there is reason to believe that inter-
action may be present at all levels, the factorial arrangement was
selected (see Appendix 11.2, Tablell.U. The values of each level were
chosen as will be indicated below.
The intent was to attempt to'deter-
mine the relative significance of the various interactions as an
-e
46
indication of the factors to include in the model for L
q2
There was no
particular interest in a model in the distribution variables per
as an end result.
~
Ten repetitions were run at each of these 32 parameter
sets.
6.3
Design for the Operational Variables
For any fixed set of parameter values in the distribution variables
there exists a corresponding transformation set in the operational
variables.
However the regression model postulates that at least the
second order effects of p and k should be considered, implying that
more than two levels of p and k are necessary for the estimation procedure.
.
e
The device used here is to establish choices of p and k in such
a way that the different levels are achieved by a particular 25 factorial
arrangement in the distributional variables, since the simulation process
requires specification of these variables.
First in importance was the traffic density:
Due to the strong
curvature in the L vs p forms, see the equations of Appendix 11.1 for
q
example, it was felt that three levels of P1 and three of P2 at the very
least would be required, preferably more.
Since 3 is not a common divisor
of 32, four levels each of P1 and P were selected. Now 4.4·n • 32
2
parameter values, so n • 2, which was hardly enough design points to
enable estimation of both k l and k 2 effects.
Therefore a fractional
factorial arrangement was devised as shown in Appendix 11.2, Table 1102 0
The exact values selected for the various parameter levels are given in
Appendix 11.3 and the method of selection of these values is described
next.
_e
47
6.4
Selection of Design Parameter Values -
e
Since the traffic density p appears to be the most important
variable its values are chosen first.
From our knowledge of the single
channel M/M/l queue system we can say that for values of p above about
0.85 the queue length increases extremely rapidly and the variance of
queue length even more rapidly.
The average queue length here would be
2.4 for constant service, 4.8 for negative exponential service.
server would only be unoccupied 15 percent of the time.
in queue length in M/M/l would be about 28.
The
The variance
For a traffic density of
p • 0.9 queue length would be 8.1 for M/M/l and the variance 73.7.
Only
a slight drop, to 10 percent from 15 percent in the server vacancy factor
would result.
.
e
We conclude that p • 0.85 is a practical upper limit for
most queueing situations •
If P • 0.6 is selected, then M/M/l gives an average queue length
of 0.9 and the server is vacant 40 percent of the time.
Variance of
queue length is 1.71, contrasted with 28 for p • 0.85.
The constant
service case gives a queue length average of 0.45.
A drop of p from
0.6 to 0.55 now reduces M/M/l queue length to 0.67 (0.33 for M/D/l).
These seem to be fairly low queue lengths and rather inefficient server
utilizations.
It seems probable that in most practical cases, when
average queue lengths drop below these values their careful prediction
is of less importance.
This is partially due to the much more stable
system and partly due to the very inefficient use of the server.
are exceptions of course.
There
In some cases it may be critically important
to reduce waiting time or queue length.
It may be impossible to provide
space for more than a limited number of customers, for example.
.e
48
When p is below 0.6 the difference in average queue length between
constant and negative exponential service is only the difference between
1.71 and 0.85 customers, while at p • 0.85 the difference is between
4.8 and 2.4 customers.
This demonstrates that an error in estimation
of service time distribution at low traffic density is not nearly as
severe in absolute terms as it is at high traffic density.
For these reasons the range of P was taken to be approximately 0.6
to 0.85.
Since it was expected the arrival rate distribution at the
second rank would be less variable than at the first rank, the higher
values of p were increased slightly for the second rank in order to
maintain the queue lengths at a level approximately the same as the
first rank.
,.
e
From the general form of the model postulated, it was known that
at least p and p2 terms would be required.
use at least three values of p.
Thus it was necessary to
It was not known A priori if the third
power of p would be required in order to obtain an adequate fit.
To
provide for this eventuality it was desirable to use four values of p,
approximately evenly spaced.
The coefficient of variation of service time, k, can take values
between zero and one for cases of interest, corresponding to a range
between constant service and negative exponential service, respectively.
If we select four values of p, say, in increasing size, PA' p' '
B
PC' and Pn for a single given value of A, this implies four corresponding values of PA, PB, PC' and Pn through the definition that p • A/pc,
assuming a given number of servers, c.
must always have pc
> A.
In order that 0 < p < l, we
And since the delayed negative exponential
.e
49
distribution must have a positive mode, defined as
1 - a,
P
restriction imposed that a must always be less than
there is a
~.
The use of the 25 factorial design for the distribution variables
influences the choice of values for the operational variables in the
following manner.
Whereas above it was stated that a choice of four P
values led to four P values for a given
P values.
of A.
~,
we are now restricted to two
In order to retain four values of p we· must select two values
Then'we have these relations.
(6.1)
and (for equal spacing a)
(6.2)
and one of two ways to use two values each of
PA • liB'
Pc • PD,
AA • ~c'
P
and A is to let
AB • ~D
(6.3)
so that, for any given value of c, say c • 1,
AA
PA .liA
AB
.PB P
A
AA
Pc .Pc
AB
.PD
Pc
(6.4)
There now being two values each of A and P, the next step is to choose
two values of a to go with the P to form the desired values of k, of
which there must now be four.
..
(6.5)
50
The earlier restriction was made that 0.60 < P < 0.85.
6 • 0.08 approximately.
From this,
Letting PA • 0.60 we have P - 0.68, Pc • 0.76,
B
PD- 0.84.
The arbitrary selection of 6A • 6 • 6 is an impossible condition
B
C
when taken in conjunction with the other constraints on the re1ationships of these variables.
However, if we allow the value of either
pair of the p's to shift in accordance with the value of A derived from
the other pair of p's, then we have a consistent set of values.
We
arbitrarily adjust slightly, to get P • 0.61607 ••• , PB • 0.69,
A
Pc • 0.75 and PD • 0.84, with AB • 0.1 and AA • 0.08928 ••••
Now
6A • 0.07393 ••• , 6B • 0.06 and 6C • 0.09.
In all eight cases of c 1 , c2 the two lower levels of P1 and P2
,
e
are the same, 0.616 ••• and 0.69.
Whenever the number of servers in a
rank increased, the two higher levels of P were increased somewhat.
See Appendix 11.3.
6.5
Selection of Design Parameter Values - k
The Er1ang distribution of service times is a two parameter distribution in which the mean, l!p, and the phase constant, r, may be
selected at will.
As a device for approximation of less mathematically
tractable distributions it suffers from a considerable lack of f1exibi1ity.
Selection of the phase constant r • 1 reduces the Er1ang to the
negative exponential, and r • • to the constant,
l!p~
with intermediate
values of r representing unimodal distributions with less variation than
•
the negative exponential.
However, selection of r • 2 gives a modal
value of one half the mean and a standard deviation which is 0.707
51
times the mean.
Higher phase constants give both modal values and
standard deviations which are larger yet, the mode being (r-1!pr and
the standard deviation 1!1r;, where l!p is the mean service time.
While it is popularly supposed that the Er1ang distribution can be used
to simulate any distribution intermediate between the constant and negative exponential, closer
i~spection
shows this not to be the case.
Excepting the negative exponential, only two-parameter distributions
with a mode equal to or exceeding half the mean but not exceeding the
mean, and with standard deviation 0.707 or less of the mean can be
simula~ed.
The mode and the standard deviation are not independent of
the mean, thus greatly reducing the flexibility of the Er1ang distribution as a device for simulating other distributions.
-
e
Furthermore, the
coefficient of variation can not exceed 0.707 and the next largest value
is 0.577 (for r • 3).
What the delayed negative exponential distribution
can do which the Er1ang distribution cannot do is to enable a wide choice
of standard deviations to be made, independent of the mean.
Thus, the
mode can be made to fall anywhere between the origin and the mean, and
the coefficient of variation can be made to take on any value between
zero and one.
The coefficient of variation, the mode and the standard
deviation can only take on particular discrete values from the Er1ang
distribution, while in the delayed exponential distribution they are
continuously adjustable.
The selection of values of k should range rather evenly between
o and
1 to exploit this situation.
Dividing the four values evenly
gives
k D • 0.8 •
(6.6)
52
However, recalling the Po11aczek formula (5.20), where the coefficient
of variation only appeared as the square, we might be better advised
to spread values of k
2
evenly over the interval 0 to 1, which would
give
k A • 0.447
k B • 0.632
k C • 0.723
(6.7)
k D • 0.895.
Let kD • 0.9, for convenience, then GB • 6.21 and k
c
•
0.739 ••••
These values of k are both higher than can be obtained by the Er1ang
(excepting the negative exponential case).
The remaining two values
of k are somewhat less important since they will fall in the region (if
not at the precise values) where the Erlangian distribution also has
some coverage.
The analytical results available from studies of somewhat similar
or related queueing systems will aid in selecting the algebraic form of
the effects or interactions to be included in the model.
It would be
quite helpful if more light could be shed on this point however, since
the model is strictly an approximation to the simulation results and not
necessarily to the true functional form (unknown) which describes the
system.
There is a way to selectively avoid the confounding in the fractional factorial design for p and k, by using the proper values for k.
the Pollaczek formula 1 + k 2 appears.
Letting
In
A • 1.68 then kA • 0.2
G
and this term becomes 1 + (0.2)2 or 1.04, while k B • 0.2434 ••• leads to
1 + (0.24 ••• ) 2 • 1.059 ••••
•
If the effect on Lq is due to this factor
of 1 + k 2 then the difference between 1.04 and 1.059 is less than 2 percent and probably negligible.
These values were chosen.
The complete
·e
53
set of values for the distributional parameters is:
AA
• 0.08928 •••
AB
• 0.1
• 6.21
• 1.68
and for the operational parameters is:
(6.8)
PA
• 0.61607 •••
PB
• 0.69
Pc
• 0.75
PD
• 0.84
k
A
• 0.2
k
• 0.2434 •••
C
• 0.739 ...
kD
k
6.6
B
• 0.9
Selection of Interaction Terms
The factorial arrangement for case
c 2• 1.lis shown in Appendix
CIt
11.2, Table 11.20 Each cell representing a particular combination of the
five variables is numbered for identification.
1, 2, 5, and 6 will show that they form a 2
2
Inspection of cells
factorial arrangement ade-
quate for comparing the effects of k l and k 2 while P1 • P2 • 0.84.
A
test of interactions between k
and k2 can be made. There are eight
1
such 22 factorial arrangements buried in the larger design. They are
listed in Appendix ll.4 t Table 11011.
If k • 0.2 and k • 0.24 are taken to be approximately equal say to
e
.
k • 0.22, the design can be re-arranged in order of increasing p[ and P2
54
within k - 0.22.
Then cells 1, 3, 9, and 11 form a 2 2 factorial which
can be tested for the PI vs. P2 interaction.
sub-factorial designs.
There are two of these
And finally there are four designs such as 1,
3, 5, and 7 which test for P vs. k l interaction, and four which test
2
for PI vs. k 2 • Samples of the data involved are presented in Appendix
11.12, Tables 11.13 and 11~14.
The tests for significant interactions are not important here
excepting as confirmation of the results of the 2 5 factorial experiment.
What is important is the pattern taken by the observed values, as it
indicates the algebraic nature of interaction terms which can be inc1uded in the model.
In Appendix 11.4, Table 11015 some of the patterns of a 22 factorial
are shown which result from various combinations of the two variables
xl and x 2 •
Each variable is shown at levels of 1 and 2 for clarity,
and at 0.6 and 0.8 to demonstrate the effects at different levels.
This list is not exhaustive of course but does exhibit some of the
simpler forms.
By comparison of the data from the interaction tables
derived from the simulation results with these models in Table 110150
selection of interaction terms may be made for the regression model which
will provide a better fit to the data •
•
55
7.
EXPERIMENTAL PROCEDURE
The experimental procedure consisted first of designing the experiment and then of programming the computer to perform the simulation.
Keeping careful and detailed account of all the operations which ensued
and which had to be exactingly carried out in absolute and precise order
before the program could be made to work was an administrative rather
than scientific undertaking.
Nevertheless, it was necessarily done by
the researcher personally in order to make sure that the program written
was the program desired.
Test cases were hand calculated and double
checked for comparison with initial computer test runs.
After the
simulation results were available many hours were spent in careful checking and cross-checking, re-running some data as appropriate.
·
e
After the simulation computer program had been completely checked
out, "debugged," and tested with the aid of sample problems (hand-calculated) a test run was made to estimate computer running time.
On the
basis of this test run it was decided that each parameter set would be
run for 13,000 time period ("days"), and 10 parameter sets would be run
consecutively.
All data being read out after each 13,000 days, this
gave ten replications for each set of parameter values and thirty-two
sets of parameter values per case.
Actually, each parameter set run
was stopped at the first event after 13,000 days to avoid some complication in the program, which introduced some irregularity into the sample
sizes.
Thts was in all cases a positive error in time thus leading to
•
a very slight negative bias in
e
examined, of 2.2 extra days in 13,000.
·
~
of the order, in 100 observations
At the start of each set of 10
·e
56
observations the entire system was empty.
To avoid a bias due to this
the system was run for 200 days before any measurements were started.
At the end of the 200 days the system had some random degree of
occupancy and the history up to this time was discarded, all observations commencing with the state of the system as at that moment.
In the worst case, the lowest arrival rate was about one customer
per 11.2 days.
In cases 5, 3 and 5, 5 this means that it took an
average of 56 days to fill the fi rst rank of servers.
The average serv-
ice time was 43.5 days (worst case) so that by 100.8 days the first five
customers were all in service in the second rank of servers and the
first rank had four new customers, not considering variation.
This it
was felt was sufficient to assure an adequately representative random
system occupancy as a starting point, but to be safe the time was
doubled, to 200 days.
In every replication the initial random numbers used to establish
the arrival and service distributions were printed out.
These numbers
were examined by chi-square tests which gave no indication of departure
from randomness.
As a further test of the arrival rates, the number of
customers arriving and the number of days elapsed were read out at
regular 520 day intervals for 51 such intervals.
A chi-square test
indicated no significant departure from the Poisson distribution.
The
programmed distribution for service time, as well as arrival times,
were read out and checked for accuracy.
,
time distributions actually produced were not available from the system
performance.
e
.
More direct tests of service
57
For the single server case 249 observations were made, at 100 day
intervals, of queue length.
This distribution of queue lengths differed significantly from the
geometric distribution, as indicated by a chi-square test at the 1 percent level.
This was to be expected, as only the negative exponential
service time distribution would give the geometric queue length distribution (Cox and Smith, 1961).
The delayed exponential service resulted
in a queue length distribution with greater frequencies at zero and at
the longer queues, that is, with greater variation than the geometric
distribution.
See Appendix 11.5, Table 11.16.
We concluded from these tests that the computer simulation program
was operating properly and delivering valid results.
•
e
The results of the simulation are far too voluminous to reprodpce
in detail, so selected representative data only are included here.
Appendix 11.6, TablenJ7,includes some of the detailed output.
Each
entry is the average queue length for a given parameter set for a simulation run of a nominal 13,000 days.
Thus each cell includes ten such
observations, all for the same set of parameter values, and the average
for the ten, which represents a run of 130,000 days.
variances are clearly visible.
The very large
These were calculated cell by cell and
plotted against average queue lengths.
While no numerical verification
was attempted it was clear by inspection of the graphs that the variance
increased in a manner similar to L + L2 , as predicted from theory for
the MIMII System.
This was a further supporting confirmation of the
validity of the simulation procedure.
.e
Although there is certainly at
least some serial correlation present within each set of 10 observations,
58
tests for the auto-correlation coefficient in 15 arbitrarily chosen
parameter sets showed no appreciable effect.
The data for queue lengths was analyzed by means of a
step~wise
regression analysis computer program, BMD02R, version of 12 November
1964, by the Health Sciences Computing Facility, UCLA.
It is described
in detail by Dixon (1965, p. 233) and the general description is given
here.
e.
This program computes a sequence of multiple linear regression
equations in a stepwise manner. At each step one variable
is added to the regression equation. The variable added is
the one which makes the greatest reduction in the error sum
of squares. Equivalently it is the variable which has highest
partial correlation wi th the dependent variable partialed
on the variables which have already been added; and equivalently it is the variable which, if it were added, would
have the highest F value. In addition, variables can be
forced into the regression equation. Non-forced variables
are automatically removed when their F-values become too
low. Regression equations with or without the regression
intercept may be selected.
The F value selected for inclusion was at probability level 0.01, for
deletion the level was 0.005 •
.
e
.
59
8.
8.1
RESULTS
The First Rank Queue
In Appendix 11.6, Table
11.18 is
recorded summary data
for the queue lengths in the first rank queue.
These figures are each
the averages of ten 13,000 day simulation runs.
The data for L
,the average queue length of the first rank queue,
ql
was first analyzed by regression analysis using the model
(8.1)
This model includes the three main effects, A, l/P , and
l
possible products of the three.
0
1
and all
In Table 11.20 of Appendix 11.7 are given the variables classified
as to the step in the regression analysis at which they first entered
the solution.
Inspection of this table will show that A/PI (or PI) is
the overwhelmingly significant variable, both
in'~erms
of significance
test (F value to remove) and in contributed reduction of sum of squares
for regression, R2 •
The next factors, consistently the second in the
stepwise regression analysis, were A aI/Pl.
which has no
operat~onal
This is equivalent to PIal'
meaning, but does indicate that the traffic
density and service time variation are important.
This term is closely
related to the first term to enter, PI' so it is not surprising the A/PI
t
portion shows up strongly.
e
enter immediately.
.
The significant part is that the
should
1
In all 8 cases of c ' c 2 these two terms accQunt for
l
0
·e
60
85 percent or more of the sum of squares.
It is still indicative of
the importance of the traffic density that it should always be among
the first two variables selected, and that the standard deviation of
service time, aI' should always (with one exception) be included in the
second and third and fourth variables selected.
The conclusion is that those variables involving both A and
(obviously PI) are most influential, those involving A, aI' and
(P l and a l ) are next, and those involving a and
l
~l
~l
~l
come third.
In
terms of the operational variables these would be PI and powers of PI'
cross product terms in PI and k l , and terms in k l •
This would be ex-
pected of course, in accordance with the form of the Pollaczek equation
for the single channel system (5.20).
These results do not introduce
any novel element in themselves excepting to confirm the general nature
of the Pollaczek equation as applied to the multi-channel system.
They
are of value in indicating what may be expected in similar analysis with
L
q2
,
and act as a semi-control in this regard.
The first model used with the operational variables was:
2
0
2
al
1
L
- B + BlP 1 + B2P1 + B - + B42
ql
0
3 ~l
(8.2)
~l
0
where
1
111
was used erroneously for PIa l •
Although not the model in-
tended, the results of this second regression analysis are of some
interest.
See Appendix 11.7, Table 11.21.
Clearly, P~ is the variable
2 2
with the greatest effect, P runs a close third after a1/111'
and al/~l
1
is negligible.
.e
61
·e
This data was analyzed a third time with k
111 ale
l
properly defined as
The model now was
(8.3)
which can be re-written as
(8.4)
or
L • BO (1 +
ql
ak~)
+ Bl(l +
Bk~)Pl
+ B2 (l +
Yk~)P~
•
(8.5)
In all cases excepting 1, 3 the term for k 2 was not significant, and in
1
2
c ' c 2 • 1, 3 it only contributed 1.39 percent to the increase in R •
l
2
Thus case 1, 3 was re-run and the k term forced out. The results of
l
this third regression analysis are shown in Tables 11.22, 11.23, and
11.24 in Appendix 11.7.
Comparison with the previous second regression
2
analysis shows that, while the significance levels and changes in R
are for the most part identical for the first two steps, the third set
of regression analyses shows marked improvement in both significance
levels and fits for the last two steps in the regressions.
Table
11.24 of Appendix 11.7 gives a brief summary and averages replicate
coefficients to indicate the effect of c '
l
The coefficients tabulated in the appendix may be used for
calculation of queue lengths if desired.
•
Depending on the accuracy
desired and the amount of computation envisioned, any of the sets
may be selected, from the pair of coefficients resulting from the
first step of regression analysis to the complete set of five
62
coefficients resulting from all four steps.
They are not likely to
be much simpler to use, nor as accurate, as the Pollaczek formula, for
the single channel case, but are the only existing solutions for the
multi-channel systems, except as shown later in this paper.
An example of the residuals from the first rank queue length (for
case 1, 1) is given in Table 11.25, Appendix 11.7.
The residuals are
arranged in order of the corresponding observed queue lengths.
The
16 values corresponding to the shorter queue lengths were compared to
the other 16 by a t-test and no significant difference in the means was
found.
These and other sets of residuals were plotted on graph paper
and no indication of relationship to queue length was visible.
0_
They
were also plotted on cumulative normal probability paper and appeared
to follow the normal curve.
8.2
The Second Rank Queue
The average queue lengths obtained from the simulation for the
second rank queues are shown in Appendix 11.6, Table
11.19.
With this data available it was possible to run an analysis of variance
5
for the 2 factorial, for the second rank queue, L • The results are
q2
shown for case c ' c2 - 1, 1 in Appendix 11.8, Table 11.26. The
l
computer program was BMD02V, Analysis of Variance for Factorial Design,
Version of May 3, 1965, Dixon (1965).
This analysis was run on the mean
observations, thus there are no replications to give an error sum of
squares.
..
e
o
Complete results for all 8 cases are displayed in Tables 11.27,
11.28 and 11.29, Appendix 11.8.
63
Using the higher order interactions as error terms one geta significance levels as indicated in Table 11.30, Appendix 11.8.
As
would be
expected, all the main effects are very significant, followed by seven
of the ten two-factor sources of variation, then by three of the ten
three-factor sources, depending of course, on which significance level
The most significant of the two factor sources was A, 1/P ,
2
corresponding in operational variables to P • This was followed in order
2
is chosen.
of decreasing significance first by 02' 1/P , corresponding to k ; second
2
2
by l/P , 1/P2 which is closely related to P1P2; third by A, 02' of no
l
direct operational meaning but related to k 2 ; fourth by A, l/P l , corre-
.
e
sponding to P ; fifth by A, 0 1 , related to k l ; and then by 01' 1/P2'
l
A first regression analysis was run for L
using a model with the
q2
five distribution variables, A, 01' 02' l/P l and 1/P2 and ten combinations of two of them at a time.
Table
The model and the results are shown in
11.31, Appendix 1108 0
A second regression analysis of L
ing operational variables only.
was performed, this time involvq2
The model was:
2
• Bo + B1Pl + B2P2 + 83Ml + B4M2 + BSP1
q2
L
222
2
+ BSP1 + B6P2 + B7M1 + B8M2
where M is defined as
(8.6)
M1 •
01/ P1 and M2 • 02/P2. The results are summarized in Table 11.32, Appendix 11.8. The R2 are rather low, around
90 percent.
t
.e
This was followed by a third regression analysis of L which used
q2
the same model with four additional terms.
•
64
2
2
L • So + Sl P1 + S2 P2 + B3M1 + B4M2 + BSP1 + B6P2
q2
2
2
3
3
3
3
+ B7M1 + BSM2 + B9P1+ B10P2 + B11M1 + B12M2 •
These results are displayed in Table 11.33, Appendix 11.S.
(S.7)
The order
of appearance is practically identical with that of the previous regression but a higher power of most of the variables.
The total R2 is
consistently but only very slightly higher for the higher powers.
It
was decided not to use the third powers of the traffic density.
A fourth regression analysis for L
q2
used this model:
22222
L • Bo + B1P2 + B2P2 + B3k 2 + S4k 2P2 + BSk 2P2
q2
222
+ B6P1 + B7Pl k1 + BSP + B9Pl k l
l
(S.S)
The results are displayed in Appendix 11.S, Tables 11.34 through 11.37.
A fifth set of regression analyses for L
q2
was computed using the
model:
2
L • Bo + B1P + B2P2 + B P
l
3
1
q2
~
222
2
B P + BSPl k 1 + B6Pl k l
4 2
222
2
242
+ B7P2k2 + B P k + B9P2!P1 + B10k 1k2 + B11 k2!P 1 •
S 2 2
(S.9)
The results of these regression analyses are displayed in Appendix 11.S,
Table 11.3S.
Case S, S was not available, due to an irretrievable error
in the computation procedure.
There is some room for diversity of opinion concerning the interpretation of these findings.
.e
Table 11.39 of Appendix ll.S compared the
order of significance of the results of the four sets of regression
analyses run for L
q2
on the operational variables.
65
The selection of the order displayed for the second and third sets
of regression analyses has been explained.
The selection for the fourth
and fifth sets is largely on the basis of the increments of R2 produced
by each term.
Tables 11.40 and 11.41 of Appendix 11.8 give a summary
of F values for these two sets.
The fourth set of regression analysis
results exhibits no change in order of terms between the two criteria.
The only question, based on this data, would be whether to keep P~ and
P1k~.
Neither term shows up as significant and yet on a theoretical
basis both P1 and k
should have some effect. In three cases, (1, 1),
1
2
(3, 1), and (3, 3), P1k 1 is the third term to enter the solution, with
F values upon entrance between 8 and 16, while the previous term to
enter had entrance F values of 20 to 27 and the succeeding term to enter
had entrance F values of 8 to 11.
The results of the fifth set of regression analyses when evaluated
by the F tests are not as consistent with the order established by the
2
2
4
increments in R.
By both
criteria
k k and k 2 /P should be discarded,
1 2
2 1
2
and p~, P~k~, and P2 obviously should be kept. By the R increment
22222
2
gauge, P2k 2 , P1k1, P2/P1' P1k1, P and P are important in that decreas1
1
ing order, but by the F value tests, they fall in the order P~/P1' P1'
22222
P2k2 P1k1, P1 and P1k1. Due to the confounded nature of the operational
variable design, and the high degree of correlation between many of the
terms, and since the objective is curve fitting as much or more than to
test models, our inclination is to go by the ordering established by
increments of R2 •
e
•
The fourth term of the third set of regression
analyses for Lq , was
P1k~,
corresponding in this situation to
P2k~,
which
-e
66
reinforces this conclusion.
All the other terms are functions of PI'
thus are correlated.
8.3
Implications
The third regression analysis of L
ql
,using four terms, determined
a useful model to be
2
L • S + SlP + S2 Pl + S4k 2l PI + SSk2P2
0
1 l
l
ql
(8.10)
which may be written
2
2
2
L • So + PI [Sl + S4k l] + p 1 [S2 + SSk l ]
ql
(8.11)
2
2
2
L • S + [SIP I + 82Pl] + k 1 [S4 Pl + 8sP l ]
ql
0
(8.12)
or
e
It wo.uld appear, both on the basis of the F values and the incre-
2
2
ments in R , that the P term accounts for appreciably more of the
1
variation as the number of servers increases.
The variation is so
great that similar observations about the other three variables are not
so obvious.
However, the single server case still appears to be unique
so far as the influence of k~P~ and PI are concerned.
Ordinarily, with several independent variables in the regression
model, each coefficient, S, would be interpreted as the change in L
ql
,
when the corresponding variable is varied, holding all other terms
constant.
When the terms are functionally related, this interpretation
is clearly not valid, because the "other," correlated, terms cannot be
e
•
held constant when the variable of interest is varied.
·e
67
However the use of any given model as a prediction equation is still
a valid one.
The least squares estimation procedure assumes that the
independent variables included constitute a fixed set with respect to
the random variable being predicted and that the coefficients computed
are then adjusted for the correlations existing among the independent
variables in the fixed set.
Then the coefficients become simply a set
of weights to be used given that the fixed set is the same as that used
to compute the weights.
S.4
Analysis of Combined Cases
All the data in all eight cases of c , c was combined and regres2
1
sion analyses were run on both Lq and L using the following models:
1
q2
222
L - Bo + B1P1 + B2P1k 1 + B3P1 + B4c l
q1
+ BS•P 1k 21 + B6c 12
L - Bo +
q2
(S.13)
222
2
B1P2 + B2P2k 2 + B3P2 + B4P2k 2
2
2
+ BSP2k 1 + B6P2 !P 1 + B7c2 + BSP1 + B9k 1k 2
222
+ B10c 1 + B11 c 2 + B P1k 1 + B c 1 •
12
13
(S.14)
The results are presented in Appendix 11.9, Table 11.42, lists
the residuals from both analyses and their standard deviations.
For
both Land L the residuals were plotted on Gaussian graph paper and
q1
q2
the resulting lines appear satisfactorily straight, a rough indication
of normality.
e
.
68
The regression coefficients for the regression analysis of Lq
1
are listed in Table 11.43, Appendix 11.9, step by step, as well as their
standard errors and the F values.
2
The R values in the final column
show a good fit of the data to the model, but no appreciable improvement
after the fifth step.
Thus, neither the increment in R2 nor the value
of F indicate that c~ should be included in the model.
Table 11.44 displays the step by step summary of the regression
The adequacy of fit as measured by R2 increases
q2
little after step 7. The significance of the variables P~' k l k 2 , c '
l
analysis for L
c~, Plk~, and c~ is indicated to be doubtful by this, by the F values,
and also by the standard errors of the coefficients.
The models as
simplified on the basis of these results then become as follows:
.
e
L
L
ql
2
2 2
2
- B0 + B1P l + B2P1k l + B Pl + B4 c l + BSPlk l
3
2 2
2
- 80 + B1P2 + B2P2k 2 +
q2
2
+ B P k l + B P !P
S 2
6 2 l
81 2 +
+ Bc
7 2
(8.15)
2
B4P2k 2
•
(8.16)
The values of the coefficients in these models are as given by Step 5,
Table 11.43, for L and Step 7, Table 11.44 for L • For predicting
ql
q2
either L
or L
the values of B , B2 , and B3 are roughly equivalent,
l
ql
q2
corresponding to the contribution of the variables p2 , p 2k 2 , and p. In
addition the constant terms, Bo ' agree rather closely.
The queue length
of the second rank queue depends only weakly on c 2 ' the number of service channels in the second rank, and not significantly on c l ' the number
in the first rank.
It should be remembered however that p, the traffic
density per channel, has already been adjusted somewhat for the number
69
of servers, hence indirectly for the value of c
l
or c ' depending on
2
the rank.
Table 11.45 summarizes the sets of regression coefficients for the
selected models giving the best fit felt to be significant •
•
•
70
9.
CONCLUSIONS AND RECOMMENDATIONS
9.1
Conclusions
For the first time an extensive simulation of a multi-station
parallel-series system of queues having delayed negative exponential
distributions of service times and a Poisson distribution of the
initial customer arrival rate has been accomplished.
Two ranks of servers have been studied, with from one to five
servers in each rank.
The results for a widely useful range of param-
eter values are presented.
Traffic density was studied in the important
range of 0.61 to 0.86, and the coefficient of variation of service time
extended from 0.2 to 0.9.
Results are presented in terms of average queue length for each
queue.
These are readily convertible into average waiting time in the
queue, average number of customers in the system, and average waiting
time in the system.
Regression analyses have been used to provide sets of approximate
predictive equations having varying degrees of complexity and accuracy.
These equations have expressed average queue length as functions of the
statistical distribution parameters which describe the system, the arriva1 rate, the service rate and the standard deviation of the service
time.
They have also expressed average queue length as functions of
operationally more meaningful and descriptive parameters, the traffic
density and the coefficient of variation of service time.
These "opera-
tional" parameters are defined in terms of the statistical distribution
.e
parameters.
•
71
This makes available a group of predictive equations which have
certain advantages over the equations using the Er1ang distribution for
service times, which until now has been the only available distribution
intermediate between the constant and negative exponential for service
times (excepting for rare and special situations).
Some of these
advantages are:
a)
The delayed negative exponential distribution in a great many
cases is more realistic than the Erlang distribution.
b)
The predictive equations presented here are far simpler computa-
tional1y than the Erlang distribution formulae for parallel multiple
server systems.
c)
These equations are more flexible than the Erlang distribution
formulae, since the parameters can be varied over a far wider range of
values.
d)
The variation in selection of parameter values possible with
these predictive equations is continuous, not restricted to integral
values, as are the Erlang distribution parameters.
The effect of non-Poisson arrival rate distribution has been
clearly demonstrated by direct comparison with systems with Poisson
arrival rate distributions, and is reflected in the sets of predictive
equations.
9.2
Recommendations for Further Research
The possibilities for further fruitful research are almost endless.
They may be loosely categorized as:
extensions in breadth,
tions of the queueing systems; extensions in depth,
1.~.
!.~.
varia-
exploration of
•
72
the
sys~em
characteristics in more
deta~l;
and extensions of method,
1.£.. new developments in simulation and analysis.
9.2.1
Variations of the Queueing Systems
The possibilities for variation of the queueing systems are
literally endless, but immediate objectives for further research should
concentrate on service time distributions other than the negative exponential.
The delayed negative exponential is a highly realistic distribu-
tion; a next logical step would be a delayed Erlang distribution.
The
numbers of parallel service channels should be expanded until some
asymptotic relation could be established.
If enough channels are
paralleled, it is conjectured that the combined output becomes distrib-
'-
uted nearly as the Poisson, no matter what the service time distribution.
This point and the conditions establishing it should be verified and
determined.
The corresponding effect should be sought as a function of
increase in the number of ranks.
If the initial arrival rate is Poisson
distributed and if each rank has a non-negative-exponential service time
distribution (generally of lesser variance than the negative exponential)
the output of the first rank will have a lesser variance than the original Poisson, and queue lengths in the second rank will be smaller than
would occur with Poisson input.
It is conjectured that succeeding identi-
cal ranks will continue this queue length reduction process to some
asymptotic value.
established.
_
•
This point should be verified and the conditions
It is further conjectured that service time distributions
with variance greater than the negative exponential, such as Morse's
hyper-exponential distributions, will lead to queue lengths greater than
73
would be achieved with the negative exponential distribution.
As
customers pass through succeeding identical ranks, the queues will get
longer and longer, the system becoming a quasi-divergent process.
should be verified and the relevant conditions established.
This
In the
more distant future, queueing systems corresponding to electrical bridge
networks, and with loops and closed circuits should be explored.
This
investigation should probably first be done with Poisson inputs and negative exponential service time distributions throughout, then with identical but non-negative exponential service, finally with differing distributions.
These approaches will be essential steps in the establishment of
a queueing systems theory.
9.2.2
Exploration of Systems Characteristics
The present research concentrated on average queue length as a
measure of queueing system performance, and there is probably no better
single measure.
There are, however, many other aspects of the queue
which were not explored.
of queue lengths.
The most obvious of these is the distribution
If this were known, many other useful queue char-
acteristics could be determined.
9.2.3
Extensions of the Method
This research was performed in the era before special computer
programs designed to simulate queues were available
0
The use of these
programs in research is to some extent restricted by their built-in
inflexibility.
Those constructed for a particular purpose can be used
generally only for a very narrow range of operations.
There are some
74
useful devices which have been developed in the process of writing these
programs which can be helpful in the construction of new ones for research purposes.
Current generations of digital electronic computers have speeds
orders of magnitude greater than the Univac 1101, and vastly greater
memories.
Thh enables simulations of eauivalent quality
<!.!..
numbers
of customers put through) to be run in tiny fractions of the time.
More importantly" it enables programs to be written which can measure
distributions without using up most of the computer memory.
Every advantage should be taken of variance reduction schemes.
While few of these are applicable to queueing situations, there are
some, for example the method of antithetic variates, which it is claimed
will halve the number of observations required for a specified accuracy.
Page (1965, p. 305) summarizes the use of antithetic variates by
When variates in a simulation study of a congestion system
are generated from uniform variates t i , Ni , ••• by inverting
the relevant distribution functions, it is suggested that
such a direct run be combined with others using the variates
1-ti , l-Ni , ••• or Ni , t i , •••• In a simple queueing situation, it is shown that such techniques can result in halving
the number of observations required for a specified accuracy.
Another method which offers some promise in variance reduction
depends on an argument which involves some rather basic issues which
can arise in simulation but rarely in practice.
In a computer simulation of any stochastic process as currently
practiced, there is no true raadomness present.
The "random number
generators" produce numbers in such a complex way as to be almost
completely without any calculable serial correlations and give
;
75
sequences of such almost complete disconnectedness as to appear to occur
in that distribution with no prior visible causation.
Nevertheless, if
the program is known, each "random" number can be predicted exactly.
"real" life, this is not so.
In
Each process studied has elements of
variation which occur in a completely unexplained manner.
ignorance of the sources of "error" or variation.
There is true
Consider a queueing
system which undergoes a random process, for example, a period of time
T occurs during which N arrivals occur.
It is known, presume, that the
arrivals occur with a Poisson distribution at a "true" rate)..
During
the period T however, NIT will practically never exactly equal), and
the arrival distribution will differ from a Poisson distribution by at
least a small factor.
While we may interpret the resulting queue length,
waiting time, etc. as though they were caused by the Poisson arrival
rate A, in actuality there was some other arrival rate NIT and some
slightly different distribution.
Suppose some simulation provides us with observations from a
normally and independently distributed variable with mean p and variance
0
2 , NID (p, 0).
2
equal to p.
Assume the observation to be
y,
We wish to determine whether results calculated from this
observation y- be analyzed as though it came from N(p,
N(y,
0
2
).
which rarely is
0
2 ) or from
Clearly there will be some quantities at least which must
2
be analyzed as though the observations arose from N(p,o).
Suppose we
consider the value of
(9.1)
This depends only on
0
2•
If each observation were treated as though
76
it arose from a p equal to its
y then every value of the exponent would
appear to be zero and every average value unity, when the true value is
obviously less than one.
There are, however, other situations where it may be assumed to
y.
advantage that each p •
To do this and to avoid the difficulties
which arise by assuming y • p we use y - p instead as a covariate.
The distribution and average value of this covariate will usually be
known and the average should be zero, in most reasonable cases at least.
If it is designed to evaluate the mean of some function, say
(9.2)
of the observations Yl' Y2' ••• Yn' then we can state that the average
value of
z - B
(y -
(9.3)
p)
will be the average of
Z
as long as
B
-
is independent of y.
Then
B
can
be adjusted to minimize the variance of the estimate.
In the case of the queueing system with the arrival rate a Poisson
dis tribution, the procedure would be as follows 2.
assume that the
arrival rate is truly Poisson with parameter exactly A, then devise a
covariate such as NIT - A.
The average of this covariate is zero.
In
fact, since N follows a Poisson distribution with parameter AT we may
devise a set of covariates with zero means,
N - AT
(9.4)
(n - AT)2 - AT
(9.5)
•
e
"
77
(9.6)
(9.7)
etc.
and, to use the observed values NIT, with T fixed, we have corresponding1y
N_ A
T
(N _ A)2
T
(9.8)
--TA
(9.9)
(N _ A)3
T
.
e
(9.10)
and
(N _ A)4 _3A
T
T2
2
(9.11)
etc.
Corresponding arguments can be made about many of the measurable variab1es in a queueing system.
The variance reducing effects of using a
covariance analysis in this way are not limited to arrival rates.
In the research reported here, the average queue lengths were
obtained by keeping complete track of all customers over a simulated
very long period of time.
In the regression analyses performed on the
data, each observation was the average queue length taken over 130,000
time periods.
•
Thus, the standard errors computed in the regression
analyses were of value only for internal comparison.
It would have been
78
desirable to have had them represent the variance or standard deviation
of
~ueue
length.
In order that this could be done, it would be necessary
to get independent samples of queue length rather than averages over a
stated period.
In order that correlation between samples should be
minimal, some idea must be obtained
~
priori as to the serial correla-
tion and then samples must be taken at sufficiently wide intervals to
avoid this effect.
This will undoubtedly require longer computer runs
than with the complete queue length data method, and the decision to
use the sampling method may well hinge on whether it is economically
more attractive than alternative ways such as keeping track of the
queue length distribution.
It is possible in some situations to combine a simulation with an
.
e
analytic approach.
Many analytic attempts at solution of queueing
systems are partially successful but founder at a point where a constant
must be evaluated.
In these situations, it may be possible to evaluate
the constant from simulation and enable the analysis to be completed.
An
example is the M/G/c system where it may be shown (Saaty,l96l)
that the steady state equation is given by
w
g:(~) • A f 0+
: :(x) H{w-x) dx - A Pc-l H(w) •
(9.l2)
The Laplace transform of P{w) from this equation is
P*{s) _
!!2l + P
s
e-l
A H* (s)
s [I_A H*{s)]
(9.l3)
where both P{o) and p
•
1 are unknown. Here P{w) is the cumulative
cdistribution of waiting time in the queue in the steady state; P{o) is
the probability of not waiting in the queue; p
c-
1 is the steady state
~e
79
probability of there being c-1 items in the system; and
-
H(y)
(
1
1/11
•
fY Bc
c-1
B (y)
c
(x) dx ]
(9.14)
where B (y) is the probability that the channel occupied by the arrival
c
is occupied for a time greater than y.
If we expand H*(s) for small
values of s we get
H*(s) • a o + a 1 s + ••• where a o •
f
•
(9.15)
H(u) du
o
and with
lim
s ...
0
(9.16)
s p* (s) • 1
we then have
>. ao
P(o) • 1 - ---.;;.1->' a o
.
e
(9.17)
which leaves Pc-1 to be determined.
This can be estimated from simula-
tion.
Estimates would have to be made in each case as to whether this
would be an economical approach, as compared to direct measurement of
the values of interest, such as queue length.
ing p
The possibility of measur-
c-1 in addition to other terms in a simulation is not excluded of
course, and would probably add only moderately to the complexity of
programming and other computational costs.
This method has the signal
advantage that it does not hide the mathematical description of the
fundamental nature of the queueing process behind a power series or other
statistical approximation.
.e
Once the value of the constant is determined,
the analytic results become useful to their complete extent.
How
80
complete this is will depend of course on the particular analysis and
the stage at which analysis stopped and simulation took over.
81
10.
LIST OF REFERENCES
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Research. McGraw-Hill Book Company, Inc., New York.
Avi-Itzhak, B. 1964. A sequence of service stations with arbitrary
input and regular service times. Department of Industrial
Engineedng Report, Cornell University.
Benes, V. E. 1962& Algebraic and topological properties of connecting
networks. Bell System Technical Journal 41:1249-1274.
Benes, V. E. 1962b. Heuristic remarks and mathematical problems regarding the theory of connecting systems. Bell Systems Technical
Journal 41:1201-1247.
Benes, V. E. 1963. A thermodynamic theory of traffic in connecting
networks. Bell System Technical Journal 42:567-607.
Brockmeyer, E., Halstrom, H. L. and Jensen, Arne. 1948. The life and
works of A. K. Erlang.·· Transactions of the Danish Academy of Technical Sciences, No.2.
»
e
Burke, P. J. 1956. The output of a queueing system.
search 4:699-704.
Operations Re-
Chang, W. 1963. Output distribution of a single channel queue.
Operations Research 11:620-623.
Clark, A. B. 1956. A waiting line process of Markov type.
Math. Statistics 27(2):452:459.
Cochran, W. G. and Cox, Gertrude M. 1950.
John Wiley and Sons, Inc., New York.
Experimental Designs.
Cox, D. R. 1955. The statistical analysis of congestion.
Stat. Soc. A. 118:324-335.
Cox, D. R. and Smith, W. L.
New York.
1961.
Queues.
Annals
J. Roy.
John Wiley and Sons, Inc.,
Cronunelin, C. D. Delay probability formulae when the holding times
are constant. Post Office Electrical Engineers Journal, Vol. 25,
April 1932, p. 41, and Vol. 26, January 1934, p. 266.
Dixon, W. J. 1965. BMD Biomedical Computer Programs. Health Sciences
Computing Facility, School of Medicine, University of California,
Los Angeles.
82
Ooig, Alison. 1957. A bibliography on the theory of queues.
Vol. 44, Pts. 3 and 4, pp. 490-514.
Edie, L. C. 1954. Traffic delays at toll booths.
Research Society, 2(2):107-138.
Erlang, A. K.
sations.
Biometrika,
Journal of Operations
1909. The theory of probabilities and telephone converNyt Tidsskrift for Matematik B, Vol. 20, p. 33.
Erlang, A. K. 1917. Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges.
Elektroteknikeren, Vol. 13, p. 5.
Everett, J. L. 1953. State probabilities in congestion problems
characterized by constant holding times. Journal of the Operations
Research Society of America 1:279-285.
Finch, F. D. 1959. The output process of the queueing system M/M/l.
J. Roy. Statistical Society, Series B, 21(2):31S-380.
Ghosa1, A. 1962. Queues in series.
Series B, 24:359-380.
J. Roy. Statistical Society,
Hammersley, J. M. and Morton, K. W. 1956. A new Monte Carlo technique:
antithetic variates. Proc. Cambridge Phi1os. Soc. 52:476-481 •
e
•
Hunt, G. C. 1956. Sequential arrays of waiting lines.
Research, 4(6):674-683.
Operations
Jackson, R. R. P. 1954. Queueing systems with phase-type service.
Operations Research Quarterly, 5:109-120.
Jackson, R. R. P. 1956. Queueing processes with phase-type service.
Journal Roy. Stat. Soc., Series B, 18(1):129-132.
Johannsen, F. R. 1907. Waiting times and numbers of calls. Reprinted
in "Telephone Management in Large Cities," Post Office Electrical
Engineering Journal, London, October 1910 and January 1911.
Johannsen, F. R. 1908. Busy. Reprinted in "The Development of Telephonic COtmnUnication in Copenhagen 1881-1931," Ingeniorvidenskabelige Skrifter A, No. 32, Copenhagen, 1932, p. 150.
Kahn, H.
1954. Use of Different Monte Carlo Sampling Techniques.
Symposium on Monte Carlo Methods. John Wiley and Sons, Inc.,
New York, pp. 146-190.
Kendall, D. E. 1951. Some problems in the theory of queues.
Statistical Society, Series B, 18:151-185.
e
\.
J. Roy.
_e
83
Kendall, D. G. 1953. Stochastic processes occurring in the theory of
queues. Ann. Math. Statistics 24:338-354.
Kendall, D. G. 1964. Some recent work and further problems in the
theory of queues. Theory of Probability and Its Applications,
Vol. IX, No.1, pp. 1-13.
Khintchine, A. 1932. Mathema~sches uber die ervarting vor exirem
offent1ichem scha1ter. Mathematicheskii Sbornik, 39:73.
Kiefer, J. and Wo1fowitz, J. 1955. On the theory of queues with many
servers. Trans. Am. Math. Soc. 78:1-18.
Kingman, J. F. C. 1961. The single server queue in heavy traffic.
Proc. Cambridge Phi1os. Soc. 57:902-904.
Kingman, J. F. C. 1962. On queues, in heavy traffic.
Soc., Series B, 24:383-392.
J. Roy. Statist.
Lindley, D. V. 1952. Mathematical theory-marshalling and queueing.
Operational Research Quarterly, Vol. 3, No. 1 t pp. 4-8.
.
e
Molina, E. C. 1927. Application of the theory of probability to
te1ephone-trunking problems. Bell System Technical Journal, 6:461 •
Morse, Philip M. 1958. Queues, Inventories and Maintenance.
Wiley and Sons, New York.
O'Brien, G. G. 1954. The solution of some queueing problems.
Ind. App1. Math. 2:133-142.
John
J. Soc.
Page, E. S. 1964. Computers and congestion problems, pp. 72-85
In
W. L. Smith and W. E. Wilkinson (ed.), Congestion Theory, University of North Carolina Press, Chapel Hill, North Carolina.
Page, E. S. 19650 On Monte Carlo methods in congestion problems,
II simulation of qpeueing systems. Operations Research 13(2):
300-305.
Po11aczek, F. 1930& Uber eine aufgabe der Wahrschein1ichkeitstheorie.
Math. Zeit. 32: 64 and 729.
Po11aczek, F. 1930bo Theorie des warten vor scha1tern.
unde Fernsprech-Technik, 19:71-78.
Te1egraphen
Raymond, Haller, and Brown, Inc. 1956. Queueing theory applied to
military communication systems. State College, Pa.
e
.
Reich, E. 1957. Waiting times when queues are in tandem.
Statist. 28(3):768.
Ann. Math.
84
Reich, E. 1963.
338-341.
Note on queues in tandem.
Ann. Math. Statist. 34:
Saaty, T. L. 1961. Elements of Queueing Theory.
Company, New York.
McGraw-Hill Book
Sasieni, M., Yaspan, A., and Friedman, F. 1959. Operations Research
Methods and Problems. John Wiley and Sons, Inc., New York.
Schreider, Y. A. 1964. Method of Statistical Testing, Monte Carlo
Method. Elsevier Pub. Co., New York.
Sonnenschein, J. 1962. Elements de solution du comportement transitoire
d' une file d' attente simple lorsque l' intensite du traffic depend
du temps. Cahiers du Centre D'Etudes de Recherche Operationel1e,
No.2, 4:117-118.
Syski, R. 1960. Introduction to Congestion Theory in Telephone Systems.
Oliver and Boyd, Edinburgh.
Syski, R. 1961. Congestion exchanges, pp. 85-98. ~ E. C. Cherry
(ed.), Information Theory. Butterworth Pub. Co., Washington, D. C.
•
e
e
.
Vo1ber, o. 1939. Prob1eme de 1a queue stationnaire et
Compt. Rend. Acad. Sci., U.R.S.S. 24:657-661 •
non-sta~ionnaire.
85
11.
11.1
Formulae for Average Queue Lengths
for the MIMIc Queueing System
P (cp)c P
L
•
q
APPENDICES
c! (1_p)2
°
1
P
°
-
c-1
r
n-O
(cp)
n
n!
+
(cp)
c
c! (l-p)
In terms of the distribution parameters,
Ac+1 p2---:0l'.-_
P
_
L
q
pc+1 (CP_A)2 (c-1)!
P
-=1
_
°
In terms of the operational parameters:
Let c - 1, then
2
L
_....P...q
1-P
86
Let c - 2, then
L
_
2p
3
1_p 2
q
• (1-P) (1+P)
• 2p
3
246
(1 + P + P + P + ••• ) •
Let c • 3, then
L
•
q
32p5
9P4
• ----..;;..,;;.;...---......-2 + 2p - p2 - 3p3
(1 - p) (2 + 4p + 3p2)
3
9 4
32
• '2
p (1 - p + '2 p - ; - + ••• ) •
Let c • 4, then
5
L •
~3=;2P;...5-__=_- •
32p
q
3 + 6p + 3p2 - 4p3 - 8p4
(1 - p) (3 + 9p + 12p2 + 8p3)
• 3; p5 (1 _ 2p + 3p 2 _ ~ p3 + ••• ) •
Let c • 5, then
L •
625p6
q
24 + 72p + 84p2 + 20p3 - 75p4 - 125p5
• (1 - p) (24 + 96p + 180p2 + 200p3 + 125p4)
.e
625 6
11 2 41 3
- U P (1- 3p +2"P -(;"p + ••• ) .
87
Appendix 11.2 The Simulation Designs ..
Table 11.1.
Factorial
arrange~nt
1/J.l
(71
1/J.l
2
(72
H
H
H
H
H
H
H
H
H
H
H
H
H
H
L
L
H
L
H
L
H
H
H
H
H
H
H
H
L
L
L
L
H
H
L
L
H
L
H
L
H
H
H
H
L
L
L
L
H
H
H
H
H
H
L
L
H
L
H
L
H
H
H
H
L
L
L
L
L
L
L
L
H
H
L
L
H
L
H
L
L
L
L
L
H
H
H
H
H
H
H
H
H
H
L
L
H
L
H
L
L
H
H
H
H
L
H
H
H
L
L
L
L
L
H
L
L
H
H
L
L
L
L
H
L
L
H
H
H
H
H
L
L
L
L
L
L
L
H
H
H
L
L
L
L
L
L
L
H
L
L
L
L
-e
2
used for distribution variab1es a
L
L
L
~ and L indicate high and low 1eve 1s •
L
L
L
L
"
•
•
Table 11.2.
•
Factorial arrangement for case c l ' c2 - 1, 1 with arrangement for operational
variables superimposed
). '" 0.089
A - 0.1
l/pl-' 8.4
l/P l - 6.9
P1- 0 •69
Pl~0.84
k1-O.2
",6.21
1
k 1",0.73
02- 1 • 68
k·.0.2
.2
P2",0.84
(1)
(1'2- 6 .21
P1- 0 • 15
kl.-0.24
,,6.21
1
k 1-0.9
P2- 0084
(5)
P2",0.84
(9)
P2",0.84
P2",0.84
P2- 0 •84
k -O,.13
2
(2)
(6)
(10)
1/"2
°2",1.68
P2111O • 69
Pi,·0.69
"'~)'~
k 2",,0 •.24
9 ",0.69
2
(3)
(7)
P2.o·-69
(4)
(71-1.68
1/"2
-8 . . 4
Cf2·~,2l
k 111Q,.9
2
l/P l - 8.4
1/Pl - 6.9
P1-0 •616
01,,6.21
0 1,,1.68
01-6.21
kJ..-0.2
k 1-O.73
k 1-O.24
k 1-0.9
P2- 0 •84
(13)
P2-0.15
. (17)
PZ-0.75
(21)
P2-00 75
(Z5)
pz.-O.7S
P2-o·84
(14)
PZ-0.75
(18)
P2-<>·15
P2-<>·75
P2-0 • 75
P2-0 •61
1'.12",0.61
1'.12",0.61
(11)
1'.1 ",0.69
2
(15)
(19)
(23)
(Z7)
PZ.o-·69
. ~2·0.• 69
PZ-0.69
1'.12",0.&1
(8)
(12)
(16)
(20)
0
(71",1.68
0
0
1"'1 068
(22)
- - -
P2·~·61
(24)
(Z6)
-f)Z",O;61
(2'8)
. (29)
(30)
PZ·O~61
(31)
1)2",0.61
(32)
CIb
co
89
Appendix 11. 3 Parametric Values Used
Table 11.3.
Parameter values for c 1 , c2 • 1, 1
11P1
..
.e
111
112
113
114
115
116
117
118
119
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.1
.089
8.4 '
6.9
.84
.69
• 75
.61
.2
.73
.24
0.9
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
6.9
6.9
6.9
6.9
6.9
6.9
6.9
6.9
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
6.9
6.9
6.9
6.9
6.9
6.9
6.9
6.9
-::
-
-
-
::
0'1
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68 .
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
.1
.0892857
8.4
6.9
.84
.69
.75
.6160713
.2
.7392857
.2434782
0.9
k1
PI
1/P 2
.2
.2
.2
.2
.73
.73
.73
.73
.24
.24
.24
.24
.9
.9
.9
.9
.2
.2
.2
.2
.73
.73
.73
.73
.24
.24
.24
.24
.9
.9
.9
.9
.84
.84
.84
.84
.84
.84
.84
.84
.69
.69
.69
.69
.69
.69
.69
.69
.75
.75
.75
.75
.75
.75
.75
.75
.61
.61
.61
.61
.61
.61
.61
.61
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
8.4
8.4
6.9
6.9
0'2
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
1.68
6.21
k2
P2
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.2
.73
.24
.9
.84
.84
.69
.69
.84
.84
.69
.69
.84
.84
.69
.69
.84
.84
.69
.69
.75
.75
.61
.61
.75
.75
.61
.61
.75
.75
.61
.61
.75
.75
.61
.61
These are the exact figures
represented by the one to
three digit numbers in the table •
.-
90
Table 11.4.
Parameter values for c , c 2 • 1, 3
1
11ls1
_
.
_
.
131
132
133
134
135
136
137
138
139
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.1
.089
8.4
6.9
.84
.69
.75
.61
25.8
20.7
.86
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
6.9
6.9
6.9
6.9
6.9
6.9
6.9
6.9
8.4
8.4
8.4
8.4
8.4
8.4
8.4
8.4
6.9
6.9
6.9
6.9
6.9
6.9
6.9
6.9
-
°1
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
1.68
1.68
1.68
1.68
6.21
6.21
6.21
6.21
.1
- .0892857
: 8.4
: 6.9
- .84
- .69
- .75
- .6160713
:25.8
:20.7
- •86
k1
P1
1hs2
°2
k2
P2
.2
.2
.2
.2
.73
.73
.73
.73
.24
.24
.24
.24
.9
.9
.9
.9
.2
.2
.2
.2
.73
.73
.73
.73
.24
.24
.24
.24
.9
.9
.9
.9
.84
.84
.84
.84
.84
.84
.84
.84
.69
.69
.69
.69
.69
.69
.69
.69
.75
.75
.75
.75
.75
.75
.75
.75
.61
.61
.61
.61
.61
.61
.61
.61
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
25.8
25.8
20.7
20.7
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
5.16
18.63
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.2
.72
.25
.9
.86
.86
.69
.69
.86
.86
.69
.69
.86
.86
.69
.69
.86
.86
.69
.69
.76
.76
.61
.61
.76
.76
.61
.61
.76
.76
.61
.61
.76
.76
.61
.61
.76
.2
.73
.9
.24
.72
.25
-
-
-
-
-
.7678570
.2
.7392857
.9
.2434782
.7220930
.2492753
These are the exact figures represented by the one to three digit
numbers in the table •
.e
91
Table 11.5.
Parameter values for c l ' c2 • 3, 1
.e
93
Table 11. 7.
Parameter values for c 1 , c2 • 3, 5
•
.
e
351
352
353
354
355
356
357
358
359
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.1
.090
25.8
2{).7
.86
.69
.76
.61
43.5
e
•
-
11111
°1
k1
PI
1/112
°2
k2
P2
25.8
25.8
25.8
25.8
25.8
25.8
25.8
25.8
20'.7
20.7
20.7
20.7
20.7
20.7
20.7
20.7
25.8
25.8
25.8
25.8
25.8
25.8
25.8
25.8
20.7
20.7
20.7
20.7
20.7
20.7
20.7
20.7
5.16
5.16
5.16
5.16
18.,63
18.63
"18.63
"18.63
5.16
5.16
5.16
5.16
18.63
18.63
18.63
18.63
5.16
5.16
5.16
5.16
18.63
18.63
18.63
18.63
5.16
5.16
5.16
5.16
18.63
18.63
18.63
18.63
.2
.2
.2
.2
.72
.72
.72
.72
.24
.24
.24
.24
.9
.9
.9
.9
.2
.2
.2
.2
.72
.72
.72
.72
.24
.24
.24
.24
.9
.9
.9
.9
.86
.86
.86
.86
.86
.86
.86
.86
.69
.69
.69
.69
.69
.69
.69
.69
.76
.76
.76
.76
.76
.76
.76
.76
.61
.61
.61
.61
.61
.61
.61
.61
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.87
.87
.69
.69
.87
.87
.69
.69
.87
.87
.69
.69
.87
.87
.69
.69
.1
= .0892857
= 25.8
= 20.7
-
.86
.69
.7678570
.6160713
= 43.5
34.5
.87
.77
.2
.72
.24
.9
.71
.252
= 34.5
-
-
-
=
.87
.7767855
.2
.7220930
.2492753
.9
.7137931
.2521739
.77
.77
.61
.61
.77
.77
.61
.61
.77
.77
.61
.61
.77
.77
.61
.61
94
Table 11.8.
Parameter values for c p c2 • 4, 4
•
.
e
1111
441
442
443
444
445
446
447
448
449
4410
4411
4412
4413
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
4427
4428
4429
4430
4431
4432
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
1
34.6
34.6
34.6
34.6
34.6
34.6
34.6
34.6
27.6
27.6
27.6
27.6
27.6
27.6
27.6
27.6
34.6
34.6
34.6
34.6
34.6
34.6
34.6
34.6
27.6
27.6
27.6
27.6
27.6
27.6
27.6
27.6
(11
6.92
6.92
6.92
6.92
24.84
24.84
24.84
24.84
6.92
6.92
6.92
6.92
24.84
24.84
24.84
24.84
6.92
6.92
6.92
6.92
24.84
24.84
24.84
24.84
6.92
6.92
6.92
6.92
24.84
24.84
24.84
24.84
.1
- .1
.089 ::
.0892857
34.6
:: 34.6
27.6
:: 27.6
.86 .865
.69 .69
e
..
k1
P1
.2
.2
.2
.2
.71
.71
.71
.71
.25
.25
.25
.25
.9
.9
.9
.9
.2
.2
.2
.2
.71
.71
.71
.71
.25
.25
.25
.25
.9
.9
.9
.9
.86
.86
.86
.86
.86
.86
.86
.86
.69
.69
.69
.69
.69
.69
.69
.69
.77
.77
.77
.77
.77
.77
.77
.77
.61
.61
.61
.61
.61
.61
.61
.61
1/11 2
(12
34.6
6.92
34.6
24.84
27.6
6.92
27.6
24.84
34.6
6.92
34.6
24.84
27.6
6.92
27.6
24.84
34.6
6.92
34.6
24.84
27.6
6.92
27.6
24.84
34.6
6.92
34.6
24.84
27.6
6.92
27.6
24.84
6.92
34.6
34.6
24.84
27.6
6.92
27.6
24.84
34.6
6.92
34.6 _24.84
27.6
6.92
27.6
24.84
34.6
6.92
34.6
24.84
27.6
6.92
27.6
24.84
6.92
34.6
34.6
24.84
27.6
6.92
27.6
24.84
-
.7723213
.77
.616 :: .6160713
.2
.2
.71 - .7179190
.25 - .2507246
.9
.9
-
k2
P2
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.86
.86
.69
.69
.86
.86
.69
.69
.86
.86
.69
.69
.86
.86
.69
.69
.77
.77
.61
.61
.77
.77
.61
.61
.77
.77
.61
.61
.77
.77
.61
.61
95
Table 11.9.
Parameter values for c1' c2 • 5, 3
·e
96
Table 11.10.
Parameterva1uea for c 1 , c2 • 5, 5
1111
.
e
551
552
553
554
555
556
557
558
559
5510
5511
5512
5513
5514
5515
5516
5517
5518
5519
5520
5521
5522
5523
5524
5525
5526
5527
5528
5529
5530
5531
5532
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.1
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
.089
1
43.5
43.5
43.5
43.5
43.5
43.5
43.5
43.5
34.5
34.5
34.5
34.5
34.5
34.5
34.5
34.5
43.5
43.5
43.5
43.5
43.5
43.5
43.5
43.5
34.5
34.5
34.5
34.5
34.5
34.5
34.5
34.5
a1
k1
PI
11112
a2
k2
P2
8.7
8.7
8.7
8.7
31.05
31.05
31.05
31.05
8.7
8.7
8.7
8.7
31.05
31.05
31.05
31.05
8.7
8.7
8.7
8.7
31.05
31.05
31.05
31.05
8.7
8.7
8.7
8.7
31.05
31.05
31.05
31.05
.2
.2
.2
.2
.71
.71
.71
.71
.25
.25
.25
.25
.9
.9
.9
.9
.2
.2
.2
.2
.71
.71
.71
.71
.25
.25
.25
.25
.9
.9
.9
.9
.87
.87
.87
.87
.87
.87
.87
.87
.69
.69
.69
.69
.69
.69
.69
.69
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
43.5
43.5
34.5
34.5
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
8.7
31.05
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.2
.71
.25
.9
.87
.87
.69
.69
.87
.87
.69
.69
.87
.87
.69
.69
.87
.87
.69
.69
.1
- .1
.089 ::
.0892857
:: 43.5
43.5
:: 34.5
34.5
.87 .87
.69
.69 -
.77
.77
.71
.77
.77
.71
.77
.77
.61
.61
.61
.61
.61
.61
.61
.61
.77
.61
.2
.25
.71
.9
-
-
-
-
.1767855
.6160713
.2
.2521739
.7137931
.9
.77
.77
.61
.61
.77
.71
.61
.61
.77
.77
.61
.61
.71
.77
.61
.61
.e
98
Table 11.11 (continued)
Table
L
M
N
0
p
.
e
Q
R
Ce11 Numbe rs
9
1
10
2
27
19
28
20
11
3
PI vs. k 2 for
12
4
at P1 • 0.69, 0.84
19
23
17
21
3
7
1
5
27
31
25
29
11
15
9
13
P1 vs. k 2
at P1 - 0.69, 0.84
PI vs. k 2
at PI • 0.61, 0.75
P2 vs. k 1
at P2 - 0.61, 0.75
P2 vs. k 1
at P2 • 0.69, 0.84
P2 vs. k 1
at P2 • 0.61, 0.75
P2 vs. k 1
at P2 - 0.69, 0.84
P2 • 0.84 k 1 • 0.22
k 2 • 0.2, 0.73
P2 • 0.616 k 1 • 0.22
k 2 • 0.24, 0.9
P2 • 0.69 k 1 • 0.22
k 2 • 0.24, 0.9
P1 • 0.75 k 2 -0.22
k 1 • 0.2, 0.73
P1 • 0.84 k 2 • 0.22
k1 •
0~2,
0.73
P1 • 0.61 k 2 • 0.22
k
1
- 0.24, 0.9
P1 • 0.69 k 2 • 0.22
k 1 • 0.24, 0.9
aA-H, 8 cases, give k 1 vs. k 2 ; I, J, 2 cases, give P1 vs. P2 ;
K-N, 4 cases, give P1 vs. k 2 ; O-R, 4 cases, give P2 vs. k t •
.e
.e
99
Table 11.12.
Cell
Number a
1
2
3
4
5
6
7
8
9
10
11
.
e
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Example of observations used in interaction tables
Interaction Table
Involved
HJ L P
HL
DJ NP
DN
HP
H
DP
D
GJ L R
GL
CJ NR
CN
GR
G
CR
C
F I K0
F K
B I M0
BM
F 0
F
B0
B
ElK Q
EK
A I MQ
AM
EQ
E
AQ
A
Observed Values
L
(1,1)
Lq2
q1
,
1.9865
2.2866
2.4677
2.3478
3.4909
3.0697
4.1488
3.6054
0.7985
0.7766
0.7775
0.7472
1.3328
1.4652
1.1937
1.2687
1.1578
1.1114
1.1265
1.1926
1.6188
1. 7577
1.6089
1.9402
0.5409
0.4959
0.5369
0.5109
0.7414
0.9095
0.7755
0.8385
0.5002
1.9358
0.0629
0.7371
1.6032
2.5253
0.4836
1.1104
1.3882
2.8197
0.2223
0.9243
2.1410
3.7410
0.6525
1.2183
0.3196
1.1040
0.0431
0.4737
0.7199
1.3836
0.2765
0.8073
0.6810
1.2987
0.1467
0.5904
0.9241
1.7764
0.4085
0.8054
~very cell used at least once, 24 cells used twice, 8 cells used
four times.
.e
.-
100
Table 11.13.
Examples of interactions for L
(1, 1)
q2
Table
_
.
_
•
Difference
Data
Interaction
A
0.1467
0.5904
0.4085
0.8054
0.2618
0.2150
-0.0468
B
0.0431
0.4737
0.2765
0.8073
0.2334
0.3336
0.1002
C
0.2223
0.9243
0.6525
1..• 2183
0.4302
0.2940
-0.1362
D
0.0629
0.7371
0.4836
1.1104
0.4207
0.3733
-0.1474
E
0.6810
1.2987
0.9241
1.7164
0.2431
0.4717
0.2346
F
0.3196
1.1040
0.7199
1.3836
0.4003
0.2796
-0.1207
G
1.3882
2.8197
2.1410
3.7410
0.7528
0.9213
0.1685
H
0.5002
1.9358
1.6032
2.5253
1.1030
0.5895
-0.5135
I
0.1467
0.6810
0.0431
0.3196
-0.1036
-0.3614
0.2578
J
0.2223
1.3882
0.0629
0.5002
-0.1594
-0.8880
0.7286
K
0.6810
1.2987
0.3196
1.1040
-0.3614
-0.1947
0.1667
L
1.3882
2.8197
0.5002
1.9358
-0.8880
-0.8839
0.0041
M
0.1467
0.5904
0.0431
0.4737
-0.1036
-0.1167
-0.0131
N
0.2223
0.9243
0.0629
0.7371
-0.1594
-0.1872
-0.Q278
Table continued
.e
•
•
•
e
.
101
Table 11.13 (continued)
Data
Table
Difference
Interaction
0
0.0431
0.3196
0.2765
0.7199
0.2334
0.4003
0.1669
p
0.0629
0.5002
0.4836
1.6032
0.4207
1.1030
0.6823
Q
0.1467
0.6810
0.4085
0.9241
0.2618
0.2431
-0.0187
R
0.2223
1.3882
0.6525
2.1410
0.4302
0.7528
0.3226
.e
102
Table 11.14.
Summary of Lq
(1, 1) interaction calculations
2
Measured
Table Interaction
A
-0.0468
B
0.1002
C
-0.1362
D
-0.1474
0.616
0.24
0.9
0.24
0.9
0.75
0.616
0.2
0.73
0.24
0.9
0.69
0.69
0.24
0.9
0.24
0.9
0.69
0.2
0.73
0.24
0.9
k 1 vs. k 2 0.616
0.84
n
.e
k2
k1
P2
PI
~:'1i
B;,
E
0.2346
0.616
0.75
0.24
0.9
0.2
0.73
F
-0.1207
0.75
0.75
0.2
0.73
0.2
0.73
G
0.1685
0.69
0.84
0.24
0.9
0.2
0.73
H
-0.5135
0.84
0.84
0.2
0.73
0.2
0.73
I
0.2578
PI vs. P2 0.61
0.75
0.61
0.75
0.22
0.22
J
0.7286
0.69
0.84
0.69
0.84
0.22
0.22
K
0.1667
PI vs. k 2 0.61
0.75
0.75
0.22
0.2
0.73
L
0.0041
0.69
0.84
0.84
0.22
0.2
0'-73
M
-0.0131
0.61
0.75
0.61
0.22
0.24
0.9
N
-0.0278
0.69
0.84
0.69
0.22
0.24
0.9
0
0.1669
p
k 1 0.75
0.61
0.75
0.2
0.73
0.22
0.6823
0.84
0.69
0.84
0.2
0.73
0.22
Q
-0.0187
0.61
0.61
0.75
0.24
0.9
0.22
R
0.3226
0.69
0.69
0.84
0.24
0.9
0.22
P2
VS.
.e
103
Table 11.15.
X2
.e
Xl • 1
List of simple algebraic forms and associated interaction
tables
Xl • 2
1
1
2
2
2
4
1
1
2
2
0.5
1
1
1
0.5
2
2
1
,1
1
4
2
2
8
1
1
2
2
4
8
1
1
0.5
2
0.5
0.25
1
1
4
2
4
16
1
0
-1
2
0
-2
1
1
0.5
4
2
Algebraic
Form
X1X2
X1 !X2
X2!X1
xi~
X1X2
2
1!X1~
x21X22
(l-~)~
2
X !X
2 1
~
Xl • 0.6
Xl • 0.8
0.6
0.36
0.48
0.8
0.48
0.64
0.6
1
1.33
0.8
0.75
1
0.6
1
0.75
0.8
1.33
1
0.216
0.384
0.8
0.288
0.512
0.6
0.216
0.288
0.8
0.384
0.512
0.6
2.78
2.08
0.8
2.08
1.56
0.6
0.13
0.23
0.8
0.23
0.41
0.6
0.24
0.12
0.8
0.32
0.16
0.6
0.6
0.45
0.8
1.067
0.8
0.6
•
.e
104
Table 11.16.
Observed
Frequency
Expected
Frequency
(Geometric)
Lq1
Observed
Frequency
Expected
Frequency
(Geometric)
0
45
31.219
16
3
3.660
1
26
27.305
17
3
3.202
2
20
23.881
18
6
2.800
3
22
20.887
19
4
2.449
4
9
18.268
20
3
2.142
5
15
15.978
21
5
1.873
6
19
13.975
22
3
1.639
7
9
12.222
23
4
1.433
8
5
10.690
24
1
1.253
9
12
9.350
25
1
1.096
10
4
8.177
26
7
0.959
11
7
7.152
27
1
0.839
12
3
6.255
28
2
0.733
13
6
5.471
29
0
0.641
14
2
4.785
30
0
0.561
15
1
4.185
31
1
0.491
•
Lql
•
e
Sampled queue length distribution
a
atf = 249; average Lq1 • 6.9759; Po • 0.87462; chi-square, 17 dof.
• 34.235, significant at the P • 0.01 level •
•
.e
105
.e
Appendix 11.5
The Distribution of Queue Length
Let qn represent the queue length at the instant customer number n
finishes service, and r
represent the number of arrivals during the
n
time the nth customer is in service.
random variables.
The r
n
These quentities qn and r
n
are
are independent and identically distributed
and are independent of the qo' q1' ••• qn-1.
Define a sequence of
random variables Iqn l such that
if
qn - 0 •
Using Heaviside 's Unit Function
6(X) - 1 (X
>
0),
6(X) - 0 (X
~
0)
we have, after Cox and Smith (1961)
Now if the waiting time of the nth customer is w then q is the
n
n
number of customers arriving during wn and E(qn) -
AE(~n).
In order to
get the distribution of queue lengths we establish the generating function
q -6(qn)+r 1
M(8) _ E(8 n
n+ )
qn-6( qn)
- t(8) E(8
)
since r n+1 is independent of qn.
E(8
qn-6(~)
) -
Wo
+
However
•
n-1
t wn 8
n-1
_ (l _ p)
+
Me.e)
-
e
11 - p)
.e
106
Substituting this into the prior expression we obtain
M(e) _ (1 - p) ( 1 - e) tee)
tee) - e
The probability generating function of r
•
tee) -
(e)
t
k-O
- f•
k
n
is
prob (r - k)
n
e-Ax(l-e) d B (x)
o
- B*(A - Ae)
where
f
B*(s) -
o
•
e- st d B (t)
is the Laplace Stie1tjes transform of the distribution of service time.
Supposing the service times were negative exponential, then
tee) _ ~_.;:.1~--=,-:1 + pel - 0)
Now substituting in M(a) above
B*(s) -
p
1",.
+ s
and
M(O) _ (1 - P)(l - e) I [1 + pC1 - 8)]
1
1
+ P (l - 0) - 0
_ 1 - P
1 - p6
and this leads to
'If
n
-
pn (l - p)
which is the geometric distribution.
It is sufficient for our purpose to point out that with the delayed
e
.
negative exponential distribution we no longer have B*(s) -
p
p
+ s
as
.e
107
for the exponential, but rather
1
(a - ;)s
B*(s) • ,:,e
_
1 + sa
which will obviously not lead to the geometric distribution.
e
.
108
.e
Appendix 11.6
Table 11.17.
Simulation Results--gueue Lengths
Selected simulation results for case c , c2 • 1, 1 for
1
first rank queue length, L
q1
1 • .1
1/11 1 • 8.4
1/111 • 6.9
al • 1.68
1/11 2 •
a2 •
8.4
1.68
•
a2 •
6.21
cell 1
e
cell 5
2.5291
1.6416
1.5590
1.6690
1.2578
1.8841
2.9600
2.1486
2.0549
1.9805
a1 • 1.68
cell 9
5.4371
2.4887
3.1764
2.7591
2.1468
3.1815
3.3042
1. 7283
6.7185
3.9686
Avg. 1.96846
Avg.
cell 2
cell 6
3.49092
Avg.
Avg.
0.79851
cell 10
3.06971
a1 • 6.21
cell 13
0.8195
0.8944
0.6666
0.8193
0.6924
0.8742
0.8589
0.7818
0.7313
0.8467
. 3.6637
2.6346
2.1673
5.5839
1.8991
2.5356
4.8583
2.5963
2.3740
2.3843
2.7951
3.7188
2.3406
1.9605
1.4950
2.1801
2.0690
2.5155
2.0700
1.7202
Avg. 2.28648
.
a 1 • 6.21
1.5478
1.3977
1.4319
1.0379
1.2517
1.6373
1.1566
1. 7050
1.1250
1.0366
Avg.
cell 14
0.7563
0.6545
0.6986
0.9509
0.6037
0.7457
0.8484
0.7095
0.8179
0.9808
Avg.
0.77663
1.33275
1.5643
1.2059
1.4838
1.6458
1.5375
1.2871
1.8036
1.2078
1.3105
1.6Q55
Avg.
1.4651 8
·-
109
Table 11.18.
Average length, first rank queue
•
~e
_
.
..
Case c 1 ,c2 • 1,1
1,3
3,1
3,3
1
2
3
4
5
1. 98647
2.28658
2.46770
2.34717
3.49094
2.26504
2.06858
2.36426
2.41815
2.91495
2.44874
2.40546
2.48587
2.28270
3.29560
2.40377
2.45481
2.42918
2.41485
3.63499
6
7
8
9
10
3.06972
4.14879
3.6Q537
0.79.851
0.17664
3.64262
2.97311
3.32529
0.83463
0.78871
2.88872
4.01048
3.69413
0.57314
0.51576
4.11261
3.38935
3.17585
0.50261
0.57357
11
12
13
14
15
0.17753
0.74722
1.33217
1.46519
1.19367
0.80238
0.78294
1.34790
1.42587
1.25248
0.59563
0.65979
0.87483
1.04323
1.05539
0.54542
0.58645
0.98274
0.99177
1.07217
16
17
18
19
20
1.26870
1.15777
1.11142
1.12648
1.19265
1.25533
1.13395
1.10726
1.10066
0.83455
1.00616
1.01058
1.24934
1.00227
0.17642
1.07121
1.06886
1.18096
1.11184
21
22
23
24
25
1. 61880
1. 75765
1.60885
1.94023
0.54088
1.91419
1.78925
1. 77170
1.97120
1.54869
1.35755
1.28215
1.31298
1.37445
0.33446
1.24186
1. 71702
1.34598
1.42936
0.32320
26
27
28
29
30
0.49593
0.53694
0.51094
0.74136
0.90947
0.52039
0.52081
0.50682
0.85514
0.80186
0.30852
0.28102
0.33546
0.42887
0.51472
0.32057
0.32754
0.33753
0.67679
0.50253
31
32
0.17550
0.83846
0.81454
0.83691
0.45234
0.48727
0.50878
0.61672
1.070~1
Table continued
.e
110
Table 11.18 (continued)
Case c1 ,c2 • 3,5
4,4
5,3
5,5
5
2.44404
2.68649
2.23931
2.25525
3.65648
2.52836
2.44654
2.46827
1. 98694
2.50925
3.00650
2.43655
2.08963
2.78291
3.41909
2.28292
2.80345
2.48054
2.12652
3.40570
6
7
8
9
10
3.81167
3.88222
3.36472
0.57232
0.67032
3.51403
3.53507
3.48973
0.54637
0.55645
4.03133
4.17240
3.62632
0.48334
0.46808
3.23433
3.26549
3.17703
0.39624
0.49241
11
12
13
14
15
0.57980
0.54204
0.96375
1.03691
1.02990
0.53924
0.46415
0.78938
0.84377
0.86985
0.50314
0.45640
0.66052
0.73090
0.72365
0.43979
0.46694
0.73202
0.64815
0.69043
16
17
18
19
20
0.92982
1.12714
0.95937
1.19653
1.02320
0.77533
1.06335
0.78263
0.87012
0.86198
0.67164
1.06403
0.93183
0.94105
0.96027
0.68875
1.03584
0.94470
0.95526
0.99877
21
22
23
24
25
1. 39783
1. 73892
1. 37632
1.66036
0.34060
1.27050
1.49154
1.43336
1.60248
0.26980
1.27572
1.14772
1.44726
1.42564
0.20125
1.53946
1.26545
1.20929
1.22053
0.21753
26
27
28
29
30
0.36377
0.30518
0.31571
0.45795
0.52646
0.27077
0.24600
0.25118
0.47066
0.49861
0.21240
0.25440
0.23786
0.30428
0.39858
0.24969
0.22813
0.22842
0.46662
0.35359
31
32
0.49847
0.50077
0.49694
0.38923
0.38320
0.37006
0.39361
0.46535
1
2
3
4
--
111
Table 11.19.
~e
e
.
Average length, second rank queue-
Case c 1 ,c 2 - 1,1
1,3
3,1
3,3
1
2
3
4
5
0.50024
1. 93584
0.06292
0.73712
1.60318
0.89352
2.47610
0.05305
0.47031
1. 44447
0.72460
2.34581
0.24011
0.77681
1.56929
0.71163
2.01270
0.05809
0.48271
1.55584
6
7
8
9
10
2.52531
0.48359
1.11044
1.38817
2.81966
2.77546
0.32144
0.72277
1. 98939
2.85543
2.50368
0.50125
1.02544
1.86088
2.62370
3.08529
0.32130
0.77416
1.55416
2.74085
11
12
13
14
15
0.22234
0.92428
2.14099
3.74099
0.65247
0.22417
0.75384
2.02822
3.19717
0.49979
0.40824
1.13045
2.20825
3.65319
0.71836
0.21722
0.67440
2.11411
3.32164
0.59798
16
17
18
19
20
1.21828
0.31956
1.10404
0.04306
0.47369
0.83305
0.31263
0.87256
0.03407
0.26902
1.29400
0.43951
1.18619
0.20530
0.52493
0.75841
0.33445
1.02751
0.03911
0.30210
21
22
23
24
25
0.71994
1.38359
0.27650
0.80731
0.68099
0.78698
1.18917
0.16888
0.40772
0.83687
0.83424
1.35035
0.32750
0.69764
0.85759
0.69532
1.35586
0.17110
0.39940
0.76989
26
27
28
29
30
1.29866
0.14673
0.59037
0.92406
1.77637
1.16768
0.11567
0.33344
0.91620
1.39340
1.33055
0.26039
0.70634
0.92898
1.54868
1.33364
0.13128
0.37827
1.00470
1.26429
31
32
0.40851
0.80545
0.26677
0.47848
0.48131
0.82706
0.29823
0.44815
Table continued
112
Table 11.19 (continued)
-e
Case c 1 ,c2 - 3,5
4,4
5,3
5,5
1
2
3
4
5
0.76734
2.56620
0.04183
0.30451
1.64589
0.70615
2.15585
0.04883
0.37390
1.38946
0.85024
1.76896
0.14854
0.43135
1.59547
0.73944
1.95179
0.005196
0.36565
1.86414
6
7
8
9
10
4.03608
0.23092
0.48499
1.63852
3.43074
3.12957
0.27033
0.61933
2.19073
3.22489
3.74872
0.32716
0.73656
2.27107
3.52771
2.21098
0.22786
0.46343
1. 79217
4.71960
11
12
13
14
15
0.18574
0.44161
2.02396
3.59035
0.43739
0.20180
0.74545
2.34464
3.67293
0.44272
0.33650
0.67683
2.39104
4.15804
0.50500
0.19234
0.54933
2-.30942
3.38430
0.40908
16
17
18
19
20
0.61432
0.34602
0.96882
0.03041
0.16802
0.76482
0.33807
0.66091
0.02964
0.18135
0.85635
0.45129
1.08088
0.10705
0.33707
0.65704
0.32097
0.79620
0.02765
0.19249
21
22
23
24
25
0.71167
0.95785
0.10865
0.34188
0.77303
0.69446
1.22049
0.15733
0.42141
0.81784
0.72023
1.18656
0.20294
0.42314
0.77892
0.67568
0.99129
0.12291
0.23773
0.63863
26
27
28
29
30
1.37497
0.08958
0.21743
0.81746
1.35899
1.16983
0.11683
0.40607
1.09445
1.58030
1.33858
0.18656
0.39987
0.82007
1.52681
1.11696
0.09780
0.27370
1.04182
1.14731
31
32
0.20992
0.32135
0.27384
0.40224
0.29330
0.51810
0.24749
0.42891
113
Appendix 11.7
Table 11.20.
Case
c 1 ,c 2
1,1
F
-e
•
.e
Regression Analysis of First Queue
First regression analysis of L ; order of distribution
q1
variable entry into solution and F values to remove
Step 1
A/lli
83.92
Step 2
Step 3
Step 4
Step 5
Aal/lli
a1
a1/111
.A
33.46
5.76
3.34
0.59
Step 6
1111 1
44.53
llR2
0.7367
0.1411
0.0209
0.0112
0.0020
0.0565
R2
0.7367
0.8777
0.8986
0.9098
0.9118
0.9683
1,3
A/111
Aa1/111
a1
1111 1
A
F
104.87
53.89
3.77
2.13
55.41
a1 /P 1
0.29
2
llR
0.7776
0.1446
0.0093
0.0050
0.0433
0.0002
R2
0.7776
0.9222
0.9314
0.9365
0.9797
0.9799
3,1
A/lli
Aa1/111
a1 /11 1
Aa 1
A
F
117.95
11.66
9.43
2.81
0.42
1111
1
62.32
llR2
0.7972
0.0582
0.0364
0.0102
0.0016
0.0688
R2
0.7972
0.8554
0.8918
0.9020
0.9036
0.9724
3,3
A/lli
Aa 1 /11 1
a1111 1
Aa 1
A
F
112.12
15.28
6.70
2.89
0.11
1/11 1
78.47
llR2
0.7889
0.0728
0.0267
0.0108
0.0027
0.0744
R2
0.7889
0.8618
0.8885
0.8993
0.9020
0.9763
Aa
3,5
A/111
Aa 1/11 1
a1/111
F
114.79
19.47
9.13
1
6.16
1111 1
0.35
A
107.00
llR2
0.7928
0.0832
0.0305
0.0174
0.0010
0.0608
R2
0.7928
0.8760
0.9065
0.9239
0.9249
0.9858
Table continued
e
•
114
Table 11.20 (continued)
Case
c ,c 2
1
4,4
F
'e
Step 1
A/P
1
116.78
Step 2
Step 3
Step 4
A0
AC11/P1
C11 /P
14.19
4.29
1
2.35
1
Step 5
1/P
1
1.53
Step 6
A
51.91
6R2
0.7956
0.0672
0.0182
0.0095
0.0061
0.0698
R2
0.7956
0.8628
0.8810
0.8906
0.8967
0.9664
A0 1 /P1
C1 1 /P1
AC1
8.70
6.98
1
3.87
5,3
A/P 1
F
109.46
1/P1
0.60
A
88.65
6R2
0.7849
0.0496
0.0330
0.0166
0.0026
0.0883
2
R
0.7849
0.8345
0.8676
A.8842
0.8868
0.9751
5,5
A/Pl'
AC1 1 /P1
C1 1 /P
F
143.76
9.92
5.07
1
2.08
1
AC1
1/P1
0.62
A
157.78
6R2
0.8274
0.0440
0.0197
0.0078
0.0024
0.0852
R2
0.8274
0.8714
0.8911
0.8989
0.9013
0.9865
_
115
Table 11. 21.
Second regression analysis of L ; order of operational
q1
variable entry into solution and F values to remove
Step 1
Step 2
Step 3
a2 /p 2
1 1
PI
Step 4
Case
cl' c 2
1,1
F
92.97
33.86
21.14
aI/PI
0.67
2
AR
0.7560
0.1314
0.0484
0.0016
R2
0.7560
0.8875
0.9359
0.9375
1,3
p2
2 2
a 1 /P1
F
--
2
PI
1
115.80
61.03
P1
28.11
a1/ P1
0.88
2
AR
0.7942
0.1395
0.0332
0.0011
R2
0.7942
0.9337
0.9669
0.9680
3,1
2
P1
PI
2 2
a 1 / P1
F
138.11
15.61
22.54
a 1/ P
1
0.13
AR2
0.8215
0.0625
0.0517
0.0003
R2
0.8215
0.8840
0.9357
0.9361
3,3
2
PI
PI
2· 2
aI/PI
F
131.14
16.16
35.92
a 1 /P1
0.16
IlR2
0.8138
0.0666
0.0672
0.0003
R2
0.8138
0.8805
0.9476
0.9480
Table continued
116
Table 11.21 (continued)
Step 1
Step 2
Step 3
Step 4
2 2
°1 / 1.11
31.96
°l!l.ll
1.22
Case
c ,c 2
1
3,5
F
132.81
PI
19.52
2
AR
0.8157
0.0741
0.0587
0.0022
R2
0.8157
0.8899
0.9486
0.9508
4,4
F
"-
2
PI
2
PI
136.62
PI
15.19
2 2
°1 / 1.11
33.54
°1 / 1.11
0.76
2
AR
0.8200
0.0619
0.0644
0.0015
R2
0.8200
0.8819
0.9462
0.9477
PI
°1 /P 1
1.36
5,3
F
2
PI
130.13
2 2
°1 / 1.1 1
22.83
22.06
AR2
0.8127
0.0825
0.0462
0.0028
R2
0.8127
0.8952
0.9414
0.9442
5,5
F
2
PI
176.20
PI
30.09
2 2
°1 /P 1
38.11
AR2
0.8545
0.0741
0.0412
R2
0.8545
0.9286
0.9698
°1 / 1.1 1
0.54
0.0006
0.9704
·e
117
Table 11.22.
Third regression analysis of L ; order of operational
ql
variable entry into solution and F values to remove
Step 1
Step 2
Step 3
Step 4
2
PI
k 2p 2
1 1
38.68
PI
2
k 1P1
Case
c1 ,c 2
1,1
F
20.77
22.56
2
AR
2
R
- 0.7560
0.1394
0.0445
0.0273
0.7560
0.8955
0.9400
0.9673
1,3
2
PI
115.80
F
"e
92.97
2 2
k1Pl
60.90
PI
22.96
2
k 1Pl
16.65
AR2
0.7942
0.1394
0.0299
0.0139
R2
0.7942
0.9336
0.9635
0.9774
2 2
k 1P1
2
k 1P1
. 3,1
F
2
PI
138.11
PI
15.61
31.40
13.26
AR2
0.8215
0.0625
0.0613
0.0180
R2
0.8215
0.8840
0.9453
0.9633
3,3
F
p2
1
131.14
2 2
k 1P
1
19.33
PI
40.90
2
k 1P1
16.41
2
AR
0.8138
0.0745
0.0663
0.0172
R2
0.8138
0.8883
0.9546
0.9718
Table continued
e
..
_
118
Table 11.22 (continued)
Step 1
Step 2
Step 3
Step 4
2 2
k 1P1
PI
2
k1P1
24.12
33.14
Case
c 1 ,c2
3,5
F
40.68
2
I>.R
0.8174
0.0829
0.0540
0.0274
R2
0.8174
0.9003
0.9544
0.9818
4,4
F
"-
2
PI
134.28
2
I>.R
R2
5,3
F
2
PI
136.62
2 2
k 1P1
PI
16.92
33.02
k 2P1
1
10.44
0.8200
0.0663
0.0615
0.0146
0.8200
0.8863
0.9478
0.9624
2
P
1
130.13
PI
22.83
2 2
k1P
1
25.35
2
k 1Pl
20.11
I>.R2
0.8127
0.0825
0.0498
0.0235
R2
0.8127
0.8952
0.9450
0.9685
5,5
F
2
PI
176.20
PI
30.09
k 2P12
1
45.15
2
k P
1 1
18.37
I>.R2
0.8545
0.0741
0.0441
0.0111
R2
0.8545
0.9286
0.9727
0.9837
.e
119
Table 11. 23.
Third regression analysis of L • coefficients and
Q1 '
their standard errors
22222
LQ1 • So + Sl Pl + S2 P1 + S3k l + S4k l PI + S5 k l PI
•
So
S2
S5
SI
1,1
Step 1
Std. error
-2.1585
6.9276
0.7184
Step 2
Std. error
-2.4642
6.7356
0.4793
2.1920
0.3524
Step 3
Std. error
15.8563
41.4677
7.6299
2.1919
0.2717
-50.7735
11.1408
Step 4
Std. error
17.4798
41.4752
5.7345
10.8349
1.8307
-53.0077
8.3864
Step 1
Std. error
-1.8924
6.3737
0.5923
Step 2
Std. error
-2.1667
6.2014
0.3428
1.9673
0.2521
Step 3
Std. error
11.3126
31.7556
5.3390
1.9673
0.1902
-37.3568
7.7957
Step 4
Std. error
12.3525
31.7604
4.2760
7.5032
1.3651
-38.7878
6.2534
Step 1
Std. error
-2.4770
6.9609
0.5923
Step 2
Std. error
18.4134
45.6038
9.7900
Step 3
Std. error
18.1333
45.4006
6.8403
1.5740
0.2808
-57.1250
10.1228
Step 4
Std. error
19.4489
45.4091
5.7038
8.5429
1. 9274
-58.9141
-5.0976
8.4553
1.3994
Table continued
-6.2558
1.3168
1,3
-4.0069
0.9819
3,1
•
-57.2579
14.4883
120
Table 11.23 (continued)
2
80 + 8l Pl + 82Pl2
+ 832
k1 + 2
84k1P2
l + 85k l Pl
B
O
8
2
8
5
81
3,3
Step 1
Std. error
-2.4819
7.0513
0.6157
Step 2
Std. error
-2.7417
6.9240
0.4859
1.7655
0.4015
Step 3
Std. error
19.1635
47.4432
6.3434
1. 7616
0.2604
-60.0377
9.3875
Step 4
Std. error
20.4517
47.4517
5.0945
8.6839
1. 7215
-61.8148
7.5521
Step 1
Std. error
-2.5635
7.2366
0.6245
Step 2
Std. error
-2.8443
7.0990
0.4701
1.9076
0.3884
Step 3
Std. error
17.4070
44.5589
6.5147
1.9040
0.2675
-55.5046
9.6410
Step 4
Std. error
19.0993
44.5698
4.1901
10.8685
1.4159
-57.8059
6.2114
Step 1
Std. error
-2.3635
6.5765
0.5626
Step 2
Std. error
-2.6005
6.4669
0.4555
1.5921
0.3870
Step 3
Std. error
17 .0208
42.4343
6.2855
1.5876
0.2668
-53.6330
9.3326
Step 4
18.1489
42.5513
5.4353
7.5551
1.8609
- 4.3763
1.3541
Table continued
-5.0634
1.2499
3,5
-6.5573
1.0280
4,4
121
Table 11.23 (continued)
Case
c1 ,c
2
2
2
2 2
2
Lq1 • So + Sl P1 + S2 P1 + S3k 1 + S4k 1Pi + S5k 1P1
81
So
S2
·Step 1
Std. error
-2.8974
7.5891
0.6652
Step 2
Std. error
23.4654
55.7733
10.0955
Step 3
Std. error
23.1475
55.5360
7.4428
1.6434
0.3264
-71.6783
11.0875
Step 4
Std. error
24.8227
55.5483
5.7379
10.4948
1.9897
-73.9.429
8.5626
Step 1
Std. error
-2.4649
6.6554
0.5014
Step 2
Std. error
18.8970
45.6993
7.1259
Step 3
Std. error
18.6413
45.5084
4.4867
1.3220
0.1967
-58.0825
6.6839
Step 4
Std. error
19.6249
45.5156
3.5245
6.5192
1.2222
-59.4121
5.2595
S5
S4
5,3
"_
e
.
-71.8709
15.0394
-6.5078
1.4512
5,5
-58.2374
10.6156
-3.8211
0.8914
·-
.-
_
.
122
Table 11.24.
Third regression analysis of L ; coefficients
q1
Case
130
131 (P1)
2
13 2 (P1)
2
13 (k 1P )
4
1
2 2
135 (k 1P1)
1,1
17.4798
-53.0077
41.4752
-6.2558
10.8349
1,3
12.3525
-38.7878
31.7604
-4.0069
7.5032
1,X
14.9162.
-45.8977
36.6178
-5.1313
9.1690
3,1
19.4489
-58.9141
45.4091
-5.0976
8.5429
3,3
20.4703
-61.8148
47.4517
-5.0634
8.6839
3,5
19.0993
-57.8059
44.5698
-6.5573
10.8685
3,X
19.6728
-59.5116
45.8102
-5.5725
9.1217
4,4
18.1489
-55.1625
42.5513
-4.3763
7.5551
5,3
24.8227
-73.9429
55.5483
-6.5078
10.4948
5,5
19.6249
-59.4121
45.5156
-3.8211
6.5192
5,X
22.2238
-66.6775
55.5319
-5.1644
8.5070
.e
'e
123
Table 11.23.
ql
,case 1, 1
Residual a
Cell Number by Order
of Queue Length
Queue Length Observed
(Abbreviated)
26
28
27
25
29
0.495
0.510
0.536
0.540
0.741
-0.0843
-0.0692
-0.0432
-0.0393
-0.0327
12
31
10
11
9
0.747
0.775
0.776
0.777
0.798
0.0465
0.0013
0.0759
1>.0768
0.0977
32
30
18
19
17
0.838
0.909
1.111
1.126
1.157
0.0643
0.1353
0.0015
0.0165
0.0478
20
15
16
14
1.192
1.193
1.268
1.332
1.465
0.0827
-0.1391
-0.0641
-0.0000
0.1323
23
21
22
24
1
1.608
1.618
1. 757
1.940
1.986
-0.2116
-0.2016
-0.0628
0.1197
':'0.3273
2
4
3
6
5
2.286
2.347
2.467
3.069
3.490
-0.0272
0.0339
0.1538
-0.4548
-0.0336
8
7
3.605
4.148
0.0807
0.6242
13
e
Compilation of residuals for L
~esting the means of the top half of the residuals against the
bottom half, t • 0.756 which is not significant at the 20 percent level.
--
124
Appendix 11.8
Table 11.26.
-_
Regression Analysis of Second Queue
Analysis of variance for factorial arrangement L for
q2
case 1, 1
Source of
Variation
Sum of
Squares
A
3.31982
A, lIllI' 1/11 2
0.13715
1111 1
0.99828
0.01709
°1
1.67872
A, 1111 1 , °2
A, °1,1/11 2
1/11 2
7.89888
A, 01' °2
0.01513
°2
5.02297
A, 1/P2' °2
0.09077
A, 1/P 1
0.21864
1/P 1 , °1 ' 1/P2
1/P 1 , ~1' °2
0.00104
A, °1
A, 1/P
2
Source of
Variation
Sum of
Squares
0.06953
0.02009
0.18606
1/P 1 , 1/11 2 , 02
0.02257
1.35603
01' 1/11 2 , 02
0.00032
A, °2
0.32762
0.00020
1111 1 , °1
0.00059
A, 1/P , °1' 1/11
2
1
A, 1111 1 , °1' °2
1111 1 , 1/11 2
0.42883
0.00995
lIllI' a2
0.00916
A, 1111 1 , 1/P 2 , °2
A, aI' l/1l2' a
2
a1' 1/p2
0.15197
l/P 1 , ai' 1/P2' °2
0.05065
°1' °2
0.00408
A, lllll' aI' l/P2' a2
0.00225
1/11 2 ,a 2
0.48425
Total
A, l/P , a l
1
0.00005
0.00462
0.00154
22.52942
125
Table 11.27.
·e
.e
Analysis of variance of factorial arrangement for L
q2
for all 8 cases, list of mean squares a
Variable
1,1
1
3.319
4.491
2
0.998
3
1,3
3,1
3,3
3,5
4,4
5,3
5,5
3.834
3.799
5.817
5.053
6.087
5.721
0.687
0.974
0.572
0.454
1.554
1.307
1.884
1.678
0.444
0.734
0.910
0.646
0.816
0.884
0.210
4
7.898
11.499
7.841
11.080
16.215
13.695
14.752
13.985
5
5.022
2.704
3.753
2.992
3.864
2.887
3.598
2.380
6
0.218
0.096
0.251
0.087
0.017
0.234
0.442
0.635
7
0.186
0.006
0.110
0.237
0.249
0.023
0.339
0.002
8
1.356
2.194
1.509
1.802
3.875
2.976
3.684
3.750
9
0.327
0.489
0.337
0.417
L.0-78
0.653
0.558
0.691
10
0.000
0.037
0.000
0.059
0.138
0.090
0.153
0.138
11
0.428
0.207
0.200
0.175
0.151
0.646
0.591
0.922
12
0.009
0.033
0.001
0.054
0.018
0.013
0.000
0.175
13
0.151
0.024
1.080
0.184
0.125
0.213
0.252
0.007
14
0.004
0.000
0.002
0.002
0.000
0.014
0.109
0.378
15
0.484
0.534
0.335
0.787
2.000
0.739
1.195
0.827
16
0.000
0.000
0.009
0.003
0.060
0.020
0.125
0.151
17
0.137
0.025
0.108
0.040
0.000
0.109
0.272
0.511
18
0.017
0.012
0.000
0.000
0.034
0.012
0.001
0.215
19
0.069
0.000
0.043
0.085
0.150
0.006
0.187
0.010
20
0.015
0.000
0.000
0.006
0.012
0.000
0.112
0.169
21
0.090
0.180
0.073
0.186
0.843
0.311
0.327
0.420
22
0.001
0.020
0.000
0.046
0.145
0.023
0.128
0.200
23
0.020
0.013
0.066
0.023
0.014
0.010
0.008
0.043
Table continued
126
.e
Table 11.27 (continued)
•
Variable
1,1
24
,
1,3
3,1
3,3
3,5
4,4
5,3
5,5
0.022
0.033
0.022
0.011
0.003
0.033
0.000
0.138
25
0.000
0.002
0.000
0.003
0.000
0.042
0.078
0.284
26
0.000
0.001
0.016
0.000
0.076
0.021
0.085
0.121
27
0.004
0.000
0.031
0.001
0.037
0.000
0.042
0.023
28
0.009
0.020
0.003
0.002
0.041
0.031
0.001
0.223
29
0.001
0.002
0.001
0.016
0.029
0.004
0.082
0.130
30
0.050
0.034
0.128
0.000
0.003
0.006
0.003
0.046
31
0.002
0.005
0.029
0.003
0.043
0.000
0.025
0.014
Total
22.529
23.806
20.503
23.598
36.151
30.247
35.441
34.418
~e
aOrder of variables as in Table 11.26.
e
.
.e
•
127
Order of mean square in factorial analysis of variance for
Table 11.28.
L
q
2
Variable
No.
1,1
1,3
3,1
3,3
3,5
4,4
5,3
5,5
1
3
2
2
2
2
2
2
2
A
2
6
5
5
7
9
5
5
5
lllli
3
4
7
6
5
8
6
7
4
1
1
1
1
1
1
1
1
1/11 2
5
2
3
3
3
4
4
4
4
(12
6
10
11
9
13
11
10
9
1/11 1 , A
7
11
12
9
10
8
5
4
4
4
3
3
3
3
I/P2' A
9
9
8
7
8
6
8
9
8
(12' A
15
14
14
12
11
9
10
12
Variable
(11
11
(11' A
11P , (11
1
•
11
8
12
13
9
10
8
15
12
14
11
15
12
14
(11 ' I/P 2
12
7
6
8
6
5
7
6
7
13
13
13
13
18
19
e
..
20
(11 ' (12
1/11 2 , (12
A, 1/111 , (11
16
17
IIP 1 , 1/11 2
lIllI' (12
14
15
6
15
14
12
15
10
A, lIllI' 1 III 2
15
A, 11P 1 , (12
A, 01' 1/P2
A, 01' (12
Table continued
128
Table 11.28 (continued)
Variable
No.
1,1
1,3
3,1
3,3
3,5
4,4
5,3
5,5
Variance
21
14
10
IS
10
7
10
12
11
A, 1/112' a 2
22
13
lIlli' 01' I/ll 2
23
24
lIlli' "I' "2
14
25
IS
lIlli' 1/112'
°2
"I' I/ll 2 ' "2
26
27
A, lIlli' "I' I/ll 2
A, lIlli' "I' "2
28
A, I/lll' I/ll 2 , "2
29
A, "I' I/ll 2 ' "2
30
31
13
lIlli' "I' I/ll 2 ' "2
A, lIlli' ai' I/ll2' " 2
129
Table 11.29.
Order
1
3
4
a
5
6
7
"e
Variable
Factor
Analysis. (1.
lh1
2
a
2
A
2
8
9
10
11
12
13
14
15
Order of significance of variables from analysis of
variance of factorial arrangement, by mean square
l
A, 1/P (P )
2 2
l/P
l
1/P 2 ' a2(k 2 )
i>
First
Regression Analysis
Factorial Analysis
(all 8 cases)
A/P 2 • P2
1/P
1/P P
1 2
a 2 / P2
A
a A
2
a/ P2
a2
A
1/P , l/p
2
1
A, a2
1/P
A, l/Pl(P )
l
A, a l
a
a l , 1/P
2
A, l/P , 1/P2
l
A, 1/P , a2
2
A, 1!P , a
2
l
2
l/Pl
l
A/P l - Pl
A al
al/P l
a 2!P l
a a
l 2
2
a
2
A, 1/P (P )
2 2
l/P
l
1/P , a (k )
2
2 2
a
l
A, a
2
l/P , 1/P
2
l
A, l/P (P )
l l
A, 1/P 2 , a
2
A, a
1
l/P , a (k )
l l
l
A, l!P , 1!P 2
l
a , 1!P
l
2
130
Table 11.30.
Analysis of variance for factorial arrangement, L
for case 1,1, significance tests
q2
Error Terms
5 Factor interaction
Mean
SQuare
0.00225
FO•05
161
Significant
MS
>0.362
F •01
O
4052
Significant
MS
>9.12
1 d. f.
~e
5 Factor interaction
and 4 Factor interaction
0.01153
6 d.f.
5.99
>0.0692
13.74
>0.158
5 Factor interaction,
4 Factor interaction
and 3 Factor interaction
0.02772
16 d. f.
4.49
>0.124
8.53
>0.236
Significant
Sources of Variation
Mean
Squares
1!1I 2
7.898
a
5.002
2
A
3.319
a
1.678
1
A, 1!1I 2
1.356
1!1I
0.998
2
1!1I 2 ' a 2
0.484
1!1I 1 , 1!1I 2
0.428
A, a
0.327
2
A, 1!1I
0.218
1
A, a 1
0.186
al' 1!1I 2
0.151
A, l!lI l , l!1I 2
A, l!1I , a 2
2
A, 1!P2' a
0.137
1
e
0.090
0.069
.-
131
Order of variables from first regression for L a
q2
Table 11.31.
Model:
Lq2 • BO + B1
+ B7
+ B2 1/~2 + B3 0 1 + B4 O2 + B A + B6 A/~l
5
+ B8 A0 1 · + 89 A0 2 + B10 1/Jl1~2 + BU 01/~1
A/~2
+ B12 °2/~1 + 813 °1/~2 + B14 °2/~2 + B15 ~102
Case
1,1
1,3
3,1
3,3
4,4
5,3
3,5
Step
0_
1/~1
1
2
3
4
5
6
7
8
7
15
10
14
13
4
9
5
2
1
6
3
8
12
9
10
11
12
13
14
15
11
7
10
7
9
13
10
14
4
13
14
4
5
2
1
6
3
8
15
12
9
5
2
1
6
12
3
11
8
15
11
7
14
8
10
4
7
14
10
11
4
9
9
5
2
3
13
1
6
5
2
3
8
13
1
12
6
11
12
15
7
5,5
7
7
9
9
10
13
14
4
5
2
1
6
3
11
8
15
12
10
14
4
1
6
2
5
9
13
3
8
11
10
11
15
14
4
5
2
1
6
12
8
13
3
&the numbers in this table refer to the subscript of the B which
enters the solution on the regression step shown.
The order of appearance and significance of these variables inferre~ from
the table and the other regression analysis data such as F tests, R , etc.
is as follows:
_
•
Order
Variable
Order
1
A/~ 2 • P2
6
A
2
1/~1~2
7
1/~2
3
°2/~2
8
1/~1
4
O
9
A/~l • PI
5
A O
2
2
10
Variable
°1/~2
.e
•
132
Table 11. 32.
Model:
Lq • 8 + 81Pl + 8 P + 8 M + 8 M +
3 l
2 2
4 2
0
2
Case
1.1
1.3
3.1
3.3
3.5
Step
'e
Order of operational variables from second regression
analysis of L a
q2
2
~5P1
2
2
2
+ 86P2 + 87M + B8M
2
1
4.4
5.3
5.5
1
6
6
6
6
6
6
6
6
2
8
8
8
8
8
8
8
8
3
3
2
2
2
2
2
2
2
4
5
5
5
3
4
5
5
5
5
2
3
3
5
3
3
7
1
6
1
1
1
4
5
1
1
7
7
7
4
7
1
7
4
4
4
8
4
7
2
R
0.8970
0.9202
3
0.9000 0.9086
0.8855
0.8875
0.8644
0.8115
8.rhe numbers in this table refer to the subscripts of the B which
enters the solution on the regression step shown.
The order of appearance and significance of these variables inferre~ from
the table and the other regression analysis data such as F tests, R , etc.
is as follows:
Order
Variable
2
P2
Order
Variable
5
M
l
2
~
6
P
l
3
P2
2
Pl
7
~
1
4
•
e
•
8
~
133
Table 11.33.
Order of operational variables from third regression
analysis of L a
q2
j
Model :
Case
1,1
1,3
1
10
10
2
8
3
3,3
3,5
4.4
5.3
5,5
10
10
10
10
10
10
12
8
12
12
12
12
12
3
6
6
6
6
6
6
6
4
9
9
9
3
4
9
9
9
5
6
3
3
9
5
3
7
1
6
11
1
11
4
7
1
1
7
1
11
1
1
11
11
4
4
4
8
12
11
4
11
11
Step
'e
0.8999
0.9226 0.9033
0.9099
0.8876
0.8901
0.8668 0.8145
~e numbers in this table refer to the subscript of the B which enters
the solution on the regression step shown.
The order of appearance and significance of these variables inferre~ from
the table and the other regression analysis data such as F tests, R , etc.
is as follows:
Order
Variable
Order
Variable
1
3
P2
5
HI
6
PI
7
H
2
3
M
1
2
3
4
•
e
~
2
P2
3
PI
8
134
Table 11.34.
Step
.-
8
2
p2
2
Step by step summary of fourth regression analysis of L
q2
8
5
2 2
P2k 2
81
P2
83
2
k2
88
2
PI
89
87
2
P1k 1 P1k1
86
PI
(l,1)
1
2
3
4
5
6
7
8
5.08
4.86
4.79
5.14
30.81
32.12
32.12
32.14
2.49
2.48
2.53
- 1.37
2.53 -37.53
- 1.37
6.22 ~41.34 -1.91 - 1.37
6.22 -41.34 -1.91 -12.37
6.22 -41.34 -1.91 -12.35
(1,3)
1
2
3
4
5
6
7
8
5.42
5.29
34.28
35.65
36.06
36.01
36.01
36.03
1.89
1.88
5.84
5.88
5.88
5.88
5.88
(3,1)
1
2
3
4
5
6
7
8
5.18
4.98 2.18
0.78
4.94 2.17
0.78
32.58 2.17 -40.40
- 1.16 0.72
32.88 2.21 -40.41
34.06 5.52 -43.83 -1. 71 - 1.16 0.72
15.24
34.07 5.52 -43.83 -1.71 -11.45 0.72
further
calculation.
F level insufficient fo
80
-1.64
-42.96
-46.99
-47.27
-47.23
-47.23
-47.24
-2.08
-2.08
-2.08
-2.08
-2.08
- 1.14
- 1.18
1.10
1.04
1.04
1.04
1.05
0.27
-1.99
-2.23
-1.68
11.86
13.93
16.08 8.13
0.82 15.08 8.20
0.7253
0.8711
0.9201
0.9392
0.9558
0.9692
0.9722
0.9724
-1.99
-2.27
13.41
15.62
16.19
16.01
9.59 12.55
9.32 12.63
0.8432
0.9148
0.9418
0.9591
0.9713
0.9819
0.9829
0.9831
-1.62
-1.92
-2.10
12.48
12.97
14.82
9.26
0.7748
0.8923
0.9188
0.9403
0.9591
0.9709
0.9746
0.57
0.57
- 7.72 -0.76 1.37
- 7.:14
R
Table continued
135
Table 11.34 (continued)
Step
B2
85
2 p2 2
k
P2
2 2
81
P2
B3
k2
2
8a
2
P1
B9
B7
B6
2
P1k1 P1k 1 P1
B
O
(3,3)
2
3
4
5
6
7
8
5.23
5.09
5.05
30.46
32.01
32.34
32.42
32.43
1.96
1.95
1.95
6.39
6.34
6 •.44
6.44
(3,5)
1
2
3
4
5
6
7
8
6.06
5.91
43.38
45.78
45.72
46.03
46.14
46.14
2.28
2.28
9.29
9 028
9.31
9.32
9.32
(4,4)
1
2
3
4
5
6
7
8
5.73
5.60
39.00
39.61
41.19
41.10
41.11
41.16
1.96
1.96
2.02
6.60
6.59
6.60
6.60
1
-37.66
-42.20
-42.42
-42.47
-42.47
-55.89
-63.00
-62.94
-63.20
-63.29
-63.29
-49.65
-50011
-54.77
-54.71
-54.71
-54.75
R
0.81
0.81
0.81
-2.34
-2.34 - 0.89 0.77
-2.34 - 1.39 -1.53 2.41
-2.34 - 6.48 -1.44 2.32
-1.89
-2.18
-2.36
11.38
13.87
14.34
14.29
7.56 11.54
0.8167
0.8964
0.9213
0.9421
0.9641
0.9736
0.9759
0.9767
-3.73
0.68
-3.73
-3.73 - 0.74 0.64
-3.73 - 1.38 -2.32 3.11
-3.73 - 2.20 -2.30 3.09
-2.38
-2.73
17.78
21. 71
21.53
21.96
21.90
1.22 21.46
0.8072
0.8789
0.9139
0.9531
0.9638
0.9681
0.9706
0.9706
-2016
-2.45
15.72
16055
19.12
18.85
0.79
0.79 12.61 14.24
-1.36 2.20 12.10 14.40
0.8125
0.8758
0.9071
0.9318
0.9510
009666
0.9685
0.9691
-2.43
-2.43
-2.43
-2.43
- 1.54
- 1.54
- 1.58
-10.07
-10.01
Table continued
.e
.e
136
Table 11.34 (continued)
Step
.-
B
B2
5
2 p2k 2
P2
2 2
(5,3)
1
2
3
4
5
6
7
8
6.21
6.06
45.29
47.17
47.70
47.61
47.62
47.69
2.18
2.17
7.60
7.66
7.65
7.65
7.66
(5,5)
1
2
3
4
5
6
7
8
5.73
5.61
43.20
43.88
45.52
45.54
45.49
45.55
1.82
1.81
1.88
6.64
6.64
6.64
6.65
B1
P2
-58.13
-63.67
-64.02
-63.96
-63.97
-64.01
-56.08
-56.63
-61.46
-61.48
-61.44
-61. 49
I
Avg
e
.
-2.86
-2.86
-2.86
-2.87
-2.87
-2.54
-2.54
-2.54
-2.54
B8
2
P1
B7
B9
2
Plk l Plk l
B6
Pl
BO
R
- 1.38
- 1.42
-14.37
-14.29
-2.31
-2.63
18.58
21.63
22.35
22.07
0.84
0.84 19.30 14.98
-2.02 2.94 18.54 15.23
0.7920
0.8599
0.8951
0.9181
0.9358
0.9512
0.9552
0.9565
- 1.61
- 1.61
-17.00
-17.05
-16.99
-2.23
-2.50
18.07
18.98
21.66
22.95 13.24
0.42 22.99 13.07
-1.56 2.04 22.41 13.27
0.7816
0.8314
0.8704
0.8966
0.9162
0.9223
0.9264
0.9272
Table of Increments in R2
Case
1,1
1,3
3,1
3,3
3,5
4,4
5,3
5,5
B3
2
k
2
0.5261
0.7110
0.6003
0.6670
0.6516
0.6601
0.6272
0.6109
0.2327
0.1259
0.1959
0.1366
0.1209
0.1068
0.1123
0.0803
0.0315
0.0502
0.0401
0.0389
0.0627
0.0559
0.0617
0.0663
0.0257
0.0328
0.0227
0.0418
0.0733
0.0361
0.0416
0.0356
0.0356
0.0236
0.0357
0.0184
0.0082
0.0453
0.0329
0.0463
0.0004 0.0058
0.0206 0.0019
- 0.0073
0.0045 0.0016
0.0049
0.0299 0.0036
0.00l5 0.0290 0.0076
0.0015 0.0077 0.0111
0.0877
0.0004
0.0480
0.0451
0.0206
0.0013
5.0542 1.1114 0.4073 0.3096 0.2460 0.2071 0.0970 0.0389
p.6318 0.1389 0.0509 0.0387 0.0307 0.0259 0.0121 0.0048 ~ • 0.9338
137
Table 11.350
Summary of final coefficients from fourth regression
analyses for L
q2
Case
c 1c 2
.e
60
61
62
63
65
1,1
8.20 -41.34 32.14 -1.91
1,3
12.63 -47.24 36.03 -2.08
3,1
9.26 -43.83 34.07 -1. 71
3,3
11.54 -42.47 32.43 -2.34
6.44
3,5
21.46 -63.29 46.14 -3.73
9.32
4,4
66
67
68
69
R
6.22 15,80
0.82 -12.35
0.27
0.9724
5.88
1.37 - 7.72 -0.76
0.9831
9.32
5.52 15.24
--
0.72
0.9746
7.56
2.32 - 6.48 -1.44
0.9767
1.22
3.09 - 2.20 -2.30
0.9706
14.40 -54.75 41.16 -2.43
6.60 12.10
2.20 -10.01 -1.36
0.9691
5,3
15.23 -64.01 47.69 -2.87
7.66 18.54
2.94 -14.29 -2 •.02
0.9565
5,5
13.27 -61.49 45.55 -2.54
6.65 22.41
2.04 -16.99 -1.56
0.9272
1,1
8.20 -41.34 32.14 -1.91
6.22 15.80
0.82 -12.35
0.27
0.9724
3,1
9.26 -43.83 34.07 -1. 71
5.52 15.24
0.72
0.9746
1,3
12.63 -47.24 36.03 -2.08
5.88
9.32
1.37 - 7.72 -0.76
0.9831
3,3
11.54 -42.47 32.43 -2.34
6.44
7.56
2.32
-6.48 -1.44
0.9167
5,3
15.23 -64.01 47.69 -2.87
7.66 18.54
2.94 -14.29 -2.02
0.9565
4,4
14.40 -54.75 41.16 -2.43
6.60 12.10 2.20 -10.01 -1.36
0.%91
3,5
21.46 -63.29 46.14 -3.73
9.32
5,5
13.27 -61.49 45.55 -2.54
6.65 22.41
1.22
--
-11.45
-11.45
3.09 - 2.20 -2.30
0.9706
2.04
0.9272
16.99 -1.56
e
,
138
Table 11.36.
Coefficients from fourth regression analysis for L
arranged by number of service channels
q2
•
c1ci~
...
1
2
3
4
5
80
1
2
3
4
5
8.20
12.63
9.26
11.54
21.46
15.23
6.22
5.88
5.52
6.44
13.27
5
9.32
7.66
6.65
86
r-41. 34
-47.24
~43.83
-42.47
-63.29
15.80
9.32
15.24
7.56
-54.75
-64.01
1.22
12.10
-61.49
22.41
18.54
87
32.14
36.03
34.07
32.43
46.14
41.16
47.69
0.82
1.37
--
2.32
45.55
3.09
2.20
2.94
2.04
88
... 1.91
- 2.08
... 1.71
- 2.34
-12.35
- 7.72
- 3.73 -11.45
- 6.48
.. 2.43
- 2.54
- 2.87
2.20
-10.01
16.99
-14.29
(R"')
R
89
1
4
6.60
83
1
2
3
4
5
3
14.40
82
1
2
3
4
5
2
85
81
1
2
3
4
5
1
0.27
- 0.76
(.9455)
.9724
(.9664)
.9831
0.72
- 1.44
- 2.30 (.9499)
.9746
(.9539)
.9767
2
3
4
5
(.9392)
.9691
- 1.36
- 2.02
.9421)
.9706
- 1.56
(.9148)
.9565
.8598)
.9272
.e
139
Table 11.37.
c1 c 2-+
.
1
Increments of R2 from fourth regression analysis for Lq
arranged by number of service channels
2
2
3
4
5
82
1
2
3
0.5261
0.7110
0.6003
0.6670
0.6516
!J
4
5
8
1
1
2
3
4
5
•
8
8
0.0356
0.0236
0.0357
0.0184
0.. :2327
0.1259
'.llS'
0.1366
0.1209
8
9
0.0877
0.0004
0.0480
0.0451
S
0.0082
0.0463
0.0206
0.0013
0.1068
0.0015
0.0025
0.0803
0.1123
4
0.0329
0.6109
0.6272
87
0.0315
0.0502
0.0401
0.0389
0.0004
0.0627
0.0559
0.0617
--
0.0206
0.0049
0.0045
0.0299
0.0077
0.0290
0.0663
86
83
1
2
3
4
5
3
0.0453
8
5
1
2
0.6601
4
S
2
1
0.0257
0.0328
P.0227
0.0418
0.0733
0.0058
0.0019
0.0073
0.0016
0.0036
0.0361
0.0416
0.0356
0.0076
-0.0111
e
140
Table 11.38.
•
Step by step summary of fifth set of regression analyses
for Lq
2
BO
B4
2
P2
B8
2 2
P2 k 2
-1.639
-1. 986
-2.233
-1. 683
11.858
13.263
15.661
7.365
7.922
7.975
6.912
5.079
4.861
4.794
5.136
30.807
30.812
30.794
30.789
30.788
30.911
29.850
2.489
2.479
2.528
2.528
10.022
10.021
10.021
10.021
9.588
11.414
(1,3)
Step 1 -1.990
2 -2.267
3 13.405
4 14.924
5 16.138
6 16.056
7 15.912
8 11. 835
9 12.141
10 12.755
11 12.732
5.421
5.285
34.278
34.288
34.718
34.638
34.588
34.594
34.596
35.190
35.194
(3,1)
Step 1
2
3
4
5
6
7
8
9
10
11
5.175
4.984
4.938
32.576
32.890
32.895
32.865
32.878
32.878
32.967
32.672
Constant
(1,1)
Step 1
2
3
4
5
6
7
8
9
10
11
r
e
e
•
-1.620
-1.923
-2.098
12.480
13.626
14.880
14.769
8.598
8.916
8.956
8.660
B2
B7
2
P2k 2
2
P1k 1
-37.527
-39.464
-53.037
-56.779
-56.778
-56.779
-56.085
-5.423
-5.425
-5.425
-5.425
-4.928
-7.314
1.097
1.044
1.044
1.045
1.042
1.044
-1.121
-0.984
-0.967
6.721
8.574
8.574
8.490
8.864
1.888
1.885
9.929
9.946
9.943
9.941
9.941
9.941
8.915
8.823
-42.958
-45.024
-48.763
-48.864
-55.953
-55.963
-55.966
-56.362
-56.335
-5.884
-5.863
-5.867
-5.869
-5.869
-5.869
-4.496
-4.403
-1.181
-1.181
-1.130
1.694
1. 767 0.776
5.248 0.770
5.252 0.771
5.252 2.403
5.040 2.403
5.024 2.389
2.178
2.171
2.171
2.215
8.908
8.907
8.907
8.907
8.595
9.100
-40.403
-44.030
-45.750
-52.977
-52.992
-52.992
-52.992
-52.798
-4.844
-4.846
-4.845
-4.845
-4.486
-5.147
0.783
0.783
0.717 1.808
0.717 1.803
0.721 5.420
0.722 5.428
-0.522 5.428
-0.426 5.367
-0.420 5.471
P2
B5
B6
B9
222
p/P 1 P1k1
3.030
2.992
2.988
1. 727
1.705
1.703
Table continued
e
.
141
Table 11.38 (continued)
•
B1
B3
2
P1
P1
(1,1)
Step 1
2
3
4
5
6
7
8
9
10
11
.-
(1,3)
Step 1
2
3
4
5
6
7
8
9
10
11
(3,1)
Step 1
2
3
4
5
6
7
8
9
10
11
e
•
B10
2 4
k 1k 2
2
k 2 !P 1
R
R2
S2
0.558
0.7253
0.8711
0.9201
0.9392
0.9558
0.9696
0.9767
0.9832
0.9854
0.9859
0.9864
0.5261
0.7588
0.8466
0.8821
0.9136
0.9402
0.9540
0.9667
0.9710
0.9720
0.9731
0.5966
0.4329
0.3514
0.3136
0.2736
0.2322
0.2079
0.1806
0.1724
0.1735
0.1742
7.766
7.765
-0.322
7.457
70462 - 0.107 -0.300
0.8432
0.9148
0.9418
0.9598
0.9742
0.9851
0.9873
0.9887
0.9893
0.9895
0.9895
0.7110
0.8369
0.8870
0.9213
0.9490
0.9704
0.9748
0.9775
0.9787
0.9791
0.9792
0.4789
0.3659
0.3099
0.2635
0.2161
0.1680
0.1581
0.1525
0.1518
0.1541
0.1574
- 1. 369
- 1. 369
- 1.364
3.304
24.820 -12.387
24.046 -12.387
23.891 -12.343 - 0.284
25.448 -12.880 - 0.319
3.770
15.100
14.679
13.783
13.787
3.704
20.650
20.214
20.101
20.527
-
Btl
-11.461
-11.462
-11.431 - 0.206
-11.576 - 0.219
0.157
0.7748 0.6003 0.5227
0.8923 0.7961 0.3796
0.9188 0.8441 0.3378
0.9403 0.8842 0.2965
0.9614 0.9242 0.2445
0.9734 0.9475 0.2075
0.9760 0.9526 0.2011
0.9807 0.9617 0.1847
0.9816 0.9635 0.1844
0.9819 0.9641 0.1873
0.9819 0.9642 0.1916
Table continued
142
Table 11.38 (continued)
143
Table 11.38 (continued)
.
B1
PI
B3
2
PI
(3,3)
Step 1
2
3
4
5
6
7
8
9
1. 775
10 11. 370 -6.488
11 10.783 -6.288
'e
•
(3,5)
Step 1
2
3
4
5
6
7
8
9
10
11
1.316
1.2508
4.612 -2.273
4.970 -2.394
(4,4)
Step 1
2
3
4
5
6
7
3.855
8
5.394
9 20.493
10 19.831
11 19.831
e
•
-10.050
-10.047
-10.047
B10
k2k 4
1 2
-0.529
-0.509
-0.504
-0.486
-0.212
-0.210
-0.221
B11
2
k2 /P 1
R
R2
s
-0.219
0.8167
0.8964
0.9213
0.9421
0.9650
0.9762
0.9831
0.9847
0.9853
0.9866
0.9867
0.6670
0.8036
0.8487
0.8876
0.9313
0.9530
0.9666
0.9696
0.9708
0.9734
0.9735
0.5118
0.3998
0.3571
0.3134
0.2497
0.2105
0.1813
0.1765
0.1769
0.1730
0.1767
0.135
0.8072
0.8789
0.9173
0.9552
0.9659
0.9711
0.9776
0.9779
0.9780
0.9781
0.9782
0.6516
0.7725
0.8415
0.9124
0.9331
0.9431
0.9557
0.9562
0.9565
0.9567
0.9568
0.6480
0.5326
0.4524
0.3424
0.3051
0.2869
0.2582
0.2623
0.2672
0.2729
0.2795
-1.478
-1.475
-0.137
-0.226
-8.423
-8.440
-8.440
0.8125
0.8758
0.9071
0.9374
0.9580
0.9487
0.9766
0.9792
0.9817
0.9832
0.9832
0.6601
0.7670
0.8229
0.8786
0.9177
0.9487
0.9537
0.9588
0.9638
0.9667
0.9667
0.5854
0.4930
0.4374
0.3687
0.3095
0.2492
0.2417
0.2327
0.2230
0.2190
0.2190
2
Table continued
.e
144
Table 11.38 (continued)
•
BO
Constant
B4
B8
B
2
B7
B5
2
P2
2 2
P2k 2
P2
2
P2k 2
2
P1k 1
B9
B6
2
2 2
k
P2 ' 1 P1 1
(5,3)
Step
1
2
3
4
5
6
7
8
9
10
11
-2.309
-2.629
18.578
20.667
22.215
22.074
23.097
22.766
14.912
14.165
14.200
6.213
6.056
2.175
45.286
2.171 -58.157
45.300 13.232 -60.967 -8.090
45.849 13.254 -65.738 -8.064
45.723 13.249 -65.830 -8.070
45.708 13.248 -68.621 -8.071
45.646 13.246 -76.693 -8.074
45.676 13.247 -76.735 -8.072
44.952 14.500 -76.252 -9.747
44.955 14.605 -76.291 -9.850
-4.223
-3.608
-3.607
-3.607
-3.669
2.194
2.280
3.671
7.698
7.709
7.973
7.990
1.155
6.897
6.055
6.056
6.057
6.070
B
9
B
6
0.0043
0.0214
0.0018
0.0135
0.0126
0.0310
0.0315
~e
2
Table of Increments in R
Case
B
2
B7
B
5
0.0315
0.0502
0.0401
0.0437
0.0710
0.0559
0.0617
0.0266 0.0877
0.0342 0.0012
0.0233 0.0480
0.0389 0.0451
0.0690 0.0206
0.0052 0.0029
0.0435 0.0133
0.0138
0.0277
0.0400
0.0217
0.0100
0.0558
0.0398
4.4433
1. 0311
0.3541
0.2407
0.2188
0.2088 0.1161
Average
0.6347
0.1473
0.0505
0.0344
0.0312
0.0298 0.0166
Cum. Avg.
0.6347
0.7820
0.8325
0.8669
0.8981
0.9279 0.9445
t
•
B8
0.5261 0.2327
0.7110 0.1259
0.6003 0.1959
0.6670 0.1366
0.6516 0.1209
0.6601 0.1068
0.6272 0.1123
(1,1)
(1,3)
(3,1)
(3,3)
(3,5)
(4,4)
(5,3)
e
B
4
Table continued
e
•
145
Table 11.38 (continued)
_
.
146
Table 11.39.
Comparison of order of significance of terms from second,
third, fourth and fifth sets of regression analyses for
Lq , using operational variables
2
1
2
P2
Variables
Third
Set
3
P2
2
m2
2
m3
2
2 2
P2 k 2
2 2
P k
2 2
3
P2
2
P2
P2
P2
4
2
PI
3
PI
k2
2
2
P2k2
5
m
l
m
l
2
PI
2
Plk l
6
PI
PI
2
Plk l
2
P2!Pl
7
~
~
Plkl
2 2
Plk l
8
2
m
1
3
m
l
P
l
Pl
Order
"-
9
•
Second
Set
Fourth
Set
2
P2
Fifth
Set
2
P2
2
PI
10
2 4
k k
1 2
11
2
k 2Pl
_
•
141
Table 11.40.
F values to remove variables _fOurth set of regression
analyses for L
q2
Coefficient
B2
B5
B1
B
3
B8
B9
87
6
Variable
2
P2
p22k 22
P2
k2
2
2
PI
2
P1k 1
P1k1
PI
Case
"-
U,l)
20.72
a
29.41
a
15.93
a
10.86
a
3.06
0.02
0.18
2.35
U,3)
52.72
a
46.95
a
40.99
a
22.51
a
1.84
0.27
0.80
1.25
(3,1)
29.02
a
28.86
a
22.32
a
10.86
a
41>33
19.45
a
(3,3)
31.39
a
41.38
a
24.35
a
20.91
a
1.26
0.86
2.05
0.77
(3,5)
37.52
a
48.39
a
31.50
a
29.13
a
0.07
1.15
1.90
0.01
(4,4)
31.88
a
26.65
a
25.34
a
13.65
a
1.89
0.47
1.14
1.25
(5,3)
24.45
a
21.11
a
19.93
a
11.26
a
2.50
0.66
1.29
1.89
(5,5)
15.85
a
10.66
a
12.89
5.84
··:b
2.21
0.25
0.39
1.73
243.55
253.41
193.25
~25.02
17.16
23.13
7.75
~2. 76
34.79
36.20
27.61
17.86
2.45
3.30
1.11
1.82
tF
Average F
&aigh1y significant.
bSianificant.
_
•
•
3.51
e
,.
148
Table 11.41.
F values to remove variables.
analyses for L
q2
P1
83
2
PI
3.09
9.46
5fJ79 0.89 0.82
8.16
2.42
3.45
2.38 0.12 0.28
0.11
6 10
6
0.98
6fJ52
5.10 0.34 0.06
3.17
3.99
51)07
8.20
2.15
1.77 1.91 0.14
32.84
6.35
3.28
1.95
5.02
0.18
0.10 0.15 0.02
13.25
50.81
3.59
1.61 13.20
3.38
4.81
3.01 0.08 0.01
12.01
30.95
4 48
3.40
7.00
5 05
4 81
3.55 0.05 0.13
104.04 285.12 29.43 14.04
57~33
87
85
P21t22 P1k 12
P2
(1,1)
30.04
18.08
39.06
6.10
0.62 15.85
(1,3)
66.77
14.90
61.94
2.86
1.03
(3,1)
29.97
10.51
30.11
2.88
(3,3)
42.55
15.48
39.41
(3,5)
36.87
19.81
(4,4)
45.01
(5,3)
28.00
a
a
a
a
a
a
a
a
a
a
a
a
a
a
tF •
I
82
a
a
a
a
a
a
a
b
b
6
a
39.88
14.86
40.73
aRigh1y significanto
bSign1ficant•
•
4.20
a
a
Average F •
•
89
86
2
2 2
P2!P 1 P1k 1
88
p 2k 2
2 2
84
2
P2
279.21
fif.th set of regression
2.00
b
8.12
a
b
6
81
a
b
6
810 8
11
2 4 2
k k k/ P
1 2
1
b
28.14 31.38 21.70 3.54 1.46
4.02
4.48
3.10 0.50 0.21
149
e
Appendix 11.9
•
Table 11.42.
Combined Regression Analysis
Distribution of residuals of L andL
from regression
q1
q2
analysis of all eight cases combined
Lq2
Lq1
Residuals
of Lq
.•
1.6
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.0
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
-1.1
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
1.7
1.6
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
Frequency
_
"
Frequency
1.0000
0.9961
0.9961
0.9883
0.9766
0.9727
0.9560
0.8947
0.7735
0.5040
0.2030
0.1132
0.0703
0.0195
0.0078
0.0039
0.0039
0.0039
0.0039
0.0000
1
0
1
0
0
1
1
0
4
1
3
20
48
56
48
30
19
9
8
1
2
3
0
0
0
0
0
0
0
0
1
0
2
3
1
4
16
31
69
17
23
11
13
3
1
0
0
0
1
0
s
•
Cumulative
Fraction
=
0.1964
Cumulative
Fraction
1.0000
0.9961
0.9961
0.9922
0.9922
0.9922
0.9883
0.9844
0.9844
0.9688
0.9650
0.9530
0.8750
0.6880
0.4690
0.2812
0.1629
0.0899
0.0547
0.0234
0.0196
0.0117
0.0000
s • 0.2535
e
•
150
Table 11.43.
Model:
13 0
2
2
222
Lq1 • 130 + Bi p 1+ B2P1k 1 + B3P1 + B4c 1 + B5P1k 1 + B6c 1
131
Step 1 -2.381
sl3
6.865
0.222
953
Step 2 -2.637
sl3
6.738
0.179
1423
Step 3 16.169
sB
Step 4
sB
F
13 3
134
135
136
R2
0.7896
0.8653
41.526
2.795
220
1.748 -51.543
0.116
4.136
228
155
0.9167
18.065
44.494
2.272
383
1. 746
0.093
350
-55.815
3.361
276
-0.122
0.010
135
19.401
44.528 8.783
1.857 0.632
575
193
-57.659
2.751
439
-0.121
0.009
199
-5.142
0.458
126
0.9640
44.609
1.838
589
-57.773
2.723
450
-0.217
0.039
30.6
-5.139 0.016
0.453 0.006
128 6.29
0.9649
F
F
Step 5
sB
132
1. 749
0.147
142
F
"_
Step by step summary of regression analysis of L , all
eight cases combined
q1
F
0.9459
It
Step 6 19.549
sB
F
e
.
8.779
0.625
197
e
•
151
Table 11.44.
Model:
Step by step summary of regression analysis of L ,
all eight cases combined
q2
2
2 2
2
2
Lq2 • BO + B1P2 + B2P2k 2 + B3 P2 + B4 P2k 2 + B5P2k 1 + B6P2/P1
2
222
+ B7c 2 + B8P1 + B9k 1k 2 + B10c 1 + B11 c 2 + B12 P1k1 + B13c 1
So
5.567
0.273
F
414
Step 2 -2.325
sa
5.414
0.223
2.094
0.183
590
131
Step 3 14.222
Ss
36.024
3.991
F
81.5
Step 4 16.073
sa
36.110
3.578
F
Step 5 16.071
sB
F
Step 6 17 .211
sa
F
Step 7 18.426
sa
F
Step 8 19.588
sa
e
B2
Step 1 -2.018
sB
F
'-
B1
F
101
36.120
3.116
134
36.435
2.774
172
38.355
2.577
221
38.260
2.510
232
B3
B
4
B5
B6
R!
0.6200
0.7497
0.7972
2.093 -45.351
0.165
5.905
160
58.9
11. 701 -47.926
1.225
5.305
-7.020
0.889
91.1
62.4
81.6
11. 700 -48.378
1.067
4.620
120
-7.020
0.774
110
823
11.720 -51.837
0.950
4.134
-7.002
0.689
152
157
103
11.647 -54.586
0.877
3.839
-6.950
0.636
202
119
11.646 -61.378
0.854
4.140
-6.953
0.619
176
186
219
126
0.8375
0.847
0.094
0.8773
81.0
0.728
0.085
73.4
0.727
0.078
85.8
0.729
0.076
91.1
1.686 0.9032
0.207
66.6
1.688 0.9178
0.191
78.3
5.088 0.9224
0.910
31.3
.
Table continued
e
.,
152
Table 11.44 (continued)
87
88
89
810
8n
812
Step 7 -0.079
0.012
44.0
81 3
R2
0.9178
Step 8 -0.084
0.012
51.4
2.438
0.639
14.6
0.9224
Step 9 -0.083
0.012
51.6
2.363 -0.321
0.637
0.174
13.8
3.41
0.9234
Step 10 -0.096
0.013
50.2
2.274
0.636
12.8
-0.322 0.024
0.173 0.013
3.47 3.13
0.9244
"_
Step
n -0.168
0.054
9.75
2.281
0.635
12.9
-0.322 0.024
0.173 0.013
3.48 3.28
0.012
0.009
1.91
,
Step 12 -0.168
0.054
9.73
2.278
0.635
12.8
-0.276 0.024
0.183 0.013
2.27 3.20
0.121
0.009
1.91
-0.550
0.706
0.607
0.9252
Step 13 -0.166
0.054
9.43
2.275
0.637
12.8
-0.276 0.038
0.183 0.054
2.26 0.482
0.012
0.009
1.80
-0.549
0.707
0.602
-0.002 0.9252
0.009
0.0661
0.9250
Table continued
e
&
e
•
153
Table 11.44 (continued)
80
-,
•
e
'"
81
82
83
84
.85
86
R2
Step 9
19.646 38.336
2.498
235
11.581 -61.430
0.850
4.120
185
222
-6.686
0.633
111
0.950
0.142
45.0
4.987
0.907
30.2
0.9234
Step 10
19.578 38.342
2.487
238
11.581 -61.210
0.847
4.104
222
187
-6.684
0.630
.112
0.951
0.141
45.4
4.877
0.905
29.0
0.9244
Step 11
19.691 38.402
2.483
239
11.578 -61.316
0.845
4.097
188
224
-6.683
0.629
113
0.950
0.141
45.6
4.887
0.904
29.2
0.9250
Step 12
19.733 38.410
2.485
239
11.589 -61.342
0.846
4.101
188
224
-6.721
0.632
113
1.476
0.689
4.58
4.771
0.917
27.1
0.9252
Step 13
19.716 38.410
2.490
238
11.589 -61. 336
0.848
4.109
187
223
-6.720
0.633
113
1.475
0.691
4.56
4.768
0.918
26.9
0.9252
e
154
•
Table 11.45.
Comparison of regression coefficients for first and
second rank queues
First Rank
Coefficient
Constant
19.401
18.426
Constant
2
P1
44.528
38.355
2
P
2
8.783
11.647
P1
-57.659
-54.586
P
2
c
- 0.121
- 0.079
c
- 5.142
- 6.950
2
P2k 2
0.727
P2k1
1.688
pi p1
2 2
P1k1
e_
..
•
e
•
Second Rank
Coefficient
Variable
Variable
1
2
P1k 1
~2
P2 2
2
2