•
UNIVERSITY OF NORTH CAROLINA
Depe,rtment of Statj.st;i.cs
Chapel Hill, N. C.
PRENvWTIVE DISCIPLINES FOR QUEUES
~.
p~
STORES
by
Ronald Emerson Thomas
April 1962
This research Vias supported by the Office of Naval
Research w1der contract No. Nonr-855(09) for research
in probability and statistics at the University of
North Carolina, Chapel Hill, N. C. Reproduction in
whole or in part is permitted for any purpose of the
United States Gover~~ent.
Institute of Statistics
IUmeo Series No. 321
ii
.,
It is a pleasure to acknowledge my indebtedness to Professor
Halter L. Smith, for suggesting the problem considered herein and
for providing encouragement and many pithy suggestions.
I am extremely grateful to the Department of Statistics and
to the Offj.ce of Naval Research for financial assistance without
vrhich this research 'ivould not have been undertaken.
I am most grateful to Mrs. Doris Gardner for her skillful
translation of the manuscript into typed form.
thanl~s
A180, my sincerest
to Martha Jordan for her unfailing efforts in guidingllle
through the labyrinth of rules, regulations and assorted pitfalls.
TABIJE OF CONTBNTS
Page
CHA.FTER
11
AcknovTledgments
I
e
II
INTRODUCTION TO QUEUES 1tJITH PREElIl)TIVE
PRIORITY DISCIPLINES
. ..
1
1.
Introduction
2.
Arrivals
...
2
3·
The Service I''lechanism
3
4.
The Queue Discipline
5.
Theory of Queues
6.
Theory of Priority Queues
7·
Queues and Stores • • • • •
13
R
v.
Service 1,1echanisms Subject to Breakdovms ~
14
..... . ...
...
7
, • 10
15
THE PREEMPTIVE STORAGE PROBLBJI
1.
Introduction. • • • • • • •
.......
. 15
2.
The Preemptive Storage Problem for 1:=2..
• 16
3.
Limiting Behavior of the 1):r.eemptiYe S+,ore
for
k=2
. . . . . " .
~
. . . . .
. . . 30
J+.
Moments of the Limiting Distribution of
Store Content • • • • • • • • • • • • • • • • • 37
5.
~loments
of the Limiting Distribution of
Store Content for
III
1
• . • • 48
>2
k
55
QUEUEING HITH BREAKDO'ltJN
. . 55
1.
Introduction
2.
Unexpired Repair and Clearing Times • , • • • •
"
•
•
•
•
•
•
•
for Unit Breakdovill. • • • • • • • •
q.
• 60
iv
Chapter
IV
Page
),
Unexpired Repair and Clearing Times for
Component Breakdown . • . . . • •
4.
Completion Times and Busy Periods
5,
Queueing Times for Active Breakdown
70
60
Queueing Times for Independent Brealtdovm
73
THE E1.'ISTENCE OF LTIUTING DISTRIBUTIONS. •
1.
Introduction
2.
Necessal~
77
77
and Sufficient Conditions for
Existence of Limiting Distributions of
Poisson Reduction Processes. . • . . •
).
79
Necessary and Sufficient Conditions for a
Finite Mean Busy Period of Certain Poisson
Reduction Processes
v
Or-7
U(
T1ill NUI':IBER OF BREAKDOvJNS
1.
93
Number of BreakdmIDs During Service of One
Customer
2.
93
Number of Brealtdowns Durh1g Queueing lilime for
Preemptive
Resun~
3. Miscellaneous
Behavior
Bibliography
Discipline • • • •
Remarl~s
Regarding Transient
..••..••••...•..
102
• • • • . . . • • • . . • • • • . . • . 108
CHAPTER I
INTRODUCTION TO QUEUES WITH PREEMPTIVE PRIORITY DISCIPLINES
1.
Introdu.ction.
1
A queue is the term used to describe
the line of customers that forms at a service mechanism according to specific rules of:
(i)
arrival:
which is the input of customers
to a. line of customers that demand service
at a service mecnanism;
(i1)
the service mechanism:
which supplies a service
to each customer during a period of time called
the service time and limits the number of
customers that can receive service simultaneously;
the queue discipline:
vlhich is the set of rules
governing the manner each customer joins other
customers in the queue, vlai ts for service and
receives service.
The customers '-rho have arrived at the queue prior to time t but
have not conlpleted service by time t are said to be in the queuej.ng system at this time.
Thus the queue disdpline governs the
behavior and the order of service of each customer with respect to
other customers at all times vlhile in the queueing system.
lThis research v~s supported by the Office of Naval Research under
contract No, Nonr-855(09) for research in probability and statistics
at the University of North Carolina, Chapel Hill, N. C. Reproduction
in whole or in part is pernlitted for any purpose of the United States
Government.
2
The study of the analytic properties of queues, called
queueing theory, has interested many authors because of the variety
of
indu~trial
and military applications.
Certainly a preponderance
of the literature has dealt with variations in the rules
gov~Tn-
ing arrivals and the service mechanism under a strict head-of-thelin~
queue discipline.
This type of discipline requires that
customers enter the service mechanism in exactly the same order
in time as their respective arrivals at the queueing system.
In
this dissertation we will consider other possible queue disciplines
where k different classes or types of customers, say c ' c , c ,
l
2
3
..• , c ' are serviced by the srune service mechanism.
k
Arrivals.
2.
The arrivals of customers may be formulated
mathematically in terms of the r-th customer to arrive after
time zero.
Let t
l'
be the time of arrival of the r-th customer,
irrespective of class.
at times t , t , t , •..
2
l
3
betvleen the (1'-1) - th and
Define t
i
only, let t l ' t 2 ' •••
= O.
Hence customers arrive
The time interval,
1'.'
time of the r-th customer.
i
o
t
- t _ ,
r l
r
th arrivals is called the interarr1val
Considering the class c. customers
~
be thej.r arrival times.
If t i = 0,
o
then the interarriva1 time of the r-th class c. customer is
J.
t 1 _ t
I'
(
l.
t
i
r
]
i
I'
1'-1 '
, i
=
=
1, 2,
It is clear that the k sequences
1, 2, ••• , k of arrival times for the k classes of
customers are subsequences of the sequence of arrival times (t
for the queueing system.
We will assume that the interarrival
r
1
3
times of class c. customers is a sequence of independent and
l
identically distributed random variables.
We \'1ill also assume
that the sequence of interarrival times for anyone class of
customers is independent of the sequence of interarrival times
for each other class of customers.
Thus the terms of the se-
quence { t / - t;_l} are independently distributed vii th distribution function A.(t) and mean interarrival time l/~., i =
l
1, 2, •.• , k.
l
In general, the distribution functions and mean
interarrival times will be diffelf;ent for different classes of
customers.
However,
",e
"Till make certain restricting assump-
tiona which will be described in Section 6.
3.
The Service Mechanism.
Throughout this dissertation
it will be assumed that the service mechanism prOVides service
for only one customer at anyone time.
Some of the queue dis-
ciplines to be discussed involve interruptions in service of
certain customers.
That is, under circumstances specified by
the queue discipline, a customer will enter the service mechanism, remain a period of time and return to the waiting line
"l1.thout completion of service.
For a customer \>lho is not
interrupted while in the service mechanism, the service time
can be formulated mathematically in terms of the r-th customer
to arrive after time zero.
Let S
r
i
be the time in the service
mechanism of the r-th class c. customer to arrive after time
l
4
zero, provided service of this customer is not interrupted.
We vlill call this the uninterrupted service time of the r-th
,.
arrival in class c ..
1.
( Sri ~ in class c
i
Suppose the uninterrupted service times
is a sequence of independent and identically
distdbuted random variables vIi th distribution function B.J_ (s)
and mean l/~., i = I, 2, ••• , k.
1.
In general, the uninter~
rupted service times of different classes of customers will be
independent with different distributions.
Discussion of the
total service times of customers with interrupted service is
deferred to the next section.
4.
The Q~eue Disciplineo
We will be considering queue
disciplines for the k classes, c l ' c2 ' "', c k ' of customers
where some classes have a priority to service with respect to
other classes.
For definiteness, suppose that the subscript
i of the class c. indicates the order of priorityo
1.
Thus c.
l
customers have priority over customers of classes c. I'
l+
c i +2 , "', ck '
Rriori~
Cobham
["1_7
queue discipline.
1
i-Btro0.1:.lc'ed the
l~~~C:'·~of- the-line
Under this rule, customers of the
same class are served according to a head-of-the-line dj.scipline.
1.
Customers of class c
i
enter the service mechanism at
Numbers in square brackets refer to bibliography.
5
those mom.ents only when service of another customer is completed
and customer classes c l ' c ' "', c _ l are empty.
2
i
Thus if a
customer of class c. arrives during the service of a customer of
~
class c. and i < j, we say that a higher priority customer has
J
arrived.
The higher priority customer must wait at least until
the lower priority customer completes service.
In fact, the
c. customer must wait until the service mechanism is empty
~
and there are no customers of classes c l ' c2 ' "', c i _ l and
no prior arrivals of class c. in the waiting line.
l
,['14_7
White and Christie
priori ty Queue discipline.
introduced the pr~:tiv~
Under this rule, customel",s of
the same class are serviced according to a head-of-the-line
discipline ~
Customers of diffe:cent priori t Jf claf3ses enter
the service mechanism; when it is free, i.n accordance vdth
their respective priority classes.
However, if a customer of
class c. arrives during service of a customer of 10vrer priority
~
class c .(Le.
J
i < j) then the higher priority cU,jtomer
diately enters the service mechanism.
imme-
The lower priority
customer returns to the head of the waiting line of c. customers.
J
Thus customers of class c. are eligible for service only
J
during those time periods that the queueing system contains no
higher priority customers, i,e. no customers of classes c ' c ,
l
2
•• ', c. l'
J-
For simplicity, we refer to this as the
~~eew~tive queu~
6
discipline.
Other authors have employed the more descriptive and
unwieldy phrase "preemptive priority queue discipline".
We have assumed that the uninterrupted service time of each
customer in class c. is a random variable S with distribution
~
tion Bi(s).
If the service time of a class c
i
func~
customer is inter-
rupted then the total time U that the customer is in the service
mechanism is governed by one of the following disciplines:
(i)
Preemptive Resume where the total service time U is
independent of the number of interruptions.
Hence a
preempted. customer returns to the service mechanism
to complete that
~ortion
of the service time remain-
ing at the time of preemption and U = S.
(ii)
Preemptive Repeat (identical) means that a class c.
1.
customer, upon initial entry to the queueing system,
is assigned an uninterrupted service time S with
distribution function B.(s).
1
The customer must re-
main in the service mechanism ''1i thout interruption a
period S in order to complete service.
Suppose a
customer is interrupted N times and is in the service
mechanism a period V. before the j-th interruption,
J
j = 1, 2, ••• , N.
U=
(iii)
N
~j=l
Thus the total service time
Vj + S where Vj < S, j = 1,2, ... , N.
Preemptive Repeat (independent) specifies that each
class c. customer is assigned an uninterrupted ser1
vice time S(j) with distribution function Bi(s)
7
upon the j-th entry to the service mechanism.
Thus
the customer must remain in the service mechanism
without interruption a time period S(j) after the
j .. th entry in order to complete service.
the c
i
Suppose
customer is i.nterrupted N times "rhile j.n ser-
vice and is in the service mechanism a period V. beJ
fore the j-th interruption, j
~
1, 2, ••• , N.
Thus
the total service tj.me U == t:.N 1 V. + S(N + 1 ) where
J=.
V. < S(j) and S(j), j
J
==
J
1,2, "., N + 1, are inde-
pendent and identically distributed random variables.
It is clear that the preemptive repeat (identical) and (independent)
disciplines are equivalent if the uninterrupted service times are
constants for each class of customers.
White and Christie
,["14_7
,compared, in a general discussion, the preemptive resume and repeat
disciplines.
Gaver
,["2_7
introduced the terminology employed here
to differentiate betvreen the tvlO preemptive repeat disciplines.
5.
Theory of Queues.
The theory of queues deals with the
study of the properties of the probability distribution of a
stochastic process
t Zt'
entire positive time axis.
t
E
T} where T is a subset of or the
The process variable Zt is either a
real valued or a vector of real valued random variables defined over
the set of time points T in such a way as to describe some aspect of
interest of the queue.
In the application of queueing theory for different priority
classes of customers many aspects of the queue may be of interest.
Thus the study of such queues could deal "l'li th the probability distribution of any of the following processes:
(a)
The queueing time of a customer of class c. arriving
1
at time t where queueing time is defined as the
elapsed time between arrival and initial entry to
the service mechanism.
(b)
The \·miting. time of a customer of class c
i
arriving
at time t where waiting time is the total time
elapsed while in the queueing system.
(Some confu-
sian exists in the literature on Queueing Theory due
to the interchange, by some authors, of the definitions of queueing time and waiting time.
use the nomenclature defined here
We will
when referring to
other authors' results regardless of whether or not
this agrees with the terminology of the cited work).
( c)
The time-in-service of a customer of class c. arriv1
ing at time t \lhere the time-in-service, as defined
by Miller
["8_7,
is the total elapsed time Ivhile the
customer is actually in the service mechanism.
The completion time of a customer of class c., arriv1
ing at time t, where the completion time, as defined
by Gaver
.['Z},
is the elapsed time from initial
entry into the service mechanism until completion of
service.
( e)
The queue size at time t is a k term vector of the
number of customers in the queueing system of each
9
class of customer (excluding the customer in the service mechanism).
(f)
The busy period comrnencing at time t with arrival of
a c. customer at an empty queue "There the busy period
1
is the shortest elapsed time between two consecutive
instants when the queueing system contains no
customers.
(g)
The clearing time of the c. customers of a queue at
1
time t is the total time required to service all c.
J.
customers already in the queueing system prOVided
no additional customers arrive and the queue contains
no c 1 ' c 2 ' "', c i _ 1 customers.
The number of preemptions of a class c. customer
1
while in some state of the queueing system.
The
number of preemptions is defined as the number of
higher priority customers who arrive after the c
i
customer and complete service prior to his departure from this state.
For a given queue discipline pertaining to priority classes of
customers some of these aspects of the queueing system are closely
related to other aspects and the properties of the probability d.istributions involved are derivable one from the other.
For example,
the completion time plus the queueing time of a customer is the
,.,ai ting time of that customer.
In the most general situation, it
is not possible to say that a study of one aspect of the queueing
10
system is more important than the study of some other aspect.
This
depends entirely on the field of application of the queueing theory.
The study of the properties of the probability distribution
of a stochastic process {Zt' t
€
T1
is divided. into tvlO areas.
If
the set T is restricted to a finite interval, or su:bset thereof, of
the positive time axis, the transient behavior of the process is
studied.
t --->
The limi ting behavior is the study of the Hmi t, as
00,
of the probability distribution for ZtO
The study of
the limit of the probability distribution involves t"TO questions:
(i)
(ii)
Is the limit a proper probability distribution?
what are its properties?
If so,
In this dissertation we will concern
ourselves primarily with the limiting behavior of certain stochastic
processes arising in the theory of queues with k priority classes
of customers under preemptive resume or repeat disciplineso
shall consider also some closely related problems.
We
In Chapters II
and III we will assume the existence of the limiting distributions
under investigation and obtain certain properties of these distributions.
In Chapter IV we will derive the necessary and suffi-
cient conditions for the existence of these limiting dif,t:L'ibutions,
6.
T~eory
of Priority Queues.
In all studies of queues with
priori ty classes of customers , arrivals are assumed to follovT an independent Poisson process for each class of customers.
This means
that the probability of n arrivals of class c. customers in a time
-'A. t
~
period t is e ~ ('A.t)n/n~ where 'A. is called the arrival rate.
~
~
This is equivalent to saying that the interarrival times of class
c
are independent and identically distributed random variables
- Ai t
vlith the negative exponential distribution, i.e. Ai(t) ::: 1- e
i
with mean l/i\ .•
J.
This type of arrivals is often referred to as
random arrivals. with rate i\..
1.
-
We shall adopt this terminology.
The moment generating function (m.g.f.) for a Poisson process is
-i\.(l- Q)t
e
1.
, from which it is apparent that the m.g.f. of tne sum
of k independent Poisson processes with rates AI' A , " ' , i\k is
2
k
th m.g. f . 0 f a pOlsson
'
e - i\( 1 - Q) t , e
process Wl. th ra't e A = Z. 1 i\ .•
1.:::
1.
It vlould be difficult to summarize completely the knovlD resuIts and their derivations for
customers.
valved..
g~eues
with priority classes of
The derivations, in many cases., are quite long and in-
Hovlever" the major results are listed in Table 1.1 for
the limiting behavior of descriptive aspects of the queue.
In this dissertation we develop a powerful method for investigating certain multivariate limiting distributions of aspects
describing single server queues with k priority
tomers.
cl~sses
of cus-
The utility and generality of the method become apparent
as we employ it to consider some hitherto unsolved queueing and
storage problems.
12
TABLE 1.1
MAJOR RESULTS FOR PREEMPTIVE AND PRIORITY QUEUES
Ii
Author and
Reference
Service Time
Distribution
Queue
Discipline
negative
exponential
Mean queueing times for
head- of- theline priori ty each of k classes of
customers.
Results
-.-----~--------_+-------l__------------,---
negative
exponential
m.g.f. of queue size and
head- of- theline priori ty associated moments.
Kesten and
Runnenberg
general
independent
Functional form of Laplace
head- of- theline priority -Stieltjes transform of
queueing time and the associated moments for low
priority customers.
White and
Christie
1-14_7
negative
exponential
Preemptive
Resume
ill.g.f. of queue size;
moments of queueing time
and waiting time.
general
independent
Preemptive
Resume
results with respect to
the queueing time, timein-service, busy periods,
and number of customers
served during a busy
period.
["7_7
convolutions
of finite
number of
negative
exponential
distributions
Preemptive
Resume
I
-lI
m.g.f. of queue size and
associated moments,
13
7.
Queues and Stores.
a store or warehouse.
Suppose loads of a material arrive at
The supply in the store of this material is
subject to a constant rate of depletion and the store has infinite
capacity.
The analytic study of the amount of material in the
store, (called the store content) with respect to time is the
Smith ~lo_7 noted that the storage problem is
storage problem.
mathematlcaJ.ly equivalent to the study of the queueing time of a
single server queue ivi th one type of customer.
The arrivals of
customers coincide with the arrivals of loads of material.
service times of customers coincide with the load sizes.
The
The
uniform rate of reduction of the stox'e content is eqUivalent to the
operation of the service mechanism at a uniform rate,
Thus the
store content, at any time, is eqUivalent to the queueing time of
a customer arriving at that time.
Suppose the store receives loads of k different types of
material, say' c l ' c 2 ' ••• , c k ' ,.,here each load is homogeneous of
one type only.
Further suppose the store content is diminished
at a uniform rate.
The store content of material c
i
is reduced
only during those time periods when the store contains no c ' c '
l
2
••
Q
J
c.1.- 1 material.
emptive rule.
This rule of operation vJe shall call a
p~
It is evident that the store content of material c
i
is equivalent to the clearing time of class c. customers for a pre1.
emptive resume queue.
In Chapter II vJe investigate the transient
and limiting behavior of the joint distribution of the store content for k different types of material.
We vJill then employ these
results to investigate the distribution of the queueing times for a
14
queue with k classes of
ciplines:
custon~rs
and three preemptive queue dis-
resume, repeat (identical), and repeat (independent).
Miller L8~] obtained the Iaplace- Stieltje's
transform and the first
tvro moments of the queueing time for the preemptive reSUIJ1e discipline
only by a more direct approach.
Therefore we will only indicate
briefly the applicability to this case of our approach in order to
a.emonstrate :L ts generality.
The same method will be applied to
consider the hitherto unsolved problems of the queueing time for the
tvlO preemptive repeat disciplines.
In the course of this we vlill
also consider certain problems involved in a head-of-the-line queue
with a single service mechanism which is subject to breakdowns.
8.
Christie
Service Mechanisms Subject to Breakdowns.
White and
["14_7 noted the analogy between a preemptive queue dis-
cipline for two classes of customers and a head-of-the-line queue
with a single service mechanism which is subject to breakdowns.
The
customers with preemptive rights are considered as breakdowns of the
service mechanism.
In Chapter III vie consider fou:(' possible types
of occurence of breakdowns.
A variation of the method developed in
Chapter II is employed to investigate queues under these types of
breakdown.
In Chapter IV the necessary and sufficient conditions
are a.eterminecl for the exi.stence of a limiting distribution for a
Poisson Reduction Process.
The Poisson Reduction Process is defined
in this chapter and shovrn to include, as special cases, all processes
considered in Chapters II and III.
The numbers of breakdowns during
queuing time and during service time of a customer are considered in
in Chapter V.
CHAPTER II
Trill PREEI>1FTIV:8 STORAGE PROBLIN
Introduction.
1.
of material, say
A store receives loads of
c I' c 2' ... , ck'
He,terial
c.(l
< i -<k) arrives at
1
-
the store in homogeneous loads at random rate
secutive loads of material
c
i
<'Llfferent types
l~
i\. ••
1
The sizes of con-
are a sequence of independent and iden-
tically distributed random variables with distribution function
l/~.,
and mean
types of
J..
i
w~terial
= 1,2,
... , k.
for
.1
I
•
T J.
are independent but not necessarily identically disThus) in general,
B. ( s)¥ B. ( s )
J..
J
The contents of the store are diminished at unit rate in
accordance with a preemptive rule.
That is,
w~terial
c
i
for depletion only when the store is empty of waterials
c _ .
i 1
1
The load sizes for two different
tributed sequences of random variables.
~
B. (s)
vIe call this the
.R..~eemptive
storage system vdth
Suppose the store has infinite capacity.
is eligible
c ' c , ... ,
l
2
l~
w~terials.
'\tIe no;·,ed in Section
7,
Chapter I that the preemptive storage problem is mathematically equivalent to a consideration of the clearing times for a single server queue
with random arrivals} independent service times, preemptive resume
queue discipline and
k
classes of customers.
For the purposes of definiteness and simplicity the preemptive
storage problem is consj.dered for
11:
in Section 5 for larger values of k.
= 2.
Then the results are extended
16
2.
The Preemptive storage ))roblem for
store content of material
over
c
at time
l
t
Ie "" 2.
Let
X
t
be the
which has preemptive rights
Y , the store content of material
t
at time t. Define the
2
following components of the bivariate distribution of the store content
r(x,
y; t)
G(y; t)
-
o < Yt :5
p(O < Xt .:: x,
p(O < X
t
;:
J(z, t)
:::
rt(t)
_. p(X
t
p(X + Y
t
t
;;:
;
y)
o < Yt -< y)
"" p(Xt ;: 0,
H(x; t)
c
:5
;
x, Y ;;: 0)
t
:5
0, Y
t
f-
z, Xt
;:
,.
0, Y (z 0)
t
0) .
Thus
Also
J(z;t) + H(z;t) + G(z;t) + rt(t) ;;: p(Xt + Yt
Loads of material
c.
in loads of sizes
Sli,, 82i,
1
arrive at random rate
s;,,./
i
1 2.
... ,:::::,
."5 z) .
A..
1
at
times
The sequence
sequence of independent and identically distributed random variables '1-11th
mean
l/~.
1
and distribution function
is independent of the sequence
Bi(S), i ::: 1, 2.
fp,2 1
"-r'
The sequence
It is assumed. that each
load of llEterial is greater than zero with probability one, i.e.,
B.(O+) ;: 0,
1
i::::: 1, 2.
The store content
Zt::: (X , Y ) is sUllIDErized in Figure
t
t
a specific realization of the process.
2.1
for
17
'--__sJ~
I
t Il
s_~~'---""7-
"__
t';;.
t
t';3
I
t'\l.l-
tl).
I
t
__
E4~~_. ;l~.1
The process
Zt:: (xtJ Y )
t
is a continuous parameter Narlwv process.
This fact suggests an investigation of the time derivatives of the
components of the instantaneous distribution function.
lYe vlill prove
first the following useful lemma.
Lemnn 2.1.
S and
G)
Consider a time interval [OJ
0 < G<
T.
27
and two ~on-negative numbers
Define a process
:: Hax (0, g - 0),
o
where
eo
(2,1)
Eo
is any prescribed non-negative number.
+ S - T
<
e (S)
-T
0<0 < Q
Then
0) -<- E
(SJ G) <T
g (SJ T) <0
e +
T
s.
JL
Proof.
for
NOvi
0,
Q* --
Q,
~
T
Thus, from the definitioD,
T.
(S, 0) : .: Max (0,
~
a
+
S - T)
~ T ( S , Q) ::.: 11ax (0, ~ a + S -
S -
l' ,
T
+ A)
and
sT (S ,
T) ::: ):.lax (0, ga + S - T, S)
The lel.11IDa is a direct result of these three equations and the fact that
~o'
S, and
T
are all non-negative ntunbers.
Consider the left open, right closed time interval
'Vihere
T
> O.
The store content process
(t, t + 27
Zt+T ::.: (X +T 'Yt +1' ) is at
t
the point (0, 0) if one of the following mutually exclusive and exhaustive events occurs:
(i)
Zero arrivals occur in the interval (t, t+!.7
o -<
(it)
X + Y
t
t
.s:
(iii)
ivhere
Jet + Yt
::.:
(;0
Q,
and
(; T (S, Q) satisfies the definition of lemwB 2.1.
Two or more arrivals of wBterial occur inth certain restrictions on
X + Y and the sum of the load shes.
t
t
In an interval of length
l
S occurs at time t +
Q :5 T, and ~ T(S, Q) ::: 0
the function
exp( -A
To
One arrival of load size
o<
and
T -A
(A l + A )T
2
2
T the probability of zero arrivals is
T) and the probability of one arrival is
The probability of two or more arrivals is
19
Thus
and, by lemma 2.1
Hence vre may "rri te, as
T
-> 0 ,
+ 0(1) .
Define the Iaplace-Stieltjes transforms of the components of the distribution of store content as follows:
x
e -ux dx H( x;t )
H (u;t)
e-VY d
y
,.
G(yjt)
;
F(x,y;t)
,.
o
e
-ux
d
x
o
co
xy
F
(u,v;t)
r -vy d
::: J e
x
y F (u,y;t)
;
o
where
u and v
numbers.
are restricted to the class of real non-negative
20
The store content
Zt+T
is in the region
< Xt-+T < ::,
(0
Y
t +T
'" 0)
if one of the follm.Q.ng mutually exclusive and exhaustive events occurs:
(i)
(ii)
There are no
arrivals of waterial in the time intel~al
(t,t+!7
'T
and
A load of
o<
G
:5
c
l
< Xt :: x +
t
waterial of size
= Yt
t + G,
8 arrives at time
G< 8 < x +
(a)
X
(b)
0 < X :: x +
(c)
0 < X + Y :: G, Y > 0, 0
t
o.
::;
and;
T,
:::: 0,
T -
Y
T,
t
:::: 0, 0
t
t
t
T -
G,
< 1;/8,
G)
t
<
~T(8,
or
:5
x,
or
Q):5 x,
So (S, G) :::: X + Y .
t
t
vrhere 'Ive clefine
(iii)
'T, Y
Two or more arrivals occur in the time interval
with certain restrictions on
Thus, employing LerJJ.rrJa 2.1
Zt
and the load sizes.
,
x
+
A1T\.{ H(x-z;t) d B1 (zt.7
T
and
X+T
+ A
1
T,!
H(X+T-z;t) d B (z)
l
o
+ AlT (J(T;t) + G(T;t)} B (X+T)_7
l
2
+ O( T ) •
(t,t+!7
21
VIe note that
J
T
0:5 Al T e-
AT
L1!(t) Bl(T) +
H(x-z;t) d BICzt7
o
uniformly in t. Similarly the terms in (2.3) and (2.6) designated
2
O(T ) are probabilities that are dominated by the probability that
two or more arrivals of :tJ1.aterial occur in an interval of length
Thus both these terms are dominated by a term which is
in
t
and in the case of
(2.6),uniformly in
OCT)
T.
uniformly
x.
In order to take Iaplace-Stieltjes transforms of (2.5) and (2.6);
vre require the following lemma.
Lemma 2.2.
Jf
Hl(X) ~ H (X), for all
2
then the LapJace Stieltjes transforms
x
and
Hl(O+)
~(u) and i;Cu),
respectively;
c:
..L.
satisfy
1\ (u)
:5 H~(U)
where the transfonn variable
J
co
u
is real and positive.
co
e -me H (x) dx
l
o
:5.J
e -ux
H (X) dx
2
0
This implies, upon integmting by parts,
<
and the lemma is proved.
= H2 (O+) = 0
22
Inequalities (2.5) and (2.6) become, after taking Laplacestieltjes transforms with respect to
x,
J
T
(2.7)
Il(u;t+T)::: e(u-A)T LHx(u;t) -
e-
ux
d H(x;tt7
x
a
r () + Hx( u;t )_7 Bx(l u )
+ e -I\, T A 1" _ n t
1
and
r
T
( 2.8)
J
e
-ux
d H(x;t)
x
-
7
o
+ A 1" e(u-A)T
1
Ln(t)
+ J(T;t) + G(T;t) +
~(u;t17
'B~(U)
,
+ o( T)
"There the terms
o( 1" )
are uniformly
o( 1" )
in
t.
= 0, 0 < Yt +1" --< y)
is in the region (X
+1"
t+1"
i.f one of the follOi'ring mutually exclusive and exhaustive events occurs:
The store content
(i)
Zt
There are zero arrivals of material in the interval
T
(ii)
One load of material
(iii)
~ T, T
(t,t+~7 at time
(a)
~o
Xt
arrives in the interval
< Xt + Yt + S :5 y +
One load of material
where
l
t --
t
t + G, 0 < Q ~ 1", of size
at time
Xt + s
c
< X -:- Y < y + 1"
= Xt
t +
+
- 0
-- y t -
Yt ·
,
c
1"
G, 0
of size
<
G
:5
Sand S < T - G,
T.
S arrives in the interval
T, and
0
This requires either
or
(t,t+~7
< ~T(S, Q) ~
Y
23
(iv)
(b)
0 < X ~ ~, Yt = 0,
t
or
(c)
0 < Xt + Yt ~ y +
Xt ~
T,
T,
Yt >
o.
Two or more loads of material arrive in the interval
(t, t+27 with certain restricti ons on sum of load sizes
and Z-,- •
v
The probabilities arising under (ii) and (iv) are dominated, uniformly
in both
lemma
t
2.1
and y,
by probabilities that are
OCT).
Thus, employing
and other elementary :i.nequal:i.ties v1e have;
(2.9) G(y;t+T)
~
e -i'.Tr
_ G( Y+T;t) + F(T,y;t)-G(T;t) - J(T;t)_7
y
+ e-i'.T i'.2TLTC(t) B (y)
2
e
+.1
G(y-z;t)
o.
B'2(Z )_7
0
, ,
O~T)
and
( 2.10)
G(y;t+T) ~
e-i'.TfG(Y+T;t) + F(T;Y+T;t) - G(T;t) - J(T;tt7
y+T
+ e-i'.T A2 TL1!(t) B2 (Y+T)
J(
00
-I.J
GCY+T-z;t) d B2 (zt7
o
+ e -AT A2T; F T,
o
+
where the terms
OCT)
OCT) are Lrr1iform in t. Upon taking Laplace-Stieltjes
transforms with respect to
(2.9) and (2.10) become;
Y and
upon applying lelillTIa 2,2, inequalities
T
-Jr
e-VY
.
(2.11)
d G(y;t)
y
-
7
o
+ e -A.T Fy( T,V;t)
e-A.T A.
+
2
'f
r
1r(t) + GY(v;t17 B~(V)
,
- o( T)
and
T
-
f
e
-vy
o
T
e(v-A.)TrFY(T,v;t)
-
+
-Je-VY dy
F(T,y;t)7
o
e(v-A.)T A. Tr1r(t) + GY(v;"c)7 BY (v)
+
2 -
-
2
+
Again the terms
OCT) are uniform in t.
The store content
is in the region
Z
t+T
(0
<
X
<
t +T -
x,
o < Yt + :5 y) if one of the following mutually exclusive and ey..hausT
tive events occurs:
(i)
No loads of material arrive in the interval (t,t+!7 and
T
(ii)
< Xt
:5 x
+ T, 0
One load of
o<Q<T
(a) X
t
c
= 0,
Q
:5
y.
material of size
l
and
< Yt -l- T
S arrives at time
either
< Yt
:5 y
+ Q, T - Q
<S <x
-I-
(b) 0 < Jet ~ Ql' Q - Ql < Yt ~ y + 9 - Ql'
T - Q
<S <x
(c) Q < Xt ~ x +
t + Q,
+ T - Q, 0
T -
<
S, 0 < Yt
Q
l
:5 y
~
Q
,
or
T - Q,
or
25
(iii)
One load of
c
2
waterial of size
S arrives at time
t + G, 0 < 9 :5 T and T < Xt :5 x + T, 0 < Yt + S :5 y.
(iv)
Two or more loads of material arrive in the time interval
(t,t+!7 with certain restrictions on Zt
As before; the application to these four events of
and the load sizes.
J.eIDl11B~
2.1 leads to
the results
+
e-~TA.ITLG(y;t)
x
B (x) +
l
j F(x-z,y;t) d B1(z).}
o
y
+
j F'(x,y<.z;t)
o
- 0(")
and
(2.14)
F(X,y;t+T):5 e-A.T .LF(X+T,y;t) - F(T,y;tt7
X+T
+ e-A.TA.IT .LG(Y+T;t) B (X+T) + jF(X+T-Z;Y+T;t)dBl (z)]
1
o
T
+ e-A. A. TLH(X+T;t)B (y)
2
2
if
+j l?(x+T,y-z;t)
d B (z_27
2
o
+ o( T) •
The Laplace-Stieltjes transforms of (2.13) and (2.14) reduce to, after
applying lemwa
2.2 and some elementary inequalities,
26
(2.15)
J
T
~Y(U,v;t+T):: e(u-1-.)TI-FXY (u,v;t) -
e- ux d ?(x,v;tt7
x
o
r
+
y
T
1-. T e -1-. _ Hx( u;t ) + Fx ( u,v;t ) _7 BY( v )
-
o( T)
2
2
and
(2.16) FXY (U,'V;t+T)
~
J
T
e (u-1-.)T.rFXY(u,v;t) -
e- ux d pY(x,v; t t7
x
o
+ 1-.
1
T
y
e (U+V-1-.)T lGY(v:t)
,
+ Fx ( u,v; t)7
_ Bx( u )
1
+ OCT)
As before, the terms
OCT)
00
Thus
and
t.
00
=J1
0- 0-
are uniform in
-ux-vy
e
'
d
()
x dY P.t x,Y
Define
27
U~on
combining inequalities (2.2), (2.7), (2.11) and (2.15) we obtain
r e'(u-A.)T 7r x( u; t )
-1_ _ H
+ _
+ e-I-.T 1-.
1
-71
y
+ Fx ( u ,v; t )_
B~(U) .LJr(t) + rr(u;t)
T
+ GY(v;t)
+ pXY(u,v;tL7
+ fe(V"I-.)T-l_7 GY(v;t)1 T
J
T
e(U-'f...)Tf
e-
o
- e
(v-'f...)T
T
ux
d
x
H(x;t) +'-.{ e-
ux
d
x
F'Y(x,v;t)_71 T
0
T
,.I
o
- o( 1) •
Moreover we observe that
J
T
- e VT
o
::: 0(1)
Similarly
T
e-'f...T .LG(T;t) - e
VT
J"'e-VY
o
d
y
G(YitL71 T
< 0 .
28
Thus
T
e-'A.T_rG(T;t) _ eVTJe- VY d
Y
G(y;t)
7/
--
T
0(1);
:=;
o
as
T
->
0+ .
Li1\:.e,.,ise
T
UT
!
e-
UX
~ H(x;t)
7/
T
:=;
e-'A.Tfrl(T;v;t) - eUT (e- ux d pY(x,v;t)
7/
e
...
x
--
0(1)
o
and.
T
v
x
-
T
:=;
0(1) .
o
Let us now omit the non-negative term
hand side of
e-'A.T J(T;t)/,.
from the right
(2.18) and then allow T to decrease to zero.
lle then
discover that
Inequalities
(2.3); (2.8); (2.12) and (2.16) are combined to form an
upper bound for
29
By an argument analogous to that employed for the lower bound it can
be shol'll that this upper bound has a limit equal to the right hand
side of inequality (2.19).
Theorem 2.1.
The double Laplace-Stieltjes transform p~(u,v) has a
partial derivative with respect to time which satisfies the equation
(2.20)
where
Proof.
~ ~ ~l + ~2 - ~1 B~(U) - ~2 B~(V)
That the right hand derivative exists and satisfies (2.20) has
just been demonstrated.
The existence of the left hand derivative, which
also satisfies (2.20), follows by considering the upper and lower
bounds for
These bounds can be derived from
(2.12), (2.15) and (2.16)
(2.2), (2.3), (2.7), (2.8), (2.11),
merely by replacing
that these eight inequalities also imply
u(t)
= U(t-T)
lr(u;t)
=
+
0(1)
HX(U;t_T) + 0(1)
GY(v;t) ~ cY(V;t-T) +
0(1)
,
t
with
t-T.
We note
30
which are employed in a
straightfo~~rd
manner to complete the proof
of the theorem.
3.
Limiting
~ehavior
of the Preemptive store for
1~.
The
necessary and sufficient conditions for the existence of the limiting
+ A /1l < 1
2 2
probability one. It is recalled
bivariate distribution of the store content are
and finite initial store content
~nth
is mean load size and
ci ' i
= 1,
2.
A.
J.
Al/Il
l
is the arrival rate of material
This is proved in Chapter IV.
Here we assume that this
condition is satisfied and obtain four functional equations involving
the limits with respect to time of distribution components
F(x,y;t) , G(y;t)
e
and
~
H(x;t).
=
-:M
G(y)
=
lim G(y;t )
t ->00
,
H(x)
-
lim H(x;t)
t ->00
,
J(z)
:::::
lim J(z;t)
t ->00
,
F(x,y)
:::::
F(x,y;t)
lim
t ->00
,
p(x,y)
:::::
t --> co
pt(x,y)
lim
t
The limit as
Define
~(t)
lim
t
~(t),
...->
co
of the individual terms of inequalities (2.2)
and (2.3) allows us to write
31
OCT)
since the terms designated
by terms vlhich are
o( T )
in these inequalities are dominated
uniformly in
t.
From (2.21) it is clear
that
(2.22)
(i\l +
i\,)
J\
:::
c.
/T
lim f(G(T) + R(T) + J(T»)
T,!Jo+
_7
We denote the Laplace-Stieltjes transforms of the following components
of the limiting distribution as
00
J
!feu)
o
J
e
-ux
0. R(X)
,
0. G(y)
,
.
00
GY(v)
:::
e-
vy
o
and
00
00
('
('
: : ,J , j
o
e-
ux vy
0.
x
0.
Y
F(x,y)
0
The uniformity vii th respect to
t, of the terms included in
o( T) in
the inequalities (2.5) and (2.6), allows us to write the Laplacet ->
Stieltjes transforms of the limits, as
00,
0::' the inequalities
as
(2.23)
~(u):: e(u-i\)TfIt(u)
J
T
-
e-
ux
0. H(xL7
o
and
(2.24.)
J
T
It(u):5 e(u-i\)T[-Hx(U) -
e-
ux
0. H(x) + i\lT
J\
B~(U)
o
+ o( T) •
32
Therefore
lim
T
inf
e-ux
,J, 0+
d
H(X~
> lim sup
tJ.. 0+
The existence of the following limit bRs been demonstrated
,
1 lm
T.J,-
e
0+ [
(U-A)-r
---
T
!
e-
ux
d H(Xl]
=
h, say.
Furthermore, since
e
-AT
R(T)
T
<
e
(U-A)T
T
T
J
e-
ux
d H(x)
<
e
(U-A)T
R(T)
T
0
it follows that
lim
;-H(T)/T 7
TJ, 0+Hence
=
h
( 2.2 5) reduces to
( 2.26)
As before, we can write the Laplace-Stieltjes transforms of the
limits, as
t -->~) of inequalities
(2.13) and (2.14) as
33
(2.27)
FXY(u)v)
~
T
e(u-;\.)T .LFxy(u)v) -
,.f e- ux ctx r(x)vt7
o
and
J
T
( 2.28)
y )
Fx (
u)v
-ux dx FY( x)v )_7
e (u-;\.)T _rFXY( u)v ) - . e.
5
o
+ e(u+v-;\.)TAlT ["GY(v) + FX:Y(u)v L7
B~( u)
+
e(U+V-;\')T;\'2T £~(U) + F,~J(u,v)_7 B~(V)
+
o( T)
Division of (2,27) and (2,28) by
and the taking of limits as
T
produces the bounds
.->
e
sup
-l.- 0+
lim
T
T
(u-;\.)T
J
."."".-=-......=.=---=
T
e-
ux
d
x
?(X,V)]
0
Thus we have verified the existence of
lim
T
.J. 0+ [
(u-;\.)T
_e_ _
T
j
o
e- ux
dx
?(X'V)]
say)
T
t 0+
34
and
(2.29)
may be rewritten
Consider the limit of the terms in (2.9) and (2.10) as
Due to the uniformit;Y' of convergence "d.th respect to
OCT)
t
t
-->
00
•
of the terms
the laplace-Stieltjes transforms of the limits are
(2.31)
GY(v)
~
J
T
e(v-A.)T .LGY(v) -
e- vy d G(YL7
a
...
o( T)
and
J'
T
(2.32)
GY(v):5 e(V-A.)T {GY(V) -
e-V'J
d G(y) + JiY(T,V)j e-VYdyFl(T,y)
a
+ A.
+
2
T
r
-
Jr
OCT)
He note that
e=kr J?Y'(T,V)
<
T
Hence
lim
~FY(T,V)/T_7 ~
T,J.. 0+
fY(v)
+ GY( v)
-
7 B2Y( v) }
35
Thus, from
(2.31)
(2.32)
and
lim inf
T~ 0+
::
lim
sup
'[.~_._-(v-r-.h
T,~O+
JT e -vy-
-J
d G(y)
To
from i'lhich inequalities i'le can deduce the existence of
lim ~
T~ 0+
t
Now
o
-
JT e- vy
(v-r-.h
T
0. G(y)
]
o
<T(
.... I\
v \
( 2. 32)
and so from
I
we obtain
< e ~r-.T F ( T,T) I T
T
J'
T
o
e- vy d
Y
F(T,y)
e
(v-r-.h
~
T
J
e-
vy
d G(Y)
o
where the limit, as
(2.34).
T.j,
0+, of this upper bound is zero by virtue of
It is now simple to deduce that
lim
T J...
0+
~J(T)/T_7 =
O.
36
Thus
eql~tion
(2.22) reduces to
We summarize the results contained in equations (2.26), (2.3°),
(2.34)
and
Theorem
(2. 35)
2.2.
in the follmving theorem.
The components of the limiting distribution of the
store content, under a preemE!ive rule of operation, satisfy the four
functional equations ;
,
Corollary to Theorem 2.2 .
The components of the l:!..m.i:ting distribu-
tion of the store content are independent of the components of the
initial distribution of the store content.
Proof of the Corollary.
of store content are
The components of' the iniUal d:J.strjJ:lUtion
j(t), H(x;t), G(y;t) and
F(x,y;t) at t
~
0.
The proof of the corollary is completed by noting that these functions
do not appear, either explicitly or implicitly, in the Iaplacestieltjes transforms of the components of the limiting distribution,
as given in equations
(2.36) to (2.39).
37
The result of clearing of fractions in (2.36) to (2.39) and
adding is
(2.40)
4. f.loments of the
(2.40)
Equation
J;.imH~ng
Distriht;tion .of store Content.
will be employed in this section to obtain the first
and second order morrents of the limiting
store content.
~istribution
of the vector of
It is possible to obtain the derivatives and hence the
moments of the components of the distribution directly from the four
(2.36) to (2.39).
equations
h
and the un1(nOiin function
gebraically quite involved.
However, the unlmovffi parameters
_v
I
\.
f~{v)
g
and
make this approach lengthy and al-
Therefore vle shall enrploy a less direct
approach which employs equation
(2.40)
and si!nplifies the algebra
immensely.
In order to shorten the notation let
and
"A;j
X
II (n) + F
Thus vre may vTrite
(2.41)
(u,v)
(2.40)
;; M(u,v)
;; H .
as
M(u - ~) + N(v - ~)
j(~
;;
o.
Also, in the remainder of this study, we will designate the order of
partial derivatives with respect to
u and v
by post superscript
numerals.
For example
ll
M
_.
d~
dUdv
and
rvr
12
""
d\J1
d'UdV 2
The function, without a post superscript, will be the function
out any operators.
with~
1\.lso note that
as opposed to
Evaluation of any function of
u and v
at
u:::; v :::; 0 will be desig-
nated by a post subscript zero, e.g.
, evaluated at
u:::; v
He recall that the load size of material
variable
i :::; 1, 2.
Si
with distribution function
c
Bi(s)
:::;
0
<!J20
-A,~
J.
E(S~)
.J..
:::;
A,
1
E(S')
1
:::;
~1,2 E(S2)
:::;
0
<!J'O
0
¢l01
0
¢l02
:::;
0
¢l 0,
0
11,1 :8(Sl) :::; A,I/~1
:::;
-A. E(S2)
2
2
A.
2
E(S')
2
:::;
-ill
2
:::;
and mean
say;
,.
say
11)' say
:::;
A. /1l
:::;
-n
:::;
,
,
m
1
;=:
2
2
2
,
:::;
n
say
n, , say
•
0 •
is the random
i
Thus
¢l10
«
l ' say
l/~i'
39
Note that
i
=::
0
for both
i and
>
j
and that
1
i'
N
o
J
for
> 1 .
Theorem
2.3.
The limit1ng distribution of store content satisfies the
equations:
(2.42) ?(x
=::
Y
(2.43 ) p(x
=::
0)
=::
1 .. m
p(y ::: 0)
=::
1C
(2.44)
w'here
u*
:::
0)
:::
rt
1 .. m - n
=::
l
l
l
u*/f.. 2
is the unique positive real root of the equation
Proof.
The first order partial derivatives of the terms of equation
(2.41),
with respect to
( u ..
<l>
u
and v
successively, are
1 0 )M .. 4> lO( N +rt)
) £'1.10 + (1-4>
=::
0
and
-M + ( 1.. 4> 01) N + ( v - 4> ) N01
( u .. 4> )M01 .. 4> ol.
These equations, evaluated at
u
=::
v
=::
4>0lrt
=::
0 .
0, are
and
Also,
vle
have
Mo + N0 +
1C
:::
1,
because we have assumed that the necessary and sufficient conditions
for the existence of the limiting distribution are satisfied.
Addition
of the respective sides of these three equations shows that
Next we rewark that
J
00
p(X
=
0) _.
rt
+
d G(Y)
o
since putting
~2;:;:
0
has no effect on the store content of material
Thus
Consider equation (2.
~l
u -
-
7).
The denominator of the right hand side
x.)
u ;:;: D()
+ ~l Blt
u )
~2
say)
has derivatives
00
IO
D
;:;:
J
1 - ~l
e -ux x d Bl(X)
0
and
00
20
=
D
~l
.f
e-
ux
x
2
d Bl(X) > 0
for
u>O
0
Horeover
D
0
.-
-~l
creasing function as
and D(oo)
u
.
=00
goes from
a unique positive real root) say
Thus
0
U*.
to
D(u)
00
is a monotone inHence
D(u)
=
0
has
Now iC(u) is the generating
41
ftmction of a component of a bivariate distribution and therefore is
bounded for
u >
o.
Thus substitution of
u
~
u*
in (2.37) must
yield an indeterminate form, in other words
Thus
Therefore
p(y
~
0)
~
rr +
HX(O)
= rr + Al
rr
fB~(U*)
- }]/ -A2
;::: rr u* / A
2
This completes the proof of' the theorem.
Corollary to Theorem 2.3
depends on the load size distribution of non-priority material
through the first moment.
H(X)
The Laplace-Stieltjes transform of
The function
H(x)
W3Y
c
2
be obtained in ex-
plicit form by inversion of its transfornl when the form of B (s) is
2
unknovm.
Proof.
The proof of the corollary is an evident result of the
equation
"There
u*
is the unique positive root of the denominator.
He desire the moments of the store content (X,Y) under the
only
42
limiting distribution.
evaluated at
u
=v
~
It is clear that
O.
Consider the second order partial derivative
of (2.41) with respect to
u.
This is
1
and on evaluation at
u
=v =0
;: m /2
2
it reduces
(1 -
to
m)
1
The reader will recognize this formula as the expected vmiting time,
under the limiting distribution, of an arriving customer in a single
server head-of-the-line queue with random arrivals, a result that is
easily obtained from formula (19) Takacs
Iii}
He also derive from (2.41) the equation
1\1
02
(u-<1»
+ 2J'/I
and
These equations give
01
(-<1>
01
02
) + l\I
(v-<1»
page 110 •
43
E(Y)
:::
:::
Theorem 2.4.
The first and second order moments of the limiting
Cl:~.:
tribution of store content are:
(2.46)
E(Y) ::: ~nlm2/(l-ml) + n2-7/2~
(2.48)
Proof.
Var(X)
;
= ~/3(1-ml) + ["ln2 /2(1-ml L72 ;
Equations (2.46)
and
(2.47)
have been verified already.
third order partial o.erivative of equation (2.41),
evaluated at
u::: V ::: 0
reduces to
The
44
The third order partial derivative of
evaluated at
u
= v = 0,
(2.41),
reduces to
Consider the two rew~ining third order r~rtial derivatives of
(2.41),
and
. 03
01 02
02 01
03.
(u-4»r,1
+ 3(-<p )M
+ 3(-<1> )H
+ (-4> )(l\1+N+1!)
Addition of the first
and evaluation at
u
to one-third of the second equation
e~lation
=v = 0
results in the equation
=
2n E(XY) + n E(X+Y) + n / 3
l
2
3
StJ;'aightforward w.anipulation of results (2.51), (2.52) and (2·53)
establishes the three results (2.48), (2.49) and (2.50) and the
theorem is proved.
Theorem 2.5.
If the first
tribution functions
Bl(s)
r
moments exist for the load size dis-
and
B (S), then the first
2
(r-l)-th
45
E ( Xi-l Y1'-1) ; i = 1; 2; .•• ,
order moments
exist for the limiting distribution.
order moments of the
(2.41)
equation
~tore
of the store content
l'
Also, if the first (r-l):th
content exist, tEey are derivable from
in a manner analogous to the derivation of the first
and second order moments.
Proof.
He i'fill prove this theorem by induction on
the theorem is valicl for
The
l' -
2( r- 2>0) .
?
Hrite
(1'+1) partial derivatives with respect to
r.
Assume that
(2.41) as
u and
v
of the
order are
(2.54)
1
i:
1'-1
. , (.l-J. ) ".
J.
j=o k=o
=
for
that
k ~ 1,
u
= v = 0,
and for
i
=1
'.'J
Ni,r-i
i ::
r
i:
1\.=2
oui-javr-i-k
u Hi,r-i + v Ni,r-i + i Ni-l,r-i + (r_i)Ni,r-i-l ,
= 0, I, 2,
i
or-l~-jLN+N+!!..7
• I
~.
~
°
i-Te
=
r.
°
have
Recall that
for
1 >1
~jk
=°
and that
when both
¢
o
:: 0.
j?- 1,
Thus
for
r-th
1~6
1'-1
E
k==l
(r-l)!
(r.. h:-l)!
k!
Division of the first equation by
r
and addition of the tvlO results
in the equations leads to
where
1'1
is a function which is linear in the moments of store
content of order
r-2
or less and linear in the first
r
moments
of the load size distributions.
Evaluation at
(i
= 2,3,
u
=v =0
of the last
r-l
equations
... , r) of the set (2.54) leads to
i
E
j=l
r- i
+ E
k==l
11::
i == 2,
After transferring all store content moments of order
3,
(r-l) to the
right hand side, this set of equations together vdth the immediately
preceeding single equation may be written in
L~trix
form as
r
.
47
o
__ (_l)r-l
o
The functions
are linear in the moments of store
, f , ... , f
r
2
content of order r-2 or less and linear in the first r moments
f
l
of the load size distributions.
f
r
are completely' determined.
Thus, by assumption
f , £2' .•. ,
l
The coefficient matrix on the right
hand side of (2,56) is non-singular provided the diagonal elements,
which are the latent roots, are non-zero.
That is, provided
and
DatIl of these conditions are satisfied if
m +
1
1.1
1
~~
1.
Tl1is is
it
necessary condition for the limiting distribution of store content
to exist, thus, we are assuming that this condition is satisfied
throughout this chapter.
proved in Chapter IV.
The necessity of this condition will be
The mathematical induction is completed by
48
reference to the results
(ct. Theorem 2.4) for the first and second
order moments of the store content.
This completes the proof of
Theorem 2.5.
L~>loments
1:
>
of the Limiting Distribution of store Content for
The extension of the methods of this chapter, Sections 1 to
2.
k > 2, is straight-
4, to a storage system 'Ilith k types of material,
forvlard.
However, the number of equations involving the laplace-
stieltjes transforms of the limiting distribution components (cf.
equations (2.36) to (2.39»
for a k-level preemptive storage
is
system.
The volume of vmrk involved in rigorouslJr deriVing these e-
quations
rapid]~
becomes impractical.
Even the algebra involved in
obtaining moments of the limiting distribution from an equation similar to (2. 41) i-muld prove lengthy and tedious.
HOi-leVer, a method
exists for obtaining the moments of the limiting distribution of store
content for
k > 2, by employing the results for
k::: 2.
section 'I-le l'lill describe briefly the method for gene!'al
lustrate it for the case
k:::
In this
k) and 11-
4.
Consider a store with random arrivals at rates
of k
types of material
operation.
c ' c , ... , c
l
2
k
The load sizes have distribution functions
B2 (s), .•. , Bk(S)
Define
AI' A , ... ) ~
2
with a preemptive l~.le of
i-nth eJ~ected load sizes
B (s),
1
l/~l' 1/~2' ... ) l/~k .
49
and
j
< i-I, i;;:; 2,3, ... k.
Also, let
j ;;:; 1, 2,
. ..,. i
;;:; 1,2, ... ; k .
The essential point, in considering the store content of material
c., is that only two groups of material types need be considered.
1
These are
cl ' c2 ,
c i ' and (b) rra terial type
(a) the group of materials
emptive rights over
••• J
C.J.- 1
c .•
Let
1
the random variable of a load size of material in group
c _ ·
i l
Thus
Si
1
Si be
c l ' c2 '
... ,
is a random variable with distribution function
~lBl(s) + ~2B2(s) + .•. + ~~J.-.L~B~;L-.L~(s)
..
;". .
vlith pre-
and random occurrence rate
Thus
,I"i-~l
r E(Sj)1
+
~'2 E?(Sj)
2 + ... +
~i-l
E(Sj)
i-l-7
+ m2J. + ... + m.1- ]-, J.
m
ij
Let
Zk
,say
X , .•. , ~t) be the random vector of the store con2
types of material. Define X. as the random variable,
= (Xl'
tent for Ie
1
under the limiting distribution, of the store content of the group
C
l
, c ' ... , c _
2
i l
of materials.
i-l
;;:;
l:
j=l
X
j
Therefore
50
Theorem 2,6.
X,1and
.-
X"
1.
The first and second moments of the store content,
under the limiting distribution, are:
'
,
(2,58)
E(-2)
X..;~
::::
-mi~
~
I3(1 - -)
m'l
1.
1
+ -2
E(X,)
: : -rm'lm'Q/(l
- m'l)
1.
1.
1.c
1.
2
r -m'2 I( 1 - -)
m'l 7
-
1.
+ m~~7/2(1
lc-
1.
-
'
- m'l
- m'l)
1.
1.
(2,60)
- I I
72
+ '01c -f- m'lm'Q
- mil ) ( 1 - -lli-ilL
1. 1~ 2\1 - m'l
1.
~
+
t
1
2
-2 rmir)/(l - m~l
- m'l)
1.
1 . -7
.-
c;.
E(X~)/2(1
- m'l)
+ E(X.)
E(X
1.
1.
1.
i
(2,61)
E(X,X,)::::
m'l
1. 1
1.
Proof,
Formulae (2,57) and (2.58) are obvious extensions of formu-
) ,
lae (2,46) and (2.47), respectively, when arrivals in the group
e , c , ... , c _ are considered as having a load size distribution
2
i l
l
function
TIBI(S) + T B2 (S) + .•. + Ti_IBi_l(s). Similarly, formulae
2
51
(2.58), (2.60) and (2.61) are obtained directly from (2.52), (2.53)
and (2.51), respectively.
Suppose we now consider arrivals in the group
vdth load size S.(i)
with distribution function
J
T.
1+
lB. l( s) + .•• +
rate
c i ' c i +l '···, c j _ l
T.B.(s) +
1
).
lB.. l( s ) . The arrivals occur 'vdth random
JAi + A + + .•. + Aj _ l ::: Xj(i), say.
Thus
i l
1+
T.
J-.
X.(i)
E r(S.(i»r 7
J
.J
-
:::
m.lr + m.l+,r
I
+ .•. + m.J- ].,r
_.
::: ill
jr
-
-TIl
ir
::: m.Jr (i) ,
say.
Therefore the store content
::: X.(i),
J
say ,
has moments;
E(X.(i»
=
J
and
t
~m'l(i)
mi0~.. /(I-m~l)
+
J
1
m·J 2(i)_7/2(1 - m'l)
,
J
52
(2,64)
E rCX,(i»2
-
J
-
(m'l(i»2 m,~/3(1-m'1)(1 - m. l )2
J
~~
J
~
7 ::;
+
~-mjl(i) IDi2 /(1 - mjl)(l - IDil )_72
-I-
m'l(i)
m'2
J
1
-I-
m'1 2 m'2(i)/2(1
J
m~2(i)/2(1
- m·J l )2(1 J
m· l )
1
- ill.])2
J .
These are a direct consequence of Theorem 2.6,
Formula (2.64) me.y be
written more simply, from equation (2.53), as
-7
ErCX.(:t)2
--
J
::; Lr2iii·J l (i)
E(X.X.(i»
]. J
+ m'2(i) E(X. 1)
J
J-I-.
We note that all cross product second moments of the limiting distributton of store content, for a k-level preemptive storage system, may
be obtained directly from formula (2.64).
,
vie have
•
For example" consider
53
:::: ErX.(i) X.
-J
J
-
X.(i+l) X.
7
J--
J
+ (x. 1(i+l))2 - (X.(i+l))2 7
J+.
For the case
k =
J-
4, the 6 cross products second moments are,
from (2. 65 ) ;
,
E(XI X2 )
:=
~ E L(XI + X2 )2 •
E(xI X )
::
~ E L(xl + X2 + X3)2_(Xl+x2)2_2~ + (X2+x )2.
3
::
~
3
E
xi .
~ _7
r8-7
L(Xl+x2+x3+x4)2.(Xl+X2+x3)2.2X~+(X2+~+x4)2
-(X2+X3)~7
,
,
The reader 'W-1l1 recognize that this is not the only algorithm that is
valid for obtaining all of the cross product second moments for k ::::
For example,
E(X X )
l 2
4.
could be evaluated directly from equation (2.51).
It will also be noticed that the terms involving only
X
3
Xl' X and
2
are valid for k ::: 3 and all six terms are valid for k > 4.
These are direct results of the fact that the store content
of material
C.J.+ l' C.J.+2'
c.
:l.
Xi
is not affected by the store content of materials
CHAPTER III
QUEUEING WITH BRF..JI.KDO\AlN
1.
Introduction.
In Section 8 of Chapter I we noted the
analogy between a two level queue with a preemptive discipline and
a head-of-the-line queue with a single server which is subject to
breakdovillS.
vJhite and Christie 1-~~7 derived the stationary moment
generating function of the queue size distribution for random breakdOvills and arrivals, and negative exponential repair and service times.
In this Chapter we will employ the methods developed in Chapter II
to consider a single server queue with random breakdovills and arrivals,
and general repair and service times.
iJ."'he types of breakdovill of a
service mechanism considered will be:
(a)
Active Unit Breakdowns.
These are breakdowns that occur
only when the service mechanism is actively servicing a
customer.
(b) !\-ctive Component Breakdowns.
These are breakdovills of com-
ponents of the service mechanism and occur either during
operation of the service mechanism (servicing a customer)
or during repair time of another component.
The service
mechanism is considered inoperative while any component
is being repaired.
( c)
Independent Unit Breakdovms.
Thes e are breakdovms that
occur at any time except when the service mechanism is
being repaired.
(d)
.Ind.ependent Component
Bre(;;~dov~.
that can occur at any time.
These are breakdo"ms
The service mechanism is
said to be in a state of breakdown vlhen one or more
components is being repaired.
A customer whose service is interrupted by a breakdown can reenter
the service mechanism according to one of the three preemptive queue
disciplines (defined in Chapter I):
(i)
preemptive resume.;
(11)
preenrptive repeat (identical) ;
(iii)
preemptive repeat ( different)
.
vie v1ill consider the three preemptive disciplines with respect
to each of the four types of queue breakdJwns.
In the study of com-
ponent breakdovm it "Till be assumed that the service mechanism has
an infinite number of components which are subject to breakdovm.
This assumption is not absurd when considering some of the modern
electronic equipment such as the large electronic computers.
A
vTOrd about the preemptive repeat disciplines also seems appropriate
here.
A program that is being processed on a computer may need to
be repeated entirely if a crnrrputer breakdown occurs.
This example
of a preemptive repeat (identical) discipline was noted by Gaver ~~7.
The dialing of a telephone number ,nth random access to a circuit
for each digit dialed, where the circuit equipment is subject to
51
breakdovffi, is an example of a preemptive repeat (different) discipline (provided the dialer does not hang up the telephone in disgust when a breakdovm occurs),
Independent component
breakdo1~
vTith a preemptive resume disci-
pline is mathematically equivalent to the storage problem of Chapter
II.
This equivalence is:
(i)
the arrivals of loads of material
Al
c
l
at random rate
with load size distribution function
BI(S)
are
considered as occurrences of component breakdovms with
repair time distribution function
(1i)
Bl(S) .
the arrivals of loads of material
a t random rate
A w"ith load size distribution function B (s) are
2
2
considered as the arrivals of customers with service
B (S) .
2
X - , the store content of msterial
t
time distribution function
(iii)
c ' is equivalent
l
to the lUlinterrupted repair time required to mske the
service mechanism operative provided no additional
breakdovms occur.
\tIe
call
the unexpired repair
time.
(iv)
Y - , the store content of msterial c , is equivalent
t
2
to the clearing time of all customers in the queue, i.e.
the time required to serve all the customers in the
system provided no additional breakdowns occur and the
service mechanism is immediately repaired.
Thus the results of Chapter II apply, I'd thout change, to this type
of breal{dovln.
Suppose we consider independent component breakdoiill .nth a preemptive resume discipline.
Let
Zt
= (Xt ,
pair and clearing times of the queue.
Y ) be the unexpired ret
Further, suppose that the ran-
dom rate of breakdo"m and the repair time distribution both depend on
the status of the stochastic process
if
~t
= 0,
time
if
Yt
d.f.
Xt > 0, Y
t
=0
breakdowns occur at rate
X
t
= 0,
Y
t
1J.1hat is;
~10
with repair
BlOCs),
=0
breakdowns occur at random rate
repair time d.f.
if
Zt;::; (X , Y ).
t
t
~ll
I'd th
Bll(s),
> 0 breakdowns occur at random rate A12 with
repair time d.f, B (s) , and
12
if
X > 0, Y > 0 breakdovnls occur at random rate
t
t
A with
13
repair t:i.me d. f. B (s) .
13
At all times, customers arrive at random rate
d.f. B (s).
2
A0
c:.
vl::.th service time
Assume that
Then a completely analogous derivation to that employed for equation
(2.20) shows that the transient behavior of the bivariate luplaceStieltjes transform of the distribution function of Zt
satisfies
= (Xt ,
Y )
t
59
(3.1)
The notation is identical to that defined in Chapter II.
This equation
does not appear particularly enlightening or useful in this fOTIm, except to indicate the complexity of the transient behavior.
Suppose that the stochastic process
bivariate distribution and Z
distribution.
=
We then speak of
Zt
= (Xt ,
Y ) has a limiting
t
(X, Y) is a random vector with this
(X, Y) as the unexpired repair time
and clearing time of the queueing system.
A completely analogous argu.-
ment to that employed in the derivation of equations
(2.36) to (2.39)
demonstrates that the components of the distribution function of
Z
= (X,
Y) satisfy the functional equations;
The equation analogous to (2.40) is therefore
60
.Lu-1\11-A.2+A.IIB~1(U) + 1\2B~(vL7
+ !feu)
(3.6)
r
lv-A12-A2+A12B~2lu) A2B~(v)3
+
+ GY(v)
xy (u,v) LU-/l.13"/l.2+r-'13B13(U)
x
+ F
-I-
:::
0
y)
/l.2B2(vt7_ ..
(3.6) apply to the distribution of unexpired re-
gquations (:3.1) to
pair time and clearing time for the four different types of breakdOvm, under a preemptive resume discipline, if
B10(S) : : : B (s) : : : B (s) : : : B (S) : : : B (S) and the rates of occur11
12
1
13
rence of brealtdovm, /1.10' /1.11 , /1.12 and 1\13 take on values in
accordance with table
3.1.
,
!Type of BreakdO'~m.
/1.
11
(a) Active Unit
0
0
/1.
(b) Active Component
0
0
1\1
/1.
(c) Independent Unit
1\1
0
/1.
0
(d) Ind.ependent Component
/1.
/1.
A.1
1\1
TABLE
2.
Unexpire~
Consider equation
is
Repair
1
1
1
1
0
1
3.1
an_~Clearing ~imes
for Unit Breakdown.
(3.6) for the case of active unit breakdovm. This
61
For this type of breakdovm i t is clear that
Y> 0
since breakdm·ms only occur when
vlhile holding
Y constant.
If( u)
Thus
p (X > 0, Y == O}
==
0
and
X is reduced to zero
=: 0
and may be omitted
from equation (3.7).
T~eorem
Th~ random vector
3.1.
(X, Y) of the
~}.rpired repair and
clearing times for active unit breakdovm ''i1ith a preemptive resume
discipline has the following properties;
rr
1 - n (1 + ~)
1
(3.8)
p(x
==
y
(3·9)
p(X
==
0, Y > 0)
==
n1 '
(3.10) p(X > 0, Y == 0)
==
0 ,
(3.11) p(X > 0, Y > 0)
==
m1 n1 '
0)
==
==
==
(3.12) E(X)
==
nrm2/2
(3.13) E(Y)
==
~n~ m2 + (1 + m1 )m2-7/ 2rr ,
m, / 3
(3.14) E(X2 )
==
(3·15) E(XY)
== ;-n,~~/6
n1
2
-
.J....)
7
~
3
+ -~m~(n0+~nl)/2rr
&:;,
&:;,
..)
-'-
7,
~
62
The notation is the same as defined in the previous chapter.
repair time
time
82
B (S) and the service
1
Sl has distribution function
has distribution function
A E(sj)
-
m.
A. E(sj)
2
2
=:
n.
1
1
B (s).
2
Thus
"" 1, 2,
3
j :: 1, 2,
3
j
J
The
and
Proof.
Equations
,
J
(3.8) to (3,16) are derived employing the method
used to prove Theorems 2.3 and 2.4.
tives, with respect to
1, 2 and
u and v, of equation (3.7) of orders
3 are evaluated at u
quations are solved for the nmue
to the constraint that
That is, all partial deriva-
=:
V
=:
O.
um~nowns
The resulting nine ein the theorem subject
~(O,O) + GY(O) + ~
=:
1.
This constraint
is due to the asstuuption of the existence of a proper lliniting distribution.
The complete details of these steps required to prove
the theorem are omitted since the details add nothing new.
For the case of Independent Unit Breakdovm equation
(3.6) re-
duces to
::;
0
j
That is, breakdowns may occur any time that the unexpired repair
time is zero .
...
Theorem 3.2.
The randop vector (X,Y) of the
une)~ired
repair
a~d
clearing times, for Independent Unit Breakdovffi vTith a preemptive
resume discipline, has the following properties;
- n
(3.20)
r(x
> 0, Y ~ 0)
p(X> 0, Y
1
,
= ml/(l + illl ) ,
= 0) = 7\l rr
~l - E(e
-7\2 81
)_71
7\2 '
(3.22)
E(X)
= ill2!2(1 + ml ) ,
(3.23)
E(Y)
= ~nlE(X)!rr(l + ml )_7 + ~(1-nl)n2!2~_7 ,
(3.24)
2
E(X )
= ~!3(1 + illl )
E(XY) = f(1-n l )n2!4rr(1+nl t7+LniE(X)!2rr(1+illlt7
2
+ Ln E(x )!2_7 ,
l
(3.26)
E(y2) = .Ln3!3rr_7+~n2E(X)+2nlE(XY17!rr(1+ml) .
Proof.
The proof is exactly the same as for Theorem 3.1 employing
equation (3.17) instead of (3.7) •
3.
Unexpired Repair and Clearing Times for Component
Breali:O:~
In the case of active component breakdovm equation (3.6) reduces to
(3.27)
rrE7\2 + 7\2B~(vt7
+ GY(v)
f
+ J1CY(u,v)
v ... Al ~ A2+7\lB~(u) + 7\2B~(vt7
i
u - 7\1 - J-' + 7\lB~(u) + A2B~(vt7
2
= 0.
64
Breakdowns may occur, in this case, only vlhen the clearing time is
greater than zero.
Theorem 3.3.
une~ired
The random vector (X, Y) of the
repair and
clearing times, for active component breakdown vnth a preemptive
resume discipline, has the following
pro~erties;
(3.28) p(x = Y = 0) = JL = (1 - ml - nl )/(1 .. ml ) ,
(3.29) p(X
~
0, Y > 0)
(3.30) p(X > 0, Y > 0)
= nl
'
== ~nl/(l
(3.31) p(X > 0, Y = 0) = 0
.. lUI)
,
,
(3.32) E(X) = nl~2/2(1 - lUl )2 ,
e
(3.33) E(Y) = ~ni m2 /(1 - ml )2 + n~_7/2(1 .. ml - nl )
,
(3.34)
2/2 ( 1 .. m~ )3 + n_ rlL. /'( 1 - ~ )2
E (. X2). = n~ m_
"
.1
c"
.1 .
.1)' •
.1 '
(3.35)
E(XY)
(3.36)
E(Y )
Proof.
The proof of this theorem is a direct result of the same
2
= ~nlE(X2 )
= ~2nlE(XY)
+ m E(Y)_7/ 2(1 ..
2
.'
l) ,
ill
+ n2E(X + Y) + n,/3_7/(1 - ~ .. nl ) •
operations employ"eo. in the proof of Theorem 3.1 but applied to
equation (3.27), in this case.
Independent component breakdovm with a preemptive resume discipline has been discussed in Chapter II under the identity of a two
level storage problem.
The mathematical equivalence of the two
blems has been discussed in Section 1 of this chapter.
results for this case are not repeated here.
Thus the
1)1'0-
4. Completion
Ti~es
and Busy Periods.
Gaver ~~7 defined the
period of time between initial entry into a service mechanism (subject to interruptions) and completion of service as the completion
time
C, of a customer.
He obtained the functi onal form of the
Laplace-Stieltjes transform of the distribution of C, when interruptions (only while the customer is receiving service, i.e., unit
breakdown) occur at rand.om and the lengths of the interruption periods
are independent random variables.
This fLlnctional form was obtained
for the completion time C and is different for each of the three
preemptive disciplines, resume, repeat (identical) and repeat (inde=
pendent).
The results obtained in the next section depend heavily on
Gaver's 'work so, for completeness, ive repeat here
h~.s
argument
garding completion times for a preemptive resume discipline.
re~
The
approach, for each of the two preemptive repeat disciplines, is
qUite similar and the results only are quoted here.
Consider a customer with service time
8 vdth distribution
2
Sl(i) be the duration of the i-th interrv.p-
function
B (s). Let
2
tion of the service time
C
:=
8 ,
2
N
+2:
S2
i=l
ivhere
N
is a random variable.
Thus, the completion time is
81 (1)
Define
,
Bi*(u), i
:=
1, 2,
as the
Iaplace-Stieltjes transforms of the repair time and service time distr1bution functions
Bl(S)
and B (S), respectively.
2
tion of conditional expectations provides;
A considera-
66
Thus
E ( e -UCjSr:> )
-u8
:::
e
<..
00
2
r.
e
-1-. 8
1 2
n!
n=o
and
which is equation (4.6) of' Gaver
{57,
page 10.
emptive resume discipline and unit breakdovm.
c,
This is f'or a preThe completion time
for a preemptive repeat (identical) discipline and unit break-
down satisfies
00
=1
o
This is equation
(4.17)
of Gaver, in our notation.
For a preemp-
tive repeat (independent) discipline Gaver's equation (4.26) is,
in our notation,
1 -
Consider the busy period of a single server queue with random
arrivals at rate
function
Bl(s)
~l}
independent service times with distribution
and a head-of-the-line discipline,
A busy period
begins when a customer arrives to find the server free; that busy
period enils when the server is next free.
rJet the random variable
D represent the length of a busy period.
Kendall ~~
the Laplace-Stieltjes transibrm,
showed that
E(e- uD )} of the busy period distri-
bution function satisfies the equ.ation
E ( e -UD) ::::
Table 3.2
lists the f~unctions ~lE(Dj)} j :::: I} 2} 3}
first three moments of the busy period distribution.
direct result of equation (3.40) due to Kendall
3.2
we list the functions
f§].
~2E(Cj), j :::: I, 2) 3}
of the
These are a
Also in table
of the completion
time for each of the three preemptive disciplines and unit breakdown.
The completion time moments involved are contained in Gaver
L?:.7,
ex-
cept for the third moments under the two preemptive repeat disciplines.
These were obtained directly from (3.38) and (3.39) by differentiation.
All these functions appear repeatedly throughout Section 5.
68
Define:
-'A. S
-1
12)"7
P
=
q
=
1 _ E(e
r
_
1 _ 'A. E(S
s
= 1 _ 'A.1E(S~ e 1
t
= (e
r'A.E(e
-
1
)
-
-'A. S
1 2)
1
2
e
-1\ S
1 2)
-'A. S
2)
AS
1 2
- 1 ) If,.
1
TABI,E
3.2
Busy Period
of queue
Random Unit Brealtdovffi
,nth random
~reempv~ve
arrivals
Resume
i----+-~::..::..:..~....::..._----'c------i---·
-------,
i
I
I
t
ill1/(1 -
1'A.1E(D)
m1 )
A2E(C)
i
--------.------
I
2
n1 (1 + ill1 )
--------1
'A E(D )
1
__
l
IA1E(D)~
,-..
,
~.
I A2E(C)
"3/(1 - W])4
+ 3m~/(1
-
--i
ill
1
)5
___2_
_'__
MOMENTS OJ? BUSY PERIOD AND COlf.PLETION TIME
•
e
e
e
RANDON UNIT BREAKDm\TN AT RATE
Al
AND REPAIR
Preemptive Repeat (identical)
TTIilE
d. f.
B (s)
1
1-3
@
l:"l
Preemptive Repeat (independent)
t:S
\.>I
,
]
I
A E £1 + m )t_7
1
2
I
-.
f\)
A.~ q(l + ~)
()
o
~
1-'.
I
A E f2(1+llJ.)
2
{tA~l +
+
I + 3~ { -v
IA2E
I
(.
2
m
2
2
+ 2u( 4t +
A~I) }
I A2P 1. m2q + 2:P(1+~)(~q2 + q + r
I
---i-I
I
I
j
i
t -v +
'
2u(2t + A.~J.) - 2t( t + A~
332
- 6m1 i11 t
2
6~ t
6~2 t ( 2u +
3t
2
l:S
r.::
-
CD
p.
.
"-'
-
2
-1 f
- ~q + 3m2 (PQ ~ pr - AI)
I
+ 6!tJ.pr( 2Ilq - A~ -L + Ilr)
; - 3~P( ~_qA~l + 38)
-1
-1
6~t t + Al )(3t + Al )
+ 3
~
I
- u}
t_7
- til.-1)
1
- rn.,. t + 3m ( u - 2t
:J
~)t2
(1 +
I
-1
+ 2t 11.1 )
A.~ I +
1 2
)}j
JI
2
-1
3Il(2pr - 2rA
- 38)
1
+ 6pqm~ ( 2pr + pQ2- 211.1 )
1
1
i_i + 6~3 p 2 q + 6~m2 Ilq2
.
J
I
j
1
MOHEIilTS OF COMPLETION TTh'lE
0'\
\.0
70
5.
Queueing Times for Active Breakdmffi.
The queueing time of
a customer is defined as the time period from arrival at the queue
up to the time of his initial entry to the service mechanism.
The
queueing time of a customer is zero if and only if the customer
arrives to find the unexpired repair and clearing times both zero.
Thus the probability that the queueing time is zero, under the
limiting distribution, has already been derived in Sections 2 and
3 for the four types of breakdown.
In this section and Secti.on 6
we will obtain a functional equation involving the Laplace-Stieltjes
transform for the limiting distribution function of the queueing time
for each of the four types of breakdoiffi.
These lead directly to the
moments.
Consider the stochastic process
Zt ~ (X , Yt ) for which the
t
limiting distribution satisfies equation (3.6).
AIO = All = Al2
and
X == 0.
t
= Al3 = 0,
Suppose that
which means that there are no breakdovms
Further, suppose that the increments to
occur at random vlith rate
A0
,
Y , 'l'lhich
t
are completion times of customers
C'-
whose service is subject to
interr~Dtions
or breakdowns.
The cus-
tomeI'S' completion times depend on the preemptive discipline adopted.
However, at present we will merely assume that the completion times
are independent and the laplace-Stieltjes transform of the completion time distribution function is
CY(v).
From (3.6) it is easy
to see that, under these conditions, the limiting distribution of
Yt
satisfies the equation
71
.
The accumulation of completion times at the service m.echanism at time
t
is
Y;
t
this is the total tim.e until the queueing sJTstem is empty
of customers, provided no add.itional customers arrive.
actually the queueing time, say
t.
Thus
Y
t
is
Qt' of a customer arriving at
time
Therefore
(3.42)
where
aY(v) + n ~ E(en
= p(Q = 0).
content of material
material
c .
1
vQ
) = v~f[-v
- ~2
+
~2cY(v)_7
vTe have noted in Chapter II that the store
cQc.;
has no effect on the store content of
Hence the results of Chapter II can be employed to
consider the limiting distribution of the queueing time
Q~,
u
which
is subject to random increases at rate
A of an amount C ( the
2
completion time for the appropriate discipline) and a uniform rate
of reduction.
This is accomplished by an identification of the
queueing time
Q"
content
under the limiting distribution, with the store
X of waterial
c ' under the limiting distribution.
l
a renaming of the functions involved in equations
Thus
(2.43), (2.46)
and (2.52) proves the following theorem.
'I'heorem 3.4.
The queueing time
of a customer of a
emptiy~
q~eue
Q, under the limi ting distri~ution,
subject to active unit breakdoiiU and a
discipline has the
propert~es;
,
pr~­
72
(3.44)
~ ~2E(C2)/2(1 - ~2E(C»
E(Q)
,
E(Q2) ~ ~2E(C3)/3(1 - ~2E(C»
Moments of the completion time
C are listed in table
3.2 for each
of the three preenrptive disciplines.
The difference between unit and component breakdO'l'll1 is that in
the latter case a breakdo1'll1 may also occur while the service mechanism is being repaired.
The moments of the completion time Imve been
derived for unit breakdovm.
If the unit repair time of the service
mechanism is considered as a busy period of a queue with random
arrivals at rate
~l
and service time distribution function
Bl(S)
then the completion time actually allows for random component breakd01'll1S at rate
~l
and repair time distribution function
Bl(S).
The commencement of "busy periods" of repair occur only during the
service time of customers at random 1.,ith rate
~l'
The completion
time for component breakdown vrill be des:l.gnated by C.
~2E(Cj), j ::: I, 2, 3, may be read directly from table
values for
3·2
upon replacing
.j ::: 1) 2,
Thus the
mj
by
~lE(Dj) in the formulas for ~2E(Cj),
3,
Theorem 3.5.
The queueing time
Q, under the limiting distribution,
of a customer of a queue subject to active component breakdovm under
a preemptive discipline has the properti.es;
•
(3.46) p(Q
~
0) ~ ~ ~ 1 ~ A E(G)
2
and
I,loments of the comp~etion time C dep~nd on !.he preemptive discip~ine
involved.
This theorem is a straightforward extension of Theorem 3.4.
Proof.
6.
Queueing
chastic process
(i)
X
t
~imes
Zt
for Independent Breakdovffi.
= (]~t'
Y ) is subject to the following rules;
t
is subject to random increases at rate
the time periods that
Zt
= (0,
0).
occurrence is a random variable
function
(ii)
Y
t
A .
2
If
?-'l
only (turing
The increase at each
Sl with distribution
B (s ) .
l
is subject, at all times, to random increases at rate
Each increase is the random variable
~letion
(iii)
Suppose the sto-
C of a com-
time with unit breakdovm.
X > 0, then it is decreasing at unit rate;
t
if
X = 0, Y > 0, then Y is decreasing at unit rate;
t
t
t
if X = Yt =
then both are held constant tmtil
t
a random increase occurs.
°
74
The lirrlting distribution of Zt; (X , Y ) satisfies the special
t
t
case of eqt~tion (3.6) where
and
x
BJ.O (u) --- BX('J.)
1 c ,
BY(V)
2
= cY(v)
Thus
1\ L-A l
- ""2 +
Y
G (v)
..
Ul;,..II
""""U,J..."",,"'"
+ .Di"(u) + J1'Y(u,vt]fu -
"2
fA, )'9\
\.",1"-,,
"I
Y
A2C (vL7
_7
Lv -
> :=
+ l..{Y(v).7
0
J
X and Y, for which the distribution func-
The separate cmnponents
V.J,.,,L
-I-
Y
""2 + A c ( v)
2
-I-
+-lOYl ,.,,,,t-l,.,-P-lo,.,
""lB~(u)
o
'h",v
...
,l\,oA,l
'"'"
rath 0
1"
oIo_.&.
,...,'h"''''''r'''' -iYl+e",nr""'h,,+-io""s
v
oIoS:.' _v(,."v ..
..., .... t.,.j
~
...........
it should be clear from the previous section that
J..L.
(X + Y)
i
H01.,.r. e1•r e'Y'_,
__
is the
random variable which satisfies the limiting distribution function
of the queueing time of a customer arriving at a queue with independent unit breakdown.
_~heorem
is
3.6.
sUb~ect
Tl~e
queueing time
to independent
Q of a custo~~r of s. que~ vlhic!:
~nit breakdov~
bas the following proper-
ties, under the limiting distribution;
Ll - A2E(CL7/f l
(3.50)
p(Q:= 0) :=
(3.51)
E(Q)
:=
E(Q2)
= L3A2E(C2)E(Q)+A2E(C3)+1\~7/
J!
L1\
=
m2 + A2E( C2 t7/
+ nJ.-7
)
2Ll - 'A.2E(Ct7,
and
(3.52)
3Ll - ""2E(C)_7.
,
Proof.
These expressions are obtained by setting
u:;;: v
in
(3·49), then evaluating its first, second and third derivatives
with respect to u
Equations
at
u
= O.
(3.i.t-9) to (3.52) are transformed to the case of inde-
pendent component breakdovm by replacing the repair time distribution
by tre busy period distribution of the repair time.
Therefore
:;;:
It is possible to obtain information about
from this equation.
X and
Y
0
separately
However, because of their obscure interpretation,
we pass
inm~diately
to the following theorem.
Theorem
3.7. The queueing time Q, under the limiting distribution,
of a customer of a queue which is subject to independent component
breakdO'lVll has the
follO'lvin~ properties;
(3.54)
p(Q = 0) =
(3·55)
E(Q):;;:
L1!
1!
= .[1
m2 /(1 ~
- "'2E(CL7Ll
- ml _7 ,
2
l )3 + "'2E(c )_7/ ~l - "'2E{C)_7,
m
and
31\
2
ID
2
(1_~)5
Moments of the completion time C depend on the preemptive discipline involved.
Proof.
These results are obtainable directly from equation
Cs. 53) .
It is also possible to obtain them from Theorem 3.6 by replacing
the moments of the repair time bJr the moments of the busy period,
and the moments of the complet:i.on time
the moments of the completion time
C :E'or unit breakd01m by
C for
component breakdovID.
As a result of the equivalence of queueing with independent
component breakdovID and queueing of two classes of customers (those
GUS tomers
Vii th and those vii thout preenrption rights), theorli4Il."). 7
applies to the 2-level queue with any one of the three preemptive
disciplines.
Thus (3.55) and (3.56) are the first two momerrts of
the queueing time of the customers without preemption rights for
the three different preemptive disciplines.
Miller
["'§.7 derived
these moments for a preemptive resume discipline employing a more
specialized argument.
These results for the two preemptive repeat
disciplines are new.
irle note that the moments of the queueing time
for the preemptive repeat disciplines depend on the form of B (S)
2
but only on the moments of Bl(S) .
The transient behavior of the different stochastic processes
considered in this chapter are chaIacterized, in each case, by a
special case of equation (3.1).
A1r,0, the e:ctension of the results
contained in this chapter regarding queueing times to three or more
classes of customers (when this has meaning) is not difficult.
is omitted here because it is tedious.
It
CHAPTER IV
THE EXISTENCE OF LIMITING DISTRIBUTIONS
1.
Introduction.
tent of k
In Chapter II we considered the store con-
materials under a preemptive rule of operation.
functional form of the limit, as
function of the store content for
t --> w
k
=2
,of the distribution
was obtained under the
assumption that such a limit actually existed.
assumed that the limit, as
of store content
Zt
= (Xt ,
t ->
w ,
The
Specifically,"I'le
of the bivariate distribution
Y ) was a proper bivariate distribution.
t
In Chapter III we have considered a queue when the service
mechanism is subject to breakdo"l'ffis, active or independent, unit or
component.
as
Under a preemptive
resv~e
queue discipline, the limit,
t --> w , of the distribution function of the unexpired repair
and service time was derived in functional form.
He also obtained
the functional form of the limit of the queueing time distribution
function for the four different types of breakdolvu, each subject to
the different preemptive queue disciplines (resume, repeat (identical),
and repeat (independent)). In each case it
existed as a pToper distribution.
"I'~s
assumed that the limit
The necessary and stli'ficient
con~
ditions for the existence, as proper distributions, of all these
limits will be derived in this chapter as special cases of a more
general theory.
Let
Zt ~ (X , Y ) be a stochastic process, for
t
t
°< t
<
OOJ
defined in the four regions:
R
Xt : : : Yt : : :
Rl
X > 0, Yt
t
R
2
Xt :::: 0, Yt
R
X > 0, Yt
t
0
3
°
;
-
°
>°
>°
:::::
J
This stochastic process is subject to the following rules;
(i)
In region
and P
there are two Poisson processes
Zt
P
li
; the rate of occurrence of events in them
and A J i :::: 0, 1, 2, 32i
vlhen an event in Pli occurs the X
t
is
(ii)
2i
R.J.
Ali
increases instantly by an amount
coordinate of
Sli' a random
sample from a population with distribution function
B (s)
li
(iii)
vlhere
Bli (0+) :::: 0, i :: 0, 1, 2, 3.
vJhen an event in
Zt
occurs the
P2i
Yt
increases instantly by an amount
coordinate of
S2i' a random
sample from a population with distribution function
( iV)
In regions
R
I
and
its upv~rd juraps and
R ,
3
X
t
y,
v
is held constant bet,.,een
decreases uniformly (a unit
amount per unit of time) between its upward jumps.
79
(v)
In region
R , X is held constant until its next upward
t
2
jump and Yt decreases uniformly between its upward
jumps until the next upv~rd jump of
Zt
(Vi)
jumps to region
In region
R
3
Ro , both Xt
Xt (i.e.
until
).
are held constant until
and Yt
the next occurrence of an event in either P
IO
(i.e. until the process
We will call
Zt
Zt
ju~)S
or P
20
out of region
a Poisson Reduction Process.
R).
o
Consideration of
some generalizations of this process would be interesting.
However,
this would be an excursus since all stochastic processes considered
in this dissertation are special cases of a Poisson Reduction Process,
as ile define it.
2.
~imiting
Necessary and Sufficient Conditions for Existence of
Distributions of Poisson Reduction Processes.
the Poisson Reduction Process
Zt'- (X , Y ).
t
t
Consider
He v1ish to obtain the
necessary and sufficient conditions for the existence of
(4.1)
lim
t
v1here
a
Theorem
->
p(Zt
€
A),
for all A
€
a,
!Xl
is the class of sets
2 proved by Smith
(0:5
L 11_7
Xt
:5
x,
0:5
Y
t
:5
y}
page 11~ is extremely useful for
proving the existence of limiting distributions for Equilibrium
Processes.
In the sequel 'tle vJill show that a Poisson Reduction
Process is a special case of an Equilibrituu Process, which we now
define.
80
Let
i :::: 1) 2) ..•
Ct. }
~
be an infinite sequence of inde-
pendent) non-negative) identically distributed randcm variables)
each of which is not zero with probability one.
Let
t
o
be a
non-negative random variable) independent of and not necessarily
distributed like
t ) i ~ 1.
i
i :::: 0) 1) 2) .•.
a general renevm,l process.
t
t >0
For each
k == 0) 1) 2) ...
n
Smith ~-1~7 calls the sequence
as the largest integer k
:F'or any general
define the random variable
such that
Suppose
is a random variable) defined over the space
initial condition of a stochastic process
z)
Zt'
Z
o
€
z)
A
€
a ) there
Z
o
specifying the
This process
is called an Equilibrium Prucess idth respect to
if) for all
Ct i })
Zo
Zt
over the sets
¢A('»)
exists a function
depending only on A such that
(4.2)
p(Zt
€
Alz o
zj n
€
t
> 0; Tn )::::
¢A(t -
Tn )
t
t
is a valid representation of the conditional probability.
Equilibrium Process
Zt j.s abbreviated
J~(Z)
The
a ) \ctil),
The
definitions of this paragraph are due to Smith ll~7 pages 9) 12
and 13.
Now consider the Poisson Reduction Process
let the time points
To) T ) T )
l
2
'qq
be the instants when the pro-
cess enters the region
R.
o
Thus
that
X
+ Yt -€ > 0
Xt :: Yt :::: 0) but
t
_€
Zt:::: (X ) Y ) and
t
t
is the least value of
t
such
for all sufficiently small
81
€
> 0;
POl'
T
l
is the second smallest such value of
t, and so on.
example, in the store content process of Chapter II the time
points
are the instants vn1en the store becomes
To , T"_ T2 ,
empty.
llrite
Plainly
t
~
(t )
i
~
T , .•. , t + t + .•. + t
o .l
n
l
~
T .
n
forms a general renewal process as described above.
Zo
Let us fix
T ,t + t l
000
€
z) where
z
is the non-negative quadrant of two-
dimensional Euclidean space, and define K (t) as the distribution
zo
function of t . Let F(t) be the distribution flillction of t.,
o
~
for
Smith
i > 1.
We now apply the following, a part of Theorem 2,
frJ=7.
Theorem.
If
l:\
(i)
(ii)
.~ (z,
is
F(t)
(A,
(t i ) ) ,
is not a step function
SUC!l
that all jumps
are over any subset of (mf)
n
(iii)
K
Z
(+00)
.-
o
E( t.) ~ !l
(iv)
~
(v)
for any w> 0,
0, 1, 2, ..•
~
1
< IXJ,
,
>
i
1 ,
the k-th convolution of
F(t)
with itself has
an absolutely continuous component for some
finite
~A(t)
(vi)
then for all
Zo
It,
€
z,
A
is measurable,
€
a.
00
lim
t
->
p(Zt
00
€
Alz o ) =
1
!l
J
o
¢A (v)
Ll - F(vt7
dv •
82
He are considering thr Poisson Reduction Process
over the sets
A
C\ ,vThere (1.
€
where
is the class of sets
x
and
yare finite.
It is
clear that the rules of the process completely determine the future
T"
1
of the process at times
i :::
0, 1, 2, ...
The future is also
independent of the past history at these times except for the fact
that the process entered state
i > 1
(0,0) at such times.
(t.}
Thus
j
1
are independent, identically distributed and independent of
t.
o
Hence equation
for
Zt
isE(z,
e::t .
(4.2) holds for the class of sets
(t ,
(t } )
i
and condition
There-
(i) of the theorem
holds.
Suppose the process
Zt
enters state (0,0) at time
and
T.
l
later. The length
next departs from this state a time period P.
1+1
of the period, Pi+l' is a random variable with the negative exponential distribution.
For example, this time period is an idle period
of the service mechanism for the Poisson Reduction Process of Chapter
III.
Define
D •.
l
that segment for which
Thus, during the interval
X + Y > 0.
t
t
t. ,
is
l
For those cases considered in
Chapter III this would be called a busy period (repair and service)
of the service mechanism.
We will call the periods
Poisson Reduction Process the busy periods.
stantaneous increment to
dent of
P.,
l
D
i
for all
The amount of the in-
X + Y at time
t
t
T _ + Pi is indepeni l
The future of the process (after time T. 1 + P. )
l-
1
depends only on the size of the increment and the rules of the
process.
Thus
D
J.
i
is independent of
D ..
1
Hence the distribution
functi. on of
of
D.
P..
J.
t
is the convolution of the distribution function
i
and the absolutely continuous distribution function of
1.
Therefore
t.,
1.
j. ==
0, 1, ..•
has an absolutely continuous
distribution function and conditions
(ii)
and. (v) of the theorem
hold.
NOVT
Let
consider the sequence
PI' Dl , P2' D2 , P , D ) ...
3 3
(m-l) be the largest integer for which
Sm(P_l)
l
L:r:rJ.:::: l
==
P.
1.
and. 6
< E(P i) < 00
0
Smith
115.7 p.
PI' P ' P ,
2
3
and, for finite
245).
Upon setting
,.,CD).
m-l
<
i:) -
<
6., vThere
:lis any finite positive number.
sequence of random variables
with
s~:i
'II
<
m-
6,
X
o
Now the
is a reneVial process
0:5 E(m-l)
+ Yo ;: 6
< 00
(cf.
we have
,.,Cp).
i:) -
m
A direct application of vlald's equation provides
E(D) + r;(m-l) -rEel?) + E(D). 7
< E(T.)
< E(m) -rE(p) + E(D)_)11-
(4.4)
Thus
E(T )
m
is finite if and only if
E(D)
< 00
7.
Hence the functi.on
K (+ (0) == 1 if E(D) < 00.
This modification of an argument found
Z
o
in Smith 1~7 p. 260 establishes condition (iii) of the theorem if
B(D)
.;
.......
--
00
•
Thus the conditions of the theorem are known to hold (when
E(D)
< (0) provided ¢A(t) is a measurable function.
this by sho't·ring that i t is a continuous function of
We will verify
t (1. e.
a
Baire function of class zero) and appealing to the theorsm that all
84
Baire functions are measurable (cf. Goftman ~~7 Theorem 1, p. 183) ,
'f
r i\l'
r
+ i\2'r 7 and i\2 == min - i\l'1 + i\2'r 7,
> 0, elementary probability considerations allOi<T us to vTrite
Let
i\l
==
max
-
1
For
and
i<There A' == (0
¢A(t)
<
for fixed
X ~
t
x +
'f,
0:5 Y
t
:5
y +
'f)
t is a measure over the sets
•
By definition,
A and must be a
continuous set function from both above and below (cf. Halmos
p, 39).
l!+...]
Thus
The above two inequalities as
'f
-> 0 allow us to write
Manipulation of these two inequalities also allows us to "Trite, for
'f
> 0,
>
and
i<There All
Thus
::;: (X-T
< X < x,
t-
y-rr
< Yt <
-
y)
Therefore
¢A(t)
tion of t.
is continuous and, as a result, a measurable
Condition (vi) of the theorem holds.
Note:
func~
Smith ~1~7
has sho't-m that conditions (v) and (vi) of Theorem 2 w.ay be replaccCi.
by the condition that
any finite
t
¢A (t) Ll - F(tL7 is of bounded variation in
interval.
to employ Lemma
2
Thus an alternative approach here would be
Smith
Ll"};7
p.
17 to demonstrate that the
bounded variation property holds for Poisson Reduction pr.ocesses.
He have now shown that all the conditions of Theorem 2 are
satisfied if E(D) <
(x + Y ) < 00 with probability one.
o
0
Thus we have completed the proof of the sufficiency part of the
00
and
following theorem.
Theorem 4.1.
~ecessary
and sufficient conditions for the existence
of the limiting distribution of a Poisson Reduction Process are;
(i)
BCD)
<~
,where
D is the duration of the busy period,
and
(i1)
the initial status
,nth probability
(x + Yo)
o
of the process is finite
on~
Proof. (of the necessary part).
Consider the regeneration points
To' Tl , T , .•. of the process and define nt as the largest inte2
ger k
such that T _l :5 t. Smith ~l"J:.7 p. 19 defines any set
k
A as an a-set if there are real numbers It, € > 0 such that
for sufficiently large
t
and Zt
any equilibrium process.
As we
86
have seen, a Poisson Reduction Process is an equilibrium process.
-f.. (x+y) .
For a Poisson Reduction Process, e l
~s a lower bound for the
probability that zero Poisson events occur in a tliae period x + y.
= [0 ~ Xt ~ x,
Thus, for any set A
o~
Yt ~ y }
A)
> e 1
-')I.
p(nt +T > nt
(4·5)
for finite
Z,
€
Zo
Zt
and y and 'r > x
x
€
+ y.
we have
(x+y)
>
The set A
0
is said to
be a null set if
lim
p(Zt
-> 00
t
Theorem (Smit.h Ll~7 p. 19).
K
Z
J+
00)
>
0
and.
< 00
1.1.
Alz o ) =
€
,
0
If there exists a non-null a-set then
B( t
i) :::
1.1.
o
By virtue of (11·.5) the set A is an a-set which is non-null
only if B(D):5
tion that
=
B(D)
lim p(Zt
€
<
1.1.
00
00.
Thus the set A is null under our assurn:P-
and
A/Z o )::
lim L1C(t)+G(y;t)+H(x;t)+F(X,y;tt7
t .... >co
t ->co
=0 .
That is, the limiting distribution does not exist as a proper distribu~ion
if B(D):::
Assume
00
X + Y :;::
o
0
P (Xt + Yt '" 00)
~
p
•
with probability p < 1.
00
for all finite
t.
Thus
Thus
1C(t) + G(y;t) + H(x;t) + F(x,y;t) :5 1 - P < 1
for all finite
x, y
and
t.
Therefore by taking limits as
87
x -> co, Y -> co
~(t) + G(+co;t) + H(+cojt) + F(+co,+~;t)
(4,6)
all finite
side of
t.
Hence the limit, as
t -->
co}
<
1 - P <1
for
of the left hand
(4.6) is bounded a'tvay from one and tb.e limiting distribu-
tion does not exist as a proper distribution.
proof of theorem
This completes the
4.1.
It i.'3 not difficult to see hOi? one would extend the notion of
a Poisson Reduction Process to
n
dimensions and prove results
corresponding to those of this section.
3.
Necessary and Sufficient Conditions for ~~,fini~Mean
Busy Period of Certain Poisson Reduction Process.es.
The different
Poisson Reduction Processes discussed in Chapter II and III will
be considered individually in this section.
Talcacs
fr~7
et. al.
developed a certain method for establishing necessary and 8ufficient conditions for the finiteness of the mean busy period for
the head-of-the-line
qUque with one server; we employ a similar
method in this section.
Suppose we consider the special case of a Poisson Reduction
Ie)
\.d
(b)
Yt
is subject to instantaneous ,Jumps at Poisson rate
in all regions, of an amount
C;
where
"'2'
C is the com-
pletion time of a customer inth either unit or component
88
blEeakdown and a preemptive queue discipline specified,
(c) if
Thus
X + Y
t
t
at time
t,
Y~
> 0, then it decreases at unit rate.
"
= Qt
in a
is the queueing time of the customer arriving
queue with active breakdo'Wrls.
The type of break-
downs considered (unit or component) and the preem9tive discipline
are allovTed for by the com91etion time distribution.
periods of the process
~,
The busy
with a preemptive resume discipline,
are identical to the busy periods for the Poisson Reduction Process
Zt
= (Xt ,
Yt ) of the unexpired repair and service times with active
breakdovffis and a preemptive resume discipline.
time
~
In fact the queueing
for any breakdo'Wrl system is zero if and only if the unex-
pired repair and service times are both zero.
Thus the conditions
for finiteness of the mean busy period may be investigated under
different guises.
We will investigate the finiteness conditions
with respect to queueing times, for the different types of breakdOvffi and preemptive queue disciplines.
These finiteness conditions
alSo apply, vlhen meaningful, to the unexpired repair and service
times.
Consider a random variable
D with distribution function
¢(d).
Suppose
= 1,2,
... , N
are independent with the same distribution function
¢2(d)
and N
is the number of Poisson events occurring, vnth rate
A,
where
D
1
has distribution function
¢l(d), D , i
2i
1n a time
period D .
I
That is
-AD
=
peN = niDI)
e
l(AD I )n/n!
A consideration of conditional expectations leads to
E(e-
"There
*
¢2(u)
uD
=
f DI , N) = e
B(e
-UD 2
-uD1
*
N
f¢2(uL7
).
Thus
-uD
=
e
1
00
I:
e
-ADI
,
n=o
and
Upon letting u -> 0
in
(4.7) it
is clear that
¢*(u)
is the
Laplace-stieltjes transform of a proper distribution function if
and only if ¢l(d)
and ¢2(d)
(He are assuming that
are proper distribution functions.
¢1(O+) < 1).
Equation
(4.7) also implies
that
Let
C be the random variable of a completion time with dis-
tribution function
G(c)
where the service mechanism is subject to
unit breakdown and an unspecified preemptive queue discipline.
C and G(e)
Let
represent the same quantities for component breakdovm.
The first three moments of the completion times
C
and Care
listed in Table 3.2 for each of the three preemptive disciplines.
90
Consider D, the random variable of the duration of a busy
period of the queueing time process for active unit breakdoims.
Continuing the notation and definitions of Chapter III we have,
(4.7),
as a result of
(4.8)
Thus
E(D)
and E(D) <
00
if 4nd only if A E(C) <
2
1.
Takacs ~127 derived a
similar result for a head-of-the-line single server queue.
apparent that a reference here to Takacs would suffice.
It is
However the
inclusion of this argu-IDent should wake the following more readable.
NOi'l consider
15, the random variable of the duration of a busy
period for the queueing time process with active component breakdovms.
The argument of the previous paragraph carries over vrith little
change to this case provided the completion time
applicable to component breah:dovm.
dis';~ribution
is made
Thus
,
(4.10)
which is finite if and only if
Suppose
~2E(C)
< 1.
D is the length of a busy period of the queueing time
process with Independent Unit breakQovm.
Let
D
l
be the random
variable of the busy period length when it comraences vrith a breakdovm.
This occurs irith probability ~l/(~l + A )·
2
Let
D be the
2
random variable of the busy period duration vlhen it connnences vTith
an arrival of a customer.
This occurs with probability
91
/1.2/(71.
1
por independent unit breal'~dovm) the length
+ 71. ),
2
D of
a busy period has mean
From
(4.7)
(4.11)
E(e
-uD1
)
=
and
(4.12)
Hence
(4.13 )
and
e
(4.14)
Thus the mean busy period,
A E(C)
2
<
E(D), is finite if and only if
1.
For Independent Component
breal~do\m
( 4 .11) to (4.14) are
easily transformed to
(4.15 )
-uD
E(e
1)
=
*
¢I(u + 71.2 -
71.
_7-.L
2 C (u))
~
.- "G" (u + 71.2 -
(4.16)
11.
2
B( e
-uD2
»),
(4.17)
and
(4.18)
where
=
¢I(d)
E(e)/(l - "-2E(e))
)
is the distribution function of a busy period of repair
92
times in isolation.
A E(C) < 1
2
only if
3.2).
Both E(D )
l
and
E(D )
2
are finite if and
A1E(Sl) < 1, see Table
(which implies that
Thus the mean busy period of the queueing process with in-
dependent component breakdovm is finite if and only if
AoE(C)
<
c.
1
These results are strrmnarized in Table
4.1.
In all cases finite-
ness of the busy period of completion times in isolation guarantees
finiteness of the mean busy period of the associated queueing time
process.
The f:i.niteness condition relating to Independent Component
breakdown and a preemptive
Miller
["'9.7.
Gaver
L?::.7
resume discipline 1ms derived by
obtained the results for unit breakdown for
all three preemptive disciplines,
IBreakdovm
unit
I
:
Component
IA2E(C)< 1 if and only if
I
--Re-s-:--n]-.(-l-~m-l-)-<-l-----~~l
Preemptive
Discipline
A E(C) < 1 if and only if
2
,
r--.------:----------------+-Repeat
(identical)
A2
f
AlLE(e
II and
-.,.-'---~-.-----------__!-
A1 S2
)-}]/f1-~7< 1
~ < 1
II
1
II
----I
Repeat
(indepenw
dent)
NecestJ~ry
TABLE 4.1
and Su:fiicient Cono.itions 1"or Finiteness of Hean Busy Periods.
CHAPTER V
THE m1BER OF BREAKDOViNS
1.
Number of Breakdovms During Service of One Customer.
In
Chapter III we have discussed four different types of breakdovms
of the service mechanism for single server queues.
He also noted
the equivalence of the mathematical model for inclependent component breakdovm and the 2-level single server queue vnth preemptive disciplines.
of times,
service.
N,
In this section we investigate the number
that one customer is interrupted while
r£~eiving
An interruption (occurrence of a breakdovm or arrival
of a higher priority customer) will be considered for the three
different preemptive disciplines.
The number of interruptions
N will be the same for both unit and component breakdown, active
or indepeno.ent.
This follows from the fact that for unit break-
dOvm N is the number of breakdowns while the custoiller is in
the service mechanism, vlhile on the other hand, for component
breakdovm,
N is the number of interruptions by busy periods of
the component repair tilae.
The hreakclo'Yms or interruptions occur at random 'Vlith rate
Al
during the actual service time
8
with distribution func2
It is assumed that the customer returns to the ser-
tion B (S).
2
vice mechanism in accordance vnth a preemptive resume discipline.
94
For a given uninterrupted service time
8
2
,
Thus the probability generating function of N is
J
00
z'n
n!
o
where
B;(U)
is the laplace-8tieltjes transform of the uninterrupted
service time distribution function.
Theorem 5.1.
The number, N, of interruptions of a customer's ser-
vice, under a preemptive resume
(5.2)
P(N=O)
= B*2 (A. l )
(5.3 )
B(N)
= A.1B(82 )
(5.4)
E(N )
2
where
:::
AiE(8~)
discipl~ne,
satisfies;
,
and
+ A. l E(82 )
,
8
is the uninterrupted service time vrlth distribution f~nc­
2
tion B (S).
2
Proof. This theorem is an obvious consequence of (5.1) by differentiation vrlth respect to
Z and then evaluated at
Z
Suppose interruptions occur at random with rate
tomer 't·rlth uninterrupted service time
=1
•
for a cusl
The customer returns to
A.
8 ,
2
service according to a preemptive repeat (identical) discipline.
For
95
a given
S2;
P (N :::: n)
(1 ..
::
Thus
e- A1S2 )
00
E(ZN)
(5.5)
:::
J
()
Theorem 5.2.
e
n
e
-A1S2
-Als
0. B (S)
2
-A. s
1 - Z(l _ e 1)
The number, N, of
in~errupt:i:~ns
of one cust01!1er I s ser-
vice under a preem~tive repeat (identical) discip11ne~~~i8fies;
,
peN : : 0) : : B;(A1)
i\ S
E(N)
:: E(e 1 2) .. 1
- 4B*
(-A)
2
1
Proof.
+
2,
The theorem is a direct consequence of equation
(505) .
Suppose interruptions occur at random rate A for a customer
l
with uninterrupted service time S and the customer returns to ser2
vice according' to the preemptive repeat (independent) discipline.
For a given sequence
rupted service times
S(l), 8(2), .•• , S(n+1)
of independent uninter-
n
peN
= n)
=:
TT
i=l
Thus
The following theorem is a direct consequence of (5.9).
Theorem 5.3.
The ntunber, N, of interruptions of one customer's ser-
vice with return to service according to a preemptive repeat (independent) discipline satisfies:
P (N
,
= 0)
E(N)
7~
=
£1 - B2
=
2
(7-'lt7/ B2*(Al )
{1-1-B;Cl-.l )_7/
e,nd
B;(I-. )} 2
l
Consider the queue 'Hi th independent component breakdovm or
equivalently the queue i'1ith trlO classes of customers and a preemp..
tive discipline.
The interruption of service for a low priority
customer is due to arrival of a high priority customer.
The
10'1'1
priority customer re-enters the service lnechanism at the end of a
"busy period" of the high priority customers (busy period in the
sense that only high priority customers are served).
Let M be
the number of high priority customers served during such a "busy
periodlt •
Takacs £l"~7 has shown that M has a probability generating
function which satisfies
97
Equation
(5.13) may be obtained as follows: -- Let Sl be the
service time of the first customer served during a busy peri-od of
high priority customers only.
Suppose M
.l
arrive during the period
Define
probability that
busy periods.
Each of the
m-lU
Sl'
customers (high priority)
(m )
p
l
(m~ml)
customers are served during
l
as the
lli
l
indepenc3.ent
Now
M
a busy period.
l
customers that arrive during
Sl
gives rise to
Thus
ill
::::
E
~::::o
Therefore
00
E(ZM /Sl) :::: E
)
m==o
00
=
Z L
ill
1
Z
illl=o
:::: Z
and
(5.13) follows upon taking expectations with respect to 81
The fOllovnng theorem is a direct consequence of
(5.13).
Theorem 5.4.
The number, M, of customers served during a busy
period of a queue with arrivals at random rate
=0
=
E(M)
:=
2
E(M )
=.[-{l
)
(1
.~ervice
l
Bl(S) and no break~?vms satisfiesr
distribution function
P (M
A. ,
M
p
where p
o
mlrl
,
+ ml)/(l i
= 1,
time
o
and
ml)~7
+ I-m /(1 - m1 )P__7,
2
2
Thus the total number of top priority customers that arrive
after a low priority customer enters the service mechanism and. complete service earlier than the low priority customer, is M' N •
The number, Ni, of top priority customers serviced during a "busy
period" of top priority customers only, is independent of the
number, N, of such busy periods.
Thus
where the factors on the right side are given by Theorems
5.1 to
5.4, depending on the preemptive discipline.
2.
Number of Breakdoims During Queueing Time for the Preemptive
Resume Discipline.
In this section we will restrict our considera-
tion to the preemptive resume discipline.
The limiting distribution
of the unexpired repair and service times (X, Y)
i~.S
in Chapters II and III for four types of breakdown.
characterized
Suppose a cus-
99
tomer arrives at the queue vlhen the unexpired repair and service
times satisfy the limiting distribution.
v~iting
This customer joins the
line for a period X + Y plus a period of time equal to
the repair times of all breakdOi'inS that may arise in obtaining a
period X, of repair and a period Y of useful service.
Let
Nl
be the nmnber of occurrences during a period X of
a random event with rate
Al and let N
2
be the nUlllber of occur..
rences of this random event during a peri.od Y.
'rhus
Therefore the joint probability gener.s,ting fvnction of N
l
is
and N
2
~Jhence
and
Suppose '\'1e consider a queue '1vi th unit breakdovm.
The breakdowns
during the queueing time of a customer occur only during the period
Y of unexpired service time.
Thus, for unit breakdovm, the number
of breakdovms during queueing time is simply N vlith moments given
2
by
(5.18) and (5.19). The moments of the unexpired service Yare
100
given by (3.13) and (3.16) for active unit and by (3.23) and (3.26)
for independent unit breakdown.
Under a component breakdoiin system, the number of breakdovID
busy periods which occur during the queueing time of a customer is
N + N vdth moments given by (5.18) to (5.20). The moments of
1
2
X and Yare given by (3.32) to (3.36) for active component breakdown and by (2.46) to (2.50) for independent component
breal~down.
N + N busy periods of repair times in isolation
1
2
is composed of M breakdovIDs where the moments of M are given by
Each of these
(5.15) and (5.16). Thus the total number of breakdowns that occur
during the queueing time of a customer is the random variable
(Nl + N2 )M for component breakdown. Each of these busy periods of
repair times in isolation, is composed of M breakdowns wh€re the
moments of H way be evaluated from (5.15) and (5.16).
Thus the
total number of breakdovIDs that occur during the queueing time of
one customer is the random variable (N + N )M for component break2
l
dovID. Clearly (N + N ) and H are independent.
2
1
Let N be the number of breal:dO'''ms that occur during the
queueing time of one customer, for a preemptive resume discipline.
By virtue of the arguments advanced in the preceeding two paragraphs,
it is straightforvrard to obtain moments of the random variable N
from ImovID formulae.
Theorem 5.5.
Hence ive have the follOivj.ng theorem.
The first two moments of N, the number of breakdowns
during the queueing time of one customer, are listed in table 5.1.
e
e
e
TABIE 5.1
M01-ffiNTS OF
N, TEE NUl1BER OF BREA.KDOlINS DURING Q,UEli'EING T:rME
\
I
UNIT
Active
COl{PONEl\JT
I
,
~i
Independent
-
B(N)
~E(Y)
A,B(Y)
-'-
(3.13')*
(3.23 )
I
I
(3.32)
I
E(N )
AiE(i' )+A1E(Y)
I
Independent
I
A1E(X+Y)£"1-!IJ._7-
I
(2.46)
I
--r--
2 y2')+AIE(Y)
AIE(
,
AIE(X+Y)£"1-~_7-.l.-
(3.33)
2
i
1
Active
I
(3.13), (3.16)
I
(3.23), (3.26)
(2.47)
I
I L AiE(X+y)2+A1E(X+Y)J I L A~~X+y)2+"J.~(X+Ylj
1+m2-Il).2'I
[2 I
"lc1~m)3J
I· . (1-"'1):5J
1+m2-~
. rI
,
1
I
(3.32) to C3,36)
I[2.46)
to (2.50)
I
-lE-
The value of' each entry is given by a general formula containing moments of unexpired repair
and service times.
the
eq~~tions
For each entry, the values of the terms of the general formula are given by
for which the rrumbers are listed.
I--'
o
I--'
102
A similar argument for the preemptive repeat disciplines is not
valid vThen using (X, Y), the unexpired repair and completion times.
This follows from the fact that the completion time for a customer
contains the breakdolYl1 repair times for breakdolms that occur during
service of that customer.
Thus the length of the unexpired comple-
tion time is not independent of the number of breakdovms that occur
during that period.
3.
Miscellaneous Remarks Regarding
Transie~t
Behavior.
Con-
sider (Xt , Yt ), the stochastic vector of unexpired repair and service times of a queue with independent component breakdovm and a
preemptive resume discipline.
Recall that this stochastic vector
was investigated in Chapter II as the store content of a mathematically equivalent storage process.
The time dependent components
of the distribution function of the store content were defined as:
3r(t)
:::;;
H(x;t)
:;;
G(y;t)
:::
=:
~
p(a < Xt
p(Xt "" 0,
a
)
,
x, Yt : :; 0)
o < Yt :5
y)
,
}
= p(a < Xt :5 x, o < Yt :5 y)
F(x,y;t)
pt(x,y)
:::;;
The respective
defined as
p(X : :; Y
t
t
,
3r(t) + H(x;t) + G(y;t) + F(x,y;t)
e~ponent8
of the limiting distribution function were
rc, H(Jc), G(y), p(x,y) and p(x,y)
havior of the process is characterized in
tel~s
dent components of the distribution function.
The transient beof the time depen-
103
In Chapter II, probability considerations led to a set of four
inequalities for each of the time dependent components (for example,
see
(2.5)
and
(2.6».
These inequalities were combined to provide
a characterization of the transient behavior in equation (2.20).
This is
where
u and v
pect to
are Laplace-Stieltjes
transfol~
variables with resx
y
x and y, respectively and <l>:: A + A - A1B l (U)-A B (V) .
l
1 2
2
Each of the four sets of inequalities were employed individually
to derive a functional equation involving components of the limiting
distribution (see
(2.36)
to
(2.39».
It might be thought that each
set of inequalities should lead to a characterization of the transient behavior for components of the distribution function.
In
general this is not true and in this section we illustrate reasons
why this is not so.
Consider (2.7) and (2.8) where
~(Ujt+T)
=
HX(u;~l <
T
+
0(1) ,
T
> O.
Thus
104
and
- 0(1)
suppose we proceed
forn~lly
tion that all limits exist.
to limits as
T
1 0+, under the assump-
Then the riGht hand time derivative
satisfj.es :
(5. 21 )
where
h(O;t)
by assumption.
Since
e( U-Ah He T ;t)
T
thus we have assumed that
lim
;-H(T;t)/T
I =
h(O;t)
-> 0+
Thus equation (5.21) IDa valid for all t
T
T -> 0+, of LH(T;t)h_1
only if the limit, as
exists for all t.
this limit does not exist for all
t
He 'Vlill show that
by two counterexamples.
105
Since
#
Bl(S) is a distribution function, the right hapd deri-
vative exists
a.e. (s).
Suppose that the right hand. derivative
is infinite at the countable sequence of points
* s *0'
*J) •••
sl'
s'7..'
c:;
Further suppose that vTe consider the process (Xt , Y ) started at
t
time zero 'l-Tith the arrival at an empty store of
load size
° < Xt
~ T,
(a)
8
with distribution function
1
Yt
=0
material of
l
Thus
if:
there are zero arrivals in the time interval
and
(b)
Bl(s).
c
t < Sl
~
t + T,
(0,~7
o~
there is at least one arrival in the time interval
(o,t) with appropriate limitations on the load sizes
and order of arrivals.
Thus
and
From this it is clear that the limit of lH(T;t)/T_7, as
does not exist for
T
->
0+
'h
* 3 *, ....
= s1'
s2'
3
e
t
For the second counterexample consider the initial distribution
function of (X , Y ).
t
t
Suppose its component
H(X;O) has an infinite
right hand derivative at the countable set of points
106
(a)
Zero arrivals occur in the time interval
t
(b)
< X0 -<
t + f,
Y0
(0,~7 and
=0 .
One or more arrivals in t:tme interval
(O,.y:..7
with
certain appropriate lilnitations on the process.
Thus
H(T;t)
T
Hence the limit of l-H(T;t)/T_7, as
t
* x2*
= xl'
TJ.. 0+
does not exist for
' x *, ...
3
Therefore we have shovm that necessary conditions for the
validity of (5.21),
for all
t, are; right hand derivatives
exist ever~fhere for the load size distribution function
and the component
n(x;o)
Bl(S)
of the initial distribution function.
It is not knovm whether these are sufficierrt conditions.
vle have shovm by counterexample that (5.21) :may be invalid on
certain sets of measure zero on the time axis.
It is not known
whether or not (5.21) actually is valid a.e. (t).
The investiga-
tion of these questions is interesting; however, the resulting
four equations characterizing the transient behavior are most
tractable.
He list below the other three equations analogous to
(5.21) without any claim to their validity.
as
J
in~
Suppose the limits,
T --> 0+, are assumed to exist and defined as follows:
107
~G(T;t)/T_7
lim
T
->
T
->
T
->
=
g(o;t)
)
0+
lim
;-J(T;t)/T 7
0+ -
,j(o;t)
-
;-JiI(T)V;t)/T
lim
0+ --
7
=
-
Then the right hand derivatj.ves i'lith respect to time satisfy;
(5. 22 )
+
fY(OJv;t) - g(o;t) - j(O;t)
and
(5.24)
The sum of the respective sides of the four equations (5.21)
to (5.24) is equation (2.20), which is rigorously derived in
Chapter II .
J
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•
•