UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
ON DISCRETIONARY PRIORITY QUEUEING
by
B. AVi-Itzhak, I. Brosh
Technion, Israel Institute of Technology, Haifa, Israel
and
P. Naor
Technion, Israel Institute of Technology, Haifa, Israel
and
University of North Carolina, Chapel Hill, N. C.
November, 1962
This research was 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 permitted for any purpose of the
United States Government.
Institute of Statistics
Mimeo Series No. 338
,
ON DISCRETIONARY PRIORITY QUb1JEING I
B. Avi-Itzhak) I. Brosh
Technion) Israel Institute of Technology) Haifa) Israel
and
P. Naor
Technion) Israel Institute of Technology) Haifa, Israel
and
University of North Carolina) Chapel Hill) N. C.
ABSTRACT
A priority regime is envisaged for single server queueing systems composed
of two customer populations with Poisson arrivals which is intermediate between
the two extreme doctrines:
a)
head of the line priority) b) pre-emptive priority.
The state of intermediacy is represented by discretionary powers vested in the
server to interrupt recently initiated - and not to interrupt alraost completed service to a low priority customer upon the arrival of a high priority customer.
For the case of constant service times the discretionary rule is defined and the
ensuing
queueing characteristics analyzed; in particular, the average total queue
lengths of both high and low priority customers are derived for two different
cases:
a)
the resume situation where service renewed to a low priority customer
starts at the point of interruption;
b)
the repeat situation where service given
to a low priority customer before an interruption, is completely lost.
Optimisa-
tion procedures are outlined and for the resume situation a simple optimal discretionary rule is obtained.
~his research ~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 permitted
for any purpose of the United States Government.
ON DISCRETIONARY PRIORITY
QU~UEING
B. AVi-Itzhak, I. Brosh
Technion, Israel Institute of Technology, Haifa, Isra.el
and
P. Naor
Technion, Israel Institute of Technology, Haifa, Israel
and
University of North Carolina, Chapel Hill, N. C.
I.
Introduction
In the literature on queueing theory mention is made of two types of priorities:
a) head of the line priority, and b)
pre-emptive priority.
The rule per-
taining to type a) is that the serv:i.ce rendered to a low priority* customer is never
interrupted if a high priority* customer arrives; rather the
HP
is placed
cust~ler
at the head of the waiting line and is dealt with only after completion of service
to the
IJ?
customer at the station.
In other words, the newly arriving
tomer does not take precedence over the
taining to type
IP
IP
customer in service.
b) is that a newly arriving IW
HP
cus-
The rule per-
customer always displaces an
customer from the station.
Now in many practical situations in the business, industrial and military
fields neither of these two extreme rules is called for, even though some priority
regime must prevail.
Intuitively, one is inclined to prefer the
times when the service to the
hand, service to the
IP
IP
t~~e
a) rule at
customer is almost completed; if, on the other
customer has hardly started application of the type
b)
rule seems to be the proper course of. action.
e
*In
and
this study "low priorityll and "high priority" will be shortened to
HP , respectively.
IF
2
Thus it appears desirable to leave some discretion to the server (or to whoever is in charge of the system) as to the administration of the priority regime.
To exercise his discretion the server must possess some further knowledge regarding
the character of the serving process)Qctual past performance on the customer in
service and future expectation of the service duration.
Furthermore) the various
cost components should be known to him so that the cost implications of each of
the possible decisions may be evaluated.
The purpose of this study is to analyze discretionary powers with which we
endow the server for the particular case of constant service times (for each
priority class).
A dis cretionary rule is easily defined and s imply parametrized
in such priority queueing models.
ferent situations.
The rule defined will
be applied to two dif-
One of them is characterized by the (more common but frequently
less realistic) assumption that interrupted service may be resumed at convenience
~
without any loss.
In the second situation it is assumed that a customer) whose
service has been interrupted and later renewed) starts allover again) i.e.)
service must be repeated.
~he
analysis of the resume situation leads - on intro-
troducing reasonable costs - to a simple optiwal rule regarding the use of the
discretionary powers vested in the server.
The repeat situation is much more
complex and no simple optimisation rule can be provided.
Hovlever) the analysis
of the repeat situation is carried to a point where knOWledge of the numerical
values of the operative parameters and of the costs enables the attainment of the
optimum.
II.
Model A:
Two Priority Classes) Constant Service Times)
Resume Situation
Consider a service station rendering service to two Poisson
LP)
of incoming customers with parameters
/\1
and
streaw~
/\2' respectively.
(HP
and
Service
times are constant within each class of customers and their designation is
Sl and
3
Now a customer of cJBss l(HP) takes precedence over one of class
S2' respectively.
2 (LP); however an
incoming
Hanner:
HP
LP
customer in service is not necessarily displaced by an
customer.
Rather the server uses his discretion in the following
If the service time already devoted to the
is equal to or exceeds a value of
(0 ~ ~
constant
:5
1), the
HP
¢S2' where
~
LP
customer at the station
is an arbitrary, discretionary
custcmer is placed at the head of the waiting
line; if, on the other hand, the partial service time already elapsed falls short
of
~
8
2
the
LP
customer is replaced by the incoming
to the head of the waiting line (of his ovm class).
On
HP
returni~g
station his service is resumed at the point of interruption.
the first stage (of duration
~82)
of service to an
LP
customer and moves
later to the
To sum up, during
customer the pre-emptive
priority rule is follovled; during the second stage (of duration (1 - ~)82)
service to an
LP
customer the head of the line priority rule is followed.
Now the fraction of time
to
lIP
b , during which the station is busy giving service
l
customel's is clearly equal to
=
and the
of
(1)
LP - busy fraction is given by
(2)
Without any
fOl~l
proof we equate the existence of steady state conditions
with non-saturation the criterion of which is obviously
1 - b1 - b 2 > 0
Let the number of HP
customers in the line waiting for initiation of ser-
vice be denoted by WI; the expectation of the waiting time
QlW' (up to initia-
tion of service) of a customer is connected with the expected value of WI by
4
(4)
A further relation bet'ween E(9
following line of reasoning.
) and E(W ) may be obtained by the
lw
l
An incoming HP customer will encounter an average
waiting line of length E(W ); this will contribute E(Wl)Sl to his average
l
waiting time. In addition, with probability b , he will find an HI? customer
l
(1 - ¢)b , he encounters an
2
being serviced and, with probability
in the second stage of his service.
Iii?
customer
The average contributions of these two events
if they occur, to the average v~iting time are
8 /2
1
and
(1 - ¢)8 /2, respective2
l.y.
Thus we establish
8
2
(1 - 'Pni)b 2 (1 - ¢) - 2
(4)
Combination of
and
(5)
yields
(6)
The total expected queue, ql' of
=
ql
b l + E(Wl )
= bl
customers at the station equals
HP
b
+
2
l
+
2(1 - b_)
J.
Choosing
¢
regime and the
=1
IW
is equivalent to the establishment of a pre-emptive priority
customers' average queue length should be equal to that of a
simple queueing model with Poisson input and constant service times.
This is .
indeed the case.
Selecting
priority regime.
¢
=0
is equivalent to the establishment of a head of the line
The average queue length reduces to
b
2
l
+ AS b
1 2 l
2(1-b )
l
(8)
5
and this may also be obtained from Cobham's (1954) study who deals with arbitrary distributions of service times in a head of the line priority regime.
For the evaluation of the average queue length of
LP
customers we note,
first, that a relation parallel to (4) holds
where
w and Q
2
2w
time of a single
LP
represent the waiting line of
LP
customers and the waiting
customer, respectively.
E(Q2w)
Next we observe that
This must be the case since
LP
is invariant under the various priority rules.
customers whose service has not started yet -
and only these are defined as waiting - are completely unaffected by changing the
regime from one extreme through states of intermediacy (¢
extreme.
¢=0
~
0, 1) to the other
Indeed Cobham (1954) and Miller (1960) were concerned with the cases
and
¢ = 1,
respectively, and both of them obtained the first moment* as
2
=
2
A.1S l + A. 2S2
(10)
2(1-b )(1-b l -b )
l
2
Obviously (10) must hold also in the intermediate model under consideration.
Combination of (9) and (10) generates the average waiting line
2
=
2
A. 2A.1Sl + A. 2S2
(11)
2(1-b l )(1-b l -b2 )
Consider now the !esidence time,
T, of an
LP
customer at the station;
this is defined as the time which elapses from initiation to termination of service.
The residence time is made up of the time devoted to the service of the
*Miller
(1960) also obtained higher moments.
6
LP customer himself and of the service times of
the time at which it is possible to displace the
his displaced position.
station a period
t
l
Now whenever an
LP
HP
LP
customers who arrive during
customer or to keep him in
customer is displaced from the
elapses until service to him is renewed.
and Avi-Itzhak and Naor (1962) obtained the expected value of
Miller (1960)
t
l
as
51
E(t l )
If an
LP
= I-b
(12)
l
customer is displaced k
times from the service station the
eArpected value of his residence time equals
(13 )
The number of displacements, k, is, of course, a Poisson-distributed random variable with parameter
A ¢S2
l
(14)
p(k) =
k!
The unconditional expected residence time is then equal to
E(T)
=~
(15)
p(k) E(Tlk)
k=o
The average queue length of
LP
customers, g2' is now obtained as
(16)
Again, setting
¢=0
and
¢ = 1,
respectively, leads to well-known re-
sults relating to the two extreme priority regimes.
We can now turn to economic considerations and arrive - on making cost
assumptions - at an optimal value of
only will be dealt with here:
¢, ¢'*
say.
One reasonable cost structure
it will be assumed that the cost of delaying a
7
single
HP
customer in the queue equals
delaying a single
average total cost,
LP
c
l
per unit time; and the cost of
c,
Thus the
unit time.
customer is taken to be
per unit time is equal to
¢
b
I-b
l
1
) +
(17)
+
On differentiating and setting the derivative equal to zero
,1.*
~
is
evaluated as
=
e.
(18)
1
This result is meaningful if, and only if, the following inequality holds
If inequality (19) is not realized it may be shown by elementary but lengthy
considerations that the optimum is reached if class 1 and class 2
exchange their priority categories.
The optimum value of
¢
customers
is still given by
(18) with indices interchanged.
,1.*
It is interesting to note that
.111.
Model B:
~
is independent of the arrival rates,
Two Priority Classes, Constant Service Times, Repeat
Situation
The priority queueing model with which we deal in this Section is identical
with Model A in all but one respect.
interrupted service is virtually lost.
In the following it will be assumed
Whenever an
LP
tha~
customer is displaced
8
in the midst of his service he will - on renewal of service - require the attention of the station during an uninterrupted period of duration
8 ; prior ser-
2
vice must be repeated.
We observe that the average queue length (and, indeed, all other properties)
HP custcrners is the same in both resume and repeat situations.
of the
relation (7)
Thus
holds in Model B as well.
To evaluate properties of the queue of
LP
customers the concept of gross
service time, 8 , must be developed.
g
An
LP customer is served by the station and his service is subjected to
k(=O, 1, ••• ) interruptions by HP
customers where k
is a geometrically distri-
buted random variable
(20)
p(k)
the expected value of which equals
= ~ k p(k)
k=o
E(k)
(21)
Now whenever displacement has taken place the partial service time
~(o ~
~ ~
¢ 82 )
expended is lost.
This partial service time,
~,
is a random
variable with a density
=
and an
(22)
eA~ectation
~ 82
E(~)
=
J
~ h(~)
dT)
(2; )
o
An
LP
customer occupies the service station during a period of duration
9
S
where both k
and
1" i
k
L:
g
(24)
1".
J..
i=l
are random variables; the expected value of
Sg
is
obtained as
co
E(S g )
=
co
L:
k:::o
p(k) E(S g fk)
::
L:
k=o
p(k) fS2 + k E( 1"
>-7
::
.
This is the expected value of the gross service time.
The fraction of time, b , during which the station is engaged in giving
2
service to I.P customers equals
b
e
2 = A2 E(8 g )
= A2S2 (1
- ¢) +
A2
(e
A ¢ 82
l
(26)
- 1)
"'1
The busy fraction, b , relating to HP customers is given, of course, by
l
relation (1). Non-saturation is again assumed to be equivalent to the existence
of steady state conditions; the criterion continues to be inequality (3) with
b
2
redefined by (26).
To gain further physical insight into the model we wish to determine the
average residence time, E(T), of an
I.P
customer at the station and the average
time interval, E(U), which elapses from the initiation of service to an
I.P
customer to that epoch at which the station is prepared to give service (if requested) to the next
Consider an
times.
I.P
I.P
customer.
customer whose service is interrupted (and
The drrival of (and interruption by) an
k
customer causes the station
customers during an average time interval E(t ) given by
l
to be engaged to
lIP
ej~ression
Clearly the
(12).
HP
repeated)
e~ected
residence time of the
I.P
customer who is
10
interrupted k
times equals
(27)
and the expected unconditional residence time is obtained as
E(T)
=
~
p(k) E(Tlk)
k=o
=
~
k=o
)7
p(k) ;-E(8 !k) + k E(t 1
g
-
=
E(k)81
+
(28)
l-b 1
Further rearrangement yields
+
The first term on the right hand side of (29) measures the average time
which elapses frow the initiation of service to an
LP
customer up to the time
at which he overcomes his inferior standing by completing the part
service.
HP
number of
Hence) on the average) the
customers in the waiting line at the time of termination of serLP
customer equals
E(Wll at termination of service to
As a single new HP
an
At the beginning of this latter
customers are present at the station.
HP
vice to the
duration
of his
The second term represents the remaining part of his service during
which it is no longer possible to displace him.
part no
¢ 82
LP
customer)
customer gives rise to an
E(t ), completion of service to an
l
LP
= ~l(l
-¢)8 2
(30)
HP-busy period of average
customer will be followed by
HP-busy period the average length of which is given by the product of the
right hand sides of (12) and
(30).
added to the average residence time.
For the evaluation of
E(U)
this must be
11
1 - b
1 - b1
1
=
I-b1
An
state
E(k) + Al (1-¢)S2
"'1(l-b l )
E(S )
g
=
alternative definition of non-saturation (and the existence of steady
condition~
is then given by
A E(U)
2
<
It is not difficult to see that (3)
To evaluate the average queue of
1
(32)
and
liP
are equivalent.
customers we pro:eed by considering the
possible states of the system in some more detailed fashion.
Whenever the system is in an ullprimed state
customers and n
I.iP
mn
E
n1ll
customers in the queue; if
being served, and if m
=0
(and
n > 0) an
liP
mn
(0 ~~, 05n~ }--.there are
m> 0
an
HP
0
HP
customer is
customer is being served but the
future service required of the station exceeds (1-¢)S2'
Whenever the system is
(0 < m < 00,1 < n <00) there are again m HP customers
mn
customers in the queue but the customer in service is of the liP
in a primed state
I.iP
In the following
E ,and primed states, E' .
we shall distinguish between unprimed states,
and n
(31)
E'
variety.
In other vlords, a state is considered 1.lUprimed if the shortest future service
time of an
liP
customer exceeds (1-~)S2; it is considered primed if this·
shortest future service time of an
liP
customer falls short of (or is equal to)
The stationary probabilities pertaining to these events will be designated as
,...,J:'mn and P ' ,respectively,
mn
Clearly the following must hold
12
co
co
co
co
E
E
E
1:
m=o n=o
Pmn +
p'
mn
m=o n=l
= 1
It 'is useful to introduce further definitions
co
=
Pm.
E
n=o
Pmn
co
=
p.n
E
m=o
Pmn
(35)
p'
(36)
00
p'
=
m.
E
ron
n=l
co
p'
.n
=
E
pi
mn
m=o
The fraction of time during which the station is
lIP-busy, b , must equal
l
(38)
It may be shown in a straightforward manner that
A.
2
A.
1
co
=
E(k)
and
00
(40)
p'
.n
The fraction of time during which the station is
LP-busy, b , is equal
2
to the sum of (39) and (40)
A.
b
2
=
co
00
,2
~l
E(k) + A. 8 (1 - ¢)
2 2
=
E P + E pi
n=1 on n=l · n
In principle, the average queue length of LP
(41)
customers obtained is given
by
00
~
=
E(n)
=
E n (p
n=l~
.n
+ p' )
.n
()+2)
13
In practice we proceed as follows:
Consider an
at the service station while service is given to an
LP
HP
customer who arrives
customer; in other
words his arrival occurs when the system is in an unprimed state (excluding
His
E
on
).
(average) total queueing time is made up of three components:
a)
On the average the time he waits for termination of service to
the HP customers who are physically present at the time of his arrival equals ~ 8 (for the HP customer in service) plus
1 00
00
Sl( E p )-1
E (m-l)p
(for the HP customers in the waiting
m.
m=1 m.
m=1
line). However, by an ar~entation similar to that leading to (12)
and (31)
this average is increased by a factor of ( I-b )
l
factor is introduced to take into account that further
mers arrive while service is rendered to
at the time of the LP customer's arrival.
HP
p)
m=l m.
(E
.b)
-1
E
custo-
Thus the contribution of
co
m=l
HP
; this
customers present
this term to average total queueing time of an
00
-1
(m-l) p
m. -
LP
customer equals
7.
Each LP customer who is in the queue at the time of the new
IP
customer's arrival delays service for an average period E(U);
this quantity was evaluated in equation (31).
of
IP
00
00
The average number
customers present in the queue at that time equals
E
E n p • Thus the contribution of this term to the average
00
00
00
m=l n=o
mn
total queueing time equals E(U) E E n Pmn (E P )-1.
m=ln=o
m=l m.
c)
The third term contributing to the average total queueing time is
the residence time,
E(T), of the new
IP customer.
Thus we obtain the average of the total queueing time of an
who arrives while service is rendered to an HP
customer as
IP
customer
14
'"
S
E(~2ql
BInn )
m>o
n>o
:::
8
1
+
2(1-b l )
L
1
(rn-.l.)p
m.
m=l
+
co
1-b l
I:
m=l
Pm,
co
E(U)
+
n p
Z
.L
m=l
n=o
rnn
I:
(43 )
E(T)
+
co
P
m= 1 m.
Now if a.n
liP
customer arrives while the system is in a primed state (that.
is) service in its terminal stage is rendered to an
IP
customer) a relation
para.llel to (43) may be set up by a similar line of argumentation
0)
E(92
J
\j'
q ni>O
EI
=
)
Inn
+
--
co
I:
m=o
E(U)
.:0
<Xl
I:
I:
p'
m
ron
+
co
I:
(44)
E(T)
pI
.n
customer who arrives when the system is in state
diately into "residence",
+
(n-l)p'
m-=o n=l
n=l
An
Z m p'
m.
m=o
1
I-b
1
2(1-b l )
n>o
+
8
(1-¢)S2
i
E
00
goes imme-
Thus his average total queueing time equals
=
E(T)
(45)
The last case to be considered is the encounter of the newly arriving
LP
E(Q2 q IE 00 )
customer with a state
bility that the
liP
E (n > 0). Such a state is chara.cterized by the possion
customer in service will be displaced from the service
station; this contingency will occur if an
of partial service time
¢S2'
HP
customer arrives before completion
Consider, then, the two subcases:
customer in service is able to complete it;
b)
the
a)
the
liP
IP customer in service is
15
~
displaced by an incoming
~,
The probability,
HP
customer.
of subcase
to occur may be obtained by observing
a)
I}
toot the average time the system spends in "'-' Eon
within the average gross
n>O
service time to an LP customer equals E(k) E(T) + ~ 8 , The first of these
2
two terms indicates, of course, the occasions on which service to the LP customer is interrupted; the second term represents the successful "breakthrough" of
the
~
LP
customer towards unhampered completion of service.
Hence the probability
equals
~ 82
~
(46)
;::
~ 8
2 + E(k)E(T)
The average total queueing time for subcase
a) is derived in the usual
fashion
,
I
E(g2
I t~
'1 >0
i
11
Eon' subcase
a) )
=
~ 82
(1-¢)8
+
2
I-b
2
+
1
co
E(U)
+
L:
n=l
L
Pon
b) we note that the new LP
and the "future tail",
customer arrives during a
,.-interval
T* , of this interval possesses a density which is the
random modification of the original density k("),
the eX]?ected value of
(47)
EeT)
+
00
n=l
In subcase
(n-l) Pon
The quantity of interest is
,. * and for its derivation the second moment of the
original density is required.
On making use of (22) we obtain
~
8
2
(~
Al
-
E(k)
~
82 )
(48)
16
'T *
The expected value of
is then equal to
~
2
E( .r*)
8 (;
2 /\,1
"/1.2
=
E('T2)
2E('T)
8 )
2
E(k)
1
=
~
-
2
A. l
The average total queueing time is derived
E(Q
2q
LJ
I n>o
Eon} subcase
b»
+
co
E(U)
:E
n=l
+
n Pon
(50)
co
:E
n=l
Pon
We are now in a position to establish an equation evaluating the unconditional average total queueing time of an
LP
00
co
:E
.E
m=l n=o
00
E' )
mn
.
+ r
1(
-
:E
m=o
Pmn
+
0:>
:E
n=l
P~ + E(Q2g
I Eoo)p oo
I
I
I!
Et G I '.__)
2 q'
n>o
custorrJcr
E
on
) subcase
+
co
a)+ (l-:n: )E( Q I\"_ l E ) subcase b)7 :E p
2 q n>o
on
- n=l on
(51)
On using relations obtained earlier in this paper e~w.tion (51) is simplified to
17
- (1 - ~)S 2 (b1 +
Under conditions of statistical equilibrium the average queue and the
average queueing time are connected by
(53 )
Thus mUltiplication of (52) by
~1+~2
b2 ~
q
2
=
1
+
~2
and rearrclngement leads to
2
~2S2
- ~~E(k)+~7 - (1-~)~2S~bl+Ql-¢)A2S2_7
1
_......('_:-
(54)
1 - b1 - b 2
If the value
~::: 0
is chosen and inserted in
(54)
Cobham's
(1954)
for the head of the line priority case is obtained as in Section II.
value
~:::
1
formula
If the
is selected the model reduces to a special case of pre-emptive
repeat priority queueing.
Indeed the relation generated from
(54)
is equivalent
to a specialization of a result of Avi-Itzhak (1962).
The selection of an optimal
¢ - assuming
that some cost structure similar
to (17) properly describes the state of affairs - is not simple and no compact
formula of type (18) is attainable.
However, for any given set
numerical computation of
of a simple desk calculating machine.
*
~
is not beyond the reach
10
IV.
Conclusion
The models discussed in this paper represent special (and relatively simple)
cases of a wider field designated here as "discretionary priority queueing".
In
actual practice a priority regime prevails in most queueing situations, and
some discretion is allowed to the server (or to the agency which actually controls
the situation)
with respect to the exercise of the priority doctrine.
The purpose of this study was to define possible (discretionary) courses
of action and to analyze the consequences of the actual choice taken.
However, as
Slid before, two specialized models only were considered and future research will
have to generalize the analysis in (at least) three directions:
a)
The replacement of the two customer populations assumption by a
many customer populations assumption.
b)
The introduction of variability into service times.
c)
The exercise of the server's discretionary powers not only on the
basis of actual past and expected future attention time to the
customer in service but also consideration of the state of the
whole system.
Furthermore, the problems may be posed when many servers are available. Also
cost structures other than (11) may reveal the true state of affairs and a wide
range of such alternatives may be worth investigation.
Some
.e
~lrther
study is in progress •
19
REFERENCES
£];7
Avi-Itzhak, B. (1962).
Pre-emptive Repeat Priority Queues as a Special
Case of the Multi-Purpose Server Problem - II, Technion, Israel Institute
of Technology.
(To be published)
..r~7 Avi-Itzhak, B. and Naor, P. (to be published in Operations Research, VoL 11
(1963) ) •
Some Queueing Prob1eIl1S lJhen the Service Station Is Subject to
Breakdovm.
~27
Cobham, A. (1954).
f!:.7
Miller, Ro G. (1960).
Operations Research
g,
70-76 •
Ann. Math. stat. 31, 86-103 •