Average waiting time of customers in a priority M/D/k queue with finite buffers

*Corresponding author. Tel.: #1-352-392-0648; fax: #1-352-392-5438.E-mail addresses: bosei@u#.edu (I. Bose), raktim}[email protected] (R. Pal).

Computers & Operations Research 29 (2002) 327}339

Average waiting time of customers in a priorityM/D/k queue with "nite bu!ers

Indranil Bose��*, Raktim Pal�

�Department of Decision and Information Sciences, Warrington College of Business Administration, University of Florida,351 Stuzin Hall, P.O. Box 117169, Gainesville, FL 32611, USA

�i2 Technologies, 909 East Las Colinas Blvd., 16th Floor, Mail drop 13 Unit 1, Irving, TX 75039, USA

Received 1 July 1999; received in revised form 1 March 2000; accepted 1 April 2000

Abstract

Analysis of priority multiserver queues is known to be a di$cult problem. In this paper, a priority M/D/kqueue with "nite bu!ers is analyzed and an approximate closed-form expression for the average waiting timeof customers belonging to di!erent priority classes is obtained. The key assumption is the presence of heavytra$c in the network leading to loss of customers from the service stations. Several interesting numericalresults are reported that show the interplay between arrival and service rate of customers, capacity of thebu!er, proportion of tra$c belonging to each class and probability of loss as experienced by incomingcustomers.

Scope and purpose

Priority queues with "nite bu!ers are used to model various types of telecommunications and manufacturingsituations. These queues are specially useful for modeling high-speed asynchronous transfer mode (ATM)networks that are subject to loss due to over"lling of the bu!ers. In general, it is di$cult to obtainclosed-form solutions of the various performance measures for these queues. Analysis of such queues with thedeterministic service discipline can help provide useful guidelines for designers and developers of telecommu-nication and manufacturing networks. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Average waiting time; Deterministic service; Finite capacity; Priority queue

Multiserver queues with priority classes and "nite capacities arise in the design of computernetworks, and manufacturing systems. In the area of computer networks one of the popular

0305-0548/02/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 5 2 - 6

switching protocols is asynchronous transfer mode (ATM). In ATM networks the messages aretransmitted as &cells' each having a "xed size of 53 bytes. At each switch of the ATM network thereexists a bu!er which has a "xed capacity and acts as a "lter to the cells arriving at the switch.Whenever the number of arriving cells exceeds a threshold value the bu!er gets full and the cells arediscarded and hence lost from the system. ATM is suitable for simultaneous transmission ofmultimedia tra$c that can be assumed to belong to di!erent priority classes according to theirdelay requirements. M/D/k/N queues are useful for representing ATM networks. In a manufactur-ing situation, M/D/k/N queues occur when jobs that require the same service time and havedi!erent priorities (according to their deadline dates) wait to be serviced in front of machines.Whenever the storage bin in front of the service area becomes full, no more jobs are allowed to bestored. In this paper the mean waiting time for customers belonging to di!erent priority classes inanM/D/k queue with "nite bu!er size is analytically obtained. This waiting time expression can beused for computation of average delays for di!erent tra$c types in a "nite-capacity ATM switch orin a job shop-type manufacturing environment.

1. Problem statement

Analysis of "nite-capacity multiserver queues with priorities is known to be a di$cult problem.The di$culty arises from two facts * the presence of more than one server and the presence ofpriorities in the service discipline. In this paper, an approximate analytical result on mean waitingtime of a customer in a "nite-capacity multiserver queue with priorities is presented. The arrivalprocess of customers is assumed to be Poisson, the service is assumed to be deterministic andprovided to customers on a "rst-come}"rst-served basis. It is assumed that the priority discipline isof nonpreemptive head-of-the-line type. The number of priority classes is assumed to be &n', wherea lower number indicates a higher priority class (i.e. class 1 is higher in priority than class 2). It isassumed that there is a bu!er of "nite capacity before the service area. Whenever the bu!ercapacity is exceeded the customers are rejected from the system irrespective of the priority classthey belong to. Finally, we assume that there is heavy tra$c in the system and the servers remainbusy most of the time.The organization of the paper is as follows. Section 2 is a brief review of literature on

"nite-capacity priority queues. In Section 3 the assumptions of the model are listed and theexpression for mean waiting time of customers belonging to di!erent priority classes is derived.Section 4 describes the numerical experiments that are conducted to show how waiting time ofcustomers is a!ected by di!erent system parameters. The conclusion and direction for futureresearch appear in Section 5.

2. Literature review

Researchers have studied "nite-capacity queueing systems for a long time. There is a signi"cantamount of literature in this area and a detailed bibliography may be obtained from Gross andHarris [5], Kleinrock [8], Ross [10], and Tijms [11]. However, priority queues with "nitecapacities have not been studied much in the queueing literature. This is largely due to the fact that

328 I. Bose, R. Pal / Computers & Operations Research 29 (2002) 327}339

priority queues are di$cult [8] to analyze in general and hence only approximate techniques arefavored for gaining insights about "nite-capacity priority queues. Davis [3] obtained an explicitexpression for the expected waiting time of customers in an M/M/k queue with nonpreemptivepriorities. Federgruen and Groenevelt [4] obtained approximate expressions for average waitingtimes of customers in an M/G/c queue under various priority rules. Their results are shown to beexact for the M/M/c and M/G/1 queues. Kella and Yechiali [6] derived the Laplace}Stieltjestransform of the waiting time of the customers for an M/M/k queue with nonpreemptive prioritydiscipline. Venkataramani et al. [12] obtained the probability distribution of the queue lengths fortwo priority classes in case of a priority MMPP/D/1 queue with "nite bu!ers, which is a closerepresentation of current ATM mutiplexers.

3. Derivation of average waiting time

In this section we list the assumptions of the model, de"ne the necessary notations and derive anapproximate expression for the mean waiting time of customers in an M/D/k/N queue. Thederivation is similar to that in [1]. We assume that the service is deterministic and is provided bythe identical servers on a "rst-come}"rst-served basis within a priority class. The priority disciplineis of nonpreemptive head-of-the-line type. The waiting room is assumed to be "nite (i.e. customersare lost from the system if the capacity of the waiting room is exceeded). When a new customerenters the system the remaining service times of all the servers are independent of each other. Thisassumption is valid under the conditions of heavy tra$c. In case of heavy tra$c the great majorityof arrivals will encounter a large queue for a large period of time and hence the departure processthat they will observe at each server will be approximately independent delayed renewal processes[9]. The notation for this derivation appears in Table 1.It follows from Little's Law that

M̧��

"��(1!P

�)=M

��, (1)

MM��

"��(1!P

�)=M

��, (2)

where ��(1!P

�) is the e!ective arrival rate due to the presence of the waiting room with "nite

capacity in front of the service area.The average waiting time of a customer belonging to priority class &n' will consist of three

components. The "rst component is the time that the customer on an average has to wait beforeany of the servers become free (note that the queuing discipline under consideration is of thenonpreemptive head-of-the-line type). The second component is the waiting time due to customersof priority classes &n' or higher, who have arrived before the new customer and are waiting to beserved. The last component is the waiting time due to customers belonging to priority classeshigher than &n' who arrive when the customer belonging to priority class &n' waits to be served. It isto be remembered here that the e!ective service rate for customers who are being served while thecustomer belonging to class &n' waits is &kR', as all the &k' servers are busy at this time [1]. In terms ofthe notation introduced before it can be written as

=M��

"=M�#

��

1kR

M̧��

#

��

1kR

MM��. (3)

I. Bose, R. Pal / Computers & Operations Research 29 (2002) 327}339 329

Table 1List of notations

=M��

Average waiting time in queue for a customer belonging to priority class &i+ (s)

=M�

Average waiting time of a new arrival until one of the &k' servers become free (s)M̧��

Average number of customers belonging to priority class &i' who are already waiting in the queue at the instantwhen the new customer arrives

MM��

Average number of customers belonging to priority class &i' who will arrive while the newly arrived customerwaits in the queue

��

Arrival rate for customers belonging to priority class &i' (units/s)R: Constant service rate for customers belonging to any priority class� (units/s)k Number of serversB Capacity of the waiting room N"k#BP�

Probability that there are &N' customers in the system (i.e. the probability that the bu!er is full and any newarrivals to the system are lost)

V Remaining service time for a serverZ Remaining service time until one of the servers become freeW Length of the gap between two successive service completions in which the new customer arrivesY Elapsed time between two successive service completions by a server

�For the M/D/k/N queue the service rate is taken to be the same for all priority classes (i.e. R�"R

�" 2 "R

�"R)

From (1) and (2) it follows that

=M��

"=M�#

��

1kR

��(1!P

�)=M

��#

��

1kR

��(1!P

�)=M

��(4)

or

=M��

"=M�#

��

��=M

��#

��

��=M

��,

where

��"

��(1!P

�)

kR. (5)

Generalizing

=M��

"=M�#

��

��=M

��#

��

��=M

��for j"1, 2,2, n!1. (5A)

From (5) it follows that

=M��

"

=M�#��

��=M

��1!��

��

. (6)


This equation may be solved recursively as discussed in Kleinrock [8, pp. 119}121]. The strategy isto "rst "nd =M

��and from this "nd =M

��. At this stage it is convenient to de"ne

��"

��

��. (7)

Solving Eq. (6) recursively the solution is obtained as

=M��

"

=M�

(1!��

)(1!��). (8)

The method for computing=M�, which is the average time that a new customer has to wait till any of

the &k' servers become free, is now described. The probability density function for V is given by

f�(v)"

1!F�(v)

E(>). (9)

For &k' servers let their respective remaining service times be <�, <

�, ... , <

�. Therefore, the time

that a new arrival has to wait till any one of the servers become free is given by

Z"min(<�, <

�, 2 , <

�). (10)

Now,

F�(z)"P(Z)z)"1!P(Z'z) (11)

or

F�(z)"1!P(<

�'z, <

�'z, 2 ,<

�'z) (12)

or

F�(z)"1!P(<

�'z)P(<

�'z) ... P(<

�'z). (13)

(This follows from the assumption that remaining service times for the &k' servers are independentwhich is true when there is heavy tra$c in the system.) Now,

P(<�'z)"�

f��(v

�) dv

��

1!F��(v

�)

E(>�)

dv�. (14)

But it is known that

F��(v

�)"�

0, <�(¹,

1, <�*¹,

where ¹"

1R

and E(>�)"¹. (15)

Therefore,

P(<�'z)"�

1!F��(v

�)

E(>�)

dv�#�

1!F��(v

�)

E(>�)

dv�"

¹!z¹

. (16)

From (13) and (16) it follows that

F�(z)"1!

(¹!z)�¹�

. (17)


So,

f�(z)"

ddz

F�(z)"

1¹�

k(¹!z)��. (18)

Once the pdf of Z has been obtained this will lead to the calculation of E�(z):

E�(z)"�

�

zf�(z) dz"

k¹��

�

z(¹!z)�� dz"

¹

k#1. (19)

Now the average waiting time for the new customer until one of the servers become free (i.e.=M�) is

given by

=M�"

��

P�E

�(z)"

¹

k#1��

P�, (20)

where P�is the probability that there are &s' customers in the system.

Substituting the value of =M�obtained from (20) in (8) it follows that

=M��

"

(¹/(k#1))��

P�

(1!��

)(1!��). (21)

The expression for the delay probability ��

P�is nonstandard and is obtained from a recent

paper by Kimura [7]. It must be remembered at this stage that calculation of the steady-stateprobability is not dependent on priority considerations as service time is not dependent on priorityclass and so the result for nonpriority queue can be used here. Hence,

P�"�

(k�)�P�

s!, s"0, 1,2, k!1,

(k�)�k!

1!�1!�

��P�, s"k, k#1,2, k#B!1,

(k�)�k!

��P�, s"k#B,

(22)

where

�"

�kR

, (23)

�"

��

��, (24)

P�"�

��

(k�)�s!

#

(k�)�k!

1!��

1!� ��, (25)

�"

�R

1!�#�R

, (26)

R

"

Expectedwaiting time for deterministic service systemExpectedwaiting time for exponential service system

. (27)


Fig. 1. (a) Waiting time for high-priority class (k"4, R"2, B"5), (b) waiting time for low-priority class (k"4, R"2,B"5).

It is to be noted here that in order to evaluate P�numerically we need to obtain an expression for

R . For high tra$c intensity the following approximation is derived in Barcello and Paradells [2]

after starting with a known expression from Tijms [11, pp. 286}310]:

R

"

�2

#

(1!�)kk#1

. (28)

We substitute the Tijm's approximation (28) in (27) to "ndR for heavy tra$c. Using Eqs. (21)}(28)

we get a closed-form expression for the average waiting time of a customer belonging to priorityclass &n' in the case of a priority M/D/k/N queue with nonpreemptive head-of-the-line prioritydiscipline.

4. Numerical experiments

In this section the dependence of the average waiting time of a customer belonging to a particu-lar priority class with the various parameters associated with the system is shown. The parametersunder consideration are arrival rate (�), service rate (R), number of servers (k), bu!er capacity (B)and proportion of customers belonging to a speci"c priority class.In Figs. 1(a) and (b) the variation of the average waiting time with arrival rate for the high- and

the low-priority class, respectively, is shown. It is assumed in these experiments k"4, R"2 andB"5. Also, the total arrival to the system is equally divided between the two priority classes. Theexperiments are repeated for a larger bu!er capacity (i.e. 20), the same service rate and the samenumber of servers. The results are shown in Figs. 2(a) and (b).Comparison of "gure sets 1 and 2 illustrate that the waiting time reduces signi"cantly for both

high- and low-priority customers with increase in the bu!er capacity. Fig. 3 reports a comparisonof the loss probability (i.e. probability that the customers of a particular class are refused entry intothe system as the bu!er becomes full) as the bu!er capacity is increased from 5 to 20 units.The second set of experiments show the variation of the average waiting time with service rate for

the two priority classes. For these experiments k"4, �"8 and B"5 units. The results of theseexperiments are shown in Figs. 4(a) and (b). The results of the experiments with the same arrival


Fig. 2. (a) Waiting time for high-priority class (k"4, R"2, B"20), (b) waiting time for low-priority class (k"4, R"2,B"20).

Fig. 3. Variation of loss probability with arrival rate (k"4, R"2).

Fig. 4. (a) Waiting time for high-priority class (k"4, �"8, B"5), (b) waiting time for low-priority class (k"4, �"8,B"5).

and service rates, number of servers, and bu!er capacity equal to 20 are shown in Figs. 5(a) and (b).It is clear from these "gures that with an increase in the bu!er capacity the service improves &more'for the high-priority customers than for the low-priority customers. Next, the loss probability isplotted against service rate for B"5 and 20 in Fig. 6.The third set of experiments demonstrates the variation of the average waiting time of the two

priority classes with number of servers. In Figs. 7(a) and (b) the bu!er capacity is taken to be B"5


Fig. 5. (a) Waiting time for high-priority class (k"4, �"8, B"20), (b) waiting time for low-priority class (k"4, �"8,B"20).

Fig. 6. Variation of loss probability with service rate (k"4, �"20).

Fig. 7. (a) Waiting time of both priority classes vs. number of servers (�"9, R"1, B"5), (b) waiting time of bothpriority classes vs. number of servers (�"9, R"1, B"20).

and 20, respectively. In Fig. 8, the loss probability for any priority class is plotted against thenumber of servers. It is observed that when number of servers in a system is su$ciently high bu!ercapacity has little e!ect on the loss probability of a priority class.In the next set of experiments the variation of the average waiting time for the priority classes is

shown against variation of the bu!er capacity. From Figs. 9(a) and (b) it appears that as the bu!ercapacity increases the average waiting time increases for both priority classes. This is because the


Fig. 8. Loss probability of both priority classes vs. number of servers (�"8, R"1).

Fig. 9. (a) Waiting time vs. bu!er capacity for high-priority class (k"10, R"1, �"9), (b) waiting time vs. bu!er capacityfor high-priority class (k"10, R"1, �"9).

bu!er acts as a "lter in front of the service area. When the bu!er capacity is low, very few customersbelonging to either class is allowed entry into the system. The overall load on the network remainslow and the average waiting time for both priority classes is small. As the bu!er capacity increasesthe net load on the system increases as more customers are allowed entry into the system. Hence,the average waiting time for customers belonging to either priority class increases. Again, we notethat increasing the bu!er capacity beyond the threshold value does not improve the performance ofthe system in any way.In Fig. 10 the loss probability for both priority classes is plotted against bu!er capacity when

arrival rate, service rate and number of servers remains same. The "gure shows that for thehigh-priority class the loss probability remains small for any choice of the bu!er capacity whereasfor the low-priority class the loss probability decreases as the bu!er capacity increases.In the sets of experiments conducted so far we have tacitly assumed that the arrival rate for the

high- and the low-priority customers is equal. In the following, set of experiments we vary theproportion of the high-priority customers. In Figs. 11(a) and (b) it is observed that the averagewaiting time of both priority classes increases as the proportion of the high-priority customersincreases. This is true irrespective of the capacity of the bu!er. A plausible explanation for this is asmore high-priority customers arrive, on an average a high-priority customer has to wait more andhence the average waiting time increases. It is also observed that the average waiting time of the


Fig. 10. Loss probability vs. bu!er capacity for both priority classes (k"10, R"1, �"9).

Fig. 11. (a) Waiting time vs. proportion of high-priority customers (�"7, R"2, k"4, B"5), (b) waiting time vs.proportion of high-priority customers (�"7, R"2, k"4, B"20).

low-priority customer increases much more in proportion than that of the high-priority customerbecause the low-priority customer is served after all entering high-priority customers have beenserviced. In Figs. 11(a) and (b) we also plot the overall average waiting which is de"ned as overallaverage waiting time"proportion of high-priority customers�average waiting time of high-priority customers#proportion of low-priority cusotmers�average waiting time of low-prioritycustomers.It is observed that the overall waiting time remains almost constant for any choice of proportion

for high-priority customers. This shows that the system performance remains the same as morehigh-priority customers are allowed entry into the system though the performance worsens (i.e. theaverage waiting time increases) for each individual class of customers.In Figs. 12(a) and (b) the loss probability for both priority classes is plotted against proportion of

high-priority tra$c for B"5 and 20, respectively. As bu!er capacity increases the loss probabilitydecreases for both priority classes, for the same mix of priority tra$c. It can be seen that for thehigh-priority customer the loss probability increases almost exponentially with increase in theproportion of the high-priority customers arriving to the system. This pattern holds for bothchoices of the bu!er capacity. Similarly, the loss probability for the low-priority customersdecreases almost negative exponentially as the proportion of the high-priority customers increases.


Fig. 12. (a) Loss probability vs. proportion of high-priority customers (�"7, R"2, k"4, B"5), (b) loss probability vs.proportion of high-priority customers (�"7, R"2, k"4, B"20).

5. Conclusion

In this paper, the M/D/k/N queue with nonpreemptive priorities is analyzed. An expression forthe mean waiting time of customers in such a queuing discipline is obtained. The motivation forthis type of a queue comes from problems that arise in design of ATM networks with "nite bu!ersand multiple service and in job shop-type manufacturing environments with "xed storage bins andmultiple job classes. Extensive numerical results that illustrate the variation of the waiting timewith variation of system parameters such as arrival rate, service rate, number of servers, bu!ercapacity, proportion of priority classes are presented. A key assumption in the derivation is that theremaining service times for the servers are independent at the time of a new arrival. This is a validassumption since we are considering heavy tra$c in the system and the servers remain occupiedmost of the time.

References

[1] Altinkemer K, Bose I, Pal R. Average waiting time of customers in an M/D/k queue with nonpreemptive priorities.Computers and Operations Research 1998;25:317}28.

[2] Barcello F, Paradells J. Performance evaluation of public access mobile radio (PAMR) systems with priority calls.Proceedings of the Fifth International Conference on Telecommunications, Nashville, TN, 1997. p. 360}7.

[3] Davis RH. Waiting-time distribution of a multi-server, priority queueing system. Operations Research1966;14:133}6.

[4] FedergruenA, Groenevelt H.M/G/c queueing systems with multiple customer classes: characterization and controlof achievable performance under nonpreemptive priority rules. Management Science 1988;34:1121}38.

[5] Gross D, Harris CM. Fundamentals of queueing theory. New York: Wiley, 1974.[6] Kella O, Yechiali U.Waiting times in the non-preemptive priority M/M/c queue. StochasticModels 1985;1:257}62.[7] Kimura T. A transform-free approximation for the M/G/s queue. Operations Research 1996;44:984}8.[8] Kleinrock L. Queueing Systems. Vols. 1, 2, New York: Wiley, 1975.[9] Nozaki SA, Ross SM. Approximations in "nite capacity multi-server queues with Poisson arrivals. Journal of

Applied Probability 1978;5:826}34.[10] Ross SM. Introduction to probability models, 5th ed. San Diego: Academic Press Inc., 1993.[11] Tijms HC. Stochastic modelling and analysis: a computational approach. New York: Wiley, 1986.


[12] Venkatramani B, Bose SK, Srivathsan KR. An exact model for the queueing analysis of a non-preemptiveMMPP/D/1 priority system for ATM applications. Proceedings of the ICCS, Singapore, 1994. p. 104}8.

Indranil Bose is Assistant Professor of Decision and Information Sciences at the University of Florida. He holdsa B.Tech. (Hons.) in Electrical Engineering from the Indian Institute of Technology,Kharagpur, anM.S. in Electrical andComputer Engineering from the University of Iowa, an M.S. in Industrial Engineering and a Ph.D. in Management fromPurdue University. His publications have appeared in Computers and Operations Research, Decision Support SystemsInformation and Management and Ergonomics. His research interests are in the areas of design and pricing issues intelecommunications, data mining, and applied operations research. His teaching interests are telecommunications andnetworking, database management systems, systems analysis and design and knowledge based systems.Raktim Pal is management consultant at Technologies. He has a B.Tech. (Hons.) in Civil Engineering from the Indian

Institute of Technology, Kharagpur, M.S. in Civil Engineering, M.S. in Industrial Engineering and Ph.D. in CivilEngineering from Purdue University. His current research interests include intelligent transportation systems, transpor-tation systems analysis, logistics and infrastructure management, and supply chain management. His publications haveappeared in ASCE Journal of Transportation Engineering, Computers and Operations Research, and TransportationResearch Record.


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Average waiting time of customers in a priority M/D/k queue with finite buffers