discrete-time queues with discretionary priorities

European Journal of Operational Research 200 (2010) 473–485

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Stochastics and Statistics

Discrete-time queues with discretionary priorities

Kilhwan Kim *, Kyung C. ChaeDepartment of Industrial and Systems Engineering, KAIST, Daejeon 305-701, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 October 2008Accepted 31 December 2008Available online 24 January 2009

Keywords:QueueingDiscretionary priorityGeneral service times

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2008.12.035

* Corresponding author. Tel.: +82 42 350 2955.E-mail addresses: [email protected] (K. Kim), kcc

In this paper, we consider a discrete-time two-class discretionary priority queueing model with generallydistributed service times and per slot i.i.d. structured inputs in which preemptions are allowed only whenthe elapsed service time of a lower-class customer being served does not exceed a certain threshold. Asthe preemption mode of the discretionary priority discipline, we consider the Preemptive Resume, Pre-emptive Repeat Different, and Preemptive Repeat Identical modes. We derive the Probability GeneratingFunctions (PGFs) and first moments of queue lengths of each class in this model for all the three preemp-tion modes in a unified manner. The obtained results include all the previous works on discrete-time pri-ority queueing models with general service times and structured inputs as their special cases. Anumerical example shows that, using the discretionary priority discipline, we can more subtly adjustthe system performances than is possible using either the pure non-preemptive or the preemptive prior-ity disciplines.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, we consider a discrete-time queueing model with discretionary priorities. In this model, customers of two classes are as-sumed to arrive in a single-server queueing system with infinite buffer space, the service times of customers of each class are generallydistributed, and the numbers of arrivals are independent and identically distributed (i.i.d.) from slot to slot, but the numbers of arrivalsof each class in a slot may be dependent/structured (and this type of arrivals will be called ‘structured inputs’ according to Sidi and Segall[13]).

Two fundamental priority disciplines are the non-preemptive (NP) and preemptive disciplines. However, these two disciplines have sev-eral drawbacks in practical application. For instance, under the NP discipline, customers of a higher-class may wait even when the serviceof a customer of a lower-class has just started, while under the preemptive discipline, the almost completed service of a customer of a low-er-class may be interrupted due to the arrival of a customer of a higher-class. This rigidity of the priority disciplines may result in the severedegradation of the service quality of a particular class under certain practical conditions. To address this problem, several hybrid types ofthe NP and preemptive priority disciplines have been considered, including the discretionary priority discipline introduced by Avi-Itzhaket al. [2]. Under this discipline, the server uses its discretion to allow a higher-class arrival to preempt the service of a lower-class customerthen being served, depending on whether the elapsed service time of the lower-class customer exceeds a certain threshold. In contrast tothe two extreme cases of the NP and preemptive disciplines, under the discretionary priority discipline, the delays of customers of eachclass can be delicately adjusted by varying the value of the threshold. As a result, there have been a number of contributions on the dis-cretionary discipline and its variations [2,12,4,7,5]. However, all these priority queueing models with discretionary priorities are continu-ous-time models with independent Poisson arrivals, while there have been no contributions on discrete-time priority queueing modelswith discretionary priorities.

Generally, discrete-time priority queueing models are more suitable for the analysis of time-slotted digital communication systemsaccommodating different types of multimedia traffic that need different Quality of Service (QoS) standards. As a result, discrete-time pri-ority queueing models have recently received increasing attention. Discrete-time priority queueing models with deterministic servicetimes equal to one time slot have been studied by a number of researchers [8,9,1,13,16,19]. These models have the advantage of being moretractable to analysis than the models with generally distributed service times, and of having no distinctions among different priority dis-ciplines such as the NP, Preemptive Resume (PR), Preemptive Repeat Different (PRD) and Preemptive Repeat Identical (PRI) priority disci-plines. On the other hand, discrete-time priority queueing models with generally distributed service times have been also studied by a

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474 K. Kim, K.C. Chae / European Journal of Operational Research 200 (2010) 473–485

number of researchers [11,10,6,15,18,17,20,21]. Compared to the models with deterministic service times, in these models, different pri-ority disciplines generally result in different system performances. Thus, most of the studies on discrete-time priority queueing modelswith generally distributed service times have considered models with different priority disciplines separately [11,10,18,17,20,21]. Therehave been two attempts to analyze discrete-time priority queueing models with different priority disciplines in a unified manner. Fiemset al. [6] proposed a technique for analyzing the discrete-time priority models with PR, PRD and PRI disciplines commonly. However, theapproach in [6] cannot be extended to the case of structured inputs because it requires that arrivals of different classes in a slot be inde-pendent. In many cases, structured inputs are more suitable than independent inputs for the analysis of digital communication systems.(For example, see the analysis of an N � N output-queueing switch in [17, Section 7].) On the other hand, Takahashi and Hashida [15] ana-lyzed the discrete-time structure-input priority models with NP and PR disciplines in a unified manner by using the delay cycle analysis.However, the delay cycle analysis can be applied to the structured input priority model only with work-conserving priority disciplines suchas NP and PR. For the structured input priority model with non-work-conserving priority disciplines such as PRD and PRI, the delay cycleapproach cannot be applied because the length of a delay cycle for customers of a lower-class may depend on the arrival process of thatclass during the delay cycle.

In this paper, we use a unified approach to analyze the queue lengths of each class in the discrete-time single-server queueing modelwith general service times and structured inputs, under three different types of discretionary priority disciplines in which the preemptiblepart of the service time of a lower-class customer is the PR, PRD, or PRI mode, respectively. These results can be of practical interest for usein time-slotted packet-based networks, since the conventional NP and PR priority disciplines are too strict for adjusting the system perfor-mance to a target level in some practical conditions. Further, it will be shown in Section 5 that the discretionary priority model with thethree preemption modes presented in this paper includes all the models in the previous works on discrete-time priority queueing modelswith general service time [11,10,6,15,18,17,20,21] as their special cases. The proposed unified approach yields some general results onqueue-length distributions that hold for all the three types of discretionary priority disciplines (and therefore for all the models in the pre-vious works). By using these general results, we can reduce the analysis of the entire system to that of the effective service time of a class-2customer as in [6]. However, because the approaches in [6] cannot be applied in the case of structured inputs, the proposed approach canbe viewed as a generalization of the approach in [6] in terms of the queue lengths of each class, rather than the waiting times of each class.

This paper is organized as follows. In Section 2, the mathematical model under consideration is presented. In Sections 3 and 4, the gen-eral results on the queue-length distributions that hold for all the three types of discretionary priority disciplines are derived. In Section 5,for each discretionary priority discipline the effective service time is analyzed and the queue lengths of each class are derived by using theobtained general results. In Section 6, the obtained results are applied to an example of an N � N output-queueing packet switch.

2. Mathematical model

We consider a discrete-time single-server queueing system with a buffer of infinite capacity. Time is assumed to be slotted, and thelength of a time slot is defined to be a time unit. We assume that services to customers can be started or completed only at slot boundaries,which are denoted by times t ¼ 0;1; . . .. Customers are assumed to arrive late during a time slot, just prior to the end of the slot. Such amodel is termed a late arrival model (see [14]). Specifically, in a late arrival model, a potential arrival takes place in ðt�; tÞ, t ¼ 1;2; . . .. There-fore, customers arriving at a slot see a departing customer (if any) about to leave the system at the end of the slot.

There are 2 classes of customers arriving at the system, class 1 and class 2. Let ajðkÞ, j ¼ 1;2, denote the number of class-j customersarriving at the system during time slot k. fða1ðkÞ; a2ðkÞÞ; k ¼ 1;2; . . .g are assumed to be i.i.d. and generally distributed. However, thenumbers of class-1 and -2 customers who arrive during the same time slot, namely, a1ðkÞ and a2ðkÞ, may be dependent. The corresponding

PGFs and moments are defined as Kðz1; z2Þ ¼ E½za11 za2

2 �, K1ðzÞ ¼ Kðz;1Þ, K2ðzÞ ¼ Kð1; zÞ, kj ¼ K0jð1Þ, and kij ¼ @2Kðz1 ;z2Þ@zi@zi

h iz1¼z2¼1

, i; j ¼ 1;2. The ser-

vice times of customers of each class are independent and generally distributed discrete random variables. The service time of a class-jcustomer is denoted by Sj. The probability mass function (p.m.f.) of Sj are denoted by bjðlÞ; l ¼ 1;2; . . .. The corresponding PGFs, tail distri-butions, and moments of Sj are defined as BjðzÞ ¼

P1l¼1zlbjðlÞ, BjðiÞ ¼

P1l¼ibjðlÞ; i ¼ 1;2; . . ., lj ¼ B0jð1Þ, and ljj ¼ B00j ð1Þ. Also, the offered loads of

each class and all classes are denoted as qj ¼ kjlj and qT ¼ q1 þ q2. Throughout this paper, we assume that the system is unsaturated.Class-1 customers have priority over class-2 customers in the following manner: if the elapsed service time of the class-2 customer in

service does not exceed a certain threshold s, and the remaining service time is at least larger than a time unit when some class-1 custom-ers arrive, then the service of the class-2 customer will be preempted by the class-1 arrivals from the next slot. However, once the elapsedservice time has exceeded s or the remaining service time has become less than one, further preemptions are prevented. In other words,class-1 customers arriving during the first s slots after the service of the class-2 customer has started have the privilege to interrupt theservice of the class-2 customer as long as there remains at least one more full time slot to be preempted.

The preempted service will be restarted at the next service attempt, when the system become empty of class-1 customers, in one ofthree different preemptive modes. The first mode is where the preempted service is resumed. This case will be called the discretionarypriority models with the PR mode. The second mode is where the preempted service is repeated with a different service time resampledfrom the same service time distribution. This case will be called the discretionary priority models with the PRD mode. The third mode iswhere the preempted service is repeated with the identical service time. This case will be called the discretionary priority models with thePRI mode. In this paper, all these three cases are analyzed in a unified manner.

3. Analysis of service time structure

Throughout this paper, we employ the following standard notation: aYj denotes the class-j increment during a random time interval Y,

i.e., the number of class-j customers who arrive at but do not depart from the system during Y. The corresponding PGFs are defined as

AYðz1; z2Þ ¼ E½zaY1

1 zaY

22 �, AY

1ðzÞ ¼ AYðz;1Þ, and AY2ðzÞ ¼ AYð1; zÞ, and the corresponding moments are also defined as aY

j ¼@AY ðz1 ;z2Þ

@zj

h iz1¼z2¼1

and

aYij ¼

@2AY ðz1 ;z2Þ@zi@zj

h iz1¼z2¼1

for i; j ¼ 1;2.

K. Kim, K.C. Chae / European Journal of Operational Research 200 (2010) 473–485 475

From the point of view of class-2 customers, three different random variables can be viewed as the effective service time of a class-2customer. First, the gross service time G of a class-2 customer is defined to be the total time spent on the class-2 customer by the server.Note that G may be different from S2 under a non-work-conserving preemptive mode such as PRD and PRI. Secondly, the completion time Cof a class-2 customer is defined to be the time interval from the first service attempt of the class-2 customer until the service of the cus-tomer is completed. Lastly, the occupation time R of a class-2 customer is defined to be C plus the time interval (if any) until the server isfree to accept the next class-2 customer for service.

Under the discretionary priority disciplines, G can be further divided into two categories: one is when the class-2 customer is beingserved under the preemptible mode, which is denoted by GP , and the other is when the customer is being served under the nonpreemptiblemode, denoted by GN . In Fig. 1, the typical structures of GP , GN , C, and R of a class-2 customer are illustrated when S2 ¼ 9, s ¼ 5, and thepreempted service is repeated with the identical service time at the next service attempt after each preemption occurs. Note that allthe three discretionary priority models with different preemptive modes share this common service time structure.

According to the standard notation, aGj , aC

j , and aRj represent the class-j increments during G, C, and R, respectively. The corresponding

PGFs and moments are also denoted following the standard notation. For brevity, let aPj and aN

j , instead of aGPj and aGN

j , denote the class-jincrements during GP and GN , respectively. The corresponding PGFs and moments also follow the standard notation rule with the super-scripts P and N instead of GP and GN . In addition, we define the joint PGF of aP

j and aNj as AP;Nðz1; z2; w1;w2Þ ¼ E½zaP

11 z

aP2

2 waN

11 w

aN2

2 �. Clearly,AGðz1; z2Þ ¼ AP;Nðz1; z2; z1; z2Þ.

To analyze service interruption periods by class-1 customers in the occupation time R (see Fig. 1), we define H as the class-1 busy periodstarting with one class-1 customer. That is, the H period starts with a class-1 customer and ends when the system becomes empty of class-1 customers. Observe that a H period consists of the service time of the first class-1 customer in the H period and all H periods generatedby customers that arrive during the first service time. Thus, we have

aH2 ¼ aS1

2 þ aH2 ð1Þ þ � � � þ aH

2 ðaS11 Þ;

where faH2 ðnÞ; n ¼ 1; . . . ; aS1

1 g have the same distribution as aH2 . Transforming the above equation into PGF form gives

AH2 ðzÞ ¼ B1ðKðAH

2 ðzÞ; zÞÞ: ð1Þ

Using the above results, ACðz1; z2Þ and AR2ðzÞ are derived as follows. Observe that each class-1 customer who arrives during GP will be served

before the service of the class-2 customer is completed and generate their own H periods during the completion time C. However, class-1customers who arrive during GN will not be served during C. See Fig. 1. Therefore, we have the following relationships.

aC1 ¼ aN

1 ;

aC2 ¼ aG

2 þ aH2 ð1Þ þ � � � þ aH

2 ðaP1Þ:

Thus, it follows that

ACðz1; z2Þ ¼ AP;NðAH2 ðz2Þ; z2; z1; z2Þ: ð2Þ

Similarly, each class-1 customer who arrives during G will generate their own H periods during R. See Fig. 1. Therefore, we have the followingrelationship

AR2ðzÞ ¼ AGðAH

2 ðzÞ; zÞ: ð3Þ

Differentiating (1)–(3) we have

aC1 ¼ aN

1 ; ð4Þ

aR2 ¼ aG

2 þ aG1

k2l1

1� q1

� �: ð5Þ

Note that (1)–(3) hold for all the three discretionary priority models regardless of the preemptive mode. Thus, as long as we can calculateAP;Nðz1; z2; w1;w2Þ for each preemptive mode, we can obtain ACðz1; z2Þ and AR

2ðzÞ from (1)–(3) in a straightforward manner. ACðz1; z2Þ and AR2ðzÞ

will be the main components for the common results on the PGFs of the queue-length distributions of each class in the three discretionarypriority models.

The service of a class-2 customer Class-1

iClass-1

t

Class-1customers The service of a

class 2 customerThe server is free for the next class

is started customers arrive customersarrive

arrive class-2 customeris completed

for the next class-2 customer

Class-1 customers’ ’sremotsuc1-ssalC’sremotsuc1-ssalCbusy period busy period busy period

Fig. 1. Service time structure of a class-2 customer.


4. Queue-length distributions

Throughout this paper, the queue-length of a particular class means the number of all customers of that particular class present in thesystem, including the one in service (if any).

4.1. Class-1 queue-length distribution at departure epochs

Clearly, some class-1 customers will arrive while the server is working for other class-1 customers; hence, those class-1 customers willarrive during a class-1 busy period. Other class-1 customers will arrive while the server is idle or is working for class-2 customers; hence,those class-1 customers will initiate class-1 busy periods. Therefore, class-1 busy periods can be grouped into two categories. If class-1customers arrive during a slot while the server is idle or while the service of a class-2 customer is in GP , those class-1 customers will imme-diately initiate a class-1 busy period starting from the next slot. We will call such a class-1 busy period a ‘‘type I busy period”. On the otherhand, if class-1 customers arrive during a slot while the service of a class-2 customer is in GN , those class-1 customers will wait until theservice of the class-2 customer is completed before, together with all other class-1 customers who arrive during the GN period, they willinitiate a class-1 busy period. We will call such a class-1 busy period a ‘‘type II busy period”.

We first consider a set of class-1 customer departure epochs during type I busy periods and constitute an embedded Markov chain atthose embedded points. If we let MI

1ðnÞ denote the number of class-1 customers just after the nth embedded point, then, noting that a type Ibusy period starts whenever at least one class-1 customer arrives at any time slot in an idle period or a GP period, it is easy to see the fol-lowing relationship

MI1ðnþ 1Þ ¼

MI1ðnÞ � 1þ aS1

1 if MI1ðnÞ > 0;

ða1 j a1 > 0Þ � 1þ aS11 otherwise:

(

Thus, if we let PI1ðzÞ ¼ limn!1E½zMI

1ðnÞ�, then it follows that

PI1ðzÞ ¼

PI1ðzÞ �PI

1ð0Þz

� B1ðK1ðzÞÞ þPI

1ð0Þz�K1ðzÞ �K1ð0Þ

1�K1ð0Þ� B1ðK1ðzÞÞ:

Solving the above equation for PI1ðzÞ, together with the normalization condition, PI

1ð1Þ ¼ 1, we have

PI1ðzÞ ¼

1�K1ðzÞk1ð1� zÞ �

ð1� q1Þð1� z1ÞB1ðK1ðzÞÞB1ðK1ðzÞÞ � z

: ð6Þ

We next consider a set of class-1 customer departure epochs during type II busy periods and constitute an embedded Markov chain at thoseembedded points. If we let MII

1ðnÞ denote the number of class-1 customers just after the nth embedded point, then, noting that a type II busyperiod starts when the completion time of a class-2 customer ends with more than one class-1 customer present, it is easy to see the fol-lowing relationship

MII1ðnþ 1Þ ¼

MII1ðnÞ � 1þ aS1

1 if MII1ðnÞ > 0;

ðaC1 j aC

1 > 0Þ � 1þ aS11 otherwise:

(

Thus, if we let PII1ðzÞ ¼ limn!1E½zMII

1 ðnÞ�, then using the same method to arrive at (6) gives

PII1ðzÞ ¼

1� AC1ðzÞ

aC1ð1� zÞ �

ð1� q1Þð1� z1ÞB1ðK1ðzÞÞB1ðK1ðzÞÞ � z

: ð7Þ

Let pI and pII denote the probabilities that a random class-1 customer departs from the system during a type I busy period and during a typeII busy period, respectively. Clearly, the departure rate of class-1 customers is identical to the arrival rate k1. On the other hand, class-2 cus-tomers arrive at the system with rate k2, and at the end of service to a class-2 customer aC

1 class-1 customers will be present and initiate atype II busy period if aC

1 > 0. Each class-1 customer initiating a type II busy period generates their own independent and identical H periods.The expected number of class-1 customers served during the H period is given by 1=ð1� q1Þ (see [14, Chapter 6]). Consequently, the rate atwhich class-1 customers depart from the system during type II busy periods is given by k2aC

1=ð1� q1Þ. Therefore, we have

pII

pI þ pII ¼k2aC

1

k1ð1� q1Þ:

From pI þ pII ¼ 1, we have

pI ¼ k1ð1� q1Þ � k2aC1

k1ð1� q1Þ; ð8Þ

pII ¼ k2aC1

k1ð1� q1Þ: ð9Þ

If we let P1ðzÞ be the PGF of the queue-length of class-1 customers immediately after a class-1 customer departs from the system, then, from(6)–(9), we finally have

P1ðzÞ ¼ pI �PI1ðzÞ þ pII �PII

1ðzÞ ¼1� q1 �

k2aC1

k1

� �ð1�K1ðzÞÞ þ k2ð1� AC

1ðzÞÞh i

B1ðK1ðzÞÞk1ðB1ðK1ðzÞÞ � zÞ : ð10Þ


4.2. Class-2 queue-length distribution at departure epochs

The general cycle, defined as the duration between two consecutive idle period starting points, can be divided into three parts. The firstof these parts is the idle period, denoted by I, which is defined as the period when the server is in the idle state. The second part of thegeneral cycle is the T1 period, which is defined as the time duration from the end of the idle period until the system is first empty ofclass-1 customers. Note that if no class-1 customers arrive during the idle period, the length of T1 will be zero. The last part of the generalcycle is the T2 period, which is defined as the time duration from the end of the T1 period until the system is empty. Note that if no class-2customers arrive at the system during either the idle or the T1 periods, the length of T2 will be zero. Observe that the T1 period consists onlyof service times of class-1 customers and that the T2 period consists only of occupation times of class-2 customers (see Fig. 2).

Let kð1Þj and kð2Þj denote the number of class-j customers at the beginning of the T1 and T2 periods, respectively. If we letKð1Þðz1; z2Þ ¼ E½zkð1Þ1

1 zkð1Þ22 �, then it is easy to see that

Kð1Þðz1; z2Þ ¼Kðz1; z2Þ �Kð0;0Þ

1�Kð0; 0Þ : ð11Þ

because during the last slot of the idle period at least one customer must arrive at the system. Since the T1 period is a class-1 busy periodstarting with kð1Þ1 class-1 customers, the T1 period is equivalent to successive kð1Þ1 H periods. Thus, if we let Kð2Þ2 ðzÞ ¼ E½zkð2Þ2 �, if follows that

Kð2Þ2 ðzÞ ¼ Kð1ÞðAH2 ðzÞ; zÞ ¼

KðAH2 ðzÞ; zÞ �Kð0;0Þ1�Kð0; 0Þ : ð12Þ

We now consider a set of time points when the occupation times of class-2 customers are terminated, and constitute an embedded Markovchain at those embedded points. If we let Mt

2ðnÞ denote the number of class-2 customers just after the nth embedded point, then, noting thatall embedded points are in T2 periods consisting only of R’s and that a T2 period will be started when kð2Þ2 > 0, it is easy to see the followingrelationship.

Mt2ðnþ 1Þ ¼

Mt2ðnÞ � 1þ aR

2 if Mt2ðnÞ > 0;

kð2Þ2 jkð2Þ2 > 0

� �� 1þ aR

2 otherwise:

8<:

Thus, if we let Pt2ðzÞ ¼ limn!1E½zMt

2ðnÞ�, then using the same method to arrive at (6), we have from (12)

Pt2ðzÞ ¼

1�KðAH2 ðzÞ; zÞ

k2ð1� zÞ=ð1� q1Þ� ð1� aR

2Þð1� zÞAR2ðzÞ

AR2ðzÞ � z

: ð13Þ

Recalling that T2 consists only of R’s, let Ps2ðzÞ denote the PGF for the number of class-2 customers immediately after an occupation time R

begins, and P2ðzÞ denote the PGF for the number of class-2 customers just after a class-2 customer departs from the system. Since duringeach occupation time R the number of class-2 customers increases by aR

2 and decreases by one, we have the relationship

PtðzÞ ¼ PsðzÞAR2ðzÞz

: ð14Þ

Similarly, we have the relationship

P2ðzÞ ¼ PsðzÞAC2ðzÞz

: ð15Þ

It finally follows from (13)–(15) that

P2ðzÞ ¼1�KðAH

2 ðzÞ; zÞk2ð1� zÞ=ð1� q1Þ

� ð1� aR2Þð1� zÞAC

2ðzÞAR

2ðzÞ � z: ð16Þ

4.3. Queue-length distributions of each class at an arbitrary time

Let UjðnÞ, j ¼ 1;2, n ¼ 1;2; . . ., denote the number of class-j customers at time slot n. Specifically, UjðnÞ is the number of class-j customersin time interval ððn� 1Þþ;n�Þ because there are no changes in the numbers of customers of each class during any intervals ððn� 1Þþ;n�Þ. Ifwe let PjðzÞ ¼ limn!1E½zUjðnÞ�, then it follows from the argument in Takagi [14, p. 22–23] (which is based on Bruneel and Kim [3]) that

PjðzÞ ¼ PjðzÞ �1�KjðzÞkjð1� zÞ : ð17Þ

K(1) K(2)

S1 S1 S1 R R RR

I T TI T1 T2

General Cycle

Fig. 2. General cycle structure.


From (10), (17) and (4), we have

P1ðzÞ ¼1� q1 �

k2aN1

k1

� �þ k2

1�AC1 ðzÞ

1�K1ðzÞ

h ið1� zÞB1ðK1ðzÞÞ

B1ðK1ðzÞÞ � z: ð18Þ

Notice that, since there are no customer departures during G, aGj ¼ kjE½G� from Wald’s equation (see [22, Section 2–13]). Thus we have

aG1=aG

2 ¼ k1=k2. Therefore, from (16), (17) and (5), we have

P2ðzÞ ¼1�KðAH

2 ðzÞ; zÞ1�K2ðzÞ

� ð1� q1 � aG2Þð1� zÞAC

2ðzÞAR

2ðzÞ � z: ð19Þ

Notice that Eqs. (18) and (19) hold for all the three discretionary priority models because they share the same occupation time structure andgeneral cycle structure (see Figs. 1 and 2). In the next section, we show that the PGFs of queue lengths of each class in the three discretionarypriority models with different preemptive modes can be obtained in a straightforward manner by using the presented general results.

5. Three discretionary priority disciplines

5.1. Discretionary priority discipline with the Preemptive Resume mode

We now consider the case of the discretionary priority discipline with the PR mode. Under this discipline, each time the service of aclass-2 customer is preempted, the service will be resumed at the next service attempt.

Throughout this paper, we employ the following standard notation: If X is a random variable and A is an event, thenE½XfAg� ¼ Pr½A� � E½X j A�. Also, we let g ¼ z

aP1

1 zaP

22 w

aN1

1 waN

22 . Thus, E½g� ¼ AP;Nðz1; z2; w1;w2Þ. To derive AP;Nðz1; z2; w1;w2Þ, the following two cases

are considered.

(1) If S2 ¼ l 6 s:

E½gfS2 ¼ l 6 sg� ¼ b2ðlÞKðz1; z2Þl�1Kðw1;w2Þ;

because class-1 arrivals during the first l� 1 slots after the service of a class-2 customer is started can preempt the service of the class-2customer, while class-1 arrivals during the last slot do not preempt it.

(2) If S2 ¼ l > s:

E½gfS2 ¼ l > sg� ¼ b2ðlÞKðz1; z2ÞsKðw1;w2Þl�s;

because only class-1 customers who arrive before the elapsed service time exceeds the threshold s can preempt the service of the class-2customer.

Thus, we have

AP;Nðz1; z2; w1;w2Þ ¼Xs

l¼1

E½gfS2 ¼ l 6 sg� þX1

l¼sþ1

E½gfS2 ¼ l > sg�

¼Xs

l¼1

b2ðlÞKðz1; z2Þl�1Kðw1;w2Þ þX1

l¼sþ1

b2ðlÞKðz1; z2ÞsKðw1;w2Þl�s ð20Þ

and

AGðz1; z2Þ ¼ AP;Nðz1; z2; z1; z2Þ ¼Xs

l¼1

b2ðlÞKðz1; z2Þl þX1

l¼sþ1

b2ðlÞKðz1; z2Þl ¼ B2ðKðz1; z2ÞÞ: ð21Þ

Differentiating (20) and (21) and letting z1 ¼ z2 ¼ w1 ¼ w2 ¼ 1, we have

aNj ¼ kj 1þ

X1m¼sþ2

B2ðmÞ !

; ð22Þ

aGj ¼ kjl2: ð23Þ

To this point, we have derived the specific results related to the discretionary priority model with the PR mode. We will combine these spe-cific results with the general results obtained in Sections 3 and 4. We first derive the PGFs of the increments of C and R. From (2), (3), (20) and(21), we can obtain

ACðz1; z2Þ ¼Xs

l¼1

b2ðlÞKðAH2 ðz2Þ; z2Þl�1Kðz1; z2Þ þ

X1l¼sþ1

b2ðlÞKðAH2 ðz2Þ; z2ÞsKðz1; z2Þl�s ð24Þ

and

ARðzÞ ¼ B2ðKðAH2 ðzÞ; zÞÞ: ð25Þ

We next derive the PGFs of queue-length of each class in the discretionary priority model with the PR mode. From (18), (24) and (22), wehave


P1ðzÞ ¼ð1� zÞB1ðK1ðzÞÞ

B1ðK1ðzÞÞ � z

26641� q1 � k2 1þ

X1

m¼sþ2B2ðmÞ

� �:þk2

1�Ps

l¼1b2ðlÞK1ðzÞ �P1

l¼1þsb2ðlÞK1ðzÞl�s

1�K1ðzÞ

0BB@

1CCA3775: ð26Þ

From (19), (24), (25) and (23), we have

P2ðzÞ ¼1�KðAH


� ð1� qTÞð1� zÞB2ðKðAH

2 ðzÞ; zÞÞ � zK2ðzÞ

Xs

l¼1

b2ðlÞKðAH2 ðzÞ; zÞ

l�1 þKðAH2 ðzÞ; zÞ

s X1l¼sþ1

b2ðlÞK2ðzÞl�s" #

: ð27Þ

If we let L1 ¼ P01ð1Þ and L2 ¼ P02ð1Þ, then differentiating (26) and (27) and letting z ¼ 1 yields

L1 ¼k2

1l11 þ k11l1

2ð1� q1Þþ q1 þ

k1k2P1

l¼sþ2b2ðlÞðl� sÞðl� s� 1Þ2ð1� q1Þ

; ð28Þ

L2 ¼k2ðk1l11 þ k11l2

1Þ2ð1� q1Þð1� qTÞ

þ k22l22

2ð1� q1Þð1� qTÞþ k12l1

1� qTþ k22l2

2ð1� qTÞþ q2 þ

k2q1

1� q1

Xsþ1

i¼2

B2ðiÞ: ð29Þ

The results of (26)–(29) include the corresponding results under the NP and PR priority disciplines as their special cases. If we let s ¼ 0 in(26)–(29), then we can derive the PGFs and moments of queue lengths of each class under the NP discipline. By using Little’s law and simplecalculation, we can show that Eqs. (17) and (18) in [18] are identical to the results from letting s ¼ 0 in (28) and (29) here. On the other hand,if we let s!1 in (26)–(29), then we can derive the corresponding results under the PR discipline. Given that the first and second moments

of S2 is finite, it follows that, as s!1, the terms B2ðsþ 1Þ,P1

l¼1þsb2ðlÞK1ðzÞl�s, KðAH2 ðzÞ; zÞ

sP1l¼sþ1b2ðlÞK2ðzÞl�s, and

P1l¼sþ2b2ðlÞðl� sÞ

ðl� s� 1Þ in (26)–(29) all go to zero. Thus, by using Little’s law and simple calculation, we can show that Eqs. (13), (14), (24) and (25) in[21] are identical to the result from letting s ! 1 in (26), (27) and (29) here.

5.2. Discretionary priority discipline with the Preemptive Repeat Different mode

We now consider the case of the discretionary priority discipline with the PRD mode. Under this discipline, each time the service of aclass-2 customer is preempted, the service will be completely repeated at the next service attempt with a different service time resampledfrom the same service time distribution. As a result, in this model, the gross service time G of a class-2 customer depends on the arrivalprocesses during the gross service time, while, in the case of the PR mode, the gross service time does not depend on the arrival processes.In order to consider this dependency of gross service time and arrival processes, we define the random variable I1 to be the time lengthfrom the beginning of the first service attempt of the class-2 customer until any class-1 customers arrive. We suppose that the serviceof the class-2 customer is started at time k and that some class-1 customers arrive in ððkþ dÞ�; kþ dÞ, d P 1, given that no class-1 custom-ers arrive during fððkþ hÞ�; kþ hÞ; 1 6 h 6 d� 1g. In this case, I1 will be d�.

To derive AP;Nðz1; z2; w1;w2Þ, the following four cases are considered.

(1) If S2 ¼ l 6 s, l� 1 < I1: In this case, no class-1 customers arrive during the first l� 1 time slots after the service of the class-2 cus-tomer is started. However, some class-1 customers may arrive during the last slot in the service of the class-2 customer, namely, inððkþ lÞ�; kþ lÞ. Hence, it implies that no preemption occurs and the service to the class-2 customer is completed at the first serviceattempt. Thus,

E½gfS2 ¼ l 6 s; l� 1 < I1g� ¼ b2ðlÞKð0; z2Þl�1Kðw1;w2Þ:

(2) If S2 ¼ l > s, s < I1: In this case, no class-1 customers arrive during the first s time slots after the service of the class-2 customer isstarted. However, some class-1 customers may arrive during the last l� s slots, namely, in ððkþ hÞ�; kþ hÞ,sþ 1 6 h 6 l. Hence, itimplies that no preemption occurs and the service to the class-2 customer is completed at this first service attempt. Thus,

E½gfS2 ¼ l > s; s < I1g� ¼ b2ðlÞKð0; z2ÞsKðw1;w2Þl�s:

(3) If S2 ¼ l 6 s, l� 1 P I1 ¼ d: In this case, no class-1 customers arrive during the first d� 1 time slots after the service of a class-2 cus-tomer is started. However, some class-1 customers arrive during the dth time slot, namely, in ððkþ dÞ�; kþ dÞ, at which the elapsedservice time does not exceed s and the remaining service time of the class-2 customer is greater than a time unit. Hence, preemptionoccurs at the first service attempt, and the service of the class-2 customer is completely repeated with another service time resam-pled from the same i.i.d. service time distribution. Thus, E½gfS2 ¼ l 6 s; l� 1 P I1 ¼ dg� is given by

b2ðlÞKð0; z2Þd�1ðKðz1; z2Þ �Kð0; z2ÞÞAP;Nðz1; z2; w1;w2Þ

for 1 6 d < l 6 s. Summing the above result over 1 6 d 6 l� 1 yields

E½gfS2 ¼ l 6 s; l� 1 P I1g� ¼ b2ðlÞ1�Kð0; z2Þl�1

1�Kð0; z2ÞðKðz1; z2Þ �Kð0; z2ÞÞAP;Nðz1; z2; w1;w2Þ:

(4) If S2 ¼ l > s P I1 ¼ d: In this case, no class-1 customers arrive during the first d� 1 time slots after the service of the class-2 cus-tomer is started. However, some class-1 customers arrive during the dth time slot, namely, in ððkþ dÞ�; kþ dÞ, when the elapsedservice time of the class-2 customer does not exceed s and the remaining service time is more than a time unit. Hence, preemptionoccurs at the first service attempt. At the second service attempt, the service of the class-2 customer is completely repeated. Thus,E½gfS2 ¼ l > s P I1 ¼ dg� is given by


b2ðlÞKð0; z2Þd�1ðKðz1; z2Þ �Kð0; z2ÞÞAP;Nðz1; z2; w1;w2Þ

for 1 6 d 6 s < l. Summing the above result over 1 6 d 6 s yields

E½gfS2 ¼ l > s P I1g� ¼ b2ðlÞ1�Kð0; z2Þs

1�Kð0; z2ÞðKðz1; z2Þ �Kð0; z2ÞÞAP;Nðz1; z2; w1;w2Þ:

Combining all these four cases, we have


l¼1

fE½gfS2 ¼ l 6 s; l� 1 < I1g� þ E½gfS2 ¼ l 6 s; l� 1 P I1g�g

þX1

l¼sþ1

fE½gfS2 ¼ l > s; s < I1g� þ E½gfS2 ¼ l > s; s P I1g�g;

which yields

AP;Nðz1; z2; w1;w2Þ ¼Ps

l¼1b2ðlÞKð0; z2Þl�1Kðw1;w2Þ þKð0; z2ÞsP1

l¼sþ1b2ðlÞKðw1;w2Þl�s

1� Kðz1 ;z2Þ�Kð0;z2Þ1�Kð0;z2Þ

1�Ps

l¼1b2ðlÞKð0; z2Þl�1 �Kð0; z2ÞsB2ðsþ 1Þn o ð30Þ

and

AGðz1; z2Þ ¼ AP;Nðz1; z2; z1; z2Þ ¼Ps

l¼1b2ðlÞKð0; z2Þl�1Kðz1; z2Þ þKð0; z2ÞsP1

l¼sþ1b2ðlÞKðz1; z2Þl�s

1� Kðz1 ;z2Þ�Kð0;z2Þ1�Kð0;z2Þ

1�Ps

l¼1b2ðlÞKð0; z2Þl�1 �Kð0; z2ÞsB2ðsþ 1Þn o : ð31Þ

By the same method used for the discretionary priority model with the PR mode, we can get the PGFs and moments of queue lengths of eachclass in the discretionary priority model with the PRD mode.


B1ðK1ðzÞÞ � z1� q1 � k2E½GN � þ

k2

1�K1ðzÞ1�

Psl¼1b2ðlÞKð0;1Þl�1K1ðzÞ þKð0;1Þs

P1l¼sþ1b2ðlÞK1ðzÞl�s

Ps

!" #; ð32Þ

P2ðzÞ ¼1�KðAH


� ð1� qT;eff Þð1� zÞPs

l¼1b2ðlÞKð0; zÞl�1K2ðzÞ þKð0; zÞsP1

l¼sþ1b2ðlÞK2ðzÞl�s

1� KðAH2 ðzÞ;zÞ�Kð0;zÞ1�Kð0;zÞ 1�

Psl¼1b2ðlÞKð0; zÞl�1 �Kð0; zÞsB2ðsþ 1Þ

n o264

375

�Ps

l¼1b2ðlÞKð0; zÞl�1KðAH2 ðzÞ; zÞ þKð0; zÞs

P1l¼sþ1b2ðlÞKðAH

2 ðzÞ; zÞl�s

1� KðAH2 ðzÞ;zÞ�Kð0;zÞ1�Kð0;zÞ 1�

Psl¼1b2ðlÞKð0; zÞl�1 �Kð0; zÞsB2ðsþ 1Þ

n o � z

264

375�1

; ð33Þ

L1 ¼k2

1l11 þ k11l1


k1k2Kð0;1ÞsP1

l¼sþ2b2ðlÞðl� sÞðl� s� 1Þ2ð1� q1ÞPs

; ð34Þ

L2 ¼ q2;eff þk2ðk1l11 þ k11l2

1Þ2ð1� q1Þð1� qT;eff Þ

þk12l1 þ k2E½GP�qT;eff

1� qT;effþ k22E½G�

2ð1� qT;eff Þþ

k22Kð0;1Þ

sP1l¼sþ2b2ðlÞðl� sÞðl� s� 1Þ

2ð1� q1Þð1� qT;eff ÞPs

þ k2kð0Þ2 fsðE½GN� � 1Þð1�Kð0;1ÞÞ þ E½GP �Kð0;1Þg

ð1� qT;eff Þð1�Kð0;1ÞÞKð0;1Þ � k2kð0Þ2 fE½G�ð1�Kð0;1ÞÞ þKð0;1Þg

ð1� qT;eff Þ

�Ps

l¼1b2ðlÞðl� 1ÞKð0;1Þl�2 þ sKð0;1Þs�1B2ðsþ 1Þð1�Kð0;1ÞÞPs

; ð35Þ

where

Ps ¼ AP;Nð0;1; 1;1Þ ¼Xs

l¼1

b2ðlÞKð0;1Þl�1 þKð0;1ÞsB2ðsþ 1Þ;

E½GP � ¼1� Ps

ð1�Kð0;1ÞÞPs;

E½GN� ¼ 1þKð0;1Þs

P1m¼sþ2

B2ðmÞ

Ps;

E½G� ¼ E½GP� þ E½GN�;

q2;eff ¼ k2E½G�;qT;eff ¼ q1 þ q2;eff and kð0Þ2 ¼@Kð0; z2Þ@z2

jz2¼1:


The results of (32)–(35) include the corresponding results under the PRD and NP priority disciplines as their special cases. If we let s ¼ 0 in(32)–(35), then we can get the same results as those from letting s ¼ 0 in (26)–(29). On the other hand, if we let s ! 1 into (32)–(35), thenwe can get the PGFs and moments of queue lengths of each class under the PRD discipline. By simple calculation, we can show that Eqs. (17)–(19) in [20] are identical to the results from letting s ! 1 in (32)–(35) here.

5.3. Discretionary priority discipline with the preemptive repeat identical mode

We now consider the case of the discretionary priority discipline with the PRI mode. Under this discipline, each time the service of aclass-2 customer is preempted, the service will be completely repeated at the next service attempt with the same service time. As inthe PRD mode, in the PRI mode the gross service time of a class-2 customer also depends on the arrival processes during the gross servicetime. Hence, using arguments similar to those in the PRD mode, we can derive AP;Nðz1; z2; w1;w2Þ.

(1) If S2 ¼ l 6 s, l� 1 < I1: Using the same argument as in the case 1 in the PRD mode, we have

E½gfS2 ¼ l 6 s; l� 1 < I1g� ¼ b2ðlÞKð0; z2Þl�1Kðw1;w2Þ:

(2) If S2 ¼ l > s, s < I1: Using the same argument as in the case 2 in the PRD mode, we have

E½gfS2 ¼ l > s; s < I1g� ¼ b2ðlÞKð0; z2ÞsKðw1;w2Þl�s:

(3) If S2 ¼ l 6 s, l� 1 P I1 ¼ d: All the arguments in case 3 of the PRD mode hold in the PRI mode except that the service of the class-2customer is completely repeated with the same service time, rather than a different service time resampled from the same distri-bution, at the second service attempt. Hence, instead of AP;Nðz1; z2; w1;w2Þ, E½g j S2 ¼ l 6 s� represents the effect of repeating here.Thus,

E½gfS2 ¼ l 6 s; l� 1 P I1g� ¼ b2ðlÞ1�Kð0; z2Þl�1

1�Kð0; z2ÞðKðz1; z2Þ �Kð0; z2ÞÞE½gjS2 ¼ l 6 s�:

(4) If S2 ¼ l > s P I1 ¼ d: All the arguments in case 4 of the PRD mode hold in the PRI mode except that the service of the class-2 cus-tomer is completely repeated with the same service time. Hence, instead of AP;Nðz1; z2; w1;w2Þ, E½g j S2 ¼ l > s� represents the effect ofrepeating here. Thus,

E½gfS2 ¼ l > s P I1g� ¼ b2ðlÞ1�Kð0; z2Þs

1�Kð0; z2ÞðKðz1; z2Þ �Kð0; z2ÞÞE½gjS2 ¼ l > s�:

Combining cases 1 and 3, we have

E½gfS2 ¼ l 6 sg� ¼ E½gfS2 ¼ l 6 s; l� 1 < I1g� þ E½gfS2 ¼ l 6 s; l� 1 P I1g�

which yields

E½gfS2 ¼ l 6 sg� ¼ ð1�Kð0; z2ÞÞb2ðlÞKð0; z2Þl�1Kðw1;w2Þ1�Kðz1; z2Þ þKð0; z2Þl�1ðKðz1; z2Þ �Kð0; z2ÞÞ

:

Similarly, combining cases 2 and 4, we have

E½gfS2 ¼ l > sg� ¼ ð1�Kð0; z2ÞÞKð0; z2Þsb2ðlÞKðw1;w2Þl�s

1�Kðz1; z2Þ þKð0; z2ÞsðKðz1; z2Þ �Kð0; z2ÞÞ:

Combining all the cases, we have


l¼1

E½gfS2 ¼ l 6 sg� þX1

l¼sþ1

E½gfS2 ¼ l > sg�

¼Xs

l¼1

ð1�Kð0; z2ÞÞKðw1;w2Þb2ðlÞKð0; z2Þl�1

1�Kðz1; z2Þ þKð0; z2Þl�1ðKðz1; z2Þ �Kð0; z2ÞÞþð1�Kð0; z2ÞÞKð0; z2Þs

P1l¼sþ1

b2ðlÞKðw1;w2Þl�s

1�Kðz1; z2Þ þKð0; z2ÞsðKðz1; z2Þ �Kð0; z2ÞÞ; ð36Þ

which yields

AGðz1; z2Þ ¼ AP;Nðz1; z2; z1; z2Þ ¼Xs

l¼1

ð1�Kð0; z2ÞÞKðz1; z2Þb2ðlÞKð0; z2Þl�1

1�Kðz1; z2Þ þKð0; z2Þl�1ðKðz1; z2Þ �Kð0; z2ÞÞþð1�Kð0; z2ÞÞKð0; z2Þs

P1l¼sþ1

b2ðlÞKðz1; z2Þl�s

1�Kðz1; z2Þ þKð0; z2ÞsðKðz1; z2Þ �Kð0; z2ÞÞ:

ð37Þ

By the same method used for the discretionary priority model with the PR mode, we can get the PGFs and moments of queue lengths of eachclass in the discretionary priority model with the PRI mode.


B1ðK1ðzÞÞ � z

"1� q1 � k2E½GN�:þ

k2

1�K1ðzÞ1�

Xs

l¼1

b2ðlÞK1ðzÞ �X1

l¼sþ1

b2ðlÞK1ðzÞl�s

!#; ð38Þ


P2ðzÞ ¼1�KðAH


� ð1� qT;eff Þð1� zÞ �Xs

l¼1

ð1�Kð0; zÞÞK2ðzÞb2ðlÞKð0; zÞl�1

1�KðAH2 ðzÞ; zÞ þKð0; zÞl�1ðKðAH

2 ðzÞ; zÞ �Kð0; zÞÞ

"

þð1�Kð0; zÞÞKð0; zÞs

P1l¼sþ1b2ðlÞK2ðzÞl�s

1�KðAH2 ðzÞ; zÞ þKð0; zÞsðKðAH


#�

Xs

l¼1

ð1�Kð0; zÞÞKðAH2 ðzÞ; zÞb2ðlÞKð0; zÞl�1

1�KðAH2 ðzÞ; zÞ þKð0; zÞl�1ðKðAH


"

þ ð1�Kð0; zÞÞKð0; zÞsP1

l¼sþ1b2ðlÞKðAH2 ðzÞ; zÞ

l�s

1�KðAH2 ðzÞ; zÞ þKð0; zÞsðKðAH

2 ðzÞ; zÞ �Kð0; zÞÞ� z

#�1

; ð39Þ

L1 ¼k2

1l11 þ k11l1


k1k2P1

l¼sþ2b2ðlÞðl� sÞðl� s� 1Þ2ð1� q1Þ

; ð40Þ

L2 ¼ q2;eff þk2E½GP �q1

1� q1þ k22E½G�

2ð1� qT;eff Þþ k12l1

1� qT;effþ k2ðk1l11 þ k11l2

1Þ2ð1� q1Þð1� qT;eff Þ

þk2

2 2E½GP� þP1

l¼sþ2b2ðlÞðl� sÞðl� s� 1Þ� �

2ð1� q1Þð1� qT;eff Þ

þ k22ð1� E½GP� � E½G� þKð0;1Þ�sðE½GN � � 1ÞÞð1� q1Þð1� qT;eff Þð1�Kð0;1ÞÞ þ

k22

Psl¼1b2ðlÞKð0;1Þ2ð1�lÞ þKð0;1Þ�2sB2ðsþ 1Þ � 1

� �ð1� q1Þð1� qT;eff Þð1�Kð0;1ÞÞ2

þk2k

ð0Þ2 1þ E½GP� �

Psl¼1l � b2ðlÞKð0;1Þ1�l � ðsþ 1ÞKð0;1Þ�sB2ðsþ 1Þ

� �ð1� qT;eff Þð1�Kð0;1ÞÞKð0;1Þ ; ð41Þ

where

E½GP � ¼Ps

l¼1Kð0;1Þ1�l b2ðlÞ þ B2ð1þ sÞKð0;1Þ�s � 1

1�Kð0;1Þ ;

E½GN� ¼ 1þX1

m¼sþ2

B2ðmÞ;

E½G� ¼ E½GP� þ E½GN�;

q2;eff ¼ k2E½G�;qT;eff ¼ q1 þ q2;eff ; and kð0Þ2 ¼@Kð0; z2Þ@z2

jz2¼1:

The results of (38)–(41) include the PGFs and first moments of queue lengths of each class under the PRI and NP priority disciplines as theirspecial cases. If we let s ¼ 0 in (38)–(41), then we obtain the same results as those from letting s ¼ 0 in (26)–(29). On the other hand, if we lets!1 in (38)–(41), then we can obtain the PGFs and moments of queue lengths of each class under the PRI discipline. By simple calculation,we can show that Eqs. (17), (18), (33) and (34) in [17] are identical to the results from letting s ! 1 in (38)–(41) here.

6. Numerical example

In this section, a numerical example is presented. We apply the results obtained in Section 5 to the example of an N � N ouput-queueingpacket switch described in [17, Section 7]. In that example, ‘‘the packet arrivals on each inlet are assumed to be i.i.d., and generated by aBernoulli process with arrival rate kT . An arriving packet is assumed to be of class j with probability kj=kT (j ¼ 1;2) ðk1 þ k2 ¼ kTÞ. Theincoming packets are then routed to the output queue corresponding to their destination, in an independent and uniform way”. Then,‘‘the arrivals of both types of packets to an output queue are generated according to a two dimensional binomial process”. Its joint PGFis given by

Kðz1; z2Þ ¼ 1� k1

Nð1� z1Þ �

k2

Nð1� z2Þ

� �N

:

We assume that N ¼ 16. The service times of class-1 customers are assumed to be deterministically equal to 5 slots. The service times ofclass-2 customers are assumed to follow one of the following three different distributions: a geometric distribution with parameter 1/20,a negative binomial distribution with parameters 1/4 and 5, or a mixture of two different geometric distributions with parameters 1/50and 1/10. Specifically, for the geometric case, the p.m.f. of the class-2 service time is defined by

b2ðlÞ ¼1

20

� �1920

� �l�1

;

for l ¼ 1;2; . . . . For the negative binomial case, the p.m.f of the class-2 service time is defined by

b2ðlÞ ¼l� 1

4

� �14

� �5 34

� �l�5

;

for l ¼ 5;6; . . . . For the geometric mixture case, the p.m.f of the class-2 service time is defined by

b2ðlÞ ¼14

150

� �4950

� �l�1

þ 34

110

� �4

10

� �l�1

;


for l ¼ 1;2; . . . . While, for these three distributions, the class-2 service times have a common mean of 20, their variances are different. Spe-cifically, the coefficients of variation are 0.95, 0.19, and 2.45 for the geometric, negative binomial, and geometric mixture cases, respectively,Hence, the variability of the geometric mixture distribution is the highest of the three distributions, while that of the negative binomial dis-tribution is the lowest. Thus, in this numerical example we can see the influence of the variability of the class-2 service time on the perfor-mance measures, as well as the influence of the preemptive mode of the discretionary priority discipline. Also, k1 ¼ 0:05 and k2 ¼ 0:0125 areassumed so that q1 ¼ q2 ¼ 0:25.

In Fig. 3, the mean class-1 and class-2 queue lengths are shown as functions of the threshold s, when the discretionary discipline isbased on the PR mode. For all the three different distributions of the class-2 service times, as s increases, the mean class-1 queue-lengthdecreases, while the mean class-2 queue-length increases. These tendencies are obvious because, as s decreases, the discretionary prioritydiscipline becomes similar to the non-preemptive priority discipline, in which class-2 customers are treated better than in the PreemptiveResume discipline. As s increases, the discretionary priority discipline becomes similar to the PR priority discipline, in which class-1 cus-tomers are treated better than in the non-preemptive priority discipline. Also, for all the values of s, the higher the variability of the class-2service time is, the greater the mean queue lengths of both classes are in the PR-mode discretionary priority model.

In Fig. 4, the mean class-1 and class-2 queue lengths are shown as functions of the threshold s, when the discretionary discipline isbased on the PRD mode. As in the PR-mode case, for the geometric and negative binomial distributions, as s increases, the mean class-1 queue-length decreases, while the mean class-2 queue-length increases. However, for the geometric mixture distribution, as s increases,the mean class-1 queue-length decreases as in the other distributions, and the mean class-2 queue-length also decreases in contrast to theother two distributions. This can be explained by the following argument: in the geometric mixture case, a class-2 service time can be(re)sampled from either the geometric distribution having a relatively long mean of 50, with a lower probability of 1/4, or the geometricdistribution having a relatively short mean of 10, with a higher probability of 3/4. As a result, a long service time can be replaced with arelatively short service time by a preemption of class-1 customers, and this can positively affect the performance measure of class-2 cus-tomers. Due to the same reason, for an even larger value of s, the negative binomial case (with the lowest variability) produces the worstperformance, while the geometric mixture case (with the highest variability) produces the best performance with respect to the meanclass-2 queue-length, of the three distributions in the PRD-mode discretionary priority model.

In Fig. 5, the mean class-1 and class-2 queue lengths are shown as functions of the threshold s, when the discretionary discipline isbased on the PRI mode. As in the PR mode, for all the three different distributions of the class-2 service time, as s increases, the mean

Fig. 3. Queue lengths versus the threshold s in the PR-mode discretionary priority model.

Fig. 4. Queue lengths versus the threshold s in the PRD-mode discretionary priority model.

Fig. 5. Queue lengths versus the threshold s in the PRI-mode discretionary priority model.

Fig. 6. Comparison of the PR-, PRD-, and PRI-mode discretionary priorities with geometric class-2 service time.

Fig. 7. Comparison of the PR-, PRD-, and PRI-mode discretionary priorities with negative binomial class-2 service time.

Fig. 8. Comparison of the PR-, PRD-, and PRI-mode discretionary priorities with geometric mixture class-2 service time.


class-1 queue-length decreases, while the mean of class-2 queue lengths increases. However, in the PRI-mode discretionary priority model,as s increases, the mean class-2 queue-length increases dramatically, much faster than in the PR- and PRD-mode discretionary prioritymodels. In this model, for the case with the geometric mixture distribution, when a class-2 service with a relatively long service time ispreempted, it will be repeated with the same long service time at the next service attempt. Thus, there is no decreasing effect on theclass-2 queue-length in the geometric mixture case, as s increases.

In Fig. 6, the mean class-1 and class-2 queue lengths in the PR-, PRD-, and PRI-mode discretionary priority models are shown as func-tions of the threshold s, when the class-2 service time distributions are geometric. In Figs. 7 and 8, the negative binomial and geometricmixture cases are presented. For all the three distributions of the class-2 service time, the PRI-mode discretionary model produces theworst performance with respect to the mean class-2 queue-length. On the other hand, for the negative binomial distribution (with the low-est variability), the PR-mode discretionary discipline produces the best performance, while, for the geometric mixture distribution (withthe highest variability), the PRD-mode discipline performs the best with respect to the mean class-2 queue-length.

7. Conclusions

In this paper, the PGFs and first moments of queue lengths in the discrete-time queueing models with PR-, PRD-, and PRI-mode discre-tionary priorities were derived. The obtained results include the previous works on discrete-time priority queueing models with generalservice times and structured inputs as their special cases. As shown in the numerical example, using the derived results, various combi-nations of the preemption mode and the threshold of the discretionary priority discipline can be investigated in practice to allow subtlemanipulation of system performance. We expect that this may help system architects make better decisions on the priority disciplinesin practice. In addition to the practical contributions of this paper, in fact, a unified approach to the analysis of discrete-time priority queue-ing models of a quite general type was proposed in this paper, which could be interesting in terms of theory. The proposed unified approachsignificantly reduces the complexity of the analysis of the entire system to that of the gross service time, which makes it possible to dealwith different priority queueing models with general service time and structured inputs in a straightforward and unified manner. Althoughwe dealt with only three different discretionary priority queueing models in this paper, we expect that this approach can be extended tothe analysis of more sophisticated discrete-time priority queueing models, including other complicated variants of the discrete-time dis-cretionary priority queueing model.


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