Matrix Geometric Methods in Priority

Queues

Karuna Ramachandran

Department of Statistical

and Actuarial Sciences

Submitted in partial fdfilment

of the requirements for the degree of

Doctor of Philosophy

Faculty of Graduate Studies

The University of Western Ontario

London, Ontario

July 1997

@Karuna Ramachandran 1997

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Abstract

Most queues we encounter in everyday life follow a first-corne, first-served discipline

in which customers are served in order of arMval. Occasionally customers are

prioritized based on their needs (such as in hospital settings) or to achieve the

sys tem desired goals (such as corn puter and communication industry) . Priority

queues can be non-preemptive or preemptive. In the non-preemptive priority model

the highest priority customer present when the semer is free will be selected for

service and will be served to completion. On the other hand, under the preemptive

priority a later arriving high-pnority customer will displace the customer in service,

if the customer presently in service is of lower priority.

The main focus of the thesis is to model and analyze various problems in the

priority setup where some of the arrival streams are non-Poisson. We initially

attempted to determine an exact solution to the problem of average delay for pri-

ority queues having general input streams. Instead we ended up developing two

approximation methods, one each for the non-preernptive and the preemptive re-

sume prionty models. We then sought for numerical rnethods which could provide

exact resd ts.

Recently the matrk-analytic methods developed by Neuts has become a fre-

quent choice as a technique to mode1 and analyze complex queues. Similar to other

researchers studying non-Markovian arrival streams nre also used matrix-geometric

methods, a subset of matriu-analytic methods, to analyze the priori& queues in-

volving non-Markovian arrival streams. Ushg this approach me initially studied a

iii

PH + hl/ P H i I l priority queue. This problem provided various insights to mode1

other complex problems. As one extension we considered a control for the service

rate For a priority queue with a Markovian arriva1 process. In a different devel-

opment we modelled the MfPHI1 -t ./PHI1 tandem priority queue. Tandem

queues have useful application in the manufacturing area involving flow shops.

This thesis i s dedicated to my loving parents f o r al1 the sacrifices they

have made jar lhezr c h i l d m

Acknowledgement s

First and foremost, 1 would like to express my sincere gratitude to my supervisor,

Dr. David A. Stanford for suggesting the thesis topic, for his generous support,

patient and valuable guidance throughout my Ph.D term.

I'rn ever thankful to Prof. Chakravarthy for providing me the relevant papers in the

area of queueing control models which gave valuable ideas for further research. This

research contributed to chapter 5 in my thesis. 1 greatly appreciate his involvement

and collaboration in the development and analysis of the model. 1 would also like

to thank his wife Jayanthi for her hospitality during my visits to Flint, MI.

My sincere thanks to Dr. Grassmann for acting as the evternal examiner and Drs.

Bell, Provost and Yu for acting as interna1 examiners. Their suggestions improved

the quality of the thesis greatly.

1 would like to express my sincere thanks to al1 the professors in the department of

Statistical and Actuarial Sciences for their guidance. I'rn thankful to al1 my fellow

graduate students, staff members - Corinne Bender and Lisa Smith who made my

years at Western very memorable. 1 would like to thank Alicia Pleasance for her

help with Latex and Golarn Kibria for his help in the preparation of this thesis.

1 greatly appreciate the help 1 received from my friends - Anila, Sridhar, Rekha,

Venu and Dipa, who made my life easier when 1 was a part time student.

Finally, I'rn grateful to my husband Balaji for his constant urging and encourage-

ment wvhich was extrernely pivotal for completing the degree. 1 do not know if

1 wodd have completed my Ph.D without his support. I'rn also thankful to my

sister and brocher who have always been there for me.

Table of Contents

Certificate of Exmination ii

Abstract iii

Dedication v

Acknowledgements vi

Table of Contents vii

List of Figures x

List of Tables xi

1 Introduction 1

2 Mathematical Preliminaries and Background Material 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Xotation 9

. . . . . . . . . . . . . . . . . . . . 2.2 Estimation of Average Workload 11

. . . . . . . . . . . . . . . . . . 2.3 X Review of Priority Queue Delays 13

. . . . . . . . . . . . . . . . . . . . . . 2.4 X Useful Conservation Law 18

. . . . . . . . . . . . . . . . . . . 2.5 Numencal Results and Discussion 22

3 Approximations for Calculating Average Delay in Priority Queues

with General and Poisson Streams 29

. . . . . . . . . . . 3.1 Method 1: An Extension of Kleinrock's Method 30

3.2 kIethod 2: An Approximation Method Based on Completion Time

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis 34

. . . . . . . 3.2.1 Completion Tirne Analysis - An Approximation 35

. . . . . . . . . . . . . . . 3.3 Non-Preemptive Priority Results for PV* 38

. . . . . . . . . . . . . . . . . . . 3.4 Numerical Results and Discussion 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions 44

4 Modelling the PH + iLl /PH/l Priority Queue Using the Matrix-

Geometric Approach 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 46

. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Mode1 Description 48

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 SolutionProcedure 53

. . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Performance Measures 56

. . . . . . . . . . . . . . . . . . . 4.5 Numerical Results and Discussion 59

5 Analysis of the M A P / P H / l Priority Queue with Service Control 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction 72

. . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Mode1 Description 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Solution Procedure 80

. . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Performance Measures 85

. . . . . . . . . . . . . . . . . . . .5 .5 Xurnerical Results and Discussion 88

6 Modelling the Two Node Priority Tandem Queueing System using

the Matrix Geometric Method for M / P H / l + . / P H I 1

viii

6.1 The Mode1 Description . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2 Basic Structure of the R Matriv . . . . . . . . . . . . . . . . . . . . 110

6.3 Algorithms for Calculating the R Matrix and O ther Related Quantities i 12

6.4 Solution for S ( 0 ) and X(1) . . . . . . . . . . . . . . . . . . . . . . 121

7 Conclusions and Scope for Further Research

A Phase-Type Distribution

B Markoviaa Arriva1 Process (MAP)

C A Brief Review of the State Reduction Method

References

Vita

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Workfoad Process 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Plot 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tirne Plot 36

-4verage Waiting time:Balanced (1:l) E2 + hl Arrivals. Cornmon

. . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Service 69

Average Waiting time:UnBalanced (12) E2 + 1b.1 Arrivals. Common

. . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Service 70

Average Waiting time:BaIanced (2:l) Ez + 1I.I Arrivais. Common

. . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Service 71

. . . . . . . . . . . . . . . . . . . . Hysteretic Semer Control Mode1 75

. . . . . . . . . . . . . . . Plot of Throughput as a Function of O1 97

Plot of Percentage of HP Customers Lost as a Function of O1 . . . 98

. . . . . . . . . . . . . . . . . . . . . . . . . Tandem Priority Queue 100

List of Tables

Exact Flow Time for the Preemptive Resume Hz + M/G + M/1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Priority Queue 24

Exact Flow Time for the Preemptive Resurne E2 + kl/G + Ml1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Priority Queue 24

Average Waiting Times for Ad + Ek/M/l Non-Preemptive Queue . 25

Average Waiting Times for M + H2/M/1 Non-Preemptive Queue . 26

Average Waiting Times for iki + Ek/hf/l Preemptive Queue . . . . 27

Average Waiting Times for M + Hz/&l/l Preemptive Queue . . . . 28

Average Waiting Time Results in E2 + M/Mi / l NP Priority Model

. . . . . . . . . . . . . . . . . . . . . . Using Kleinrock's Approach 41

Average Waiting Time Results in E2 + ~C[/hf~/l NP Priority Model

. . . . . . . . . Using the Completion Time Approximation Method 42

Average Waiting Time Results in H2 + M/ik&/l NP Prîority Model

Using the Completion Time Approximation Method . . . . . . . . . 43

Non-Preemptive Average Waiting Times Ek + M .h ival S . . . . . . 63

Non-Preemp tive Average Waiting Times Ek + M vals . . . . . . 64

Xon-Preemptive Average Waiting Times H2 + M .!mi vals . . . . . 65

Non-Preemptive Average Waiting Times H2 + M Arrivals . . . . . 66

Preemptive Average Waiting Times Ek + M Arrivals . . . . . . . . 67

Preemptive Average Waiting Times Ek + M Arrivals . . . . . . . . 68

-4verage Vqaiting Times. X = 0.3. BI = = 4 . . . . . . . . . . . . . 92

Average Waiting Times. X = 0.6. = O2 = 4 . . . . . . . . . . . . . 93

Average Waiting Times. X = 0.9. = O2 = 4 . . . . . . . . . . . . . 94

Average Waiting Times. X = 0.6. O? = 2 . . . . . . . . . . . . . . . . 95

Average FVai ting Times. X = 0.6. & = 4 . . . . . . . . . . . . . . . . 95

Average Waiting Times. X = 0.6. O2 = 2 . . . . . . . . . . . . . . . . 96

Average Waiting Times. X = 0.6. = 4 . . . . . . . . . . . . . . . . 96

Chapter 1

Introduction

Waiting is inevitable and we encounter it in one forrn or another in different

walks of our life. Starting from day-to-day life - banks, grocery store, doctor

office, to various organizations - businesses of al1 types, government, military,

experience waiting. Many of these congestion situations have benented when

suitable queueing models were used to analyze their operation.

There are several contributing factors to any given type of congestion.

The key aspects used for rnodelling are an ivd and service distributions,

number of servers, servicing mechanism and the capacity of the queue. De-

pending on the circumstance the manner in which these types of jobs are

processed becomes very important. The most common way of service en-

countered in our everyday life is first corne, first served (FCFS). Some other

types of queue disciplines are last corne, first served, random service and

priority service,

The study of priority queueing systems started as early as 1950's and the

need for doing more research is growing since we frequently encounter these

types of queues in hospitals, the telephone industry, computer networks and

other areas. Priority queues can be broadly sub-divided into two types: the

non-preemptive (NP) priority queue (also know as "head of the line ") and

the preemptive (PR) priority queue. In the non-preemptive priority mode1

the highest priority customer present is selected when the server is free for

service and is served to completion. On the other hand, under PR priority

a later arriving high-priority (HP) custorner will displace the customer in

service, if the job/person being in service is of lower priority. When the

server is able to serve the displaced job/person, the job either continues from

the point of preemption (the so-called preemptive resume case), or it starts

again (this type of preemptive priority is called preemptive repeat priority).

The number of priority classes within a priority setup is always greater than

or equal to two ( 2 2 2), and a certain queueing discipline is observed within

each class, usually FCFS.

The bulk of the research done so far focuses on the case where al1 the

arrivai streams are Poisson [9] [17] [21] [24]. The most common performance

mesures obtained are the mean waiting t h e , and the distribution of waiting

times in terms of Laplace-Stieltjes transforms (LSTs) .

Recently there has been a spurt in the gowth of the telecommunica-

tion industrv, where the arriva1 patterns for the customer classes are now

more frequently non-Poisson than before, due to the explosion of types and

arnounts of traffic being served.

The mathematical analysis of a queue gets complicated when arrival pro-

cesses are general, since one no longer h a . a memoryless process. As a result

busy cycles (consisting of service times extended by high-priority busy pe-

riods) cannot be easily characterized by LSTs. The case in which we have

mixture of Poisson and general (lower-priority) arrival streams have been

studied since the 1970's [18] [31]. Even in those cases the higher classes were

Poisson and lower classes were general.

Very recently a few papers (21 [37] (381 have appeared to analyze priority

queues tvhere the arrival streams are non-Poisson. Most of the authors em-

ploy matrix-analytic methods introduced by Neuts (261. Matrix-geometric

approach, a numerical method is a very powerful technique for providing

exact and eEcient solutions for complicated problems. They also provide

a preferred alternative to stochastic simulation for the class of models with

rnatrix-geometric form. Stochastic simulation is a cumbersome effort requir-

ing considerable time to calculate just the confidence intervals which bound

the mean of performance statistics, and it is discouraging to note that the

mid-point of the confidence interval taken as the mean of the performance

is still an approximate value. On the other hand when the problerns be-

corne complicated, product form solutions and numerical methods may fail

to provide a tractable solution. In such instances simulation is a preferred

m o d e h g technique.

TLus, the initial focus of the thesis was to study queues priority having

non-Poisson arrival processes. We initially approached the problem featur-

ing high priority non-Markovian and low priority general arrival streams by

developing two approximation rnethods for determining average delays in

priority queues with two classes of customers with the higher class being

non-Poisson and the lower class Poisson. We present these two rnethods in

chapter 3. This chap ter demonstrated the importance of designing models

to give exact analytic solution.

Chapter 4 analyses queues with non-Markovian arrivals using rnatriv geo-

metric methods. Specifically, we analyze the PH + M/PHi / l priority model.

Here, PH denotes the phase type distribution and the PH +LM/ P H i / l queue

is t herefore a queue with phase distributed high priority arrivais and Poisson

low priority arrivals. Both the arrivals have class dependent phase distributed

service t ime distributions.

Priority queues 116th service control are very useful for optimizing any

given system. Usually optimization of a queueing system may be attempted

from either of two stand points, the first to favour the management, and

the second to favour the customers. klanagement is interested in the effi-

ciency of the system and would tend to operate the seMce mechanism a t its

full capacity, i.e. to cut d o m idle times. The customers, on the contrary

would Like to cut d o m waiting time in a queue. This increases customer

satisfaction, and thus, the customers are less lîkely to balk or renege from

the queue. Therefore, any optimization of a congestion necessitates a "sys-

tems approach" with the intrinsic complex evaluation and assessment of al1

consequences for each possible decision.

In chapter 5 ive focused on the optimization aspect by modelling the

priority queueing mode1 with server control. We also generalized the mode1

further by allowing arrival processes to be Markovian arriva1 processes which

contains exponent ial and phase type distribution as speciai cases.

Queueing network modelling is one such area where pnority setups have

immense application in the performance and prediction analysis of cornputer,

communication and manufacturing systems. The motivation to study queue-

ing networks started with the telephone industry. Jackson (191 was the first

to pursue this problern and it provided a significant insight to the analysis

of open queueing nehvorks. However, the types of networks for which exact

results are known is essentiaily limited and a class which in general does not

include systems with pnorities. To facilitate the analysis of these models sev-

eral software packages are available, one such popularly-knom package, QN.4

(Queueing netlvork analyzer) was developed by Whitt's [39]. Other s o h a r e

packages are: BEST/l, CADS, P.4NACEA and one based on HEFFES. hlost

of these software packages contain algonthms for Markov models which c m

be solved exactly. Exact solutions are available only under restrictive as-

sumptions, which typically do not hold in practice. Approximation methods

for queueing networks have thus been of practical interest.

Very Little work has been done so far to study networks even approxi-

mately arising in a priority environment. Recently Reiman and Simon [29]

analyzed a priority queueing network in heavy traffic with one bottleneck

station.

We attempt to address the topic of priority queueing networks by mod-

elling a priority tandem network using matrix analytic methods. We model

a M / P H / l -t . /PHI1 priority model in chapter 6 and the proposed rnethod

is exact. This chapter is a logical extension of chapter 4. To Our knowledge

nobody has attempted to study the tandem priority model using rnatrix

analytic methods. The model can also be easily extended to have general

arrivais. Tandem queues have important application in the study of flow

shops of operation in manufacturing industry. One such appiication is where

parts sequentially require work from severaI machines.

Before proceeding to the next chapter we state in this paragraph the main

topics addressed in the later chapters. We start the next chapter by intro-

ducing the notation, definitions and mostly revîenring the existing results in

M/G/l priority queues. In chapter 3 we provide two approximate methods of

studying a priority model accommodating two classes with the higher priority

class being non-Narkovian. Chapters 4 through 6 employ matriv geometric

approach for st udying priority queueing models mostly involving general in-

put streams. In particular, we study the PH + i . f / P H i / l priority mode1 in

chapter 4. The priority queue with hysteretic semer control for MAP/PH/ l

is discussed in chapter 5. In chapter 6 we analyze a tandem network priority

mode1 for ikt /PH/1 -t . /PHI1 models. Chapters 2 through 5 have numer-

ical examples at the end of each chapter. Since chapter 6 focused more on

the careful and rigorous rnodelling and since the performance measures can

be obtained by adopting a procedure similar to those used in chapter 4 and

chapter 5, we chose not to have a numerical section. Nevertheless, complete

algorithms are developed to compute the key measures required for calculat-

ing the performance measures of interest to us - average waiting times for

both cIasses of customers at nodes 1 and 2.

Chapter 2

Mat hemat ical Preliminaries and Background Material

As we have mentioned in the introduction our interest is to study queues

requiring priority service. In particuiar, we are interested in studying models

more complex than the Poisson arriva1 priority queues. The emphasis of this

chapter is to lay the ground work for the analysis of these types of priority

queues discussed in the lat er chapters.

Therefore, we begin this chapter by introducing some notation and defini-

tions which are used in various pIaces in the thesis. The notation was chosen

to conform as closely as possible to standard queueing textbooks.

In the two subsections followhg the notation we review the known results

for average delay in FCFS queues (section 2.2) and priority queues (section

2.3). In both cases, delay results are presented for queues subjected to both

Poisson and non-Poisson strearns. These results form the departure point for

our m r k on General-arrivd priority queues.

In section 2.4, we discuss a fundamental result in queueing theory called

the conservation law and we also introduce an extension of this law not

rnentioned in Gelenbe and Mitrani [16]. The conservation law essentially

states that iinder weak conditions to be defined, the total work a server

has to do is independent of the type of service discipline. This fact can be

exploited to determine the average delay for some priority queueing systems

if the delay results are available for a similar queue in which the priorities

are reversed.

Finally, we conciude chapter 2 with some numerical results to illustrate

the results of sections 2.3 and 2.4.

2.1 Notation

The standard mode1 we consider in this chapter is a single N-priority queue.

In this and subsequent chapters we make use of the following notations.

X i = class-i arriva1 rate, i=1 ,2, . . . ,N,

a: = second moment of class-i interarrival tirne distribution i=1,2, . . . ,N

Czi =squared coefficient of vaxiation of the intermival time distribution

i=1,2, . * * , N?

hi = average class-i service tirne, i=i,2 . . . ,N,

pi = 1 /hi = Class-i service rate, i=1,2, . . . , N,

hj2) = second moment of the class-i service tirne, i=1,2, . . . ,N7

C:i =squared coefficient of variation of the service time distribution

i=i,2, .. . ,Nt

pi = class-i occupancy, i=1,2, . . . ,N,

~k = ~ f = , pi , sum of occupancies up to class k,

p = CL, pi = total server occupancy,

F i = average waiting time (prior to first service attempt) of class-i, i=1,2,

... ,N,

CVi = average system time of class-i, i=1,2, . . . ,N,

N i = average number of jobs in the queue for class-i, i=1,2, . . . , N,

Ni = average number of jobs in the system for class i, i=1,2, . . . ,N,

Ci = completion time of class i interrupted by customers above class i,

Vs(t) = ivorkload process, Le., the total of al1 unfinished work in the system

at time t ,

v = average workload for ciass i,

Qx-(s) = Laplace - Stieltjes transform (LST) for the random vaxîable X.

( Y ~ ( s ) = LST of class-i interanival time distribution,i=l,&, . . . ,N

&(s) = LST of class-i service time distribution,i=1,2, . . . ,N

~ ( s ) =LST of class-i busy cycle distribution. i=1,2, . . . ,N.

The following acronyms are used to deno te certain types of distributions.

Gr = General and independent,

P H = Phase distribution,

MAP = Markovian arrival process,

H2 = Hyperexponential of order 2

Ek = Erlang of order k.

2.2 Estimation of Average Workload

Prier to reviewing available delay results for pnority queues in the next

section, we review here the available results for FCFS single-semer queues

with competing Poisson and general mival streams. This is usually called

the il1 + GI/Gi/l FCFS queue. We arbitrarily denote the Poisson and GI

arrival streams to be class 1 and class 2 respectively.

Ott [28] has s h o m that the unîînished virtual total workload for such a

queue c m be stochastically decomposed into two parts: one correspondhg

to the unfinished workload in the M/GI/l FCFS queue and the other to a

modified workload in a GI/G;/l queue, which will be descnbed shortly.

The LST of the virtual workload (semer's unfinished work) at GI arrival

instants (#F(s)) is shown by Ott to be:

where (P[:(s) is the LST of the waiting time distribution in the iCI/G1/l

queue with Poisson stream only and QC," (s) is the LST of the waiting time

distribution at arrival instants in a GI/G;/l queue (i.e.) the queue with the

general arrival stream and whose service tirnes consists of a GI-class service

followed by a delay busy period of Poisson-class arrivals

The expected waiting time seen by the Poisson class is precisely the virtual

workload, since Poisson arrivals see time averages. A similar equation to

(2.2.1) can be written for the saiting time at Poisson arrival instants, and

when it is differentiated ive obtain

where E {V&. } denotes the average workload in the GT/G;/l queue.

Thus. the ability to determine V is contingent upon the ability to deter-

mine E{VGI.}. For example for GI arrival processes with a rationd LST,

Cohen (101 in page 330 has shown that

where 6 is the unique root > 0, and assuming that we have a "h;" interarrival

tirne distribution, 6 satisfies,

where a1(s) is a polynomial of degree at most 1 and ~ ( s ) is a polynornial of

degree 2.

,D* and & are the first and the second moments of the G; service time dis-

tribution extended by the busy periods of Poisson arrivals. Similarly pz is

the server occupancy for the GI/G;/l queue extended by the busy perîod of

Poisson arrivals.

2.3 A Review of Priority Queue Delays

Waiting times for priority queues featuring solely Poisson amivals have been

thoroughly studied since the mies. Cobham [9] and Holley [l?] determined

the mean waiting time for each priority class in single-semer N P prîority

queues. The mean waiting time formula for class k, k = 1 , 2 , . . . , N, is given

âs:

r,Yi hi^) PV+ = for k = 1,. . . , N .

2(1 - ok-*)(l - ok)

Kesten and Runnenburg [21] determined the distribution of the NP prior-

ity waiting times through Laplace-Stieltjes transforms (LSTs). The LSTs for

the waiting time distributions for preemptive resume were first determined

by Miller [24]. Thorough treatments on priority M/G/l queues can be found

in Conway, Maxwell and Miller [Il], Jaiswal [20] and Takagi [36].

The case in which there is a mixture of Poisson (higher-priority) and gen-

eral (lower-priority) arriva1 streams has received attention since the 1970's.

Such a single-semer priority queueing mode1 has been studied by Hooke (181,

Schassberger (311 and Sumita [34]. Hooke [18] denved the virtual waiting

tirne LST for the low priority class in steady state under N P priority. Schas-

sberger [31] derived LST results for the waiting time distributions of both

hi& and loi-priori8 customers in steady-state for both PR and N P disci-

plines. Sumita [34] studied the average delays under NP and PR priorities

using conservation laws for the semer's workload. Schassberger's [31] re-

sults for the low-priority waiting time distribution have been generalized by

Fischer[l3] for the priority queue with an arbitrary nurnber of high-prionty

Poisson streams and a single low-priority general stream. The high-priority

waiting time result was generalized by Schmidt !30] for an arbitrary nurnber

of priority classes.

In this section we summarize the results that have been obtained for

queues with higher-priority Poisson arrival classes and one or more general-

arrival cIasses of lower priority to the Poisson séreams.

LVe assume that there are N priority cIasses served on a non-preemptive

priority basis. The first n < N of these Eeature independent Poisson arrival

processes a t rate Ai, i=l,O, . . . ,n. It is assumed that the remaining classes

each have a general renewal arrival process, independent of the others and

1 of the higher-priority Poisson-arriva1 classes, with mean inter-arriva1 time x, 2'=71+1,72+2, . . . ,N.

Let us first consider the non-preemptive priority discipline. Schmidt [30]

has established that not only the means but even the tvaiting time distribu-

tions as seen by the classes 1 to n are the same as if .4LL of the classes 1

to N featured Poisson arrivals. In other words, the Poisson-arriva1 classes 1

to n are insensitive to the shape of the inter-arriva1 distribution of classes

(ntl) to NT beyond their mean 5, i=n+l,n+2, . . . , N. A more recent proof

of that result can be f o u d in S t d o r d (321. The average waiting time for

such a class is

xbi ~ ~ h ! ~ ) r?l,, = for k = 1,. . . , n.

2(1 - cTk-l)(l - cTk)

This result was first established for the special case of n=l Poisson stream

and N=2 (Le. one general-amval strearn) by Schassberger, it is also estab-

lished by Sumita.

Schassberger was also able to determine the waiting time distribution (in

terms of its Laplace Stieltjes transform) of the Lower-priority general-arriva1

class.

Later Fischer [13] obtained a similar result, but for the general case of

n = N - 1 higher-priority Poisson classes and a single lowest-priority general

arrival class. Fischer established that:

where E{ W,, (GI/G8 / 1) } denotes the average delay in a single server queue

with only customers from class N, the general arrival class, and modified

service times. (The rnodified service time consists of a regular class-Nservice

time e-xtended by a busy period of service times of customers from the Poisson

classes.) For further details see Fischer [l3] or Stanford 1321.

Preemptive Resume

Consider identical customers, from the lowest priority class of a single

server priority queue under the N P and PR disciplines. Under both schemes,

an arriving lowest-priori- customer must wait for its initial service atternpt

until all work in the system has been served, as well as al1 later-aniving

higher-priority work untii there is none left. The PR and NP disciplines only

differ in the order in which this work is served. Consequently the PR waiting

time distribution of a lovvest priority customer is identical to that under NP

priority. We will refer to this fact in later chapters.

Therefore, (2.4) is valid for IVqN under PR priority as well. Since higher-

priority customer preempts lower-priority ones and are thus unaware of them

the waiting time of a Poisson arrival class k, k=1,2, . . . ,N-1 can be obtained

by comparing it with a NP priority queue with k classes. Therefore CVqk

under PR discipline is given by:

The system time for class-k is given below (see Conway, Maxwell and

Miller [Il]):

The PR result can be justified as follows: in the preemptive resume prior-

ity queue, since the low priority customers get preempted on the arrivai of an

higher-priorîty customer, their total system time equals the amount of time

they spend initially (before first entry into service), the interruptions caused

by the later arriving higher-priority customers and their rnean service time.

The sum of the service time and the interruptions is cornmonly called the

tenn above. completion time and its average is represented by the

Useful Conservation Law

The order in which the server chooses to work (i.e the discipline) has no

bearing on the total amount of work a server has left to do: the server disposes

of work a t rate one second per second. Thus if service to one class improves

under a particular discipline, it must necessarily worsen for a t l e s t one other

class. If the average delays can be calculated for al1 classes under some service

strategy S (for instance, first corne first served), this conservation law can

likely help to determine the average delays under other, more analytically

cornplicated arrangements.

Figure 2.1: Workload Process

The workload process in a single-server queue Vs(t) can be defhed as the

total at time t of al1 of the service times of dl waiting customers, plus the

residual service time of the customer in service under scheduling strategy

S (see Gelenbe and Mitrani [16], pg 173). In a work-conserving situation,

Vs(t) jumps upward a t arriva1 instants by the amount of work the arriving

customer brings into the system. At ail other times, Vs(t) decreases with

slope (-1) until Vs(t) equals O (see sample realization, Fig. 2.1). Letting

Vs = iirn : ( t )d t denote the equilibriurn average workload, the discussion 7400

above irnplies the following result of Gelenbe and Mitrani (see (161 pg. 174,

theorem 6.1):

Theorem 2.1 For any single-semer queueing system in equilzbrium there

exïsts a constant V , detemzned only by the parameters of the arriva1 and

required service times processes, such that

for al1 work-conseniing scheduling strategzes S .

Gelenbe and Mitrani then proceed to relate the total average workload

V to the individual average workload v R ) for custorner classes i=1, . . . ,N,

under scheduling strategy 23

Under the assurnption that the service times of dl classes are exponen-

tially distributed with mean k, i=l, . . . ,N, Gelenbe and Mitrani show that

For non-preemptive strategies with general service, they find

where rn/ii, z=1, . . . , N denotes the mean residual service time of the class i

customer:

Note that (2.7) and (2.8) are identical in the case of exponential service.

In fact, Gelenbe and Mitrani's rnethods can be used to establish one slight

generalization of (2.7) not mentioned in [16]. For the sake of completeness

we present the theorem and proof below: (From this proof, the specific proofs

required to establish (2.1) and (2.8) become evident.)

Theorem 2.2 Consider a work-consenring single seruer queue operating un-

der a preemptiue-resume scheduling strategy in which the highest pn'ority

class has generallv-distributed semice times whereas al1 other classes have

exponentially-distributed semice times. Then

Proof: For each of classes 2 through N, since service times are memoryless,

the average workload is the average number of customers in the system Ni

times the mean service time:

1

Little's law implies that Ni = XiPt/i so that

For the highest priority class, service is never preempted. The semer

spends pl of the time serving this class, and the tirne-averaged residual service

time when serving class 1 is yl, the average residual service time. On average,

there are nl class-1 customers in the queue, and each of these represents an

average workload of hl. Thus

Little's law applied to the high-priority queue says that nl = XI Wql, so

we obtain

Substituting appropriately into (2.6) one obtains (2.9). .4s expected (2.9)

reduces to (2.7' in the case of high-p~ority exponentially distributed seince

t imes.

We employ conservation law (2.8) for N=2 classes in chapter 3 and chap

ter 4 in order to determine the average delay of either the higher or lower

priority. We discuss the method of determination of V in the next section

(also see Fischer and Stanford [14]).

2.5 Numerical Result s and Discussion

In this section we focus on presenting numerical examples based on results

established in sections 2.3 and 2.4. InitiaIly we present resuIts for exact

system (flow) times in a Gl + i\l/Gi/l PR priority queue to illustrate our

new result, theorem 2.2. Four different scenarios were considered. In table

2.1 we present the results in which the HP arrival process is hyperexponential

(C2=6.25) and the LP is Poisson. In al1 the cases the mean service time is

assumed to be 1. The service time distributions are either hyperexponential

or Erlang-2 for HP class and exponential for the LP class. Tables 2.2 present

similar results, but the HP arrival process was characterized by an Erlang

distribution of order 2. We notice from table 2.1 and table 2.2 that the

type of service time distribution of the HP class not only influences the HP

Elow times but its impact is also felt by the LP customers. When we switch

from an Erlang to a hyperexponential service time distribution, the flow

time increases almost by 50% when the t r a c intensity is low and increases

roughly by 100% for moderate and heavy t r s c .

In order to illustrate some typical values for average delay for the 11.1 +

GI/Gi / l mode1 of section 2.3, we have included Tables 2.3 through 2.6. Ta-

bles 2.3and 2.4 present average waiting tirne results for non-preemptive prior-

ity in a queue featuring Poisson HP arrivals and a low priority GI stream. The

low priority IAT distributions we consider are hyperexponential (CZ2=2.25,

CZ2=4.25, Cz2=6.25), and Erlang-k (k = 2, 3, 5). The important result that

one observes is, whether the LP is a Erlang or hyperexponential, the average

delay observed by the HP is sarne, as we would except in light of (2.3.3). For

the sake of completeness we present in table 2.5 and table 2.6 the average

waiting time results for the preemptive priority queue.

Summary of Results

In Tables 2.1-2.2 the exact system time results for the preemptive resume

priority queue are presented. These results pertain to a non-Markovian HP

arriva1 process n i th general seMce time distribution. These results are in

light of Theorem 2.2, ivhich was a generalization of the conservation law to

accornodate generd service time distribution for the HP class in a preemptive

resume queue. It is important to note that these results are for average

flow times and NOT average waiting times. In facto there are NO universal

expressions available for W, in either the NP or PR queues when the HP

stream is non-Poisson. The desire to develop results in this area is the focus

of the next two chapters.

24

Table 2.1: Exact Flow Time for the Preemptive Resume H2 + M/G + M I 1 Priority Queue

H2 + M/E2 + !LT/l Priority queue

0.6

0.9

Table 2.2: Exact Flow Time for the Preemptive Resurne E2 + iW/G + M / 1 Priority Queue

E2 + 1W/ H2 + M/1 Priority queue

H2 + M / H 2 + M/1 Priority queue

X 1 / X 2 = 1 AL/& = 0.5 XI/& = 0.25

3.694 8.314

6.167 60.031

E2 + M/E2 + M/1 Priority Queue

2.567 5.679

3.694 35.690

1.853 4.100

2.375 22.743

Table 2.3: Average Waiting Times for M + Ek/iZf/l Non-Preemptive Queue

Table 2.4: Average Waiting Times for 1W + H 2 / M / 1 Non-Pmemptive Queue

Table 2.5: Average Waiting Times for M + Ek/hI/l Preemptive Queue

28

Table 2.6: Average Waiting Times for M + H2/h1/1 Preemptive Queue

Chapter 3

Approximations for Calculat ing Average Delay in Priority Queues with General and Poisson Streams

In the previous chapter we reviewed the elristing average waiting time results

in priority queues when al1 the arrival classes were Poisson or a t least the

higher-priority classes being Poisson. The determination of exact system

time for the PR priority queue with non-Markovian HP arrival process was

also presented. The determination of the waiting time from the system time

is complicated since the completion tirne for the G I + i21/Gi/l is hard to

characterize for such a queue.

Hence, our interest was to study the waiting times for queues in mhich

the highest arrival class is general. For this type of a pnority queue there

are no set of equations readily available for calculating the average waiting

times. Hence, when concise closed form solutions are extremely difficult

or intractable to obtain, an approximation is an alternate for studying a

particular model. Non-Markovian queues and networks are typical example

where approximations are employed [39] [7] [6]. In this chapter, we present

two approximation methods, one each for the NP and PR disciplines for

finding average delay in a GI + M/Gi/l priority queue.

3.1 Method 1: An Extension of Kleinrock's Method

Vie consider first the non-preemptive (NP) discipline. One method which

suggests itself to determine the average waiting times PVql and 6Vq2 of high

and low-priority customers is to develop a set of two linear equations involving

I.V& and IVq2 which can then be solved to identify explicit solutions for W,,

i=1,2.

Equation 2.2.3 provides us wïth one linear equation, namely:

W n g I.V& and I.Vq2. TO determine a second Linear equation linking PVflI and

IY,?: we follow the approach of Kleinrock [22] in pages 108-110. Basically,

ive divide PVq2 into three parts reflecting the average delays due to 1) the

customer in service upon the anival of a typical lower priority cudomer

(whom we cal1 the "tagged7' cudomer), 2) those customers already waiting

in the system, and 3) those custorners of higher priority who will arrive later

but be served ahead of the "tagged" customer.

Since lower priority arrivals follow a Poisson distribution, we can express

t hese t hree terms as:

where I.Vo is the average residual waiting time of the customer in service, the

terrn in the summation refers to the average waiting time of those customers

waiting to be served who arrived before the "tagged" customer, and Ml is

the average number of class 1 customers to arrive during a typical class 2

waiting time. Since lower class arrivals are Poisson, CVo is given by

Due to Little's law, iV~=XiCV,, 2=1,2, so that (3.1 2) reduces to

CVe therefore have an exact system of two equations (3.1.1) and (3.1.3) in

the two unknoms CVql and K2. So long a s an expression can be found for

Ml, the solution is exact and straightforward. Intuitively, one might suspect

t hat

This can be shown to be true for Poisson high-priority arrivals by work-

ing backwards with the well-known expressions for average waiting times in

non-preernptive priority M/G/1 queues. In what follows we provide an alter-

nate prooc due to S tanford [personal communication], ivhich also shows that

(3.1.4) is NOT in general true for non-Poisson high-priority arrivals.

Theorem 3.3 Consider a stable non-preemp tive two-priority queue with Pois-

son arriva1 processes for 60th classes. The average number of high-prion'ty

arrivals during the waiting tirne of a low pn'ority customer is given by

Proof: The duration of the low-priority average waiting tirne is independent

of the order in which this work is served (it consists of the service times of

those present in the system plus al1 later-arriving high priority custorners).

LVe choose to rearrange the order of service into dependent sub-intervals as

follows. First, we seme ail work present in the systern upon the arrival of the

low-priority customer (excluding t hat customer) . Let To denote the duration

of this interval.

During To, some number of high-priorie customers (possibly O) will ar-

rive: let their nurnber be denoted by Ni and let Ti be the time to serve

them. Continuing in this fashion, we let Tj denote the time to serve the Nj

customers who arrived during T,--I (see figure 3.1). Since the queue is stable,

Figure 3.1: Time Plot

N1 HP IV, HP Nj+L HP 1 arrivals 1 arrivals 1 . . . 1 amvals 1 . . . 1 To = time to 1 Tl = time to 1 1 T j = t i m e t o 1

serve work serve NI HP serve Nj HP already t here cus tomers customers

eventually the Tj's equal O with probability 1, and we can write

Now let ml denote the number of high-priority customers to arrive during

the waiting time of a typical lower priority customer. From figure 3.1 we see

that ml = Cgl iVj and since Ml = E{mi} we find

CVe now develop a relationship between Nj and Ti-l, j 2 1. Since the num-

ber of arrivals in non-overlapping time intervals from a Poisson process are

independent, nte can condition on = t to find

Removing the conditioning on T,-l we obtain

Finally, substituting (3.1.8) into (3.1.6) yields

which complet es the proof.

In contrast, the numbers of arrivals in non-overlapping intervals are NOS

in general independent for non-Poisson processes, so that we cannot establish

the conditional independence arnong the Nj's represented by (3.1.8). In these

cases an exact determination of Nj requires knowledge not only of but

also a11 previous LV,, i=l,. . . , j - 1. In these cases, (3.1.4) can still be used

as an approximation.

3.2 Method 2: An Approximation Method Based on Completion Time Analysis

In the previous section we demonstrated a simple approximation procedure

based on Kleinrock's rnethod to estimate average delays under NP discipline

for both classes of customers. -4s mentioned in chapter 2 (section il), the

preemptive resume results for CVq2 follow irnmediately from NP and vice

versa. In this section we propose to study the GI t M/G + M/1 priorîty

model by anaiyzing the preemptive resume model tirst, from which the .NP

results readily follow.

From Theorem 2.2, it

Since we are analyzing a

foUoms when N= 2

preemptive resurne model, Wql can be easily ob-

tained using GI/G/l FCFS results enabling us to calculate LV2 exactly- The

difficulty lies in calculating Wq2 since it is not merely the flow (system) time

less the mean service time, but the system time less the completion time,

Exact analysis of completion time is extremely difficult for this model,

since the high priority IAT distribution does not possess the rnernoryless

property. In the ne* subsection we present an approximation method to

analyxe the completion time of LP customer due to the interruptions by GI

high priority arrivals, which is the vital quant ity required for calculating the

average delay for both priorities.

3.2.1 Complet ion Time Analysis - An Approximation

Let the random variable X denote a typical completion time. Here we con-

sider the time to the arriva1 of the first class-1 customer to be the fonvard

recurrence tirne (i.e. residual life) of T, the general interarrival time.

The low p r i o ~ t y service time is assumed to be exponential, so we treat every

time the system goes empty of high priority customers as a renewai instant.

This is a t least true of high priori@ Poisson arrivals. The residual service

time is identicdy distrîbuted to the entire service time. Thus X can be

mathematicaily written as

Figure 3.2: Time Plot

T*= t H.P Busy Period - Start of ciass 2

serv1cc

Where T' = Residual Interarrival Time

-Yo = Service Time

%= High Priority busy Period.

One of two possible scenarios could occur: 1) the low priority service ends

before the arriva1 of a HP customer, in which case ,Y is merely the service

time of the LP customer ( S o ) or else 2) the arrivals of the HP preempts

the LP customer in service. In the latter scenario X consists of the residual

interarrival time T', the busy period of the HP customer and the residual

service time of the LP customer XI. We treat XI and X are identically

distributed.

Let X(s) = LST for the cornpletion time X.

Therefore,

P2 [1 - (PT- ( ~ 2 + s) ] X(S) = (-) (3.11) ~2 + S [1 - @ T - ( P ~ + s)T~(s)]

Equation (3.2.11) shows that X(s) can be ivritten as the product of a ex-

ponential (pz) LST and another term. Let us denote the other term by say,

which c m be viewed as interruptions caused by high-priority customers dur-

ing class-:! service tirnes. Hence 3.2.11 can be written as

P2 X(s) = (-)Y (s) (3.12) 112 + s

If high priorîty arrivals are not Poisson then the time until the next HP

arriva1 is not a tme fonvard recurrence time, and X and X' are not identically

distributed. In such cases, we can still use 3.2.12 as an approximation. For

exponential low priority semice, we analyze Y(s) which depends upon low

pnority service only through the rate p2. For non-Poisson input recall that

therefore,

Hence,

Differentiation of 3.2.12 in the usual manner Ieads to

The determination of E(Y) is s h o m below.

Evaluating the above equation at s=O, we obtain,

CVe can notice that E{Y} depends on the high-priority busy period, high-

priority interarrival time and the LST of the high-priority UT evaluated a t

service rate p2. Methods for calculating the expected busy period in selected

GI/G/1 queues can be obtained Erom Bertsimas et.al [4].

3.3 Non-Preemptive Priority Results for Wqi

The average waiting time before service for the low priority customer is same

under both NP and PR disciplines. The average delay of the high-priority

class can be obtained using the conservation law relationship:

In the next section ive present some numerical results based on the two

approximations presented in sections 3.2 and 3.3.

3.4 Numerical Results and Discussion

Table 3.1 contrasts average waiting time results for the approximation ("Sim-

ple") based on equations 3.1 .l, 3.1 -3, and 3.1.4 with confidence intervals

obtained frorn a simulation and results frorn the exact method ("Exact") de-

scribed in the next chapter. In the table "Lower CI" and "Upper CI" denote

the lower and upper ends of the 95% confidence interval from the simulation.

In al1 configurations, 10 runs of 1,000,000 customers were considered, leading

to very tight confidence intervals.

The system under consideration has an Erlang-2 high-priority interarrival

t ime distribution. Al1 seMce times are exponentially distributed with rnean

1. For occupancies of 0.3: 0.6 and 0.9, and for arrivais of the high and low

priority streams in proportion 1:1, 12, and 2 1 respectively, values of kVqI

and kVq2 are tabulated.

The approximation method generally provides better estimates for Wq2

than for CV&. This is understandable, as equation 3.1.1 is a weighted average,

and smaller percentage errors in the larger delay iVq2 lead to larger percentage

errors, generally speaking, in the smaller delay FV& Based on these values,

the method appears to be a fairly good approximation for Wq2. The errors

for Wq2 range from +0.5% to 4.5%, but are typically in the range 2-3%. In

contrast, the errors for kVql lie in the range -4.4% to -10.9%.

One can also observe from the table that the approximation method de-

41

Table 3.1: Average Waiting Time Results in E2+lL1/M/l N P Priority Mode1 Using Kleinrock's Approach

Simple

Lower CI

Upper CI

Exact

Simple

Lower CI

Upper CI

Exact

Simple

Lower CI

Upper CI

Exact

scribed in section 3.1 for C.V,l lies outside of the 95% confidence interval in

al1 9 configurations tested. The approximation for CVq2 lies outside of the CI

for al1 configurations with p=0.3 and p=0.6, but INSIDE the CI at the 0.9

occupancy level.

By using 3.1 -4, we are irnplicitly assuming that the H P a.rrîval process

is Poisson, even though it is not tme. The degree of error obtained for Wq2

42

Table 3.2: Average Waiting Tirne Results in E2+lW/Mi/1 NP Priority Mode1 Using the Completion Time Approximation Method

-4 pprox

Lower CI

Upper CI

Exact

Approx

Lower CI

Upper CI

Exact

Approx

Lower CI

Upper CI

Exact

will depend on the type of HP amval process. If HP arrivals are hyper-

exponential, the computed value will be lower than the actual value and the

reverse would be seen for Erlangian arrivals. We see this type of behaviour

since Erlangian arrivals are more regular than Poisson and hyper-exponential

exhibits bursty arriva1 patterns than Poisson.

Tables 3.2 and 3.3 contrast the results based on the second approximation

43

Table 3.3: Average Waiting Time Results in Hz +M/kfi/i/l NP Priority Model Using the Completion Time Approximation Method

Approx

Lower CI

Upper CI

Exact

Ap prox

Lower CI

Upper CI

Exact

Approx

Lower CI

Upper CI

Exact

("Approx") method described in section 3.2 with confidence intervals and

exact results. These results are based on the equations 3.2.1 û-3.2.12 and

3.3.13. In tables 3.2 and 3.3 we present the average waiting t h e results

E2 + i\.I/Mi/l and H2 + kf/1Vli/l NP priority mode1 respectively. The same

set of configurations nrhich mere used in table 3.1 are considered here.

The degree of errors for lie between -1.4% to 2.9 % and -1.6% to

2.8% for Erlang and Hz HP arrivals respectively. Similarly, the degree of

error for Wq2 lie between -1.5% to 2.8% and -2.9% to 1.8% for Erlang and

H d P arrivals respectively.

One can also observe from tables 3.2 and 3.3 that PVq2 approximation

values lies inside the 95% confidence intervals for both Erlang and hyperex-

ponential arrivals in al1 the 9 cases tested, which provides a bench mark that

suggests it is a good approximation method for CV&. The CVqI approximation

values also are generally good in most cases with few exceptions. It can be

seen that the IVqI lies outside the 95% confidence interval for al1 configu-

rations with p = 0.3 for both E2 and Hz HP arrivals. Only a single value

lies outside the 95% confidence for both E2 and Hz HP arrivals for p = 0.6.

When p = 0.9 al1 the values lie outside the 95% confidence interval for H2

arrivals and only one value lies outside when HP arrivals are Erlang. This

type of behaviour is justified since in our analysis we assume that the time of

the arriva1 of the first HP customer to be the fonvard recurrence time, which

is not in fact true. This suggests that the Wq2 delay will be underestimated

for Erlang arrivals and overestimated for more bursty arrivals as we observe.

3.5 Conclusions

Based on the numerical results we can conclude that the results obtained

using the Kleinrock's approach is reasonably good, since the degree of errors

in most cases mere within 10% with only few outliers. The approximation

method based on the completion time presented in section 3.2 is a better

refinernent to the method presented in section 3.2. Thus the approximation

method bascd on the completion time approach provides us with good ap-

proximation results. Taken together these two methods should bound the

true value. In any case, the approximation is just chat - an approximation - and another method is needed to determine the average delays exactly in a

priority queue with a non-Poisson high priority Stream. One such mode1 is

presented in the next chapter.

ce

Chapter 4

Modelling the PH + MIPHI1 Priority Queue Using the Matrix- Geometric Approach

Introduction

The rnatriv geometric met hod is recently becoming a frequent choice of mod-

elling tool for many complex problems lacking product f o m solution. Under

this approach the rnost important and essential task involves specifying var-

ious bIock matrices completely which WU track al1 the parameters for a.ny

given model. Once these matrices are specified, we can establish using rnatriv

geometric theory, see Neuts (261, a matriz geometric relationship between

the steady state probabilities X ( i ) using

where i refers to the level of the process and R is called the "rate matrixJ7 by

Neuts. Later on in this chapter after specifying the model we describe the

46

structure and explain the meaning of a given block entry in

In the case of general hi&-priority amvals and Poisson

rivals, the waiting time analysis becomes extremely hard to

the R matrk.

low-priori ty ar-

solve, since one

no longer has a memoryless high-priority arrivai process. -4s a result busy

cycles (consisting of low-priority service times extended by high-priority busy

periods) cannot be so easily characterized by LSTs.

Very recently a few papers have appeared to analyze queues of this nature.

-4s indicated in the introduction, most of the papers employ matrix-analytic

or matrix geometric methods introduced by Neuts. Such methods were first

used for priority queues by Miller [25] to study steady-state probabilities for

the M/!Ii/l priority queue. Alfa [2] has extended Miller's work to study

the queue length and waiting time distributions of discrete MAP/PH/ l

priority queues. Wagner [37: 381 has studied a NP multi-sever priority mode1

featuring non-renewal input. Bertismas and Mourtizinou [5] study, among

others, a two-priority queue with mived generalized Erlang arrivals using

generat ing function techniques.

In this chapter we analyze the single semer priority system when the

high priority customer's interarrival t h e (MT) distribution is given by a

Phase distribution (see appendk A) of order k and is represented by (y, - A),

whereas the low-priority LAT distribution is exponential. SeMce times are

phase distributed and they are represented by ((y: T) of dimension c for HP

and (,O, S) of dimension d for LP customers. Unlike standard linear algebra -

texts in which vectors are consistently al1 row or column vectors, we adopt

the vector conventions of Neuts (261. The vectors are summarized as follows:

a, p and y are row vectors, each containing the respective probabilities of - - -

s tarting the phase process pertaining in its corresponding transient st ates.

ir' is the probability row vector and g is a colurnn vector of appropriate - dimension. We define - so = -Sc, - to = -Tg, and - A. = -Ag to be the vectors

of absorption rates from the respective transient states.

-4lthough the selection of PX HP arrivais limits the applicability some-

what, we did so because of the following reasons: PH distributions are widely

known and used in many queueing applications, they readily extend Miller's

[25] rnodel, and they contain the exponentid, Erlang and hyperexponential

models as special cases.

4.2 The Mode1 Description

CVe rnodel this system as a continuous-time Markov chain on the state space

S={(i , j), i 3 O, j 2 O), where i corresponds to the number of high priority

customers in the system. Henceforth we refer to i as the level of the process.

For i = O, [ j /k(l + d)] equals the number of low-priority customers in the

queue, and for i 2 1, [ j / k ( c + d ) ] equals the nurnber of low-priority customers

in the queue. (Here [Y] equals the integer portion of y). In what follows

below, we have maintained Miller's notations as closely as possible for clarity

of purpose.

The infinitesimal generator of this mode1 takes the form of a Quasi-Birth

and Death process (QBD); namely,

where the blocks group al1 states with a particular level i. Thus Aa correspond

to transitions involving the amval (and A2 CO the departure) of a high priority

customer, whereas AL corresponds to al1 other possible changes to the state

of the system.

The blocks ilo, Al, .A2, Bo, BI and A; are of infinite dimension and can

be partitioned into sub-blocks themselves, pertaining to al1 states with the

same number of low priority customers in the queue mhere:

... . . . . . .

M L O O O O O

. . . . . . O Ml1 O O O

A2 = O O luLL O O

O O O Mil O '-• i . . . . . .

A. and r12 are block diagonal matrices of appropriate dimension because

a high prionty arriva1 and service completion, in their cases has occured.

Thus no other events can happen simultaneously. A i is similar to rio, but

the size of the sub-blocks differ because they contain rows corresponding to

and

For a given sub-biock, the number of type 1 and type 2 customers in

system and queue, respectively, is known. To completely specify the state

within the various rows of each sub-block, we must describe who is in service,

the current phase of the high-priority interarriva1 distribution, and the phase

of service for whichever class of the customer is in service. To be consistent

with Miller's work, the upper part of each sub-block corresponds to states

having a high-priority custorner being in service, and the lower part to states

having a low-priority service.

Descri~tion of the submatrices

Whenever one of the phase-distributed quantities (be it an in terarriva1

tirnes or a service tirne) completes, another or a similar such time begins.

W e need to describe the rate a t which transitions occur in which the first

process completes from given states and the second process commences in

various states. Thus each of the matrices defined below is the product of the

column vector of termination rates for the completing process and the row

vector of starting probabilities for the process that is commencing.

To refers to the event where the service completion of one HP customer

is immediately followed by the commencement of service of another HP cus-

tomer. Likewise So refers to the corresponding event involving LP customers.

T, describes the event where the service completion of an HP customer is fol-

lowed by the commencement of service of an LP customer, and St corresponds

to the reverse situation. A. refers to the arrival of the HP customer folIowed

by the start of the next arrival phase of the HP custorner.

With respect to more general definitions, Iij indicates an identity matrix

of dimension y. In what folIows, we make extensive use of Kronecker prod-

ucts and sums. The Kronecker product of two matrices .4,,, and Bkxl is

defined in Neuts [26], p53. For an alternate definition, see Stewart. W.J.,

[35], sec 9.6, pg. 464. It is a mk x nl matrk concisely written as

where A = [aij]. The Kronecker sum of square matrices A,,, and Bkxk is

defined to be

these definitions in mind, we turn our attention to the remaining sub-

matrix specifications. rUl of the sub-matrices A,',, Aoo, ICI, L2, ftli,, ~ & i , 1w127 ~b122~

K2, K3, LZ2 take the fom:

of appropriate dimension. The submatku taking the form A corresponds

to an HP service completion leading to the commencement of another HP

service, B represents the termination of HP service followed by an LP service.

Likewise C and D corresponds to the completion of LP service followed by

an HP and LP service respectively. Unspecified blocks are assurned to be

null. We sspecify below the entries of the submatrices:

4.3 Solution Procedure

Let g be a unit column vector of infinite dimension. The stationary

probability vector ,Y of Q which satisfies XQ = 0, & = 1, can be partitioned

into subvectors (X(O), X(1), X(2), X(3) ......), mhere X ( i ) pertains to those

States associated with i high pnority customers in systern.

illatrix-geornetric theory (261 indicates that subvector X ( k ) is related to

X(1) via

where R is the minimal non-negative solution to

It also establishes that the (v, u) entry of R represents the expected amount

of time that the process spends in state (i + 1, u) prior to returning to level

i starting from state (i, v). Partitioning R into blocks of appropriate di-

mensions, the j th block row pertains to starting states with j low priority

customers in the queue. A transition of the forrn i -t i + 1 corresponds to

the arriva1 of a high priority customer. Since none of the waiting low prior-

ity cus tomers will enter sentice before the high-priority custorners are gone,

al1 biocks in R below the main diagonal will be O. -41~0 since transitions of

the form ( 2 , v) -t (i + 1 , ~ ) will depend on the diffemnce in the number of

low-priority customers involved it follows that

where

The special structure of the R matriv enables one to solve for the ele-

ments in Ra, Ri, . . . recursively. Since the recursive equations typically in-

volve numerous subtractions the recursive procedure is prone to e-uhibit some

numerical instability. Hence, we opted for a matrix iterative method which

exploited the repetitive structure to determine the £3 matriu.

This matLu iterative approach can be described as follows (see .4smussen

1 ) let r equal the largest diagonal element of Q in absolute value. Let

P'=T-~Q + 1. Al1 of the entries in P' they can be viewed as the transition

probabilities of a discrete time Markov c h a h The corresponding discrete

time Markov chain matrices are A;=rdLA2, .4O=~-~.4~, A;=T-~& + 1. The equation for R using this discrete time Markov chah is given by

This procedure will converge starting frorn & = O (see Neuts [26]).

In order to deterrnine the desired performance measures in the next sec-

tion, one needs to solve for X(0) and X(1) as well, using X(0) Bo+ X(1) Bi

=O? X(1) (RA2 + .Al)+ X(O)A;=O and X(0) g+X(l)(I - R)-lg=l. We em-

ployed the state reduction (SR) rnethod of Grassmann (see Grassmann [15]

or Stanford and Grassmann [33], ) as it is inherently stable. (SR can also

be employed to determine the R matrix; the interested reader is directed to

[33].) -4 brief description on state reduction is given in appendix C.

In the case of class dependent exponential service tirne we observe that

-4; reduces to ilo and thus X(1) can be related to X(0) by X(1) = X(0)R.

4.4 Performance Measures

Our principal measures of interest are the average waiting times IVqk, k=1,2.

However, having obtained S(O), X(1) and R, many other quantities of inter-

est (such as queue length distributions) can be obtained with little additional

ivork. For instance let Pki=Prob {k type 1 custorners in the system ). Then

Similarly, define Pi2=Pïob{i type 2 customers in the queue}, and X* =

X ( k ) = X(1)(I - R)-l. This vector can be partitioned as X* =

[XS(O), X8(1), . . . 1, where X*(n) is the vector of probabilities correspondhg

to n low priority customers in queue and a t l e s t one type 1 customer in

system. Similarly X(O) can be partitioned. Then

FVe can obtain the average number of type 1 customer in the system by

paralleling equation[l.8.4] from Neuts [26]:

One t hen immediately finds

The average delay of the low-priority customer under NP priority can be

obtained directly frorn the mode1 or altemately by using conservation laws.

Ln the latter case, we can use (2.8) to obtain VVq2 assuming that V had been

obtained from a related mode1 as described in section 2.3. Alternatively, the

average queue length Qî and waiting time CVq2 for the low priority under NP

priority class can be obtained by the relations:

Preemptive-Resume Priority

Under both N P and PR disciplines an arriving low priorîty customer must

wait until al1 work in the system has been served, as well as ail later-arriving

high-priority work until there is none left before service commences. The

PR and 'JP disciplines only differ in the order in which the work is served.

Consequently the PR waiting time distribution of a low priority customer is

identical to that under NP priority. The high-priority class under PR is not

influenced by the low-priority class, hence it can be analyzed using FCFS

queueing resuits.

Remarks: To compute R and the steady-state probability vector X, we need

to truncate the mode1 at some sub-level (the maximum number of low-priority

customers in queue, say J). If the average waiting time of the low-priority

class agrees to a decent level of accuracy from both methods (conservation

law approach and 4.8), we can be assured that the chosen value for J was

sufficient. On the other hand, if we had chosen J too small we would have

obtained a lower bound for kVql from (4.61, thus providing us with an upper

bound for Pi$? by the conservation law approach, and a lower bound for

CVq2 from equation 4.8. One could then re-run the particular configuration

with larger values of J until the desired accuracy is obtained. Before we

present the numerical results in the next section, we would like to summarize

the calculation of average waiting tirne under NP priority by the following

algorit hm.

Stepl: Calculate the rate mat rk R.

Step2: Obtain q=(I - R)-lg.

Step3: Solve X (O) and X(1) which satisfies X(0) Bo+X(l) Bi = O, X (1) (RA2+

Al)+ X(O)Ai = O and X ( 0 ) g+X(l) ( I - R)-L g = 1.

Step4: Obtain w=(I - R)-2g.

Step5: Compute the mean number and average system time of the high pri-

ori- customer in the system respectively by: L(I)=X(l)w and WL=L(l)/X1.

Step6: Calculate average waiting times CVqi and PVq2 using equations 4.6 and

4 -8 respectively

4.5 Numerical Results and Discussion

Extensive numerical tests were performed to check the accuracy of the model.

CVe considered examples where the HP IAT is governed by Erlang or hyper-

exponential time distributions and LP class is Poisson arrival. Both the

arrivais have class dependent exponential service times. The sub-matrices

--lia, AOo1 Ki, L2 Mi 1, hll2, 1blZ2, K2, K3, L22 for these cases are given by:

Common sub-matrices

L2: -4 = D = diag(X2), Mil: A = diag(pl),

LZ2: D =diag(&),

D =diag(p2),

Mi2: B = diag(pl),

Case 1: Erlann-k HP arrivais

-Aoo: The (kJ) entry of matrices A and D is kXL,

Ki : A =(diag(J, superdiag(kXL) ) , C =diag(p2) and

D =(diag(g), superdiag(kXi) ),

K2: .4 = (diag(d), superdiag(kXl)), B = diag (A2)> C = diag (1.12) and

D = (diag(g) , superdiag(kXL)),

16: -4 =(diag(-ML), superdiag(kAi)), D=(diag(g) , superdiag(kAl)).

where

f = -(k& + Al + pl), g = -(kXL + X2 + pa) and 6 = -(kAl + A2).

Case 2: Hvperexponential-k HP am'vals

Ki : .A = d i a g ( j I 1 . . . , fk) , C = d i a g ( p 2 ) and D = d i a g ( g i l . . . , g k )

16: A = d i a g ( b l , . . . , bk) , B = d i a g ( X 2 ) , C = d i a g ( p z ) and D = d i a g ( g l , . . . , g k ) ,

16: -4 = d i a g ( - y l , . . . , - y k ) , D = d i a g ( g . . . , gk) .

where

fi Fi --( + A 2 + P L ) , gi = -(% +ha + p z ) and bi = -(yi +A2) .

Tables 4.1 through 4.6 present the average waiting time results for both

classes under NP priority for different configurations. Tables 4.1 - 4.2 present

the results for the mean tvaiting time featuring Erlang high priority arrïvals of

degree 2, 3 and 5, respectively and Poisson low priority arrivals. In table 4.1

both prionty classes have exponentially-distributed service times with mean

1. Three different traffic mixes were considered, namely: arrivals in the ratio

of 1:l: 12 and 2:l. The mean delay for both customer classes decreases when

the degree of the Erlang distribution is increased, as expected.

Table 4.2 presents the results for the mean waiting time with different

service times. Mean s e ~ c e times are in the ratio of 12, 2:l and 41. Similar

trends are observed as in table 4.1.

To model the case where the HP arriva1 process is more variable than Pois-

son, we considered several examples involving hyperexponential-2 distribu-

tions with balanced means (F = h). The parameters of the hyperexponential-

2 distribution in this case would be:

where,

where, E{T) is the mean arriva1 rate and 2 is the squared coefficient of

variation of the hyperexponential-2 distribution.

These cases featured c2 equals to 2.25, 4.25 and 6.25. The results appear

in tables 4.3 - 4.4. As 2 increased the average delay increased due to the

greater variation in the HP interarrival times. The effect is generally felt more

strongly by the HP customers than the LP ones. As a further check on the

accuracy of the model, the average nmiting time results were compared with

simulation results. In al1 configurations, 10 runs of 1,000,000 customers were

considered, leading to very tight confidence intervals. There is a high degree

of agreement in al1 the cases. We present the graphs for the E2 + lW/n/I/l

-W rnodel to substantiate our claim. We present the average waiting time

results under preemptive priority discipline just for Ek + M/Mi/l in tables

4.5 - 4.6.

Figure 4.1 presents the average waiting time results against the server

occupancy for identical arriva1 rates (Le. (1:l)). Figures 4.2 and 4.3 display

results when arrivais for in the ratio of 1:2 and 2:l respectively.

Our goal was to select J (the maximum number of low-priority customers

in the queue) in such a way that the expressions for kVq2 obtained using

rnatrix analytic method and conservation law method agreed to at least 4

significant digits of accuracy. The values chosen for J were 25, 40 and 75 for

p = 0.3, p = 0.6 and p = 0.9 respectively for Erlang-k HP arrivals. These

values of J were more than adequate for p = 0.3 and p = 0.6. To achieve 4

significant digits of accuracy for the hyperexponential-2 HP arrivals we had

to increase the value of J to 35 and 50 for p = 0.3 and p = 0.6 respectively.

Table 4.1: Non-Preemptive Average Waiting Times Ek + M Amvals

Table 4.2: Non-Preemptive Average Waiting Times & + hl Arrivals

Table 4.3: Non-Preemptive .Average Waiting Times H2 + 1bf Airivals

Table 4.4: Non-Preemptive Average Waiting Times H2 + M Arrivals

Table 4.5: Preemptive Average Waiting Times Ek + !LI Arrivals

Table 4.6: Preemptive Average Waiting Times Ek + hl .4rrivals

Figure 4.1: Average Waiting time:Balanced (1:l) E2 + ib1 .hmvals, Common Exponential Service

High-Priority Class Low-Priority Class

0.0 0.2 0.4 0.6 0.8 1 .O Semer Occupancy

0.0 0.2 0.4 0.6 0.8 1 .O Server Occupancy

Figure 4.9: Average Waiting time:UnBalanced ( 1 2 ) E2 + ibl Arrivals, Com- mon Exponential Service

0.0 0.2 0.4 0.6 0.8 1.0 Server Occupancy

Low-Priority Class

0.0 0.2 0.4 0.6 0.8 1 .O Server Occupancy

Figure 4.3: Average Waiting time:Balanced (2:l) E2 + iLI Arrivais, Common Exponential Service

0.0 0.2 0.4 0.6 0.8 1.0 Server Occupancy

Low-Priority Class

0.0 0.2 0.4 0.6 0.8 1 .O Server Occupancy

Chapter 5

Analysis of the MAPI PHI l Priority Queue wit h Service Control

Introduction

Control and design models are primarily intended to improve the efficiency of

the system performance. The control models are useful to determine what the

optimal system parameters should be. One way to improve the efficiency is by

subjecting the server to work at a faster rate when the system/queue length

hits a certain threshold limit and to continue at this rate until systern/queue

length falls below a certain (lower) limit. This type of control is c d e d the

bi-level hysteretic control policy.

The interest in studying control models for single queues started in the

early 1970's [12]. In the control models when m i t c h h g costs (snritching

from normal to fast and vice versa ) are involved, determination of optimal

system parameters for efficient hnctioning of the system is difficult. Neuts

and Rao (271 have addressed this problem and many more in their paper

by developing efficient algorithms for a finite M/G/1 queue with phase-type

services. Chakravarthy [8] has extended the work of Neuts and Rao to analyze

the PiI.AP/PH/l/K queue.

In this chapter, we appIy such a service control rnechanism to the non-

preemptive priority single semer model characterized by Markovian arrival

process with phase-service, namely MAP/PH/l model with finite capacity

(say K) for the HP class. We allow the LP capacity to be infinite. When

the queue size for the HP exceeds a threshold value, Say N < K, we increase

the service rate by factors dl > 1 and e2 > 1 br HP and LP customers

respectively. Normal service is restored for both customers once the queue

size for the HP drops to M (1 $ iLI 5 N < K).

The Markovian arrival process (MAP) is a tractable class of renewal pro-

cess which is widely used in different applications of stochastic rnodelling.

Many of the well-knom distributions such as Poisson, Erlang, PH-renewd

process and Markov-moddated Poisson process are special cases of MAP.

The MAP has great use in various engineering arenas namely, telecommu-

nications, production and manufacturing because of their tractability. See

appendix B for a detailed description of M.4Ps.

The MAP for this model is govemed by three matrices Do, Dl and D2.

Do governs the transitions correspondhg to no arrivais of either type. Dl

and D2 governs the transitions corresponding to arrivals of type 1 and type

2 respectively. The service time distributions are phase distributed and they

are represented by (a,T) of dimension c for HP and (P , - S) of dimension d

for LP customers. During fast service, the service time distribution has the

representation (g, BIT) of dimension c and (P, - 02S) of dimension d for HP

and LP customer respectively, 81 > 1, > 1.

5.2 The Mode1 Description

We can model the system as a continuous time bfarkov chain. Unlike the

previous chapter the level of the process corresponds to the number of LP

customers waiting in the queue and the sub-level corresponds to the number

of HP customers waiting to be served in the queue. In the priority arrange

ment we see far less HP customers than LP custorners in the queue. By

choosing a finite capacity model for the HP customer, we loose very few HP

customers. We also know that the matriv geometric method allows for one

level to be potentially infinite. Thus, by having the number of LP customers

waiting in the queue as the level we can accommodate infinite number of low

priority customers.

The infinitesimal generator Q for this system has the form:

where the first (block) row of the Q rnatrix corresponds to a truly empty

system, rows 2,3 ,4 , . . . pertain to O, 1,2, . . . LP customers waiting in the

queue respectively. The block matrices Ao, AL, A2 C and B group al1 the

states pertaining to a particular level i (the number of LP customers waiting

in the queue).

Figure 5.1: Hysteretic Server Control Mode1

Fast

Normal LLJ

The sublevel (the number of type 1 customers waiting in the queue) is

partitioned into a 3 x 3 block matrix. The k s t row of this matrix groups

states O through N for the sublevel corresponding to a normal service, the

last row groups states N+1 through K (which necessarily corresponds to

fast service) and the second row groups states M+1 through N under fast

service, reached from the fast service states. These states are denoted by

{ ( M + l)', . . . , N') to distinguish thern frorn the slow service states. As

an illustration the graphical representation for a hysteretic service control

system involving exponential service distribution is presented in figure 5.1.

The structure of the system which describes the transitions at various

levels allows us to write:

where the block .Ao is a block diagonal matrix since the arriva1 of a LP cus-

tomer does not permit any simultaneous events to occur. The block m a t r k

F describes transitions pertaining to the LP arrivais, and we describe this

matrix later in the chapter.

The block matrix A;, corresponds to the decrease in the number of L P in the

queue. The only way such an event can happen is mhen there are no HP

customers to be served. Since the rows of A:') describe the number of HP

customers in the queue, the only non-zero entry is in row zero, Le., when no

HP customers are present in the queue to be served. Thus,

The rnatrir i\& will be described later. The block matriv Al preserves the

number of LP customers is the queue. AL contains transitions which do not

affect the number of LP customers in the queue. Thus, the various entries of

the -4 bIock contain the following transitions wit h corresponding matrices:

HP arrivals (L2), the service cornpletion of HP class followed by HP class

( ~ b f ~ ~ for normal states,Mi, for fast), arriva1 and service phase changes (K I

for normal and K: for fast states. The non-zero entry (1%') in the first

row of the AL') b n n g ~ the system operating h m a fast senrice mode to

normal service, hence we observe a transition from (ibf + 1)'th row to the

Mth column. We also observe that the K x K element denoted by Ky is

distinct, since we truncate the mode1 by not accommodating any more HP

amvals. Thus in summary we have,

...

...

. . *

. * -

a..

. . . . . . K: L2 O O

-413) = j 1)" : I'. 1:: ; j .

O f KI'

For a given sub-block, the number of class 1 and class 2 customers in the

queue is known. To completely specify the state within the various rom of

each sub-block, we must describe who is in service and the phase of service

for whichever class of the customer is in service. To be consistent with Our

previous work, the upper part of each sub-block corresponds to a high-prionty

customer being in service, and the latter part to a low-priority service.

Description of the submatrices

Before we describe the submatrices we would like to define the following

notation:

Let Iij indicates the identity rnatrk of dimension i.j, is an column

vector of 1 of appropriate dimension, 5 is a colurnn vector with 1 at the i'th

position only, $ is the transpose of ci and finally the matrices Do, DI and

D2 are assumed to be of dimension 1 .

5.3 Solution Procedure

Let g be a unit coIurnn vector of appropriate dimension. The stationary

probability vector of Q which satisfies XQ = O, & = 1, can be partitioned

into subvectors (X', X(O), X(1), . . . , . . .), where X(i) pertains to those states

associated with i low priority customers in the queue and X* corresponds to

an idle system. Thus the steady state equations are given by:

As s h o m in the previous chapter, rnatrix-geometric theory indicates that

subvector X ( i ) is related to X ( i - 1) through the equation

where R is the minimal non-negative solution to

To correspond to the structure of Ao, Ai and A2 ive choose to write the R

rnatriv as:

One of several well known methods such as state reduction or the matrix

iterative method can be used to solve this R matriu. When we assume the

arrivals to be Poisson one can obtain the R rnatrix explicitly. The determi-

nation of R and the solution of X(i) for the nl f /PH/ l is given below.

Determination of the R matrix (Poisson Arrivals)

When arrivals are Poisson the submatrices are given by:

A* = &Il

B = g , @ [ ~ ] , - ~ = & @ [ h z &e] .

XlIC 0

T - X I c O OIT - XIc s - XId 82s - XId

Ky = O 1 , where A=Al+A2.

62s - X21d

tVe can thus establish that

Postmultiplying equation (5.3) by g and substituting for Ale = -(Ale+ A2eJ

we obtain

Also note that in case of Poisson arrivals Do is -A. By induction we can

show that

Further let us define

Hence

X(i + 1)A2 = (uo(i + l)h122,Q,û,. .. ,. . . JI).

Rewriting equation (5.9) as we obtain

&X(i)g = (uOl(i + 1)-+ ,UO& + 1)-).

Postmultiplying the above equation by - P Ive get

&X(i)& - = (uol(i + l) to - + ,uo~(i + 1)a)P. (5.10)

Thus, S(i + l)Az = [(O, uoi(i + l ) toP - + ,uo& + 1)@), (0,) . . . , (Q)].

Therefore we can relate X(i + 1) in terms of X ( i ) via.,

X(i + 1)A2 = X?X(i) H (5.11)

where H is given by

ç= and represent the unit column vector of dimension c and d respectively.

Substituting X ( i + l)& = A2X(i) H in equation 5.4 me get

The above equation can be rewritten as

-X(i)[Al + X2H] = X2X(i - 1).

Thus from the above equation R is given by

R = -&[A1 + & H I - ' .

Determination of X(0)

The X(0) vector can be obtained from these equations:

AX* = X(0)B (5.14)

X' + X(O) ( I - R)-l = 1. (5.16)

Substituting X' = iX(0)B and X(0)RA2 = X2H in equation 5.15 we get

which can be wrîtten as

X(0)Y = 0.

Similarly from equation (5.16) ive get

X(0)Z = 1

where

Solve for X(0) using equations 5.15 and 5.16. Once we obtain X(0) we can

easily solve for the X ( i ) by relation 5.5.

5.4 Performance Measures

In this section we present various performance measures for the MAPIPHI1

NP priority rnodel. The results for the M / P X / l easily follow since they form

a special case of MAPf P H I l rnodel. Most of the performance measures that

are of interest to us can be obtained using R, X(0) and 7 = X(0) (1 - R)-l.

Let us partition 7 as:

where and 6, corresponds to i HP customers waiting in the queue to be

served and j = 1 , 2 represents the type of customer in service. The V ~ ~ O U S

performance measures that can be deterrnined are listed below:

1) The system is empty:

x* = -(D;~x(o)B).

2) Average Queue Length

Low priority

High Priority

3) Probability the server is serving a t a normal rate

Low priority

High Priority

4) Probability the server is serving a t a faster rate

Low priority

High Prioritv

5) Switching Rate

Fast to Normal

khr+i,i(@ 5) + 92~h,r+i.2(@ 9)-

6) Probability the arriving HP customer finds the buffer full (Pb):

1 Pb = - [ ^ 1 K i ( ~ i ~ @ Al C) f 7 1 ( 2 ( 0 1 ~ @ G)].

7) Average waiting times

Low Priority

1 PVq2 = -q(2) .

A2

High Priority

8) The throughtput( $), which is defined as the average number of departures

per unit of time can be obtained using:

or alternat ively by

5.5 Numerical Results and Discussion

In this section we present a few exarnples that provide insight on the robust-

ness of the model discussed in this chapter. As mentioned in the previous

section niimerous performance measures can be found as one desires with

the model. Since our primary quantities of interest are the average rvaiting

times, we initially present the average waiting time results for various scenar-

ios. CVe are also interested in studying the throughput of the system and the

percentage of the high priority customers lost, since we place a limit on the

number of high prionty customers adrnitted to the queue. For the examples

presented we assumed Poisson arrivals for al1 the cases, but different service

time distributions are considered, namely exponential, Erlang of order 3, and

hyperexponential (with C2 = 6.25). For al1 the evamples presented here ar-

rival streams of HP and LP are in equal ratio (1:l). The mean service time

for both classes of custorners during normal service was assumed to be of

mean 1. Light (A = 0.3): moderate (A = 0.6), and heavy (A = 0.9) traffic

intensities are tested.

Tables 5.1, 5.2 and 5.3 present average waïting time results where the

fast service rates, are held constant (O1 = O2 = 4). We Vary the threshold

levels N and M for a given value of K (the maximum dlowable HP queue

length). Table 5.1 presents results for the light occupancy case under different

scenarios. When X = 0.3 we fixed K = 6 and we O bserved that the throughput

was equal to 0.3.i.q practically no HP customers were lost. The chances of

losing a HP customer even in the worst scenario (Hz + H2 service, N = iLT =

5) was less than 0.0005. When HP service is either exponential or Erlang the

waiting time increases initially, but tends to stabilize from N=M=3. On the

other hand when we consider the extreme case, namely, H2 + H2 ive can see

that the average waiting time increases even after N=il1=3.

Similar trends can also be observed in tables 5.2 and 5.3. Table 5.2

and 5.3 present sirnilar results as 5.1 but for moderate (A = 0.6) and heavy

(A = 0.9) occupancies respectively. For moderate occupancy we fixed K=f2

and varied N=M to be 2, 4, 6, 8, 10. By fixing K=l2 when X = 0.6

the throughput for rnost cases was 0.6 and the probability of losing a HP

customer was less than 0.00055 in al1 cases. For heavy occupancy (A = 0.9)

we fked K=18 and varied N=M to be 3, 6, 9, 12 and 18. We achieved a

throughput of 0.9 for most cases and the probability of losing a HP custorner

was less than 0.0006.

If we allow K to be large, the results obtained from this model d l be

very close to that of an infinite model, i.e., the model which does not place

a bound on the HP customers in the queue.

Tables 5.4 through 5.7 present average waiting time results by varying 81

and holding K, N, M and 82 constant. This exercise is useful to see whether

the average waiting time decreases by making the semer to work faster with

6xed threshold. The results presented here are for moderate occupancy,

A = 0.6. Table 5.4-5.5 present average waiting time results when HP service

time distribution is exponential and tables 5.65.7' present similar results for

Hz service time distribution.

In table 5.4 we present the results for Bi = 2, 4, 6, 8, 10 and hold O2 = 2,

K=12, N=8 and M=4. Average waiting times for both classes of custorners

are identical to a t Ieast the second digit for Exp + Exp and Exp + E3 service

time distributions, but we notice a decrease in the waiting times for every

change in for H2 service time distribution. In table 5.5 rve show the same

configurations but we changed B2 to 4. We see that B2 has absolutely no influ-

ence on both and CVq2 when LP service tirne distribution is exponential

or Erlang, but has some influence when LP service time distribution is Hz.

The reason Oz and 61 > 2 does not exert any influence on the average waiting

times results is due to 1) the chances of finding eight HP customers in the

queue is rare, thus not influencing any impact on the threshold value and 2)

at most only one LP customer's service time (if the LP customer happens

to service when the buffer hits the threshold value) can have any bearing on

HP service time distribution.

Tables 5.6 and 5.7 present similar results as tables 5.4 and 5.5 but the

HP service time distribution is H2. For this case the influence of subjecting

the server to work faster (varying 81) certainly reduces PVql. We observe

this type of phenomenon because of the bursty nature of hyper-exponential

distribution. -4s before changiog d2=2 to 02=4 has very little impact on the

average waiting tirne resul ts.

Thus we can Say that subjecting the semer to fast service has very little

influence on average waiting time compared to changing the thresholds M

and N. Thus to operate a system efficiently, nameiy decreasing the system or

average waiting times, it would be much more advantageous first to change

the threshold limit rather than changing Bi or 02 or both.

Figures 5.1 and 5.2 plot the throughput and the probability of HP cus-

tomer being lost versus 6' (OL and O*) respectively. The case that is begin

plotted is for heavy traffic (A = 1). For al1 the cases we Ex IK = 4, N = M =

2 and Vary OL = O2 from 2 to 20. The phenomenal change is felt when we have

H2 + H2 service time distribution, compared to Exponential+Exponential or

E3 + E3. AS we had discussed earlier varying OL or O2 has very marginal

impact on decreasing the percent loss of the HP customer if the threshold is

set a t a particular level where the chances of the HP customer encountering

that level is very low. This c m be seen very much by examining the curves

for Esponential+Exponential and E3 + E 3 Thus it is better to serve faster

more often than to serve extremely fast rarely.

Sable 5.1: Average Waiting Times, h = 0.3, = O2 = 4

E x p + E x p Exp + E3 E x p + Hz

Table 5.2: Average Waiting Times, A = 0.6, el = e2 = 4

E x p + E x p Exp + E3 Exp + H 2

wqt w q 2

0.933 2.442

1.369 3.502

1.637 4.156

1.795 4.533

1.883 4.741

wq. PV** 0.708 1.882

0.836 2.107

0.855 2.139

0.857 2.142

0.857 1.143

C c v , 4 2

0.609 1.609

0.700 1.763

0.713 1.783

0.714 1.785

0.714 1.786

Table 5.3: Average Waiting Times, X = 0.9, O1 = O2 = 4

Exp + Exp w q l pvq2

1.314 11.789

Ezp + 4 Wq 1 W 9 2

1.130 10.479

1.329 12.207

1.359 13.586

1.363 13.631

1.364 13.636

Exp + H2

1 wq2

1.641 10.092

2.548 20.247

3.111 28.112

3.434 32.904

3.607 35.461

Table 5.4: Average Waiting Times, X = 0.6, O2 = 2

E x p + Exp E x p + E3 Exp + H2 & 4 2

1.797 4.499

Table 5.5: Average Waiting Times, X = 0.6, O2 = 4

E x p + E x p E x p + E3 Exp + H2 pvq1 IV, 1.758 4.417

tvql ~v, 0.714 1.785

el 2

tv,, v 0.857 2.142

Table 5.6: Average Waiting Times, X = 0.6, Oz = 2

Table 5.7: .Average Waiting Times, X = 0.6, O2 = 4

Figure 5.2: Plot of Tliroughput as a F'unction of BI

Exponential Hyperexponential Ertang

Figure 5.3: Plot of Percentage of HP Customers Lost as a Function of el

Exponential Hyperexponential Erlang

Chapter 6

Modelling the Two Node Priority Tandem Queueing System using the Matrix Geometric Met hod for M / P H / l + . /PHI1

We were able to mode1 and analyze priority queues having non-Markovian

input using matrix geometric theory in chapter 4. By ordering the level and

sub-level as the nurnber of high priority customers in the systern and the

nurnber of low priority customers in the queue respectively we obtained a

special and simple structure for the R matnc. The presence of HP customers

acts as a "barrier" preventing LP customers from being selected from service.

This leads to an upper traingular structure for the R matriu.

If we consider a tandem arrangement of priority queues this "barrier"

blocks the LP customers in the queue Erom proceeding to the second node.

Figure 6.1: Tandem Priority Queue

Thus we decided to focus in this chapter on the use of matrix rnethods to

study a two node tandem non-preernptive priority queue.

Rather than computing actual numerical results, our focus is instead to

develop workable algorithms that can be used in the calculation of various

performance statistics. As ive have seen in the earlier chapters, if one succeeds

in speciwng ail the parameters to any given problem in a block tri-diagonal

f o m , matrk geornetric theory provides the means of obtaining the steady

state probability vector K. The system being modelled is presented in figure

6.1.

As seen in chapters 4 and 5, the most important task involves the complete

specification of al1 the mode1 parameters efficiently to exploit the structure

of the block matrices for the calculation of the rate rnatriv R, and the steady

state probability vector X. Since the modelling requires us to specify all the

transitions pertaining to the problem, the six parameters that we need to

track for this model are:

1) The number of high priority customers a t nodes 1 and 2.

2) The number of low-priority customers a t nodes 1 and 2.

3) The type of service in progress a t nodes 1 and 2.

Because of the nested structure of the problem one can easily visualixe

the complexity of this problem when six factors need to be considered. This

problem can get not ationally even more cumbersome if we assume PH type

of arriva1 or service distributions rather than exponential. Hence, we develop

the model assuming exponential inter-arriva1 time distributions for both the

streams a t node 1. The service time distributions for both classes of cus-

tomers a t nodes 1 and 2 are independent and they are Phase distributed.

The structure of the block matrices are preserved even if one were to extend

this model to allow MAP or PH arrivais.

The other assumptions to the network are:

1) Each node is operated by a single semer.

2) A11 customers of either type arrive a t node 1 and are served a t both nodes

before departing.

3) Customers cannot change classes.

4) The FCFS discipline is assurned rvithin a given class.

We have chosen to rnaintain the notation as closely as possible to chapter

il. Let XI and h2 be the arrivd rates of clnss 1 and class 2 respectively to

node 1. The service time distribution of the high priority custorner a t nodes 1

and 2 are given by (ai, - Ti) i=1,2 of dimension c and e respectively. Similarly

the service time distribution of the low-priority customers at nodes 1 and 2

are represented by (Pi, Si) of dimension d and f respectively. Let Toi=toioi -

and Soi=soiA - for i = f , Z . The vectors - toi and - soi are the vectors of absorption

rates as described in chapter 4.

We choose to retain the same level and the first sub-level as in chapter

4, nameiy the number of HP customers in system at node 1 and the number

of low priority customers in queue at node 1. In so doing, the structure

of the R mat rk is identical to 4.3, since no low-priority customers waiting

to be served can enter service before al1 the high-priority customers leave

the system. Prior to speci6ng the structure of the block matrices we make

the Eollowing definitions: let HPL and H& denote the number of type 1

customers in the system at node 1 and node 2 respectively. Let LPL denote

the number of type 2 customers in the queue at node 1 and L 4 denote the

number of type 2 customers in the system a t node 2.

6.1 The Mode1 Description

CVe mode1 the system as a irreducible continuous time Markov chah whose

infinitesimal generator Q has the fom:

Q =

The first sublevel is the nurnber of type 2 customers in the queue at node

1. The blocks Ao, AL, AS, Bo, Bi are of infinite dimension and are partitioned

into blocks themselves which describe the transitions of the LP customers in

the queue at the first node, the number of type 1 and type 2 customers in the

system at node 2 and the service mechanisms at bath nodes. After specifying

the nurnber of type 2 customers in queue a t node 1, a t the second sublevel we

specify the number of type 2 customers in the system at node 2, so as to be

able to exploit the structure of the R matLu. (Recall Our previous comment

regarding the blocking effec t of HP customers at node 1 .) Further transitions

pertain, in order, to the nurnber of type 1 customers in the system a t node

2, the senice characteristics at node 2 and finally the service characteristics

Description of Blocks

The A. block which describes the arriva1 of the type 1 customer at node

1 for î >_ 1 has the form

.Ao is a block diagonal rnatrix of appropriate dimension because a HP

arrival has occurred which further precludes any other events Crom happening

simul t aneo usly.

The .Az block which describes the service completion of the type 1 cus-

tomer at node 1 is given by:

Io! is an f x (e + f ) mat rk whose first e columns are ndl . The rernaining

f x f block contains an identity rnatrix

The Al transitions preserve the nurnber of HP customers in the system

at node 1, so that no arriva1 events or service completions of type 1 occur a t

node 1. The f o m of Ai is given by:

where

where Id is an e x (e + f) matrk, where the first block ( e x e) contains an

identity mat rk and the last f columns are null.

where D =

Essentially D describes phase changes and seMce completion events occuring

at node 2. We describe below the various transtions contained in the D block

matrk.

Dl: 4, D; and DG are block matrices that form part of the D matrix, which

in turn is an entry of DT and Ds. We recall that DT and LIs track the phase

changes at node 1 and events happening a t node 2. Since we do not see

any semice completions a t node 1, we cannot expect to see any arrivals at

node 2. Thus, the only possible transitions that could happen are: service

cornpletions of both types and phase changes. That is indeed the transitions

we see in D I , D2, D; and D; . Untii this point we described the transitions

corresponding to the non-boundary blocks. We describe below the transitions

relating to the boundary blocks.

Description of the boundary Blocks

Ai is similar to Ao, Le., the arriva1 of type 1 customer occurs. The underlying

distinction is: A; increases from there being no type 1 customer in the system

to one in the system, whereas in Ao, there is already a HP customer present

in the system when an HP amval joins the system.

Bo corresponds to there being no HP customers in the ~ d e m at node 1.

Hence the possible transitions are: arrivals of type 2 customers at node 1,

service completions of type 2 customers, phase changes a t node 1 and various

transitions at node 2 tracked in the D matrix, which has been described

earlier.

Bi =

BI is similar to .A2, viz., a service completion of type 1 customer at node 1.

.12 triggers at the level when there are at least two HP customers present in

the system. Hence, when the HP finishes service another HP immediately

is taken into service. In the B1 bloc$ a LP customer is selected for service

after completing service for the HP customer, since there are no more HP

customers present in the system. If there are no other LP customers present

in the queue when the HP customer h ishes service, the system goes empty

at node 1.

It is worth pointing out that the structure of the matrices up to the first

node is identical to the matrbc structure in chapter 4. Thus, by p r e s e ~ n g

the structure of the matrices, me have preserved the resdts as well.

In order to calculate any performance statistics for this system, the next

step is to determine the R matrix and the stationary probability X of Q.

In the next section we describe the steps invoIved in the calculation of these

quantities.

6.2 Basic Structure of the R Matrix

Xdopting the similar procedure to the previous two chapters the station-

ary probability vector X of Q which satisfies XQ = O and & = 1, can

be partitioned into subvectors (X(O), X(1), S(2), X(3), . . . , . . . ), where X ( i )

pertains to those states associated with i high priority customers in system.

In order to determine the matrix-geometric solution to this QBD process,

one needs to find the R rnatrix which is the minimal non-negative solution

We can esta blish using matrk-geometric theory

Since the level and the sub-level of the process is the number of high

priority customers in the system and number of low priority in the queue a t

node 1, similar to the one described in chapter 4, we can thus establish that

the form of R is given by:

where each R, has the foilowing structure:

We choose the next sub-level as the number of type 2 customers in the system

a t node 2, because we know no further low-priority customers can reach the

second queue before the level of our Markov chah reaches O. Hence both RiA

and RD are block lower triangular and can be represented as

&z =

where the number in the superscript enclosed in the parentheses represents

the number of low-priority service completions at node 2. The first column

of &, x = -4, D, represented by prime (1 ) is unique because the server could

have gone idle. The f&= has a modified block Loiver triangular as there will

be a low-priority service completion at node 1, which increases the node 2

low-prïority queue length by one.

As ive have observed in the previous chapters the determination of R is

vital for any calcuiation. In the next section we describe a complete algorithm

for determining the various pieces of the &, x = A, C, D.

6.3 Algorithms for Calculating the R Matrix and Other Related Quantities

Algorithm for Calculating the 9,

In order to calculate the a,, x = A, C, D, we first need to develop the

recursive equations that indicate the sequence in which the Ris are calculated.

Once this recursion is developed we can break the building block matrices

into subblock matrices and further a recursion can be developed to calculate

t h e & , x = A , C, D a n d i = 1 , 2 ,..., J.

The recursive equations needed to calculate the Ri's are thus given by:

Once Ro is obtained the remaining &'s can be obtained using:

The evaluation of the current R, depends upon the prior determination of

BO, RI, . . . &+ Since R, has the form of 6.4, we can solve for Roa, hc and

RoD using:

and solve RA, Ric, RiD using:

(6.10)

( j ) W r The determination of 4, , Riz , RE)' requires appropriate substitution

of 6.5 and 6.6 in 6.9 and 6.10. We observe that and can each be

obtained on their own without reference to other &,&, X = A, C, D, but

&c requires the prior determination of and

Algorithm for the Ri,

Three operative equations involved in the cornputation of & (ROA, hC

and RoD) derived from 6.9 are stated below:

Sirnilady the three equations required for the computation of 4 (RiA, Ric

and &) denved from 6.10 are stated as follows:

Before proceeding to speci& the recursionç for RE), since DT = D @(Tl -

XI,) , Ds = D @(Si - X I d ) and (&ka + has the same structure

as D, D and hc respectively we define for ease of notations three matrices

DT, DS and ric as:

Similarly

and

Recursive equations for hl, i = 1,2 , . . . , J

Determination of R!;, j = 1 , 2 , . . . , J1

Determination of ~ g , j = 1,2,. . . ,JI

~ l o ) d ~ ~ + ~ $ - ~ ) d ~ ~ + x ~ R ~ ~ ~ ) ~ ( K ~ ) @ I d ) = O

Determination of RF;, j = 1 , 2 , . . . , J'

~ $ ' d ; , + X~ R!:!~) (K$~) @ Id) = O

As we had mentioned earlier in this chapter we notice that the RZ'S can

be obtained by just referring to the previous ~8 'S. tVe pproceed to develop

the recursion for R$'s.

. - Recursive equations for &A, z = z,2, . . . , J

Determination of R F ~ : j = 1,2 , . . . , J f

Deterrnination of ~g~ j = 1,2,. . . , Jr

Deterrnination of RE)" j = 1 ,2* Jt

Similarly we observe that the above set of equations involving R: reecursions

can be obtained Mthout any reference RF 's or R$ ?S.

Recursive equations for T i c ) i = i, 2 , . . . , J

( j ) Determination of roc, j = 1) 2 , . . . J'

T ~ ~ ( K F ) @ T ~ ~ ) + ~ r z d ~ ~ + R E ( K ; ~ ) @alal) = O

@ T ~ ~ ) + ~ t $ d ~ ~ + R,C-%= + R~ '$ (Ec ;~ ) (8) = O

Ci ) I f Determination of roc , j = 1,2? . . . , Jr

rg1 (~ - (4 ) @ T ~ ~ ) + R::' : c ~ + @zd;, = O

(dl Determination of ric , j = 1,2, . . . , Jt

W' j = 1 ~ 2 , . . . , J I Determination of ri, ,

This finally cornpletes al1 the recursions for z = A, C, D. The re-

cursive developed thus far is up to the second sublevel - the nurnber of LP

customers at node 2. The state reduction method, ma t rk analytic rnethod

or any other procedure can be applied to determine these quantities. Below

we state the expansions for the ri;).

0') Expansion of ri,

J

( j - r ) (r+1)1 u'-r) (r+L)it + C ~ k c R(i -k)A + R k ~ R( i -k )C) }

It is worthwhile to repeat at this stage that RZ'S and RE'S can be

obtained by just referring to the previous ~ 5 ' s and R,uD)'s respectively. The

(3-1 (3-1 recursions for RF's involve the knowledge of previous R$ 's, Ric 'ç and RiD 'S.

As a logical progression, in the next section we develop the recursions for the

probability vector &.

6.4 Solution for X ( 0 ) and X(1)

The solution for X(0) and X(1) can be obtained by solving the following

boundary cquations:

X(O)g + X ( l ) q - = 1, where - q = (1 - R)-le. (6.19)

Furtherrnore, we can partition X(i) as:

X ( i ) = (/Y&), Xl(i), &(i), . . . . . . , Xj( i ) ) ,

where ,Y,@) represents the probability of i high-priority custorners in the sys-

tem and j low-priority customers in the queue a t node 1. These probabilities

can be further partitioned to obtain various other probabilities pertaining to

transitions at node 2 and the service phases at both nodes.

By substituting the appropriate block matrices in equations 6.13 through

6.15 we can obtain expressions for Xj(0) and ,Y,(1). The expressions thus

obtained from equation (6.13) are:

Xo (O) C + Xl (O) K3 + X2 (0) hIZ + XL (1) K!:) = O (6.21)

Similarly the expressions derived from 6.14 are:

Finally equation 6.15 can be written as:

k=O

The solution for qk is given by: -

Equations 6.20 through 6.28 can be appropriately combined to obtain X j (O)

and -Yj(l), j = 0,1,2,. . . ,. Once these two probability vectors are deter-

mined we can take marginafs to provide us the vector of probabilities, Say,

S4( j ) corresponding to j low priority customers in the queue. Thus, X*(j)

can be further partitioned to provide probabilities corresponding to transi-

tions at node 2. These probabilities can to grouped appropriately to obtain

various measures of interest at node 2.

Chapter 7

Conclusions and Scope for Furt her Research

In this thesis a variety of problems were addressed in the area of priority

queues involving two classes of customers (high and low priority). Very

specifically, rnost of the rnodels had non-Markovian arrivais. Predominantly,

we adopted the matriv geometric approach to mode1 and analyze the prob-

Lems we addressed.

In chapter 2 we generalized the conservation law to accomodate general

service time distributions for the highest class. This result could prove useful

when one is analyzing queues where the highest amval stream requires non-

exponential service.

In chapter 3 we developed two approximation methods for estimating

average delays for queues involving general high pnority arriva1 stream and

Poisson low pnority arriva1 stream. The delay results derived in this chapter

are fairly easy to determine, and thus, they can serve as a benchmark when

one is initially trying to estimate the average delays without much effort.

In chapter 4 through chapter 6 we addressed problems in priority setups

that provided exact results for the performance measures. In particular in

chapter 4, we studied a PH + M / P H i / l priority model. The ordering of

the levels enabled a special structure, namely an upper traingular rnatrix for

the R rnatrix. This structure was very useul for the developement of chapter

6. These types of queues are helpful in modelling telecommunication traffic

that are frequently non-Poisson in nature.

Chapter 5 focussed on the control aspect of the priority problem. This

chapter brings together the rnatrk geometrîc methods as a means to study

priority queues, and its previous application by Chakravarthy [8] on semer

control rnodels. Unlike the previous chapter the Level was specifîed to be

the number of low priority customers in the queue and sublevel the number

of high priority customers in the queue. In doing so we were able to ob-

tain an explicit solution for the "rate matrix" R and the probability vector

S, when we assumed Poisson amvals. This explicit solution facilitated the

computation effort tremendously.

In Chapter 6 we developed a model for analyzing a sequence of priority

queues in tandem. We learnt from the structure of the matrices in chapter

4, that the presence of high priority customers at a node acts as a "barrie?',

which blocks low priority customers horn proceeding to subsequent node. By

an appropriate organization of the nested sublevels we were able to maintain

the structure of the R matrix obtained in chapter 4 as well as the next

sublevel.

Furt her Extensions

As a first extension to chapter 4, one can easily extend the underlying

model to accomodate PH or MAP arrivals for the low priority class and

111AP arrivals For high priority class. This mode1 can be also be estended

to accomodate more classes of customers. The special structure of the R

matrix, Le, upper traingular is preserved even if one extends the model tu

have greater than twvo priority classes.

Chapter 5 can be incorporated as a part of a larger optimization problem

to optimize the cost, Le., efficiency of the systern. We can associate different

cost constraints for different semer operating modes, viz., normal and fast

service and the customers waiting times.

Similar to the e.xtentions mentioned For chapter 4, the first extension to

chapter 6 would be to generalize the model to handle PH or MAP arrivals.

The model can be extended to accomodate more classes of customers. We

can also extend the tandem arrangemerit for more than 2 nodes. A11 these

extensions will have simiiar structure for the R m a t r k not only a t the first

node, but also a t the subsequent nodes. Fiaally, this model d s o provides

insight for developing networks involving priority arrangement.

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Appendix A

Phase-Type Distribution

The purpose of this appendk is to give a brief description and properties

of phase type distributions. The Phase-type distibutions are very widely

used for modelling queues, communication systems, dams and inventories. A

det ailed description of this family of distributions can be found in Neuts[26].

-4 brief summary of the key properties are presented here.

Consider a Markov process on the state space {1,2, . . . , m + 1) where

{1,2, . . . , fm) are transient states and m+l is an absorbing state. The in-

finitesimal generator Q of such a process has the forrn:

The m x m matrix T has negative diagonal entried Tii, i = 1,2, . . . ,m.

ri represents the exit rate from the state i. The other entries, mhich are

transition rates from state i to other states, are therefore non-negative. The

m x 1 vector to= -Te, where e is a vector of lys, contains the rates at which

transitions occur frorn the individual transient States to the absorbing state.

Let the process start in state i with probability ai, i = 1,2,. . . , m f 1, and

let a = ( a l , . . . , a,). (In many practical problerns, a,+1 = O.) Now let F ( x )

denote the distribution of the time to absorption, X, into state rn + 1. The

distribution F(.) is said to be of "Phase type with representation (a, T)" .

The probability distribution function P(x) is aven by

F ( x ) =1- a exp ((Tx) e; x 2 0 .

Assuming a,,~ = 0, its probability density function f (x) is

l ( x ) = c u exp (Tx) e; x 1 0 .

The special cases of phase-type distributions that are used are the expo-

nential, mixtures of LW exponential and Erlang-k (a special case of gamma

distributions with integer shape parameter) distributions.

1) Exponential

In the case of the exponential distribution, d l of the rnatrîx quantities

reduce to scalar results, namely:

a = [il? T = [-A], to = [A],

2) Mixtures of Exponentials

This is a generalisation of the exponential distribution. It c m be used to

mode1 processes that are more bursty in nature. Variability or burstiness is

often measured by the squared coefficient of the process c2, which is defined

by c? = V a r [ X ] / ( E [ X ] ) 2 . The c2 for the mixture of exponentials is greater

than 1. The Phase-type formulation of a mixture of hi exponentials has the

following form:

f (r) = C ~ ~ , \ ~ e ' ' ; z 2 0.

3) Erlang-k

The Erlsng-k distribution is a special case of the family of gamma distri-

butions. It has integer-valued shape parameter k since it is a kfold convolu-

tion of the evponential distribution. It is often used to mode1 distributions

which are less variable than the exponential, Le., c2 being less than 1.

The phase-type representation of the Erlang-k can be stated as a process

successively moving t hrough States 1 through m prior to absorption into state

Appendix B

Markovian Arriva1 Process (MAP)

In order to better understand the MAP, we start with the fimilar Poisson pro-

cess. Suppose there is a continuous time Markov process with one transient

state and one absorbing state. Since sojourn times in a Markov process are

exponential, a sojourn time in the transient state expires after an exponen-

tially distributed interval with rate, say A. Upon getting into the absorbing

state, the process is instantaneously started in the transient state. If we

construct a point process by associating an arriva1 Nith each transition in

the above Markov process, then the resulting process is Poisson. Now based

on this constructive definition, we can generalize Poisson process by adding

additional t ransient states to the Markov process and associating arrivals in

the point process with certain transitions in the underlying Markov process.

Suppose we have an irreducible Markov chah vith m transient states and

one absorption state. Each time the Markov chah jumps from a transient

state to absorbing state, we let this event represent an arrivai to the systern.

-4ssurne that a t an arbitrary time epoch to, the Markov chain is in state i

and its sojourn time in this state is exponentially distributed with parameter

hi. Then a t the end of the sojourn, either one of the following two events

must occur: 1) a transition from state i to state j through an absorption; let

this event, corresponding to an arrival of a packet, occur with probability p i j ,

1 i, j 5 m; 2) a transition from i to j; let this event, which corresponds

to no arrival, happen with probability qij, j # i, 1 5 i, j 5 m. Note that the

Markov chain can go from state i to state i (same state) only through an

arrival.

Thus, it is easy to see that

C P i j + C Gj = 1, for al1 1 5 i Ç m.

It is convenient to represent the evolution of the system in terms of ma-

trices. We define two square matrices C = [cij] and D = [dij] of order m

(that is the number OF the transient states of the underlying Markov chain)

S U C ~ that = -Xi, 15 i 5 m, C+ = Xiqijti, for j # i, 15 i,j,# mij and

dïj = hp. Thus, the mat rk C governs the transitions corresponding to no

arrivals and D govems those corresponding to an arrival. The C rnat6.x has

strïctly negative diagonal elements, nonnegative off-diagonal elements: D has

al1 noonegative elements such that C+D is a generator. We assume that C is

nonsingular, so that with probability one the interamval times are finite (see

Lemma 2.2.1 of Neuts [26]) and that the arriva1 process does not terminate.

In other words, C is a stable matrix (i.e., al1 of its eigenvalues have negative

real parts, sec Bellman [3]).

The generator Q of the Markov chain is then defined as

Many familar renewal and nonrenewal processes can be obtained as special

cases of MAP. Below we list a few of some commonly used processes that are

used in stochastic rnodelling.

(1). Poisson Process: Here the underlying Markov chain has only one

transient state (m=l). By setting C = -A, D = A, we get the hmilar

Poisson process.

(2). PH-renewal process: By taking interarrival times to be of PH-type

with representation (p, S) of order m, the renewal process is referred to as

PH-renewal process. The M.4P representation for this process is obtained by

taking C = S, D = so@ where so = -Se. -4s mentioned earlier, PH-renewal - - process includes Exponentiai, Erlang (generaüzed), hyperexponential, as well

as finite mixtures and finite convolutions of these.

(3). Markov-Modulated Poissom Process (MMPP): This process is also

known as doubly stochastic Poisson Process, in which arrivals occur accord-

ing to a Poisson process with parameter, say Xi, whenever the underlying

hfarkov chah is in state i. A special property of MMPP is that if an ar-

rivai occurs when the blarkov chah makes a transition from state i, then the

Markov chain rvilï corne back to the same state i with probability one. If the

underlying Markov chain has a generator Q, then the MAP representation of

MMPP is D, a diagonal matrix with entries Xi, XI , . . . , Am dong the diagonal,

and C=Q-D. It should be noted that a trvo-state rvlMPP in which only one

Ai is positive is the interrupted Poisson process. MMPP is very widely used

in data communications.

(4). Markov-Switched Poisson Process (MSPP): This process is the one

in which a realization consists of geometric runs of exponential duration

whose parameters (within each run) depend on the state of the discrete-time

Markov chain with transition probability rnatriv P. For this process, an amval

will always occur whenever the underlying MC makes a transition, therefore

the MAP representation is given by C=-il, a diagonal matriv with entries

cS1, &,.. . ,6,, and D = AP. It shodd be noted that any Markov renewal

process with exponential sojoum times is MSPP. -4n application of this type

of correlated process is very useful in telecornmunication networks.

(5). Sequence of PH-interarrivai times selected via a Markov chain: La-

touche [23] introduced a point process in which successive interasrival times,

that are assumed to be of PH-type, are chosen according to the state of a

Markov chain. For example, consider a two-state Markov chain 6 t h transi-

tion probability matrix P=pij. Let the two PH-distributions have represen-

tations ( ~ ( l ) , - L(1)) and (y(2), - L(1)) . Then the resulting point process can

be described by a MAP with representation given by

where - Lo( i ) = -L( i )g for i = 1,2. A special case of this process is an arriva1

process with Erlang inter-amval times, where the order of successive Erlang

variables form s Markov chain.

Appendix C

A Brief Review of the State Reduct ion Met hod

Let Q = { q i j } denote the generator of a continuous-time Markov chain with

3 States and p denote the stationary probaility vector. The state reduction

method for cornputing p is summarized below:

1. For n = N, !V - 1 .... 2, compute

2. Upon completion of step 1, arbitrarily set pl=l and compute

3. After normalizing the sum to one, the stationary probability vector p

is obtained by the relation:

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