matrix - library and archives canada · i'rn ever thankful to prof. chakravarthy for providing...
TRANSCRIPT
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
National tibrary I * m of Canada Bibliothèque nationale du Canada
Acquisitions and Acquisitions et Bibliographie Services services bibliographiques
395 Wellington Street 395, nie Wellington OttawaON K1A O N 4 Ottawa ON Kt A ON4 Canada Canada
The author has granted a non- L'auteur a accordé une licence non exclusive licence dowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seU reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fih, de
reproduction sur papier ou sur format électronique.
The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.
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.
Bibliography
[l] Asmussen, S., Applied Probability and Queues, Wiley, New York, 1987.
[2] Alfa, A.S., 1995. hlatrix-Geometric Solution of a Discrete Time
MAP/PH/l Queue. Submitted for publication. *
[3] Bellman, R.E.: Introductàon to matrix analysis, McGraw Hill, New York,
NY, 1960.
[4] Bertsimas, D., Keilson, J., Nakazato, D., and Zhang, H., 1991. Transient
and busy period analysis of the GI/G/l queue as a Hilbert factorization
problem. J. Appl. Prob 28, 873-885.
[j] Bertismas, D. and hlourtzinou, G., -4 unified method to analyse overtake
free queueing systems. Journal of Applied Pro bability (to appear) .
[6] Bitran, G.R. and Timpati, D.; .iIultiproduct queuing networks with
deterministic routing: Decomposition apporach and the notion of inter-
ference. Management Science, 34, 75-1 00.
[7] Buzacott, J.A. and Shantikumar, J.G., On approximate queueing mod-
els of dynamic job shop. Management Science, 31, 870-887.
[8] Chakravarthy, S., 1996. Analysis of the bIAP/PH/l/K queue with ser-
vice cont rol Preprint.
[9] Cobham, A., 1954. Priority assignrnent in waiting line problems. Opns.
Res. 2, 70-76.
[IO] Cohen, LW., 1982. The single semer queue. North-Holland, .Amsterdam.
[ I l ] Conway, R.W., W.L. Maxwell, and L.W. Miller., 1967. Theonj of
Scheduling. Addison-Wesley, Reading, MA.
[12] Crabill, T.B., Gross, D., and Magazine, M. J., 1977. A classified bibliog-
raphy of research on optimal design and control of queues. Operations
Research 25, 219-232.
[13] Fischer, W. 1989. Waiting times in priority systems with general arrivais.
INRS Technical Report 89-34, INRS-TeIecommunications, Verdun, Que-
bec, Canada.
(141 Fischer, W., and Stanford, D., 1992. Approximations for the per-class
waiting time and interdepaxture time in the x, GIi/Gli/l queue, Per-
formance Evahation 14, 65-78.
[l5] Grassmann, 1986. The PHX/h.l jc Queue, Selecta Statistica Canadiana,
~01.7, 35-52.
[16] Gelenbe, E., and Mitrani, T., 1980. Analysis and Synthesis of Cornputer
Systems, Acadernic Press, London.
[17] Holley, J.L., 1954. Waiting lines subject to priorities. Opns. Res. 2, 341-
343.
[18] Hooke, J.X.? 1972. -4 priority queue with low priority arrivals general.
Opns. Res. 20, 373-350.
[19] Jackson, J.R., 1957. Network of waiting lines. Oper. Res. 5, 518-521.
[20] Jaistval, N.K., 1968. Priority Queues. -4cademic Press, New York.
[21] Kesten, H., and J.TH. Runnenburg. 1957. Priority in waiting lines prob-
lems. Nederl. Akad. Wetensch. Indagationes Math. 60, 312-336.
(221 KIeinrock, L., Queuezng Systerns Volume II, Wiley, New York 1976.
[23] Latouche, G, 1992. Xlgorithrns for infinite Markov chains with repeating
columns. IMA workshop on linear algebra, Markov chains and queueing
.models, Janurary, 1992.
[241 Miller, R.G., 1960. Priorîty Queues. Ann. Math. Statist. 31, 86-103.
[25] Miller, D.R., 1981. Computation of steady-state probabilities for M/M/1
priority queues. Opns. Res. 29, 945-958.
[26] Neuts, M.F., Matrix-Geometric solutions in stochastic models, The John
Hopkins University Press, Baltimore, 1981.
[27] Neuts, N F , and B.M. Rao 1992. On the design of a finite-capacity queue
with phase-type service times and hysteretic control. European Journal
of Operational Research 62, 221-240.
[28] Ott, T.J., The single-server queue with independent GI/G and M/G
input streams 1987. Adv. Appl. Prob. 19, 266-286.
[29] Reiman, M.1 and Simon, B 1990. A network oipriority queues in heavy
traffic one bot tleneck station. Queuein9 Sgstems 6, 33-58.
[30] Schmidt, V., 1984. The stationary waiting time precess in single-semer
priority queus with general low priority arriva1 process. Math. Opera-
tionsforsch. u. Statist. ser. optimization 15, 301-312.
[31] Schassberger, R., 1974. A broad analysis of single server priority queues
tvith two independent input streams, one of them Poisson. Adv. Appl.
P d . 6, 666-688.
[32] Stanford, D.A., 1994. Waiting and interdepature times in priority queues
with Poisson- and general-anival streams. Revised submission to Opns.
Res. in preparation.
[33] Stanford, DA., and Grassmann, W.K., 1992. The bilingual server sys-
tem: -4 queueing mode1 featuring fully and partially qualified servers.
INFOR 31, 261-277.
(341 Sumita, S., 1986. An application of the conservation law to analyzing sin-
gle server queueing systems with two independent input streams. Trans.
IECE Japan 69, 628-637.
[35] Stewart, W.J., Introduction to the numerical solution O/ Markou chains,
Princeton University Press, Princeton, New Jersey, 1994.
[36] Takagi, H., 1991. Queuezng Analysis: A Foandatzon of Performance
Evahation, Vol 1: Vacation and Priority Systems, Part 1, North-
Holland, -Amsterdam.
[37] Wagner, D., 1994. Analysis of a multi-semer with non-preemptive prior-
ities and non renewal input. The Fundamental Role of Teletrafic in the
Evolution of Telecommunzcations Netuorks, Elsevier, Amsterdam, pp.
737-766.
[38] Wagner, D., 1995. Analysis of a finite capacity multi-server mode1 with
non-preemptive prionties and non-renewd input (to appear) .
[39] Whitt, W., The queueing network analyzer. Bell Tech. J. 62 (9) (1983),
2779-2815).
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:
APPLIED 2 IMAGE. Inc t= 1653 East Main Street - -- , ,, Rochester. NY 14609 USA
-L --= Phone: 71 6/482-0300 -- - - = Fax 71Wûû-5989