stochastic petri nets with product form equilibrium ... · contents list of figures list of tables...

e+'3..r\

University of AdelaideDepartment of Applied Mathematics

PhD Thesis

Stochastic Petri NetsWith Product Form

Equilibrium Distributions

James L. Coleman B.Sc.(Ma)Hons.15th December, 1993

Ar,,,a'.'ø\ ¡' ¿1 lcii:1 ¡.,.

Contents

List of Figures

List of Tables

Summary

Declaration

Acknowledgements

1 Introduction

1.1 Preliminaries

1.2 Petri Nets

1.3 Properties of Nets

1.3.1 Behavioural Properties

L.3.2 Structural Properties

I.4 Stochastic Petri Nets

1.5 Conflict Resolution

vlll

v

vll

x

xt

8

I

1

1

2

10

13

t7

191.5.1 Preselection Policy

L.5.2 Race Policy

1.6 Some Other Petri Net Extensions

2 Petri Net Analysis

2.I Structural Analysis

2.1.1 Reachability Tree/Coverability Tree Method

2.1.2 Matrix Equation Method

2.1.3 Reduction

2.2 SPN Analysis

2.2.L Discrete Time

2.2.2 Continuous Time

2.2.3 Remarks

2.3 Product-Form...

2.3.1 Existing Product-Form Results in Petri Nets .

3 A Class of Product-Form Nets

3.1 Instantaneous Firings

3.1.1 Examples

3.1.2 Product-Form

3.1.3 The Role of the T-invariants

3.1.4 The Role of the Function ty'

3.2 Delayed Firings

ll

3.2.I The Model .

3.2.2 Product-Form

4 Computational Algorithms

4.L S-Invariant Reachable Petri Nets

4.2 Performance Measures

5 Convolution Algorithms

5.1 The State Dependent Case .

5.2 The State Independent Case

5.3 Example

6 ComplementaryConvolutions

6.1 Loss Networks

6.2 Markovian Petri Nets

6.3 Example

7 Utilisation Recursron

7.t Markovian Petri Nets

7.2 Single Movement Queueing Networks

7.3 Examples

8 Conclusions

A Proofs

87

99

97

103

106

106

111

113

118

118

t25

127

r27

133

136

140

t42

. r22

4.1 Complementary Convolution Proof t42

A.2 Boundary Conditions for the Complementary Convolution

4.3 Finding the Summation Bounds

B Basic Notation

. t43

t44

L46

IV

List of Figures

1.1 Car Assembly Plant

1.2 Inhibitor and Escape Arcs

2.1 A Simple Marked Petri Net

2.2 The Reachability Tree

2.3 The Reachability Graph

2.4 An Unbounded Marked Petri Net

2.5 The Coverability Tree

2.6 The Dining Phìlosophers

2.7 Reachability Graph

2.8 Markov Chain for Philosopher i

2.9 Triggered MPN with Product-Form

3.1 A Live and Closed SPN without Product-Form

3.2 Reachability Graph and Routing Chain

3.3 A Simple Loss Network

3.4 Closed MPN Equivalent

3

2L

25

25

26

27

28

43

46

53

Ðb

64

65

66

67

7T3.5 Computer System Network

3.6 Addition of Firing Times

3.7 Net without Firing Times

3.8 Addition of Firing Times

5.1 Example 5.1

5.2 Graph from Table 5.1

6.1 Complementary Region .

83

90

91

113

TT7

119

t256.2 Simple Switch .

7.1 Example 7.1

7.2 Queueing Network

r37

138

VI

List of Tables

1.1 Typical Interpretations of Transitions and Places

2.1 Necessary and Sufficient Conditions for some Structural Properties . 34

5.1 Time Compartsons 115

3



6.1 Blocking for a Simple Switch

7.I Time Comparison

7.2 Time Comparisons

. 126

115

116

139

r37

vll

Sumrnary

Networks of queues with product-form equilibrium distributions are well established

and have applications in a wide range of fields. With few exceptions these networks

provide the only efficient means of analysis in large systems. The advantage that

the product-form distribution provides is the abundance of efficient computational

algorithms for calculating various performance measures either directly or through the

normalising constant. These algorithms help to overcome the state space explosion

problem which is inherent in large networks. For this reason large amounts of effort

have been invested by researchers in extending the set of queueing networks that have

this special form.

In recent years the theory of product-form queueing networks has evolved beyond the

single movement restriction to include networks of queues which allow batch movement

of customers and correlated routing. Petri nets, a modelling tool attracting increasing

interest, can be used to model such networks. Numerical algorithms based on the

product-form equilibrium distribution of these batch movement networks have not

been considered by many researchers. The state space explosion problem still exists

for these systems, as it does for Petri nets in general.

Increased effort is now being devoted to finding classes of Petri nets which possess

product-form equilibrium distributions, and associated algorithms for evaluating per-

formance measures.

The purpose of this thesis is threefold: To survey the product-form results and asso-

ciated algorithms which currently exist in Petri nets, extend the product-form results

VIII

for certain classes of Petri nets, and derive and survey algorithms for evaluating nor-

malising constants in product-form stochastic Petri nets.

Chapter 1 introduces Petri nets, their time extended counterparts, and some related

issues. The chapter concludes with a brief account of other Petri net extensions.

Chapter 2 considers the analysis of timed and untimed Petri nets. Various classes of

time extended nets are discussed followed by a survey of existing product-form results.

In Chapter 3 we extend the knowledge of a specific class of product-form Petri nets

initially proposed by other authors.

In the remaining chapters we consider several methods for evaluating the normalising

constants for product-form networks in general by identifying the important factors

for the existence of convolution type algorithms.

In the conclusion we present some ideas for further research which arises naturally

from the work contained in this thesis.

Throughout the thesis, vectors are denoted by bold face characters, matrices by bold

face capitals and sets by curly capitals. In Appendix B we summarise the notation

and symbols used.

IX

Acknowledgements

There are many people that I would like to sincerely thank for making the production

of this thesis possible.

Firstly I would like to thank both of my supervisors, Bill Henderson and Peter Taylor,

for their encouragement and direction over the past years and for providing me with

a topic which has borne much fruit, for which I am ever grateful.

In addition to my supervisors, I would also like to thank Charles Pearce for many

invaluable discussions regarding various aspects of my research.

Thanks also goes to Bruce Northcote for proof reading various portions of this thesis

as it neated completion.

Finally, I would like to thank my wife Nadia for her love and support through a busy

period in my life.

XI

Chapter 1

Introduction

Systems of complex inter-dependent information flow occur naturally in a vast and

diverse range of fields. There are numerous modelling techniques that can be applied

in order to tackle any one particular example. The aim of this thesis is to introdllce a

modelling tool which continues to find new applications in the mathematical modelling

field, and to take the reader along a path which specialises in an area which is currently

receiving much attention, the product form stochastic Petri net.

1.1 Preliminaries

petri nets (PNs) originated in the PhD dissertation of C.A. Petri [82]. They are graph

models capable of describing systems which exhibit sequential' concurrent, dependent

and conflicting behaviour and provide a graphical interface which is common ground

to both the modeller and the theoretician.

Since conception, PNs have been modified to improve their modelling power with an

ever increasing amount of research being devoted to them worldwide, resulting in an

increased understanding and awareness of the issues which still need to be addressed.

This evolution along with the basic notation and definitions is the basis of this chapter.

I

1. 2 Pef,ri Nets

L.2 Petri Nets

A mathematical model of a real-life system is an attempt to capture mathematically

the pertinent properties and features of the system by making certain assumptions

whilst discarding those which are not relevant. The model is then used to make

predictions on future behaviour based on various initial conditions. To construct an

effective model it is therefore necessary to break the system down into a set of events

and conditions which capture the features of interest.

Given any system that we wish to model one can always define a set of conditions, a set

of possible events and the event-condition dependencies, where the state of the system

at any time is the state of each condition, and the system changes state after the

occurrence of an event. For example consider a flexible manufacturing system (FMS)

such as a motor car production plant. The set of conditions is the availability of car

components, personnel, and machinery, with the set of events being the transitions

between various stages of car assembly. The flow of car components in various stages

of assembly throughout the factory, is perfectly suited to modelling through Petri nets.

This is just one of a vast number of direct applications of PNs and their extensions.

Other examples are communications protocols, compiler and operating systems, logic

programs, discrete-event systems and neural networks [68].

In the simplest of Petri nets conditions are represented as circles, events as rectangles

and dependencies as directed lines. In Petri net terminology, the circles are known as

places,the rectangles as transitionsand the directed lines as arcs. A dot or toleen in a

place indicates that the condition associated with that place is true. Nets of this type

are known simply as euent-cond,ition Petri nets. Each place in an event-condition net

can hold at most one token since clearly a condition is either true or false [70].

Event-condition nets however, are not always appropriate for modelling more complex

systems. Generally speaking, places can accommodate any number of tokens and

places and transitions are not restricted to representing only conditions and events.

2

1-2 ri l\Tets

Input Places Transitions Output Places

PreconditionsInput Data

Input SignalsResources Needed

ConditionsBuffers

EventComputation StepSignal Processor

Task or JobClause in Logic

Processor

PostconditionsOutput Data

Output SignalsResources Released

Conclusion(s)Buffers

oaoo

P P P2 3

Table 1.1: Typical Interpretations of Transitions and Places

Pr Pz P¡ P Pz P¡1I

Pq Pr 4

Pe P6 P6

A B c

Figure 1.1: Car Assembly Plant

For example, a place could signify that a certain item is present with the number of

tokens indicating how many items there are. Transitions as well as places can have

many interpretations. Some typical place and transition interpretations are given in

Table 1.1, taken from [68]. Input and output places are described in the example

below.

Consider Figure 1.1 which is a sequence of Petri net models for the above mentioned

car assembly plant. It considers one aspect of car assembly, namely the joining of

P

3

1-2 ni l\Tcts

the front windscreen to the car body. The interpretations of the transitions and the

presence of tokens in the places are as follows:

Pr: Number of Car Bodies

Pz: Number of Windscreens

P3: Units of Sealant

Pa: Operation in Progress

Ps: Machine Available

P6: Number of Cars with Windscreens Fitted

tt Operation Start

tz: Operation finish

We have assumed there are 4 necessary requirements for the operation to occur, the

presence of a car body, a piece of windscreen glass, adequate sealant, and a working

robot. The tokens in each of the places Pt, Pz,Pa indicate how many times each

requirement is satisfied. For example the 2 tokens in place Pz indicates that there are

2 windscreens available. The token in place P5 indicates that the machine is available.

When the operation takes place transition ú1 is said to fi,re upon which a single token

is removed from each of the input places Pt, Pz, and P5, two tokens are removed from

input place P3, and one token is placed in the single output place, Pa, as depicted

in Figure 1.18. An input place of a transition is any place which has directed lines

from that place to the transition. Similarly output places have directed lines from

the transition to those places. There are two directed lines from input place P3 to

the single transition ú1 implying that two units of sealant are required to complete

the task. This arc is said to have a weight of 2. The vector of the weights of arcs

connecting places to transitions and transitions to places are called input bags and

output óøgs respectively. The input and output bags of transition f1 in Figure 1.1 are

(1,Ir2,0,1,0) and (0,0,0,1,0,0) respectively. A token in place & indicates that the

operation is in progress. When the operation is completed, transition t2 frtes which

4

1.2 Pef.ri l\ets

removes a token from Pa and places a token in each of P5 and P6 indicating the machine

to be available and the presence of a car body with the window fitted, Figure 1.1C.

Note that Figure 1.1 represents a very small portion of a complete FMS. Tokens in

the resource places Pt, P, and P3 would be replenished from the firing of transitions

"outside" the Petri net shown and place P6 would be an input place for some other

"outside" transition.

Reisig [83] describes PNs as constructions consisting of two sorts of objects, places

and transitions. The relationships between places and transitions are represented as

directed arcs which always connect objects of different types. That is an arc will always

connect a place to a transition or a transition to a place but never a place to a place

or a transition to a transition.

PNs have a finite or an infinite state space depending on the net structure and place

and transition interpretatìon. A state is a possible token distribution throughout the

places and is called a marking, denoted by the vector ¡¡ : (ne(l), . . . , m(N P))r where

rn(i) denotes the number of tokens present in place 4 and ,¡/P is the number of places

in the net. The initial state is called the initial marking, ms, which for the windscreen

example may be seen in Figure 1.14 to be (1,2,2,0,1,0)t. The set of all places

(po, i: L,..., ¡úP) is denoted by P. The marking for a subset of places P' will be

denoted by m(2') : (m(i) | P¿ € P')'.

PNs evolve by changing state, which occurs each time a transition fires. A transition

has the potential to fire in a particular marking if the number of tokens in each input

place equals or exceeds the weight of each input arc. A transition with the potential

to fire is said to be enabled. If it can then also fire, it is called fi,rable. A transition

can be enabled but still unable to fire (see Section 1.5). Upon firing, the input bag of

tokens is removed from the net and the output bag corresponding to the fired transition

deposited. If the marking m' can be attained from m through a sequence of transition

firings we say that m' is reachable from m. The set of all markings reachable from

the initial marking constitutes the state space of the Petri net and is known as the

5

1_2 rl l\üets

reachability set.

These concepts and associated notation are defined formally in the following:

Definition 1.1 - Marked Petri Net

A Marked, PN, (N,ms), is a weighteil bipartite directed graph with nodes being either

places or transitions and, is d'efined by the ,-tuple N : (P,T,A',rrlo,W), where

o P: (Pr,.. ., PNp) is a finite set of places.

o T : (úr,..., ú¡rr) is a finite set of transitions.

o AÇ (P xT)U(7 x P) is a set of arcs

. nro : P -+ {0,1,2,' ''} it the initial marleing.

o W : A -+ {I,2,"'} assigns a weight to each arc in A'

Definition L.2 - Input Bag

The input bag for transitionti'I(ti): (/1(f¡), ...,Ixp(t¡)) is d'ef'ned by

o I¿(t¡):W(p¿,t¡) Vie (f,...,NP).

Definition 1.3 - Output Bag

The output bag for transition t, O(ti): (O1(¿), . . ., O¡'vp(t)) is d'ef'ned by

o I;(t¡):W(t¡,P¿) Vie (1,...,NP).

Definition L.4 - tansition Enabling

A transition t¡ is enabled, in a marlcing rn iff rr, > I(¿¡) (component-wise).

Definition 1.5 - Immediate Reachability

A marleing m' is immeiliately reachable from a marking rn if 1 a transition t¡ € T

6

\.2 Petri Nets

fi,rable in rn which, when fireil, proiluces the new marleing rn'. We write * 3 rn' and'

*3Definition 1.6 - Legal Firing Sequence (tFS)

Let o: {úr,..., úry} denote a sequence of N transitions. The sequence is a LFS from

sorne rnolking rn iff

. *5trr1, Irtl 3^r,...rmN-r3**

We write m å and rn å -N.

Definition L.7 - Firing Vector (Parikh Mapping)

The uector ã : (õt,. . . , õNr) is the firing uector of the LFS o where õ; is equal to the

number of occurrences of transition t¿ in the LFS o. The firi'ng uector is known as the

Parikh mapping of its LFS.

Definition 1.8 - ReachabilitY

A marking m' is said to be reachable from son'Ie n'¿arking rn if there eústs a LFS o

suchthatmåm'.

Definition 1.9 - ReachabilitY Set

Giuen an initial marleing rns, the set of reachable marlcings, the reachability set, is

giuen by

7?(*o) : {- e ZIP : ) a LFS d, ûrs å *} .

If we wish to relate the reachability set to a particular PN, N , we can write R(N ,,rns).

Definition 1.10 - Enabled Set

For each marleing rn in the reachability graph of a PN we call the set of enabled

transitions the enabled set tS(m) , defined by

SS(m):{tj €T z m2I(t¡)}.

I


1-.3 Properties of Nets

There are many properties of SPNs that make them an ideal modelling tool for a

number of specialist applications. For example in the design of our car assembly plant

we need. to ensure that the factory is operational for as long as possible and production

isn't restricted through poor design. In PN terminology, if no transition can fire, the

net is said to be in d,ead,lock. Deadlock is one of many net properties that we cover in

this section.

Net properties are distinguished through their marking dependence. Properties which

are marking dependent are known as behauioural properúies while those which are

independent of any marking are structural properties. Many of the definitions included

here have been taken from either Peterson [70] or Murata [68]'

To discuss net properties, we need to define the incidence matrix, which describes the

net structure independent of timings or markings.

Definition 1.11 - Incidence Matrtx

Let J{ be a PN with N P places and NT transitions. The incidence matrit, A', is an

lú? x N P matrir with entries giuen bY

a¿¡ : W(t;, P¡) - W(P¡,t¿)

where W is the weight function.

Entry ø¿¡ is the total change in the number of tokens in place j when transition i fires.

The incidence matrix is commonly used in the literature with some authors using the

transpose of our definition.

8


1.3.1 Behavioural ProPerties

The definitions of this section are for a PN ,Â/, with initial marking ms'

Definition 1.12 - ReachabilitY

Reachability is one of the funilarnental problems for PNs and is defi,ned in Definitions

1.5 and, 1.8. The reachabitity problem for Petri nets i's calculating whether or not

rn' e R(N, ms) /or a giuen marlcing rn' and marked net (Al,rns)'

There rnay be instances where one nxay only be interesteil in the marki'ngs which are

possible for a subset of places. This is known as the sub-marleing reachability problem.

It has been shown that the reachability problem is decidable, but talees at least etponen-

tial space and, time, in the number of places and transitions, to uerify in the general

case [68].

Definition 1.13 - Boundedness

In a bouniled, net the number of toleens in each place is restri,cted to be less than some

upper bounil. For erample, if the upper bounil is ilefined by an integer Ic, then m(i) 3 k

Y p, e P anil V m € R(N,rns) and the net is called k-bounded or k-safe in this case.

If k : I the net is saiil to be safe. The boundedness of a net may be natural or forced,

ft01. U it is forceil, this means that transitions may be enabled but unable to fire as the

resultant m,arleings would, uiolate the boundedness criteria: this is lcnown øs blocking.

Definition 1.14 - Liveness

A transition t¡ is liue if, for euery reachable marlcing rn, there eústs a marloing rn'

reachable frorn rn in which t¡ is firable. In other words, from any reachable marleing,

transition t¡ can ultimately fi,re again. A net is liue in some initial marking rno il

euerA transition is liue in rns. Various leuels of liueness haue been defined and, can be

founil in [65]. Necessary anil sufficient conditions forliueness haue also been giuen for

I


particular classes of nets, [30].

Definition 1.15 - Deadlock

Deaillock in a net is a rnarlcing in which no transition can fire, thereby halting erecution

of the net. Liueness anil d,eadlocle are closely related as a liue net can neuer haue

deadlock and a net with a d,eadlock marlcing cl,n neuer be liue.

Definition 1.16 - Coverability

Giuen a net (tr/, *o) and marleing rn e R(N,mo), m is couerable if there erists

another marking n' e R(N,rns) such that rn' ) rn (component-wise).

1.3.2 Structural Properties

The first structural properties that we consider are net invariants. Invariants play an

important role in both PNs and their time extended counterparts which we consider

in Section 1.4.

Definition 1.17 S-Invariant

A non-negatiue integer uector, s I 0 which is a solution to the equation

As :0

is called an S-inuariant. It is also lenown as a place inuariant, P-inuariant, or P-

semifl,ow in the literature. S-inuariants haue the property that the dot (inner) proiluct

of any S-inuariant with any marlcing is constant. They play an important role in the

work of following chapters.

S-inuariants haue a strong relati,on to bouniledness in that a net whose places are

couered, by S-inuariants rnust be naturally bounded. By couered, uJe rneûn that each

place is represented in at least one S-inuariant.

10

1.3 Prop erties of Nets

Definition 1.18 T-Invariant

A non-negatiue integer uector, t+O, which is a solution to the equation

ATt: o

is called, a T-inuariant, also lenown as a T-semifl,ow or transition inuariant. If a T-

inuariant is consiilereil to be a firi,ng uector, the marking of d net wíll be unchanged

upon the f,ring of a T-inuariant. In the notation ilefi,ned, earlier, for any marlcing

m € 7t(mo) anil firable T-inuariantt we can write * S *. They define potential

Ioops or cycles in the reachability graph and also play an important role in the work of

this thesis.

Definition 1.19 - Invariant Supports

For any S-inuariant s or T-inuariant t, the set of places or transitions respectiuely

which haue strictly positiue conxponents form the support of the inuariant and are

d,enoteil óg ll r ll and ll t ll respectiuely. As the standard notation for the norm of a

uector is not used elsewhere, this notation should not create any confusion.

Definition 1.20 - Minimal Invariants

The support of an S or T-inuariant is saiil to be minimal if no subset of the support

is the support of some other inuariant. An inuariant uector v is saiil to be minimal

if there is no other inuariant uector v" such that u.(i) < ,(i) for all i. A minimal

inuariant corresponding to a minimal support is known ¿s ¿ minimal support invariant.

The set of all minimal support inuariants form a generator for all inuariants [65], [63]

It is interesting to note that in general, the set of minimal support inuari'ants i's not

lin early in il ep en d ent.

Definition L.2L S-invariant Matrix

Let the number of linearly independent minimal support S-inuariants be ilenoted by N S .

The N S x ,^rIP matrir S whose rows are a set of linearly independent minimal support

S-inuariants is known as the S-inuariant rnatrir. We may want to refer to a sub-matrit

of S in which case ue write S(P') to represent the sub-matrfu of S restricted to the

11


columns in the set Pt. This def,nition is new and does not o,ppeo,r in the li'terature.

Definition L.22 - Structural Liveness

A net, N, is said to be structurally tiue if there exists an initial marleing for which the

net is liue.

Definition 1.23 - Structural Boundedness

A net, Al, is structurally boundeil if it is bounded for all initial marleings.

Definition 1.24 - Conservativeness

A net, Al, is conseruatiue if there eri,sts a positiue integer uector v such that

mTv: moTv : constant

for all initial marlcings rns and marleings rn e R(N, mo). If the conilition is satisfied,

for a subset of places, that is, some elements of v úre zero, then the net is partially

conseruatiue. Recall that the inner product of any S-inuariant with any marlcing is

constant. Hence a net which is couered by S-inuariants is conseruatiue.

Definition 1.25 - Repetitiveness

A net, Al, is repetitiue if from sorne marleing there erists a LFS in which euery tran-

sition fires infi,nitely often. If only a subset of transitions fires infinitely often the net

is said to be partially repetitiue.

Note that if the repetitiue property was satisfied for euery reachable marking, the net

would be liue.

Definition 1.26 - Consistency

A net, Al is consistent if fron't, sonle rnarlcing rn there erists a LFS which talees the net

baclc to rn in which euery transition fires at least once. On comparing thi's definition

with that of T-inuariants, it is clear that consistency is equiualent to a net being couered

by T-inuariants.

t2

14 Stoehastie ni l\Tets

If the consistency condition is only satisfied for a subset of transitions, the net is said

to be partially consistent.

The above structural and behavioural properties play an important role in the Petri

net models of various systems. For example in a net model of a computer network

consisting of buffers and registers it is important for the net to be bounded which

guarantees that overflow cannot occur. It is therefore important to be able to establish

which properties hold for a given Petri net. We consider the analysis of the above

structural and behavioural properties in Chapter 2.

L.4 Stochastic Petri Nets

Petri nets as conceived were limited in their modelling power because there was no

concept of time involved. Before the advent of time, PNs were used predominantly for

protocol and software verification which were time independent.

In order to gain a measure of system performance, the ability to allow for the different

time requirements of various events is essential. Knowledge of performance measures

such as throughput, waiting time, and utilisation provide crucial information in as-

sessing a system's efficiency. Such performance analysis can be based on either the

transient or equilibrium behaviour of a system but in this thesis we only consider the

equilibrium or long term behaviour of a system's performance.

In the literature, time has been incorporated mostly in the form of delays on transi-

tions, but researchers have also investigated time delays on the places, [79], [91]. It

has been shown that under certain conditions, nets with timed places are equivalent to

nets with timed transitions [S0]. In this thesis we are concerned only with transition

delays.

As the marking process of a net evolves, we consider two distinct time periods. The

time taken for a transition to fire once it is enabled is known as the enabling tirne.

13

1.4 Stochast r Pef.rt Nets

After firing, tokens are removed from the input places but are not deposited into the

output places until a period of time, the firing time, has elapsed. Non-zero firing

times imply that tokens are missing from the net for a finite period: the tokens can

be thought of as being absorbed by the transitions.

The firing and enabling times of each transition can be described in a number of

ways. They may be deterministic with a positive rational number determining the

appropriate delay, or probabilistic with the delay length drawn from either a continuous

or discrete probability distribution. PNs for which the time delays are deterministic

are known in the literature as timed PNs (TdPNs) while those with probabilistic

delays or combinations of probabilistic and deterministic delays have various names.

In this thesis we will use the term SPN (stochastic Petri net) to mean any Petri net

with some form of time delay incorporated with the transitions. (See below for a

discussion on Petri net terminology). Historically, firing times were assumed to be

zero 1701. PNs can be viewed as TdPNs or SPNs with zero delays and all transitions

firing instantaneously. Transitions with zero enabling and firing times are said to be

immediate.

Although in future chapters we deal with SPNs, many of the important properties of

nets are independent of time. Thus for the remainder of the thesis, \rye use the generic

terun net to mean either a PN or an SPN of some kind.

In general the time delays in PNs are dependent on the complete history of the net.

The higher the correlation between time delays and net history, the more difficult is

the analysis. In this thesis we are only concerned with nets where the time delays are

independent of each other and dependent on the current marking only. That is, we are

concerned with nets for which the underlying stochastic process is a Markov process

whose state space is the reachability set.

Markings in which the net resides for zero time are called uanishing rnarkings whilst

those which have a non-zero sojourn time are called tangible markings. With respect to

performance analysis, vanishing markings are normally removed from the reachability

t4

1 4 Stoehastic ni l\Icts

graph as they provide no information.

Various classes of SPNs defined in the literature assume the time delays are drawn

from particular probability distributions and they distinguish between deterministic

and probabilistic delays. We note here that deterministic delays are special cases

of probabilistic delays and can be modelled within the same framework. As such,

throughout this thesis, we will assume that SPNs incorporate both deterministic and

probabilistic time delays. We consider below some classes of SPNs and the time delays

which characterise them.

Definition L.27 - Stochastic Petri Net (SPN)

An SPN is defined by the 7-tuple

N : (P,T,A,rns,W,t,F), as giuen i,n Definition 1.1 and' where

o t is a set of marlcing dependent distribution functions for the enabling time of

each transition.

o F is a set of marlcing dependent ilistribution functions for the firing time of each

transition

Note again that the distribution functions can be chosen to rnodel both d,eterministic

anil probabilistic delays.

One of the first works to involve time delays can be attributed to Ramchandani [74].

His model has zero firing delays and deterministic enabling times, that is, a TdPN.

TdPNs were applied in the modelling of the performance and reliability of on-line

banking systems in [8a].

Merlin [64] proposed an SPN model where each enabling delay falls within an interval

[ú-i.r,úmax] and the firing times àte zero. The enabling interval can be modelled with

the enabling delay drawn from an appropriate distribution, for example the uniform

distribution corresponding to the interval.

15

1 4 Stor:hastie ni l\Tcts

Razouk and Phelps [72], [73] utilised both the enabling and firing times for the analysis

of communication protocols.

Zuberek [94] also considered SPNs with both enabling and firing delays which were ex-

ponentially distributed. He calls his type of nets M-timed nets due to their Markovian

nature.

A class of nets with zero firing times and exponentially distributed enabling times were

independently proposed by Symons [81], Natkin [69] and Molloy [66]. This class of

nets are normallyreferred to in the literature as SPNs. The word stochastic is derived

from the Greek language and means "random" or ((chance" l47l.A stochastic process

then is a process which moves from one state to one of a selection of states according to

certain probabilities. It then follows that a stochastic Petri net should be a PN which

changes markings according to probabilities. As the tenn stochasticby definition does

not refer specifically to the exponential distribution, it makes sense to use the term

SPN as we have defined it above in Definitionl.27, and the term MPN (Markovian

PN) to represent any SPN for which the underlying stochastic process is a Markov

process whose state space is the reachability set. This terminology is non standard in

the Petri net literature, but is more consistent with stochastic process terminology'

A class of MPNs, Generalised SPNs (GSPNs), first proposed in [60], allow two types

of transition enabling delays, immediate and exponential. The exponential transitions

model the delays associated with various activities while the immediate transitions

represent logical actions which require no time. GSPNs have both vanishing and

tangible markings. They reduce the complexity of analysing an MPN model since

vanishing markings can be removed from the reachability graph.

A slight generalisation of GSPNs was given by deterministic and stochastic PNs

(DSPNs), [59]. Again all firing times are set to zero but in this case transition enabling

times can be either deterministic or exponential. GSPNs can be considered a special

case of DSPNs with deterministic delays of zero.

16

1.ñ llnnfliet lrrtinn

Note that GSPNs are MPNs but DSPNs are not, as the current marking is not suf-

ficient to describe the future of the marking process when one or more delays are

deterministically non-zero.

The classes of SPNs given above are distinguished on the basis of time delays only'

There are numerous classes of nets defined by their structure alone. Three such classes

are given in the following definitions.

Definition 1.28 - OrdinarY Nets

A net, N is ordinary if the weight of each arc in A is one.

Definition I.29 - State Machines

A net, Al , is a state machine if it is ord,inary and each transition has eractly one input

and, output place. A state machine is cyclic if the net is strongly connected, that is, a

directed path exists between euerA pair of places in the net.

Definition 1.30 - Marked GraPhs

A net, N, is a marleed, graph if it is ord,inary and each place has exactly one input and,

output transition.

L.5 Conflict Resolution

As well as having no notion of time, the first PN models also had no firing rules to

decide which transition frres when more than one is enabled. The choice of which

transition fires when one or more are jointly enabled and the outcomes that result

from such a choice can completely alter the properties of a model.

As the purpose of this thesis is to concentrate on equilibrium distributions and not

net semantics, we do not cover this topic in full detail but refer the interested reader

to the various articles mentioned.

Firstly we note that conflict can be defined as a behavioural property for a PN or SPN

t7

1.5 Resolution

as follows:

Definition 1.31 - Conflict Net

Let N be a PN or SPN with initial marking rr.s o,nd, reachability graph 7?(*o) . N is

a confl"ict net if there erists a reachable marking rn such that

c ) ty,tze€E(rn) , *5m' anilt2/tS(rn')

Marsan, Balbo, Chiola, Conte and Cumani [61, 62] have addressed the issue of conflict

resolution. They note that many different underlying stochastic processes result from

different firing execution policies. The tractability of net models relies on specifying the

stochastic behaviour of a system within the framework of known stochastic processes

such as Markov and semi-Markov processes.

Whichever firing execution policy is used, the result in most cases is an explosion in

the information required to define a given net model. Models with complex firing

execution policies are difficult to implement in large systems as conflict resolution

probabilities may be required for each marking. In addition, the underlying stochastic

processes may become too complex for analysis by all methods except simulation.

To describe the firing resolution policies more we assign a clock or counter to each time

delay. As the marking process of a net evolves, the values of the clocks vary depending

on the firing execution policy. Clocks corresponding to enabled transitions are classed

as active and inactive otherwise. All active clocks count down at the same speed while

the values of all inactive clocks remain the same (they count down at zero speed). The

clocks indicate the length of time for a given delay with different clocks active at each

marking depending on which transitions are enabled. When a clock counts down to

zero the event corresponding to that clock occurs and the clock value is redrawn from

the appropriate enabling or firing time distribution when it is next active. The clocks

for firing delays are only active after the corresponding clock for the enabling delay

has expired.

18

1.5 Co Resolution

It is important to note that a transition can be enabled but unable to fire due to the

firing execution policy. This leads to the definition of the firable set which is a subset

of the enabled set (given in Definition 1.10).

Definition L.32 - Firable Set

For each marking tn in the reachability graph of a PN the set of transitions which can

fire is called the firable set FE(rn) Ç t5(m), defined' by

.rs(m) :tr, €T: *3)

We will consider two firing execution policies, preselection arrd race.

l.S.L Preselection PolicY

With this policy, the net definition, whether it be a PN or a time extended PN,

also has to include probability sets for the resolution of each conflict situation in the

reachability graph.

Let p¡(m) be the probability that transition ú¡ will fire next in marking m for a net

,Â/ with initial marking m¡. We require

Dpr(ttt) :t Vm€7?(*o)'t¡eT

If p¡(m) is the same for all markings, the policy is known as Global preselection.

Otherwise the policy is called local preselection.

The transition selected from the enabling set will fire only after the clocks correspond-

ing to both the enabling and firing times have elapsed.

L.5.2 Race Policy

The race policy differs from preselection in that the clock values themselves determine

the next transition to fire. When the net enters a marking m, the clock values, either

19

l- ã C]nnflir:t lrrtinn

deterministic or probabilistic, are calculated for the delays of each enabled transition.

Each active clock then counts down with the one reaching zero first determining the

transition to fire and activating the corresponding clock for the firing time. The firing

time clock counts down to zero before firing actually occurs. If more than one clock

reaches zero a preselection policy is then used to determine the one which will actually

fire. This complication doesn't occur of course if all transition delays are drawn from

continuous probability distributions.

The clock values can be controlled in a number of ways. Marsan et al. [61]' [62]

consider three policies, resarnplingi ûge Inelnory and enabling rnemorq.

Resampling

In this case the clocks are independent ofthe history ofthe net and at each

change of marking are re-sampled. Each change of state corresponds to a

regeneration point in a Markov process.

Age Memory

With age memory the clock values are dependent on the history of the net

and are only re-sampled after counting down to zero. The clocks corre-

sponding to transitions which were enabled but did not fire, retain their

values but are inactive until the appropriate transition is next enabled'

That is, they count down at zero speed when not active'

Enabling Memory

Enabling memory is a combination of the other two. Any enabled transition

which was enabled in the last marking but did not fire retains the values of

its clocks. If the transition actually fired or was not enabled in the previous

marking the clock values are redrawn.

Note that the above policies and combinations thereof are but a few possibilities. Any

number of firing execution policies could be devised for various situations. In fact

20

1-6 Sorne Other Petri Net enstons

Pi

t¡

Figure 1.2: Inhibitor and Escape Arcs

each transition in a given net model may obey a different policy. The important point

however is that firing policies that are too complex are almost impossible to work with.

Well known solution techniques for various stochastic processes can only be applied

to nets with special firing execution policies. The major weakness of PNs and SPNs

is the complexity of the models. In most cases there must be a compromise between

modelling power and analysis of the underlying stochastic process.

Many firing policies are impractical in any but very simple SPNs as firing rules must be

established for each different state. As such the race execution policy with resampling

or enabling memory is often assumed in order to obtain analytical results.

Petri nets augmented with time delays and conflict resolution and firing policies have

been referred to as performance Petri nets [85].

1.6 Some Other Petri Net Extensions

There have been numerous other extensions to the original PN definitions.

o Inhibitor Arcs - Model priorities of simultaneous events [70].

An inhibitor arc from place P¿ to transition Í¡ provides an additional condition

for the enabling of transition t¡. Transition ú¡ is enabled if there are sufficient

tokens in its input bag andthe number of tokens in place P; is zero. It is indicated

graphically by a directed arc with a hollow circle replacing the arrowhead as

t¡

2T

depicted for transition Ú¡ of Figure 1.2. Inhibitor arcs only effect enabling rules,

firing rules are unchanged

o Escape Arcs - Model the interruption of firing times or the pre-emption of

sefvers.

In normal untimed PNs, escape arcs are equivalent to inhibitor arcs [94]' In

sPNs, an escape arc from place P¿ to transition ú¡ provides a means for the

firing of transition f¿ to be interrupted. If at any time during the frring of ú¡

the number of tokens in place P¿ equals or exceeds the weight of the escape

arc, the firing of f¡ is stopped and its input bag which had disappeared since

firing began is re-deposited. Nets with escape arcs are known in the literature as

ertend,eil time Petri nets. Escape arcs are depicted graphically in the same way

as inhibitor alcs but with a filled circle as shown for transition Ú¡ of Figure 1'2'

o Compound Nets

For the modelling of more complex system, PNs and sPNs suffer from the prob-

lem of an exploded diagrammatic representation which makes these models dif-

ficult to use. There ate a number of extensions to standard nets all of which

carry more information in the places, transitions and tokens. some examples are

coloured nets [93], predicate transition nets [29] and high level nets [58]' Each

approach allows complex nets with various firing execution policies to be drawn

in a simple manner. A comparison of the three compound nets can be found in

1441.

In compound nets tokens, places and transitions can carry considerably more in-

formation and the firing rules can be much more complex [93] than in a standard

net of the type described above. It ìs worth pointing out that these higher order

nets are folilings or graphical simplifications of standard nets, and that they can

be unfoliled again into a standard net representation. Higher order nets allow

more complex systems to be described in a concise way' but provide the same

modelling po\4/er as SPNs and are no easier to analyse'

22

Chapter 2

Petri Net Analysis

In this thesis we consider two types of nets, PNs and SPNs. The analysis of each

type is different as one contains timing information and the other does not. Analysis

techniques for PNs also apply to SPNs as there are properties of nets which reflect

only the net structure regardless of timing delays, but the time dependent analysis

methods of SPNs is meaningless in the PN context. It was mentioned earlier that

performance analysis can be based on either the time dependent or the equilibrium

behaviour of a system. PN analysis is concerned with examining net properties such as

those defined in Section 1.3 whilst SPN analysis investigates performance aspects. We

begin by considering three types of time independent analysis, and then we examine

the equilibrium distribution of an MPN.

2.L Structural Analysis

We consider two types of time independent analysis in some detail, the matrix equation

approach, the coverability/reachability tree method, and mention briefly the reduction

method. We concentrate on the first two methods as the last approach is not relevant

to the work in later chapters. We also discuss the application of the two methods in

determining which of the net properties discussed in Section 1.3 are true for a given PN

or SPN. Neither approach can solve all of the structural and behavioural properties of

nets and each method has its limitations.

23

2.1 Structural Analysis

z.L.L Reachability Tbee/Coverability Tbee Method

The reachability set, given by Definition 1.9, is a fundamental Petri net concept [70].

It is the set of all possible markings, and is sufficient to answer many questions which

arise when dealing with Petri nets. The reachability tree contains all of the information

in the reachability set but also indicates all possible transition firing paths that will

take the marking process from one marking to another. The reachability set and the

reachability tree are only applicable in bounded nets, that is, nets with a finite number

of markings. For unbounded nets a variation of the reachability tree, the coverability

tree has been developed.

To construct a reachability tree we begin with the initial marking and consider its

firability set. Recall that a transition may be enabled but unable to fire' The resultant

markings from the firing of each transition in the firability set are known as frontier

markings and are added to the tree. Any frontier marking which duplicates an existing

marking is considered to be a dead marking which does not enable any transitions,

and the generation of the tree stops at that point. The firability set of each frontier

marking is then considered in turn producing additional frontier markings. A directed

arc labelled with the appropriate transition is drawn between each marking in the

resulting tree. The tree continues to grow in this manner until all frontier markings

are dead.

Consider the construction of the reachability tree for the Petri net given in Figure 2.1.

Transitions úr and t2 carr fire in the initial marking (1,1,1,0,0). The firing of these

transitions produces markings (0,0, 1, 1,0) and (1,0,0,0, 1) respectively. In each of

these markings only one transition can fire which in both cases reproduces the initial

marking which is already present in the tree and hence the tree is complete, as shown

in Figure 2.2.

An alternative method is to discard terminal frontier markings and redirect the labelled

arcs which would have been directed to those markings to the position in the tree where

24


ooO

P5

P¡Pl

t1

r3

Pz

12

P4

r4

Figure 2.1: A Simple Marked Petri Net

)11(1

t1 t2

( 0,0,1

t3

( 1,L,1

,01

,0,0

(1

(1

)

)

1)

4

,0,0,1

,0

1 )

Figure 2,22 The ReachabilitY Tree

25

2.1 Structural AnalYsis

1,1,

t3 tl t2 f4

0,0,1,1,0 ) ( 1,0,0,0,1)

Figure 2.3: The ReachabilitY GraPh

the duplicate markings already appear. A reachability tree of this type is known as a

reachability graph. The reachability graph for the net in Figure 2.1 is given in Figure

2.3.

Note that the reachability tree or graph described above will never be complete in an

unbounded net as there will always be more frontier markings. In these cases, that

is, when the Petri net has an infinite reachability set, each of the reachability trees

described above will also be infinite. This makes any analysis more difficult since the

reachability tree cannot be generated. The coverability tree on the other hand is finite

in size for any unbounded Petri net, and is equivalent to the reachability tree for a

bounded net. Peterson [70] points out that a finite representation of an infinite set

must result in loss of information, which limits the power of the coverability tree.

The coverability tree is constructed in essentially the same way as the reachability tree

except for the labelling and designation of markings. The coverability tree introduces

a special symbol u.r which represents infinity or unboundedness and will appear in the

markings of the tree if the net is unbounded.

Any frontier marking in the reachability tree which covers another marking already

present in the tree has the marking of the covering places replaced by the symbol to.

All subsequent markings from that point retain the ur labelling. All frontier markings

are included in the coverability tree with duplicate frontier markings considered dead

as in the case of the reachability tree. Peterson [70] proves that the coverability tree

)

26


oao

P5

P¡Pl

rt

r3

2

P2

t2

P¿

t4

Figure 2.42 An Unbounded Marked Petri Net

is always finite.

Note again that the coverability tree of a bounded net does not contain the symbol tr''

and is identical to the reachability tree of the type given in Figure 2'2'

The pN of Figure 2.1 can easily be converted into an unbounded net by changing the

weight of the arc connecting transition úa with place Pr to 2, as shown in Figure 2'4'

Continued firings of the sequence {Úr,ús} now accumulates tokens one at a time in

place Pr. The coverability tree for the altered net is shown in Figure 2'5'

The reachability and coverability trees provide immediate solutions to some of the

behavioural and structural properties of Section 1'3'

Safeness and Boundedness

The coverability tree solves the boundedness problem at a glance. For each place P¿

in a net, an examination of the coverability tree will reveal one of two things: either

p; is labelled with the symbol u at some marking in the tree in which case the place

is unbounded, or there will exist a finite number, 1ú, representing the upper bound on

the number of tokens possible in that place. Recall that a net is bounded if and only

if every place is bounded, that is, a net is bounded if and only if ?I, never appears in

27

2.L Structural Analysis

0,0,1,1,0 )

1,1,L,0,0 )

0,0,1 )

1,0,0,0,1 )

(

t1 t2

(

(

(

t3 t4

w,1,1O,0 ) ( 1,1,L,0,0 )

t t 2

1,0 ) ( w,o,( w,o ,1,

t4

( w,l,1,0,0 ) ( w,1,1,0,0 )

Figure 2.5: The CoverabilitY Tree

the coverability tree [70]. If the bound for all places is 1, the net is safe.

Coverability

Coverability was defined in Definition 1.16.

The existence of a reachable marking which completely covers another marking can

always be determined by the coverability tree. A given marking m is covered by

another marking m' such that m' > m if the marking m' is in the coverability tree.

The positions of the two markings will determine the firing sequence between the two.

Limitations of the Coverability Tree

In practice, the coverability tree approach can only be applied to relatively small

nets. This is due to the state space explosion problem which is inherent in Petri nets

and which makes the generation of coverability trees impractical in many systems.

t 3

28

2.L Structural AnalYsis

Obtaining information from an existing tree can also be computationally expensive as

a complete enumeration over the tree is necessary to analyse most net properties.

The coverability tree cannot in general solve the problems of liveness and reachability'

This is because the symbol ur doesn't specify exactly which markings are reachable,

only that the places which are marked with a w are unbounded. Peterson [70] gives

an example of two Petri nets with the same coverability tree, one which is live and

one which is not. The coverability tree can sometimes provide information regard-

ing reachability and liveness. For example, a net whose coverability tree contains a

marking which enables no transitions is clearly not live. Similarly, all markings in the

coverability tree without the tr.' symbol are reachable and any marking which is not

covered by some marking in the coverability tree is not reachable.

2.L.2 Matrix Equation Method

The second approach that we consider is the matrix equation method. It is a powerful

technique but is often restricted to special classes of nets [68], such as those defined in

Chapter 3. When it can be applied, it avoids the costly generation of the reachability

tree. The matrix approach is of particular importance to this thesis as it forms the

basis for much of the work in later chapters.

A self-loop occurs when a place is present in both the input bag and the output bag of a

transition. Since the incidence matrix cannot distinguish between two nets which only

differ through the plesence of self-loops, it is normally assumed that any net which is

analysed through matrix equations has no self-loops. This is not a strong restriction

as self-loops can be removed by the addition of an extra place and transition [68].

Assuming no self loops, all of the information regarding marking evolution in a PN

is contained in the incidence matrix A, given in Definition 1.11. Each row of the

incidence matrix determines the change in the marking as a result of the firing of each

transition and can be used to describe the dynamic behaviour of a net. Consider a

29


marking m in which transition Ú¡ is firable. The marking m' resulting from the firing

of transition ú¡ can be found through the matrix equation

m':m+Are.l (2.1)

where e¡ is a column vector of zeros with a 1 in the j¿h position, and all markings are

assumed to be column vectors.

Now consider a LFS, ø, which is firable in m, and its firing count vector ã. Extending

the result of Equation (2.1), the marking resulting from firing ø in m' is given by the

matrix equation

m':m+Arã (2.2)

which we shall refer to as the matrix state equation'

Equation (2.2) in conjunction with the s and T-invariants can determine many net

properties. s and T-invariants were defined in section I.3.2. The matrix state equation

provides insight into the role of these invariants'

Consider Equation (2.2) when the column vector ã is a T-invariant' We obtain,

m': m

since ATd: O. Hence a T-invariant indicates how many times each transition would

have to fire to reproduce the same marking. In other words, T-invariants define possible

loops or cycles within the reachability graph of a net. Note however that a T-invariant

vector may not be firable.

We can also use the state equation to provide insight into S-invariants. Consider the

result of pre-multiplying the state equation (2.2) by an S-invariant sr (invariants by

definition are column vectors).

srm': sTm * srATã. (2.3)

Recall for an S-invariant s that As : 0 which implies that srAr : 0 and hence Equa-

tion (2.3) becomes

srm': sTm: .I( K e Z¡. (2'4)

30


Which implies that the net is conservative with respect to the weighting factor s.

Safeness and Boundedness

Clearly from Equation (2.4), any place which appears in an S-invariant must be

bounded. Since the minimal support S-invariants form a generator for all S-invariants,

it is a simple matter of checking each of the minimal support S-invariants to see

whether a place is bounded or unbounded. If all places appear in one or more of the

minimal support S-invariants, then the net is bounded.

It can be shown, [68], that a net is structurally bounded if and only if there exists a

positive integer vector v such that

Av(0, (2.5)

We can use the state equation to provide some insight into the condition. Consider

pre-multiplying (2.2) by a positive integer vector such that (2.5) holds,

vTm':vTm+vTATd (2.6)

Since d consists of positive integers, vTArd I 0 and hence (2.6) becomes

m.

The interpretation is now clear. If (2.5) is satisfied, there is no marking m' reachable

from some marking rn which has a larger weighted sum with the vector v. Since the

component of every place in v is positive the net must be bounded.

Reachability

Unfortunately, the matrix state equation only provides a necessary but not sufÊcient

condition for reachability. It can be used to establish that a given marking is not

reachable, but in general cannot determine if a marking is reachable. There are nets

vTrn'( vT

31


however for which the matrix equation provides both necessary and sufficient condi-

tions for reachability [68].

Consider the net (l/, *) and some marking m for which we wish to determine if

rn e R(N,mo). If m were reachable it would satisfy (2.2). Hence to determine the

reachability of m we firstly need to find a non-negative integer solution d to

AM : rrt - rrto : ATd. (2.7)

A necessary condition for Equation (2.7) to have a non-negative solution is for it to

have a real solution. This result is given by the following theorem which appears in

Murata [68] without proof. The proof is included here for completeness.

Theorem 2.1 - ReachabilitY

A real solution to Equation (2.7) exists if and only if

srAM: o V s: As: o

Proof

To prove the result we show that a solution exists to the system of linear

equations (2.7) ifr AM is orthogonal to the kernel of A.

(+)

Assume that a solution to (2.7) exists. Then for all vectors d such that

(2.7) holds we can pre-multiply the equation by any vector sr such that

sr Ar : 0 which implies that srAM : 0.

(c).

Assume that AM is orthogonal to the kernel of A, which means

As:0+AMTs:0

This implies that the kernel of A is a subset of the kernel of aMr which

in turn implies that any constant multiple of the column vector AM is a

32


subset of the column space of AT since the column space of any matrix is

orthogonal to the kernel of its transpose. Hence AM itself can be written

as a linear combination of the columns of AT which implies that a solution

exists for the equation

Ard: aM.

tr

Theorem 2.1 guarantees the existence of a solution to (2.7),, but not a non-negative

integer solution. Even if a non negative integer solution d does exist this does not

guarantee the reachability of the marking in question since d is only a count vector

for the number of times each transition must fire to produce the desired change in

marking and may not have a LFS. In general it has been shown that the problem of

find,ing a LFS for a given change in marking is NP-complete [87].

The reachability problem can only be solved if we can find both a non-negative integer

firing count vector to (2.7) and a corresponding LFS a. It is this requirement to find

non-negative integer solutions and LFSs which makes the reachability problem very

difficult to solve.

Liveness

It has been shown that the liveness and reachability problems are equivalent [70].

Therefore it should be of no surprise that the matrix approach cannot solve the liveness

problem in general, but as in the case of the reachability problem some results can be

obtained.

It has been shown [63] that if there exists a transition vector v 2 0 such that ATv { O

then there is no initial marking for which the net is live.

33


Properties Necessary and SufficientConditions

Structurally BoundedConservative

RepetitiveConsistent

fv)0,4v(Ofv)0,4v:0lv)0,Arv20f v ) 0,Arv:0

Table 2.1: Necessary and Sufficient Conditions for some Structural Properties

Structural Properties

The incidence matrix can be used to provide necessary and sufficient conditions for

various other structural properties. Together with (2.5) these are summarised in Table

2.1 taken from [68]. Readers who would like more details can refer directly to that

paper.

2.1.3 Reduction

Very large nets are difficult to analyse due to their size. In many instances only

a few properties are of interest at any one time. Reduction is a method by which

portions of a larger net are simplified through fusion or omission in such a way that the

properties of interest are preserved. For example, if we wish to determine the liveness

of a particular transition, there are many simplifications or decompositions which can

be applied elsewhere in the net which will preserve the liveness of the transition in

question. The resulting net may pose no resemblance to the original except that the

liveness of the transition under inspection will be the same in both nets. Numerous

references to this topic can be found in [68].

As complexity is a major weakness of Petri nets, methods of transformation, which

reduce the complexity of a net whilst preserving properties of interest, are very im-

portant.

34

2.2 SPN AnalYsis

2.2 SPN AnalYsis

The many difierent classes of SPNs can be applied in the difficutt task of modelling

complex real world systems. Exact analysis of such systems is in most cases unrealistic

either because the correct model cannot be specified or because the solution to a

correct model is intractable. Therefore in modelling a given system, we either need to

find an approximate solution to an exact model, an exact solution to an approximate

model, or an approximate solution to an approximate model. There are essentially

two approaches. Either the underlying stochastic process can be mapped onto a well

known stochastic process with well known solution techniques, or simulation must be

used [31].

Accurate time dependent analysis can only be achieved through simulation in many

cases, and is not of interest here. The type of time dependent analysis in which we

are interested is calculating lhe steady-state or equilibrizrn distribution.

For the remainder of the thesis we restrict ourselves to considering only MPNs. Con-

sider an MPN, ,Â,/, with an underlying Markov process given by X(t) which gives the

state of the net at time ú where x(0) : ms' X(f) takes values from the reachability

set which we assume to be countable [49]. If the process is of a discrete nature, time

will be taken on the set of non-negative integers , z¡, and if time is continuous it will

be taken over the set of non-negative reals, 701. We also assume the process to be

time homogeneous and irreducible.

The following discussion on equilibrium probabilities for continuous and discrete time

Markov processes can be found in Kelly [49]

35

2.2 SPN Analysis

2.2,! Discrete Time

For a discrete time Markov process the one-step transition probabilities from marking

m¿ to ffij, Irli,mj € R(N,ms)' are defined by

p(m¿, mi) : P(x(ú + 1) : m¡lx(t) - *,)

The discrete time Markov process satisfying the above assumptions will have an ln-

variant measure c(m¿) satisfying the equation

c(m¿) : D c(m¡)p(m, m¿) rn¿ e 17(N, mo). (2.8)m¡e7t(,Â/,ms)

Furthermore, if the process is aperiodic and there exists a positive real number K such

that

t c(m¿):m¡eR(rt/,mo)

then the equilibrium distribution defined by

1

K(2.e)

"(m) : ,\* P(x(¿) : mlx(O) - *o)

is equal to

"(m) : I(c(m) rn e R(AI, mo). (2.10)

The equilibrium distribution also corresponds to the proportion of time spent in each

marking. A definition of aperiodicity can be found in any text on Markov processes

such as 147,77).

2.2.2 Continuous Time

For a continuous time Markov process, the infinitesimal rate at which the state changes

from state m¿ to mj € R(N,ms) is defined by

limr-r¡ry0

rn¿ # m¡mi: mJ

q(rn;,m¡) :

36

2,2 SPN Analysrs

and the invariant measure of such a process satisfies

"(m¿) D q(rn¿,m¡) : t c(m¡)q(m¡, m¿) rn¿ e R(Af, mo).m¡€Ît(.,V,ms) mreR(rv,ms)

(2.11)

As in the discrete case, if there exists a positive real number 1l such that (2.9) holds

then the equilibrium distribution is also given by (2.10). We assume the process is not

explosive, that is, it is regular. For a discussion of explosive processes see Chapter 1

of Anderso" [2].

2.2.3 Remarks

For the continuous case, if Equation (2.10) is substituted into Equation (2.11) we

obtain

n(m;) t q(rn¿,m¡) : D n(m¡)q(mj, m¿) m; € R(N, mo).

(2.t2)

The equilibrium probabilities also satisfy

t zr(m¡) :1. (2.13)m,€ß("Â/,ms)

Equations (2.12) and (2.13) are more commonly known as the global balance equations.

The global balance equations equate the flux into each state with the flux out of each

state. For the discrete case, the global balance equations are defined in a similar way

from the substitution of (2.10) into (2'8).

In order to calculate the equilibrium distribution, we must first find the invariant

measure, and then calculate the normalising constant 1( (also known as the partition

function). To calculate the invariant measure c(m) we must calculate the reachabil-

ity set and the transition rates or probabilities depending on whether the process is

discrete or continuous. Obtaining the solution to either (2.11) or (2.8) then requires

dealing with extremely large, possibly infinite, matrices. Given an invariant mea-

sure, we are still faced with the prospect of finding the normalising constant, which is

required to define a unique equilibrium distribution.

m¡eR(.Â/,m6) mreTt(.Â/,ms)

37

2.2 SPN Analysis

Needless to say, finding equilibrium distributions of MPNs through solving the global

balance equations is restricted to systems with a moderately small number of markings.

For example, the Petri net package GreatSPN [8] becomes far too slow for nets with

more than about 40000 tangible markings. In most cases, for nets with a larger

reachability set, or nets which are unbounded, exact analysis is not possible unless the

net under consideration has an underlying Markov process which has a closed-form

invariant measure.

The existence of a closed-form invariant measure means that the invariant measure

is known for all inìtial markings independent of the size of the reachability set. This

avoids the need to solve very large systems of matrix equations and reduces the prob-

lem of solving the global balance equations to that of calculating just the normalising

constant which is the only remaining unknown. Calculating the equilibrium distribu-

tion is still a difficult task however, because as we shall see, finding the normalising

constant for a closed-form invariant measure is far from trivial.

The type of closed-form invariant measure which we are interested in is the so called

proiluct-forrn invariant measure which we discuss in Section 2.3. The existence of a

closed-form and in particular a product-form equilibrium distribution is a powerful

property for a system to possess. Clearly product-form will not exist in general and

we would expect MPNs which exhibit product-form to possess special structure and/or

transition probabilities or intensities. As we shall see this is indeed the case. In the next

section we define what we mean by product-form and survey the product-form nets in

the literature to examine the sort of restrictions which are necessary for product-form

to exist.

First though, we introduce an important Petri net concept which will be used through-

out the remainder of the thesis and which arises naturally in product-form equilibrium

distributions of MPNs.

38

2-g Pro rref.-Ítorrn

Definition 2.1 - Sufficient Place Set

A subset of places P' C P is a sfficient place set if the sub-marking of each place in

P, prouid"es sfficient information to ilefine uniquely the marleing of all places.

The number of places required in a sfficient place set is calleil the dimension of the

marlcing process [27], and is giuen by

N P - d,im(Ker(Ã)).

2.3 Product-Form

Molloy [67] believes that the real success of Petri net modelling will not occur until

the fundamental problem of the reachability set explosion is solved. This is the same

problem that plagued queueing networks for many years. It was overcome in some

cases by the discovery of classes of queueing networks with product-form equilibrium

distributions.

One of the first works on product-form equilibrium distributions was by Jackson

142,, 431who established that the invariant measure for particular classes of queue-

ing networks was a product of the marginal measures for each node. Some of the

more noted results on product-form queueing theory which have appeared since the

work of Jackson are by Whittle [88, 89], Baskett, Chandy, Muntz and Palacios [3] and

Kelly [aS]. One practical appeal of product-form queueing networks is the abundance

of efficient methods for calculating their normalising constants and thereby evaluating

performance measules defined in terms of these constants. Another is that product-

form equilibrium distributions allow the exact analysis of larger networks than would

otherwise be possible.

The places in a Petri net are analogous to the nodes in a queueing network and the

tokens analogous to customers in each queue. This suggests that we should be striving

to obtain an equilibrium distribution for MPNs which is a product over the places of

the net. That is, we are interested in MPNs which have an equilibrium distribution of

39

the form

"(m) : Ka(m) f[ c¿(rn(i))

NP

i=l(2.r4)

(2.15)

where as above, .Il is the normalising constant given by

-l

K_ D z'(mme7è(rV,ms)

and c¿(rn(i)) is the element of the invariant measure corresponding to place P¿' The

function O(*) is a given arbitrary non negative function defined over the reachability

set. In this thesis, the term "product-form" will mean an equilibrium distribution

of the form given by Equation (2.14) unless stated otherwise. Most of the product-

form equilibrium distributions in queueing networks are of the form given by (2.1a).

One of the aims of this thesis is to identify classes of MPNs which also have such an

equilibrium distribution.

As previously mentioned, the only unknown in a product-form network is the normal-

ising constant 1l given by Equation (2.15). Attempts to evaluate K directly require

summing the invariant measure over every state or marking' For networks with a

finite but very large number of states this calculation is impractical. In product-form

queueing networks there are various ways to calculat e K vety efficiently by avoiding

direct application of (2.15). The first work to concentrate on normalising constants for

product-form queueing networks was by Koenigsberg [53] who derived closed form ex-

pressions for the normalising constants of queueing networks with constant rate single

server queues. Koenigsberg only considered cyclic networks without realising that his

results applied to more general queueing networks. Fifteen years after Koenigsberg's

efiorts Buzen [6] derived a number of efficient convolution algorithms which recursively

calculate normalising constants for networks of queues with both load dependent and

load independent servers. Other convolution algorithms for evaluating normalising

constants for specific product-form networks were given by Kaufman [46], Roberts

[76], Dziong and Roberts [23], Conway and Georganas [16] and Conway and Pinsky

[17].

40

2.3 P duct-Form

Recently, product-form queueing networks have evolved beyond the single movement

restriction to include batch movement of customers and correlated routing, [37, 38, 40],

precisely the features which characterise MPNs. It has been shown that a certain class

of MPNs possesses an equivalent product-form [34, 39]. Independent of queueing

theory, product-form results have also appeared for other types of MPNs which we

consider in the next section.

Given that we can find classes of Petri nets with an appropriate product-form invariant

measure we must still be able to efficiently calculate the normalising constant K. The

cardinality problem which arises in product-form queueing networks is accentuated

in Petri nets due to the more complex state description and state transitions. In

order to make use of any product-form results in MPNs we require similar algorithms

to efficiently evaluate normalising constants given by Equation (2.15). In Chapter 4

we present various algorithms which operate in the same way as those for queueing

networks and which calculate the normalising constants for various classes of product-

form MPNs such as the class described in Chapter 3.

As well as providing an efficient means of analysis for relatively large networks, whether

it be queueing networks or MPNs, product-form results may well play a more important

role in the analysis of very large systems through aggregation and disaggregation

techniques. In the first instance parts of a large non product-form network may be

aggregated into single components with specifically chosen parameters so that each

aggregated component mimics the part of the network that it replaces [7]' [33].

Alternatively, a large non product-form network can be decomposed or disaggregated

into sub-networks with adjusted firing rates [10], [41]. If required, parts of a given sub-

network could then be aggregated as mentioned above. In either case, these methods

produce smaller networks or systems of smaller networks. If these sub-networks can

be shown to possess product-form equilibrium distributions, their analysis is greatly

simplified, leading to a more efficient analysis of the original problem. In some cases

aggregation and disaggregation provide exact results [33].

4I

2.3 Product -Ítorrn

It is therefore of great interest to identify as many product-form classes of nets as

possible. In the following section we survey existing product-form results in MPNs

including work which has appeared very recently'

2.3.L Existing Product-Form Results in Petri Nets

Although it was not recognised as such by the author, one of the earlier examples in

the literature of an MpN with a product-form equilibrium distribution was given by

Peterson [70] in a Petri net version of the dining philosophers problem- The problem

was originally proposed by Dijkstra [21] as a model for resource sharing in computer

networks and has since been used as a product-form example by numerous authors'

[5], [55], [5s], [e3].

The classical problem concerns a number of philosophers seated at a round table who

alternatively eat and think with the restriction that no two adjacent philosophers can

both eat at the same time. It is a binary system in the sense that the state of each

philosopher can be represented by a zeto or a one. The problem has been considered

in a more general context by Ycart [92] with numerous applications in physics'

A Petri net version of the dining philosophers problem with five philosophers is given

in Figure 2.6. Places P1 through to P5 correspond to each philosopher thinking and

Po to Pro to each philosopher eating. Places Prr to P15 correspond to chopsticks being

present with a philosopher requiring two chopsticks, one from either side, to move

from the thinking to the eating state. Note that the set of places {Pu,.. . , Pro} form

a sufficient place set since knowledge of the markings of these places uniquely define

the markings of the remaining places. This is not the only sufficient place set for the

net, another is the first five places. Both of the sufficient place sets are apparent from

an inspection of the minimal support s-invariants which can be written in the form

sm : sm0,

m(i)+m(i+5)

42

*o(i)+rns(i+5)

2.3 Prod rref,-T'orrn

m(i -r 5) + m(i + 6) + rn(i + 10)

ræ(6) + rn(10) + rn (15)

Figure 2.6: The Dining Philosophers

: ms(i * 5) + rno(i + 6) + rno(i + to)

: -o(6) * rno(10) + rno(15)

fori:1,...,5.

The equilibrium distribution can be shown to be

,(m) : . (ffi)-"' (ffi)^n (ffi)-"' (ffi)-.,' (ffi)-"" (2 16)

where q(l¿) is the state independent firing rate of transition ú¿. Note that the places

whose markings appear in the product-form correspond to a sufficient place set. This

is always the case.

We have chosen the dining philosophers problem as it is consistent with the frameworks

43

2.3 duct-Form

of many of the existing product-form Petri nets. We will use it as a reference to

illustrate various concepts in the following discussion.

Lazar and Robertazzi

The first product-form results specifically for PNs can be attributed to Lazar and

Robertazzi [54], [SS] who consider nets at the marking level. They define a class of safe

and live Markovian Petri nets whose equilibrium probabilities satisfy partial balance

equations and have product-form equilibrium distributions when marking independent

firing rates are assumed. The restriction for the nets to be safe was lifted in a follow

up paper by Robert azzi 1751.

Following earlier work in queueing theory [54] their approach was to look for the

presence of building blocks in the reachability graph. Building blocks are specific closed

finite sub-graphs of the reachability graph. Geometrically the original reachability

graph can be reconstructed by pasting the building blocks together. Mathematically,

the sum of the global balance equations for the states within each building block,

when considered in isolation, re-constitute the global balance equations for the entire

reachability graph. The global balance equations for the states of each building block

when considered in isolation are a possible set of partial balance equations, a concept

first introduced by Whittle [88].

If each partial balance equation produces the same equilibrium probabilities, the equa-

tions are said to be consistent and moreover the solution of the partial balance equa-

tions is also a solution to the global balance equations. Partial balance has many

meanings in the literature with many authors using different terminology for the same

type of balance. The type of partial balance considered by Lazar and Robertazzi,

where the global balance equations are decomposed into partial balance equations

corresponding to subsets of markings, is just one type.

Consider again the global balance equations given by (2.I2). In any decomposition of

44

these equations into partial balance equations, each state may appear many times but

each transition rate may appear only once. If this were not the case, the partial balance

equations would not reform the global balance equations when summed together' In

terms of building blocks, this means that each directed arc may only be used in one

building block while each marking may appear in many building blocks.

We will refer to the above approach as the marking approach since it is based on the

configuration of each marking within the reachability graph. one of the limitations of

the method is that the decomposition of the reachability graph into building blocks

is not normally unique and the partial balance equations for a given set of building

blocks may not be consistent. It has been noted [3] that the partial balance method

does not always lead to a solution. Another limitation is the need to generate the

reachability graph.

The queueing network paper by Lazar and Rober tazzi 154) gives necessary and sufficient

conditions for the consistency of the partial balance equations derived from the building

blocks. These conditions require certain flow relations, based on the transition rates,

to hold for any closed path in the reachability graph. The consistency conditions are

obtained through what is called the consistency tree. The interested reader can find

more detail in the Paper'

The relationship between consistent building blocks and product-form is referred to by

Lazar and Rober tazzi asthe duality principle. This is equivalent to the long established

relationship between product-form and the partial balance of whittle.

Now consider the dining philosophers problem in the context of La"za't and Robertazzi's

approach. The net in Figure 2.6 was considered by Lazar and Robertazziin [55]. The

reachability graph when all philosophers are originally thinking, that is with initial

marking

lrls - (1, 1,1, 1, 1,0,0,0,0,0, 1, 1,1, 1,1),

is shown in Figure 2.7.

45

2.3 P rtet-[lrìrrn

(0,1,0,0,1)

f2 t5

t4 (1,0,0,0,0) 13

t9 r1tg

( 1,0,0,1,0) (1,0,1,0,0)

f7 ttorlrl

(0,0,0,0,1) r6 (0,1,0,0,0)tto t7

12 tgrg r5 (0,0,0,0,0)

t9 t8

t6 14 bt6(0,0,0,1,0) (0,0,1,0,0)

J7 tto

t3 t4

(0,0,1,0,1) r5 t2 (0,1,0,1,0)

Figure 2.7: Reachability Graph

As the places corresponding to each philosopher thinking form a sufficient place set

we only consider this sub-marking in the reachability graph, that is, each marking in

the reachability graph is of the form (rn(6),. . ., rn(10)).

It can be easily shown that the building blocks given by the single arcs between

any two states produce consistent partial balance equations. These partial balance

equations are a special kind known as detailed balance. A Markov process that exhibits

detailed balance is reversible and possesses a special kind of product-form equilibrium

distribution. The interested reader can find a detailed description of reversibility in

the book by Kelly [49].

An MPN exhibits detailed balance if

n(m¿)q(m¿, m¡) : zr(m¡)q(mj, mr) (2.t7)

46

for all markings m¿ and m¡ in the reachability graph. Repeated application of the

detailed balance equations will always give an invariant measure which is a product

of ratios of firing rates. In this example repeated application of (2.17) reproduces the

equilibrium distribution (2.16) for the current initial marking.

For example, consider the three markings (0, 1,0,0, 1), (0,0,0,0, 1) and (0,0,0,0,0) in

the reachability graph. From the detailed balance equations we have

zr (0, 1, 0, 0, , 1) : t{#Ëfiì{#n+ìl n(o, o, o, o, 1)

: ffi "(o,o, o, o, 1)

and similarly

zr(0,0,0, 0, 1) : ffi zr(0,0,0,0,0).

Combining the previous two equations gives us

zr(0,1,0,0,,1) : (ffi) (ffi) zr(0,0,0,0,0). (2.18)

Every marking in the reachability graph of Figure 2.7 can be written in a similar

manner with respect to the sub-marking (0,0,0,0,0). The equilibrium probability

zr(0,0,0,0,0) is the normalising constant in this case. Note that Equation (2.18) is

consistent with the equilibrium distribution given by Equation (2.16).

2.3 Prod Form

Although the class of nets which Lazar and Robertazzi consider is safe and therefore

bounded, the theory of building blocks can also be extended to unbounded nets, as

it is not the actual size of the reachability graph which is important, but rather the

existence of sufficient structure to make analysis possible. Indeed there are classes of

queueing networks which can have state spaces of arbitrary size but which decompose

into building blocks in such a way that product-form exists [3], [42], [43]. It was also

shown in [5a] that for the state independent case, altering the size of the state space

by the addition or subtraction of building blocks from the state transition diagram

does not change the product-form equilibrium distribution except in the value of the

normalising constant.

47

2-3^ Prod Ttorrn

Li and Georganas

Li and Georganas [57] refer to MPNs which satisfy the product-form criteria ol Lazar

and Robeúazzi as locally balanced stochastic Petri nets (LBSPNs). The concept

of local balance which they consider is equivalent to the concept of partial balance

discussed above.

The product-form of Lazat and Robertazzi is based on a decomposition of the reacha-

bility graph into building blocks. Li and Georganas extend their definition of LBSPNs

to a class of nets which are locally balanced based on a decomposition of the Petri net

[57]. Based on the initial marking, they decompose a given net according to its firable

T-invariants so that sub-nets have unique transitions but may have common places.

Since each firable T-invariant corresponds to a closed path in the reachability graph,

the decomposition of Li and Georganas is a method for selecting building blocks. If

the partial/local balance equations based on the building blocks are consistent the

nets are classed as LBSPNs.

A test procedure has been developed by Li in his Ph.D dissertation [56] which deter-

mines whether or not a particular MPN is an LBSPN. Unfortunately the test proce-

dure is not readily available. It is apparent however that Li and Georganas extend

the method of Lazar and Robertazzi to more general MPNs and select building blocks

based on the structure of the MPN'

flosch

Frosch [25] considers a class of nets which he calls closed synchronised systems of

stochastic sequential processes (CS). We require the following definition'

48

2-3 rret-Ír(lnrn

Definition 2,2 - Sequential Tþansitions

Two transitions o,re sequentiat if the output bag of the first is the i'nput bag of the

second.

Frosch's nets consist of cyclic state machines which can be joined together through

common buffer places such that the flow of tokens through the buffer places is conserved

for every pair of sequential transitions in any state machine. Each cyclic state machine

is equivalent to a closed migration process such as that considered by Kelly [a9]. When

the state machines are joined together by buffer places the result is a network of

migration processes whose state transitions are restricted in a special way. The use of

buffer tokens by one cyclic state machine may restrict the possible transition firings

within other state machines.

The product-form which Frosch obtains is both a product over each state machine and

over each place and is based on the structure of the net. UnlikeLazar and Robertazzi

he allows state dependent firing rates and does not restrict his nets to be safe. In

order to guarantee product-form, Frosch assumes that every T-invariant for each state

machine is firable whenever one the transitions in a T-invariant is enabled. He calls

this assumption "4.

Although he does not prove the result, he claims that assumption,4 corresponds to the

product-form conditions of Lazar and Robertazzi. Recall that Lazar and R.obertazzi's

product-form results are based on the existence of a consistent set of partial balance

equations derived from building blocks. Frosch's assumption ,4 ensures that every

T-invariant is represented as a closed path in the reachability graph. Thus like Li and

Georganas' results, assumption ,4, is related to building blocks. It ensures that the

building blocks for each T-invariant are repeated as often as possible. Assumption "4

however does not guarantee the existence of building blocks with consistent partial

balance equations. For these reasons assumption ,4 alone does not appear to be

equivalent to the product-form conditions of Lazar and Robertazzi.

49

2^3 Pro rref-trtorrn

The dining philosophers net is also an example of Frosch's class of nets, where the

chopstick places are buffer places and the two states of each philosopher form cyclic

state machines.

Ilenderson, Lucic and Taylor

In contrast to the approach by Lazat and Robertazzi which examines a Petri net

at the marking level, Henderson, Lucic and Taylor [34, 39] obtained product-form

criteria based only on the structure of the net, without the need to generate the

reachability graph. Their class of nets originates from results in batch movement

queueing networks.

The product-form equilibrium distribution which they obtain is not the classical prod-

uct over the places but a product of two functions one, g(m), related to routing and

the other, O(*), to transition firings,

z'(m) : ¡(O(m)g(m)' (2'19)

The standard product-form given by Equation (2.14) can be written in this form. They

require every input bag to also be an output bag and every output bag to be an input

bug. In addition no two transitions can have the same input bag. Nets which don't

satisfy the input bag criteria can often be transformed into nets which do by the use

of probabilistic routing (see page 60).

The existence of unique input bags implies a one to one relationship between input

bags and transitions which allows the transitions themselves to be states in an Markov

chain which they refer to as the routing chain. A necessary condition for a product-

form equilibrium distribution of the form (2.19) to exist is that there are no transient

states in the routing chain. The input-output bag condition is a consequence of this

requirement.

They show that finding an invariant measure for the routing chain can lead to an

invariant measure for the MPN. Except for the assumptions mentioned above, the

50

2-S Prod rref.-Ítorrn

class of nets which they consìder can have arbitrary structure, they are not necessarily

bounded, and state dependent routing and firing is allowed. As the product-form which

they obtain is dependent only on net structure, we would expect the net invariants,

which reflect net structure, to play an important role in determining the equilibrium

distribution.

The dining philosophers problem is also an example of an MPN with product-form

which satisfies the input-output bag criteria of Henderson, Lucic and Taylor. The

structural approach establishes Equation (2.16) as the equilibrium distribution of the

net in Figure 2.6 for any live initial marking.

The work of Henderson, Lucic and Taylor is covered in more detail in Chapter 3

where we examine necessaÌy and sufficient conditions for the equilibrium distribution

(2.19) to take the more familiar form of Equation (2.14) and examine the role that the

invariants play in the existence of the product-form.

Boucherie

A recent paper by Boucherie [5] discusses product-form results for MPNs using a

Markov chain approach. He considers .lú continuous time Markov chains which are

related only through their transition rates. The state space for each Markov chain is

partitioned into mutually exclusive sets each corresponding to a resource so the state

of each Markov chain determines which resources are in use. Markov chain 7¿ uses

resource i if the current state of the chain is in the partition corresponding to that

resource. It is assumed that each Markov chain is irreducible and possesses a unique

equilibrium distribution.

Dependence between the chains only occurs through competition for resources as fol-

lows. For each resource there is a subset of chains which compete for it. Only one

of the chains competing for a given resource can use that resource at any one time.

A resource cal be used by more than one chain only if the respective chains do not

51

2-3 drret-ftorrn

compete with each other for that resource.

The process whose state space is the union of the state spaces for each Markov chain

is known as the product process. The product process has a state description given by

m : (mr,. . ., m¡r)

where m¿ is the state of Markov chain i

The product process is restricted in such a way that it can only change state through

a change in state of one of its constituent Markov chains. Thus the transition rates

are defined to be

Nq(*, -') : t ø;(mr, ml) 1[*" - mí.UlY"] (2.20)

i=l

N

il,r#ir=1

where 1[.] is the indicator function and e;(rn'mi) is the transition rate within chain

i. The first indicator function ensures that only chain i can change state while the

condition V" ensures that no resource for which chain i competes is in use. The effect

of this is that whenever just one of the resources for which chain n competes is in use

Markov chain n comes to a halt and no state changes are allowed.

The equilibrium distribution of the product process is still shown however to factorise

into a product of marginal distributions for each Markov chain. That is, the product

process is shown to have an equilibrium distribution of the form given by Equation

(2.t4) with Õ(m) : 1.

The reason for the product-form equilibrium distribution is the restricted interaction

between the individual Markov chains and the fact that each chain is either operational

or stopped. Since only one of the Markov chains can change state at any one time, the

global balance equations break down into the sum of the global balance equations for

each Markov chain. Since each of the Markov chains is assumed to possess a unique

equilibrium distribution, it follows that the global balance equations for the product

process will also possess a unique solution. The restriction on the firing rates forces

the Markov chains of the product process to behave as if they were independent.

52

2-g Prod rret-Ttorrn

ti

ti+5

Figure 2.8: Markov Chain for Philosopher i

Thus the class of Petri nets which characterise Boucherie's results are similar to, but

allow more structural freedom than, those of Frosch [25]. Cyclic state machines are

not assumed, and the connection of the sub-nets is not as strict. Any number of

nets can be joined together by common places as long as only one sub-net can use a

given resource at any time. The firing rates of the transitions in the merged net must

satisfy Equation (2.20). Note however that Frosch's result does not restrict the firing

of transitions in the same way as Boucherie. In Frosch's formulation transitions can

still fire in each sub-net even if one of its buffer/resource places has tokens used by a

competing state machine. In Boucherie's case, the whole sub-net would cease to fire.

The dining philosophers example of Figure 2.6 is an example of a net which satisfies

Boucherie's criteria. The two states for each philosopher can be considered to comprise

separate Markov chains as shown in Figure 2.8, which are joined together by the

chopstick places such that any transition firing only changes the state of one Markov

chain.

Boucherie's result then tells us that the equilibrium distribution is a product of the

marginal equilibrium distributions for each Markov chain which leads trivially to Equa-

tion (2.16). To the best of our knowledge the class of nets which satisfy Boucherie's

criteria is distinct from any other.

53

2-g Prod rr r:t-trtorrn

Ilenderson, Northcote and TaYlor

Some recent product-form results in queueing theory, [28], [35], were applied to MPNs

by Henderson, Northcote and Taylor [36]. Like the closed nets of Henderson, Lucic

and Taylor their result depends only on the structure, but unlike the other work there

is no simple structural characterisation for their type of nets.

When a transition fires it can either behave normally, or it can attempt to force the

immediate firing of a second transition. If the forcing is successful the input bags of

both the forcing and forced transitions will vanish and be replaced by an output bag

which is dependent on both of the input bags, otherwise only the input bag of the

forcing transition will vanish without depositing an output bag.

They assume there to be a one to one correspondence between input bags and tran-

sitions which, as in the work of Henderson, Lucic and Taylor, defines a Markov chain

on the transitions again called the routing chain. They require the routing chain to

be irreducible which is slightly stronger than the requirement of Henderson' Lucic and

Taylor who assume non transience of the routing chain.

There is more than one way to graphically represent their nets, either as a collection of

immediate transitions and inhibitor arcs or immediate transitions with corresponding

conflict resolution sets. Remarkably, they show that under certain conditions, the nets,

which may have source or sink transitions, have product-form equilibrium distributions

for any live initial marking, that is, a structural product-form.

The net in Figure 2.9 is taken from [36]. Transitions úe and ta ate immediate which

implies that tokens do not reside in the temporary place P¿ but are absorbed instantly.

The equilibrium distribution can be shown to be

1-o)q(¿')(

where a and B define the probabilistic routing in the net (see section 3.1).

,,(m) : . (ffi)-"' (+å) -"' (#)'' ( q(t6)

54

^(4)

2-3^ Prod rrr.t-ftnrrn

t1

l-p

1-s

t6

P2

tz

p

t3 t4

P3

P4

t5

Figure 2.9: Triggered MPN with Product-Form

Florin and Natkin

Product-form results different to that of Equation (2J\ are presented by Florin and

Natkin [27]. They consider a class of synchronised queueing networks which can be

modelled by ordinary (Definition 1.28) and bounded MPNs with marking independent

transition firing rates and a reachability graph which is strongly connected. For these

nets matrix geometric product-form results are obtained where the equilibrium dis-

tribution is a sum of product-forms. As far as we are aware there exist no efficient

algorithms for normalising their type of product-form.

Remarks

The concept of building blocks and consistent partial balance equations introduced

by Lazar and Robertazzi is responsible in one form or another for each of the above

product-form equilibrium distributions. To be consistent with the terminology used

55

2.3 Pro rret-tr'orrn

to distinguish between net properties, we shall refer to product-form as behavioural

product-form when it is marking dependent, and as structural product-form when it

exists for all live initial markings. Each of the above classes of product-form Petri nets

is one of these two types.

The product-forms of Frosch, Henderson and Taylor, Boucherie, and Henderson,

Northcote and Taylor are structural product-forms. In each case the structure of

the MPN defines building blocks in the reachabilìty graph and the firing rates of the

transitions are appropriately restricted to ensure consistency of the partial balance

equations based on those building blocks for every live initial marking.

The product-form of Lazar and Robertazzi and Li and Georganas are behavioural.

Although Laza,- and Robertazzi give classes of MPNs for which the product-form is

known for all initial markings, in general applying the marking approach requires an

analysis of the reachability graph for each initial marking. It is also possible for the

structure of a net to be sufficient to provide a product-form equilibrium distribution

for some initial markings and not others. Sereno and Donnatelli give an example of

such a net 122).

It is important to realise that the structural product-form is much more powerful than

its behavioural counterpart. To illustrate this consider again the dining philosophers

problem, Figure 2.6, but this time we do not restrict the net to be safe and we change

the interpretation of the places. The tokens in each place between any two philosophers

now signifies the maximum number of courses that can be eaten by the two adjacent

philosophers at any one time. We also allow multiple levels of thinking and eating so

that each philosopher can eat and think at the same time with the number of tokens

in the eating and thinking places signifying the proportion of time devoted to each

activity.

Consider the initial marking

rïrs : (2r2r2r2r2r0r0, 0, 0, 0r2r2r212,2)

56

23 Pro r r ¡t-trtr¡rrn

which corresponds to the situation where any two adjacent philosophers can eat at

most two courses at a time and each philosopher spends l00To of their time thinking.

Since the structural product-form holds for any live initial marking, we know that the

equilibrium distribution is still given by Equation (2.16) with a different normalising

constant than before. In contrast, the reachability graph for the new initial marking

consists of 57 markings compared with the 11 in Figure 2.7. Clearly, identifying

building blocks in a graph five times the size is considerably more difficult. Solving

the subsequent partial balance equations is also much more time consuming. Given

the increased difficulty in applying ihe building block technique for the new initial

marking, which is still quite small, attempting the same for even larger initial markings

with reachability sets which contain thousands of markings would be impractical. For

example, with initial marking

1116 : (4, 4141414,,010, 0, 0, 0r4141414r4)

the reachability graph consists of 553 markings. Attempting to find the product-

form equilibrium distribution by finding building blocks and solving partial balance

equations in this case would be a nightmare. Considering that the dining philosophers

example is a very small net further emphasizes the advantages of a structural product-

form over a behavioural one.

Ð/

Chapter 3

A Class of Product-Form Nets

As already mentioned, the theory of product-form queueing networks has evolved

beyond the single movement restriction to include networks of queues which allow

batch movement of customers and correlated routing. walrand [86] studied batch

movement queueing networks in discrete time with arrivals and service following state

dependent Poisson distributions. Batch movements of a restricted type can occur in

the clustering processes d,iscussed by whittle [90] and Pollett [71]. At a clustering node

clusters of disparate particles can aggregate to form clusters, and these clusters can

disaggregate into their constituent particles. By regarding clustering nodes as quasi-

reversible Markov processes whittle discussed how such nodes can be coupled via

the migration of clusters from one clustering node to another. Pollett [71] considered

clustering processes but with nodes which are partially balanced. Extensions to more

general batch movement was provided by Henderson, Pearce, Taylor and van Dijk

[37] and further extended by Boucherie and van Dijk [a] and Henderson and Taylor

[3S], [40]. Henderson, Lucic and Taylor [34], and Henderson and Taylor [39], showed

that a particular class of SPNs exhibits the non-standard product-form equilibrium

distribution given by (2.19), which is not a product over the nodes of the network, but

is instead a product of two functions, one related to service and the other to routing'

In this chapter, we extend the product-form results of Henderson, Lucic and Taylor

[34] discussed briefly in the last chapter by finding necessary and sufficient conditions

58


for the equilibrium distribution (2.19) to take the form given by Equation (2.14).

That is, necessary and sufficient conditions for the function g(m) to assume the more

familiar structure of a product over the places of the net. Analogous to the consistency

conditions of Lazar and Robertazzi discussed in the previous chapter, our product-form

condition is a matrix condition based on the incidence matrix.

We also obtain product-form results for MPNs with non-zero enabling and firing times.

The assumptions which we require are much less restrictive than those of Jackson

networks 1421,1431.

3.1- Instantaneous Firings

We firstly consider SPNs with zero firing times (we will remove this restriction in

Section 3.2), exponentially distributed enabling times with transitions that follow the

race firing execution policy, and a particular net structure. As we are dealing only with

exponentially distributed time delays the class of nets which we are dealing with are

MPNs. Since all of the delays are drawn from continuous distributions the selection of

the next firing transition is unambiguous and no conflict resolution probability sets are

required. In the following we formally define the class of nets proposed by Henderson,

Lucic and Taylor [34, 39].

When the MPN enters state m, before firing, transition ú¡ waits for a state dependent

exponentially distributed length of time, with parameter

(3.1)

where ry' is an arbitrary non-negative function and ¡ and O are arbitrary positive

functions. The range of the function (Þ is assumed to be finite. Transition ú¡ is said to

fire at rate q(rn,f¡).

The form of the firing rates (3.1) implies that the following must hold for transition t¡

, ¡\ ,/'(m-I(t¡))y(t¡)9(m,¿jl: - O(rrÐ

59


to be able to fire,

. ú(m - I(új)) > 0.

Hence $ can be used to inhibit the firing of one or more transitions under certain

marking conditions. It is generally assumed that t/(m-I(¿¡)) : 0 whenever m ll(t¡)although this restriction is not essential [32].

We now introduce another Petri net extension used by Henderson, Lucic and Taylor

[34, Bg], that of probabilistic routing. Probabilistic routing allows a transition to have

more than one output bag, each with a certain probability of occurring. When transi-

tion ú¡ fires, its input bag of tokens I(ú¡) is released for routing, and transformed into

one of a possible set of output bags o¿(ú¡) e z*' ,i: I,...,8r¡ with probability

p(I(¿j), o¿(¿¡)) and rhe state of the MPN changes from m to m - I(úi)+ O¿(Ú¡). Here

B¿, is the number of possible output bags for transition t¡. Let

B : {(t(t j), o¿(¿¡)) : p(I(t¡),oi(új)) > 0}

be the set of distinct ordered input/output bag pairs and write g:lßl:Dt¡eTBt¡.

The routing probabilities are an additional requirement in the net definition.

We assume thatBrj

f rlIltr¡,o¡(ú¡)) :1 v t¡eTi=l

Any MPN with probabilistic routing can be transformed into one without probabilistic

routing by breaking each probabilistically routed transition into multiple transitions

each with a common clock for their enabling delays, or by replacing each probabilisti-

cally routed transition with multiple immediate transitions each followed by a timed

transition. The routing probabilities can then be thought of as conflict resolution

probabilities for that set of transitions since they will all attempt to fire at once. Each

interpretation is equivalent.

60


The structure of the MPN is restricted by the following condition.

Structural Condition

1. No two transitions have the same input bag.

MPNs that do not have this structure can often be transformed into MPNs which

do. For example, if an MPN has more than one transition with the same input

bag, provided the firing rates have the same state dependence, these transitions can,

through the use of probabilistic output bags, be merged into a single transition [34].

Probabilistic routing can still be analysed through matrix equations, but we require a

new definition for the incidence matrix, as Definition 1.11 is no longer appropriate'

Definition 3.1 - Extended Incidence Matrix

For an SPN with probabilistic routing the inciilence rnatrix A' is a B x N P matrix

with a row for each input/output bag pair and a column for each place. The entry

A(b,p), b e B, p eP represents the change in the number of tokens at place p after

the transition corresponding to the input/output bag pair, b, fires'

Let p(t¡,f¡) be the probability that the output bag of transition f¡ transforms into the

input bag of transition ú¡. More formally

p(t¡,t*):P(I(rj), o;(¿¡)) if oi(¿j) : I(¿*)

0 otherwise.

Note that p(t¡,t*) is not the probability that transition ú¡ fires after transition Ú¡'

The discrete-time Markov chain with transition probabilities p(ti,tk) t¡,t¡ e T is

known as the routing chain. The existence of the routing chain requires a one to one

relationship between input bags and transitions which is given by Structural Condition

l. By finding an invariant measure for the routing chain, an invariant measure for

61


the MPN can sometimes be found. Define the set of positive functions /, such that

X(t)f (t¡) is an invariant measure for the routing chain, as

.tr_ l(.),7 -r R: f(t¡)>0, y(t¡)f(t¡):D x(tk)l(tk)p(tx,t¡),,V t¡ eTt¡

(3.2)

The equilibrium distribution for the class of MPNs defined above is given by the

following theorem proved in Henderson and Taylor [39]'

Theorem 3.1 - Equilibriurn Distribution

If I afunction f eF and afunction gtM -+.Rsuchthat

gtm+lÍ¿¡ìì :frl!,t), vt¡,tt€T : p(t¡,tn)>0 (3.3)s(m + I(¿¿)) l(t*)

then the marking process of an MPN satisfying Structural Condition t has

a closed-form invariant measure given by

c(m) : O(m)e(m) rn € M, (3.4)

which can be normalised to give the equilibrium distribution

n(m) : /lc(m)

whenever the underlying Markov process is regular and

Dme¡r¿ O(m)g(m) < oo.

The product of two functions given by Equation (3.a) is quite general. O(*) is related

to the firing rates of the transitions through the definition of frring rates given by

Equation (3.1), while g(m) holds routing information through its relationship with

the incidence matrix (Section 3.1.2). In following sections we look a little deeper and

examine necessary conditions for g(m) itself to be a product over the places of the

net

62


For the above theorem to hold the set F must be non-empty. Since the number of

transitions is finite this means that the routing chain can have no transient states.

The structural consequences of this are:

Structural Conditions

2. For each transition ú¡ e T,and.i: I,..., Br¡, O¿(Ú¡) : I(f¡) for sometransition

tn eT. We denote the transition ú¡ by E¿(t¡).

3.Foreverytransitiontt"€Ttheremustexistatransitiont¡€Tandan

i e {L,. .. ,, Br¡}, such that I(¿fr) : O¿(ú¡).

Structural Condition 3 implies that every input bag is also an output bag, and Struc-

tural Condition 2 says that every output bag must also be an input bag.

The relationship between the routing chain and the closed-form equilibrium invariant

measure may be unclear to the reader. It is important to note that Theorem 3.1

applies only if /(t;) is positive for all transitions t¡ e T. The structural conditions

are a consequence of this. However, the routing chain of an MPN satisfying the three

structural conditions may still have transient states and Theorem 3.1 will not apply.

The following is an example of an MPN which satisfies each of the structural conditions

and for which the set .F is emPtY.

Example 3.L - Non Product-Form MPN

Consider the MPN given in Figure 3.1. This net satisfies each of the

structural conditions and is live for the initial marking shown.

The dashed arcs from transition ú2 indicate probabilistic routing and the

routing chain is defined by the following routing probabilities

p(tq,tz):p(ts,úr):1p(h,h)

63


t3t1

Pl

ItI

t4

P4 oP3

I

Figure 3.1: A Live and Closed SPN without Product-Form

p(t2,tL) : a

p(t2,,t4) : (1 -o)

where a is also a probability.

The reachability graph and the routing chain are given in Figure 3.2 A anð'

B respectively.

Consider calculation of the set .F, Equation (3.2). We have

x(¿')/(¿') ! ay(t2)f (t2)x(tn)l(tn)x(t')f (t')

(r - a)y(t")l(tr)

where k1 and k2ate constants.

Clearly the second and fourth equation are contradicting and imply that

/(ú2) and f (ta) are equal to zero. Therefore any potential product-form

equilibrium distribution given by Theorem 3.1 does not exist.

Finding the equilibrium distribution by solving the global balance equa-

tions is quite difficult and is not included here. Such an exercise however

x(t')l(tr)x(tr)r(tz)x(¿')/(¿u)x(ta)lQa)

:kt:k2:kL

t^tù2

64


h..t

(1,1,2,0)<---.--->

t4

h.,q

tl t3

tl t3b,t

(0,2,1,1)<--------> (1,1 tz,,t

r1

(0,2,2,O)

(2,0,0,2)

h.,t

r3

(1,1 1,1)t1

t4

r3

t4 h,4

(0,2,02)

A

t2 ------------> tl

r3¡4

B

Figure 3.2: Reachability Graph and Routing Chain

very quickly verifies that the net does not have a product-form equilibrium

distribution for the initial marking shown. tr

Sereno and Donnatelli [22] call nets which satisfy the three structural conditions closed

neús. From the above example it is apparent that closed MPNs do not always have a

closed-form invariant measure given by Theorem 3.1.

As we are interested in MPNs that do have product-form, for the remainder of the

thesis we shall refer to MPNs which satisfy each of the structural conditions and for

65

3.1 Instantaneous Firinqs

Node I

Link

Node 2

2

Node 3

Figure 3.3: A Simple Loss Network

which the set .F is non-empty as balanceil Marlcoui,an Petri' neús (BMPNs).

Note that the Markovian assumption is not necessary for Theorem 3.1 to hold. Hen-

derson and Taylor [39] relaxed the exponential assumption by considering the jump

chain of the Markov process defined above as the jn-p chain of a semi-Markov process

embedded in a more general Petri net framework.

3.1,.1 Examples

We will now present some examples of BMPNs which have product-form equilibrium

distributions. In each case we will give the form of the function g(m) which satisfies

the conditions of Theorem 3.1. A method to calculate the required form of g(m) is

considered later in Section 3.7.2.

Loss Networks

BMPNs can be used to model loss networks which are well known in the field of

telecommunications (see for example [52]). A loss network consists of a number of

nodes connected by links. Each link has a certain capacity or number of circuits.

When one node wishes to make a connection to a second node, the network checks to

see if sufficient capacity is available. If there is not enough capacity on every link along

the path, the call is refused and is said to be blocked, otherwise the call is connected

and uses every circuit on its route for the duration of the call.

Consider the loss network of three nodes, Figure 3.3, and its closed MPN equivalent,

66

3.1 Instantaneous Firinss

P,Pl

Po P5P

3

t3

t6

t2

t5

tt

J4

Figure 3.4: Closed MPN Equivalent

Figure 3.4. The loss netork allows each of the three possible node connections (calls),

which are explicitly included in the Petri net model. A call between node 1 and node

3 requires 1 circuit from link 1 which is represented by transition f1 in the Petri net.

The remaining two calls are represented in the same way. The top places, Pt and Pz

correspond to links 1 and 2 respectively with the number of tokens in these places

representing the number of circuits available on each link. The lower places are the

hotding places which indicate how many calls of each type are in progress.

Calls on each route arrive as independent Poisson streams with an upper transition

for each call and the call holding times are exponentially distributed. Call completion

is modelled by each lower transition which reduces the number of calls on the route

by one and releases the circuits on each link so that they can be re-used. MPNs have

been used by other authors for the modelling of loss networks [19, 41].

The minimal support S-invariants in a loss network are such that the holding places

form a sufficient place set. Let /ú.ú be the number of links in the loss network or top

places in the BMPN.

67

3.1 Instantaneous Firinss

Loss networks are then modelled by BMPNs with the following parameters,

NP

ú(*) : Õ(m) : II oi(rn(i)),j--t

if1<i<NLo¿(rn(i)) :

I

ifNL<i<NPn1.lN)l

(3.5)

X(t;) if ú¿ is a top transitionq(m, ú¿) :

m(i)y(t¿) if ú¡ is a lower transition.

Theorem 3.1 is satisfied by a function 9(m) of the form,

NPg(m) : II siØ

j=NL+r

This gives the equilibrium distribution to be

aiQ) (3.6)*(j)l

which is the well known loss network product-form.

Clustering Processes

Clustering processes are considered by Whittle [90] and Pollett [71]. They consist of

collections of clusters which can aggregate or dis-aggregate to form other clusters or

they can mutate from one type of cluster to another. Clusters are abstract entities with

multiple interpretations. For example, clusters can represent different sized groups of

individuals or elements of chemical reactions.

The state description of a clustering process is the number of each type of cluster.

Clusters can aggregate, dis-aggregate and mutate in a state dependent manner. As an

example consider the simple coupled chemical reactions

NPzr(m) : 1( II

j=NL+t

q5--l\-q2

A+ B

68

C


q4

A + Dqr

q6

B + E.q3

(3.7)

This says that clusters A and B can join to form C which in turn can break apart into

its two constituents. Clusters A and B can also mutate into D and E respectively

which can in turn mutate back into A and B. The BMPN used to model the above

loss network, Figure 3.4, can also be used to model the clustering process described

by Equation (3.7) with the tokens in the places having the following interpretations,

P1 : Number of units of A

P2 : Number of units of B

Pz : Number of units of D

Pa : Number of units of C

P5 : Number of units of E.

The rate of a chemical reaction is proportional to the number of possible collisions of

reactants, which implies that

et : q(m, Ír) : m(P1)y(ty)

ez : q(rn,tz): m(Pt)m(Pr)X(tr)

es : q(m, fs) : m(P2)y(ts)

eq : q(m, ú¿) : m(ft)y(ta)

es : q(m, fu) : m(Pa)y(t5)

eo : q(m,úo) :m(P5)y(t6).

The above rates of reaction are modelled by a BMPN with the following parameters

NP

ú(*) : o(m) : II oi(rn(i)),i=l

O¿(rn(i)) :#

69


Theorem 3.1 is satisfied by

g(m) : II aTu)NP

j=NL+r

The resulting equilibrium distribution is

As with loss networks, the communicating classes of the routing chain of clustering pro-

cesses consist of two states. The only difference between loss networks and clustering

processes lies in the type of state dependent firing that each exhibits. We believe this

to be an important distinction which will hopefully clarify the relationship between

the two processes.

Other Examples

Jin and Aoki [a5] used a closed SPN to model a computer system network consisting

of multiple workstations and a central database. The net which they used is given

in Figure 3.5. By assuming exponential delays, the underlying stochastic process is

a Markov process and, using Laplace transform techniques and a complicated proce-

dure, for a particular initial marking, they obtain numerical results for performance

measures such as mean recurrence times, equilibrium probabilities, utilisations and

average number of visits.

By incorporating probabilistic routing we analysed the net using the methods discussed

above and those which follow below to give the equilibrium distribution in the state

independent firing case for any initial marking to be

-(s)

,,(m) : " (,q #) (,=Ti., #)

]-"'[#]-"'x(t6)x tz

x(t4)(x(t6) + x(¿r))

ts)

y(t1)y(t6)y(tr)

x(tz

"(m) : I{

(¿o)(x(¿u) + x(tz))

*(a)

70

(t") x(t o) (x(¿u) + x(t z))*


P2 P3 t3

t1

t5

t6

P5r4

t 1 : Setting of a local Prooess

t2 : Initiating a local process

t3 : Terminationof alocalprocess

t4 : Setting of a remote process

t5 : Initiating a remote Process

t6 : Reservation of a remote process

t7 : Termination of a remote Process

P1 : Idle CPU

P2 : Local process number set

P3 : Executing localprocess

P4 : Remote process number set

P5 : Executing remote process

P6 : Nurnber ofprocesses left

Figure 3.5: Computer System Network

where ¡(ú¿) is the state independent component of the firing rate given by Equation

(3.1).

3.L.2 Product-Form

In single movement queueing networks, the functions g(m) and O(m) are often

product-forms over the nodes of the network and, as discussed in Section 2.2.3, the sub-

sequent product-form equilibrium distribution is amenable to convolution algorithms

for finding normalising constants (Convolution algorithms are discussed in detail in

Chapter 5). To apply the product-form result of Theorem 3.1 it would be advanta-

geous if the functional forms of g(m) and Õ(m) in equation (3.a) were such that the

equilibrium distribution was amenable to the application of convolution algorithms.

This would be possible if these functions were simple product-forms themselves either

7I

3.1 Instantaneous Firinqs

over the transitions or over the places of the SPN. Equation (3.3) suggests that a

form for g(m) could be found by tracing a finite sequence of transition firings linking

each marking to a home state (the initial marking for example), which would yield

g(m) as a product-form over the transitions of the SPN. Contrary to this we find a

product-form over the places of the SPN subject to the minimal support S-invariants.

Let/€.Fand

¿,n(#)

c(/) : (3.8)

nn(##)

Iog

where E¿(t¡) is defined in Structural condition 2 of section 3.1.

Note

There may be many functions f e F. However each one is unique up to a constant

multiple in each communicating class of the Markov chain with transition probabilities

p(tj,tk).Thus the ratio /(ú¡) lfþù is invariant over.F whenever Ú¡ and t,¡ate in the

same communicating class. By definition E¿(t¡) is in the same communicating class as

t¡rV i,ú¡. Therefore C(/) is invariant over alt / € F,, anð, without loss of generality,

we can write C(/) as C.

Theorem 3.2

For a BMPN, ,A/, with .F non-empty and zero firing times, the function Irequired to satisfy Theorem 3'1 is of the form,

72


NPg(m) : f yiØ V rn€M

i=l

if and only if

Ëønk([A]) : Rank(l¡ lCl),

where tA I C] is the matrix A augmented with the vector C

In this case {y¿, i: P¿ € P}, satisfies the matrix equation

(3.e)

(3.10)

Iog (vr)

-A -c.log (ar)

To establish when the form of g(m), given in equation (3.9)' satisfies the

conditions of Theorem 3.1, substitute equation (3.9) into equation (3.3) to

(3.11)

Proof

glve

yro(o,r)Aîo(b,2)...yñlþ,*r): ffi V t¡,t¡:p(t¡,¿o) > 0.

where b is the input/output bag pair (I(Ú¡), o¿(úr)) such that o¿(t¡): I(úr).

Taking logs gives,

-A(b,I)los(yù A(b, P)tos(Yr) : ,"nfffi)'

This is a system of B equations in lúP unknowns which can be expressed

in matrix form as equation (3.11). Equation (3.11) is solvable if and only

if

Ëønk([A]) : Rank(l¡ lcl).

73


conversely if equation (3.11) is satisfied then 9(m) given by (3.9) satisfies

equation (3.3).

Remark

Theorem 3.1 requires the existence of the functions / and g satisfying equa-

tion (3.3) to establish the existence of a product-form invariant measure

for the marking process of the MPN. Theorem 3.2 requires the existence

of the function /, and condition (3.10), as a sufficient condition for the

existence of the function g. It is then necessary and sufficient that I has

the special form (3.9). It is important to note that equation (3.10) is a

significantly easier condition to establish than that of equation (3.3).

Note that C depends on the firing rates through the definition of /. When equation

(3.10) is satisfied for any C of the form (3.8), there is no restriction on the firing rates

for the product-form (3.9) to exist. A product-form can still exist when (3.10) is satis-

fied only for certain values of X (and hence /), but in this case there is interdependence

between the firing rates of the SPN. When the rank condition is satisfred, we can then

find the functions y¿ by solving the system of equations (3.11).

As for closed queueing networks, the product-form (3.9) reflects the fact that, in

equilibrium, for some fl and P¡ €P, the only dependence between rn(i) and m(j) at

an arbitrary time point in a stationary process, lies in the normalising constant.

There may be solutions to the matrix equation (3.11) where log(A): 0 for some i. In

this case U¿ : I and no term involving y¿ appears in the product.

3.1.3 The Role of the T-invariants

It is not immediately apparent, but the minimal support T-invariants play an im-

portant role in the rank condition (3.10), and provide an alternative but equivalent

74


condition which is more intuitive and which is analogous to the consistency conditions

of Lazar and Robertazzi.

Theorem 3.3 - Equivalent Rank Condition

For a BMPN, the following statements are equivalent.

(i) Ëorzk([A]) : Bønk([Alc])

(ii) tC:0 VtsuchthattA:0.

Proof

The result is similar to that proved in Theorem 2'1.

(+)

Assume that condition (i) is true. Then a solution exists to Equation

(3.11). Then for all vectors y such that (3.11) holds we can pre-multiply

the equation by any vector t such that tA : 0 which implies that tc : 0.

(+)

Assume that condition (ii) holds. That is, assume that C is orthogonal to

the left kernel of A, which means

tA:0 + tC:0

this implies that the left kernel of A is a subset of the left kernel of C which

in turn implies that the column space of C is a subset of the column space

of A since the column space of any matrix is orthogonal to the kernel of its

transpose. Hence C can be written as a linear combination of the columns

of A which implies that a solution exists to Equation (3.11) which implies

that Ranlc([A]) : r?ønk([Alc]).

75


The equivalence of the two conditions of Theorem 3.3 allows the T-invariants to be

used to establish product-form over the places. We can however obtain a stronger

equivalent rank condition.

Corollary 3.1

The second condition of Theorem 3.3 can be replaced by

t C:0 Vminimalsupport T-invariants t.

Proof

It was proved in [1a] that when .F is non-empty the set of minimal support

T-invariants span the kernel of A" which gives the result.

Remark

As with the consistency conditions of Lazar and Robertazzi, Corollary 3.1 conserves

some kind of "flow", in this case defined by the vector C, for every T-invariant. The

rank condition is easy to verify and is a structural consistency condition unlike that of

Lazar and Robertazzi. The Lazar and Robertazzi condition conserves flow over every

closed path of the reachability graph and must be established for each initial marking.

3.L.4 The Role of the Function t/

It was mentioned earlier in this chapter that the function rþ can be used to inhibit

the firing of transitions under certain marking conditions. Unfortunately the extent

to which this can be accomplished is limited. It will become apparent in this section

that the definition of firing rates, Equation (3.1), imparts conditions on BMPNs which

affect liveness and hence reachability.

76


We define C C T to be a communicating class of the routing chain. Hence for any two

transitions tj,tx e C there exists a path of N transitions

{t¡,t¡r,...,tj*,t¡} (3.12)

such that

p(t ¡,t ¡,)p(t ¡,,t ¡") . " p(t ¡ *,¿o) > 0.

Definition 3.2 - Cycle

A cycle is a sequence or path of transitions within the same communicating class of

the routing chain such as that giuen by (3.12), where the f,rst and the last transition

are the se,rne,

{t¡,t¡r,...,tj*,tj}. (3.13)

The definition of firing rates (3.1) forces solidarity-like properties on the firability of

transitions in the same communicating class of the routing chain. Before we examine

the consequences of the definition of firing rates we require the following definition.

Definition 3.3 - Base Markings

The set of markings giuen by

BM :{m - I(f¡) : rn e R(N,mo), tj e FS(N,m)}.

are lenown as the set of base markings

Base markings are the residual markings left behind after input bags are removed

by firing transitions but before the corresponding output bags are deposited. The

following theorem shows the role that base markings play in the behaviour of the class

of nets that we are considering.

77


Theorem 3.4

For a fixed m e BM one and only one of the following is true,

(i) q(m*I(ú¡),ú¡):0 Y t¡ eT

OR

(ii) q(m*I(¿¡),ú¡) > 0 V t¡ eT

Proof

Let q(m + I(új), f¡) : 0 for some t¡ € T ' This implies that

ú(*)x(¿¡) :QÕ(m * I(¿¡))

which, since ¡(ú¡) ) 0 is equivalent to

t/(m) : 0.

Hence we can writeú(*) -0x(t*)

o(m * I(¿*))

for any transition f¡.

Since ú¡ is arbitrary this proves both parts'

¡

Related to Theorem 3.4 is the following Lemma which relates to the firability of tran-

sitions within communicating classes of the routing chain.

Lemma 3.1

If a transition f¡ of some communicating class c of the routing chain is

firable, then every transition in C is firable'

78


Proof

Let m € 7?(ms) be a marking for which transition f¡ is enabled. This

implies that

g(m, ú¡) : ,þ m- t ))x(¿¡) >0o(*)and thus

,á(*-I(¿j))>0 (3.14)

where m - I(ú¡) is the base marking resulting from the firing of transition

ú¡ in marking m.

Now consider a second transition tn € C such that p(t¡,t*) > 0' The new

marking resulting from the firing of transition ú¡ and depositing the input

bag of transition ú¡ is m _ l(t¡) + I(¿È). Note that when transition Ú¡ fires

this new marking has the same base marking as m when ú¡ fires. Transition

f¡ will fire at rate

q(m - I(új) + I(úr), tx) : ,þ(^ - I(¿¡)) >0o(*)

due to (3.14). Since there exists a firing path between any two transitions in

the communicating class, the above argument will follow for each transition

in any given path. Hence from m there exists a LFS which allows every

transition in the communicating class to fire.

tr

The transitions in each communicating class of the routing chain will form one or more

cycles. The following corollary shows that the net change in marking resulting from

firing a sequence of transitions for one of these cycles is zero. Thus the firing count

vector for any such cycle must be a T-invariant.

xQk)

79


Corollary 3.2 - T-invariants within The Routing Chain

Let õ be the firing vector for a sequence of transitions which constitutes a

cycle in a communicating class C of the routing chain. The net change in

marking due to the firing of d is zero, that i, m 4 m.

Proof

Let the sequence of lú transitions defining a cycle in C be given by (3.13)

with firing vector ã.

Recall from the proof of Lemma 3.1 that the marking resulting from the

firing of transition ú¡ in such a sequence is m - I(Úi) + I(Ú¡). Since the last

transition in a firing sequence which is a cycle is also ú¡, firing transition

f¡ will re-produce the marking m and the result is proved.

The immediate result of Corollary 3.2, as mentioned above, is that the firing vector for

every closed sequence of transitions in each communicating class of the routing chain

is a T-invariant. It then follows, by Corollary 3.1, that every T-invariant which is a

firing count vector for a cycle within a communicating class is firable if at least one

transition in the communicating class is firable.

Remark

We see the building block and consistency conditions of Lazar and Robertazzi appear

ing again. Corollary 3.2 guarantees the firability of T-invariants within communicating

classes of the routing chain. Firable T-invariants correspond to closed paths in the

reachability graph and hence to building blocks. The rank condition is a consistency

condition which is necessary and sufficient for a product-form to exist.

80

3.2 Delayed Firings

9.2 Delayed Firings

We now generalise the SPN model of the previous section by allowing exponentially

distributed enabling anil frringtimes. In previous sections we have assumed zero frring

times which means that tokens instantaneously move from input to output places.

Henderson and Taylor [3g] considered continuous time SPNs with generally distributed

enabling and firing times. They obtain various closed-form equilibrium distributions

by considering the Markov process embedded at enabling and firing points; enabling

points being the time when a transition becomes enabled and a firing point the time

when it fires. By considering the process only at particular time points the sPNs are

modelled as semi-Markov processes'

There is one very important difference between the the model which we introduce here

and that of Henderson and Taylor. The model described in [39] requires the firing of

each transition to be completed before any other transitions can fire. That is, the input

bag of tokens from a firing transition must be deposited as the appropriate output bag

before another transition may fire.

The model that we consider in this section is not so restrictive in its firing rules. We

allow transitions to fire when input bags of other transitions are in transit- Moreover

we allow the firing rates of transitions to depend both on the tokens present in the net

and on the tokens in transit. Provided there are sufÊcient tokens, enabled transitions

may continue to fire regardless of whether any output bags have been deposited'

The state of the system at any time is the current marking plus the list of input

bags waiting to be deposited. In this section we derive the product-form equilibrium

distribution for such a BMPN with exponentially distributed enabling and firing times.

81

3.2 Delayed Firinss

3.2.L The Model

Consider a continuous time BMPN, ,Â/ with exponentially distributed enabling and

firing times. Such an extended net can now model the situation where an original

transition can fire before the firing of another original transition is complete' This

is very useful in modelling as it allows for the passage times of various events. The

connection time of a telephone call in a loss network, the time for packets to travel

down a virtual link in a packet switched network, or the change over time for machines

in a FMS are possible applications of this model.

We can analyse such an extended net by making a simple observation and applying the

results that we already know from previous sections. The introduction of exponential

firing times is equivalent to inserting a state dependent single server queue between

each transition and its output places. The output bag then acts like a customer that

requires an exponential amount of service from the queue before it can be deposited

as the input bag for another transition.

Hence the incorporation of firing times into a BMPN, N, can be accomplished by

constructing a new extended BMPN, ,Â/*, *ith zero firing times but which has an

additional place and transition inserted at the output of each original transition. Thus

to incorporate exponential firing times into a net with IúP places, .fú? transitions and

zero firing times, we add lú? additional places and transitions to produce an extended

net with (NP+NT) places, 2lú? transitions and with a significantly larger reachability

set (if bounded). Note again that the extended net has zero firing times and can be

analysed through Theorem 3.1. It is the additional places and transitions which model

exponential firing times for the original net.

The additional transition and place for transition ú¡ will be labelled tj+xr and P¡.'¡vp

respectively. Examples of which are shown in Figures 3.6 A and B.

Although rvve can analyse the extended net,.Â,/*, using the product-form results es-

tablished earlier in this chapter, this requires dealing with a much larger incidence

82

3.2 Delayed Firinss

tj ,j tjtj

PiP¡+Ne q q.

P¡+Ne

+ t¡ + ttt + ti+Nt

A

matrix and ignores the fact that the extended net has special structure. In the follow-

ing we make use of this special structure by establishing a method for calculating the

product-form invariant measure of the extended net which only requires knowledge of

the original net ,Â,/.

The state of the extended BMPN, Al*, at any time is denoted by m* : (m, a) which

consists of the original marking m and the marking of the additional places a : (a(i) :

m(NP +i),i - 1,..., N?). The component cr¿ can be thought of as the number of

output bags in transit for each original transition.

Let the new set of transitions and places be denoted by T* and P* respectively, and

the new reachability graph by M*. Also define the set F* by

Pkq

B

q

Figure 3.6: Addition of Firing Times

f O,T ìT* -> R: f (t¡) > o,x(ti)l(ti) xþù l(tn)p(tx,t¡), V t¡ € T ñ T

The extended net still satisfies the three structural conditions of the previous section.

This is apparent from Figure 3.6 since each additional transition has a unique input

bag and each additional place is both an input bag and an output bag.

The transition firing rates of the extended net are given by

:tt*€TnT'

83

3.2 Delayed Firinss

where the extended input bag for each transition ú¡ in the extended net is defined by

I-(ú¡) : (I1(m, t¡),12(a,t¡))

where

11(m,f¡): I(¿¡) if t¡ e T0 ift¡€T*

and

I2(a,ú¡) :[:t :f^'.''T- l0 if t¡€Tare the portion of the extended input bag corresponding to each set of places P and

P* respectively. The extended output bags, o.(¿), are defined similarly.

The extended net has two disjoint sets of places, P and P* in the sense that the

extended input and output bags take non-zero entries from only one of these sets. We

know make use of this property by assuming that all of the relevant functions can be

partitioned as a product of two functions, one from each of the two disjoint sets of

places P and ?*. That is,

,á(*. - I.(¿¡)) : rþ(^- Ir(m, t¡),^- 12(a,l¡)) : tþt(rn - It(*, t¡))rþr(" - 12(a,f¡)),

and

Õ(*.) : O(m,a) : iÞr(m)Or(").

The following lemma establishes sufficient conditions for the function g(m.) to also

partition into a product of two functions.

Lemma 3.2

If there exists functions gr(m) and 92(a) such that

er(m * I(t : T(t¡) t¡ €T (3.15)er(m)

gz(a -+ -et) : r(t¡,) tn € T*

gz\a)(3.16)

and

84

3.2 Delayed Firings

then

satisfies Equation (3.3),

g(m.) : 9(m, a) : 91(m)Or(";

V t¡,t¡: p(t¡,¿*) > 0.

Consider the left hand side of Equation (3.3)' which becomes

er(m * Ir(m, t¡))g, (a * Iz(a, ú¡))(3.17)

9r(m * Ir(m, tn))gr(^ * Iz(a, ú¡))

Equation (3.17) produces two cases depending on the transitions t¡ and't¡.

In each case, the right hand side of (3.3) is reproduced.

Case i: t¡ €T, tn €T"

9r(m 1 t )gr("))

Proof

g{yn)gz(a* er)

Case ii: f¡ €T*, tn€T

e1(m)s=rj3l e1)r : I(:.t, v t¡,t¡,: p(t¡,¿o) > 0.er(m + I(úfr))s2(a) f(t¡l

Thus if we can find the two functions 91(m) and g2(a), their product satis-

fies the conditions of Theorem 3.1 and the product-form invariant measure

follows.

Consider now the firing rates of transitions from each of the sets 7 and T*

th (m -l(t ¡)) tþz (a) x(t ¡)@1(m) 02(a) if t¡ eT

:ffi y t¡,t¡:p(t¡,¿*) >o

,h(rn=l !z(?;ei) -xþ¡l if t¡ e T*<Þ1(m) 02(a)

tr

q(m., úi) :

85

3.2 Delayed Firings

In particular, we can model various state dependent firing rates in the extended net

by choosing the functions út(*), ,þr(^),Ot(*), Or("), X(ú) appropriately'

We can model the situation where the firing rates of the transitions in the two sets

T ard T* are dependent only on the markings of the places in the sets P and' P*

respectively by setting Q1(m) : Út(*) and Õ2(a) :'þz(a) which gives

I or (m-I(,i)) x(tr) ir t ¡ € T| .r. rm\

g(m*, ¿¡) : {

I irt¡eT..

On the other hand, if we wish to p eserve the firing rates of the original transitions,

we can set O2(a) : ,þr(u) which giv s

.rr(m:I(¿i)_) x(úi) if t¡ e T@r(m)

q(m*' úi) :úr(m)oz(a-e') x(tr)

o1(m)02(a)

i=1

if t¡ €T*

What we accomplish is a decomposition which allows .Â'/* to be split into two separate

BMPNs which are linked only through the routing chain.

The simple nature of the second condition, Equation (3.16), defines Cz(a) to be

NT

ez(a) : f[ ,f(tn*"r)'(n)

The first condition (3.15), will produce a functional form for 91(m) which is net de-

pendent, and can be found through the solution of a matrix equation. We consider a

product-form lesult for 91(m) in the next theorem.

We have established the following corollary to Theorem 3.1 for BMPNs with exponen-

tial firing times.

86

3.2 Delayed Firinss

Corollary 3.3 - BMPN with Exponential Firing Times

If lafunction ÍeF. andafunction h:M ìRsuchthat

er(m + I(¿i) : r(t¡) v t¡ eT (3.18)gr(m)

and Theorem 3.3 is satisfied for the extended net N" then the marking

process of a BMPN with exponentially distributed firing and enabling times

has an invariant measure given bY

N?

"(m.) : iÞ1(m)Õr(")gr(*)fl"f(¿n*rr)"(n) nf e M*,, (3.19)

)

i=I

which can be normalised to give the equilibrium distribution whenever the

underlying Markov process is regular and

N?

D Õ1(m)Õ2(a)ø'(*;fl "f(¿n*"r)"(n) < oo

m'eJv[. i=l

tr

3.2.2 Product-Form

The condition given in Equation (3.18) can be used to produce a matrix equation as

was done earlier in Theorem 3.2. As before, in order to apply the product-form result

of Corollary 3.3 we need to be able to find a functional form for gr(m) which leaves

the equilibrium distribution amenable to the application of convolution algorithms. In

this section we again produce a matrix equation and a rank condition which will define

a product over the places for 91(m). We require the following matrix and vector.

Definition 3.4 - Input Bag Matrix

The input bag matrb B is a N? x N P matrir with a row for the input bag of each

transition.

87

3.2 Delayed Firinss

Also let

r(/) :Ios (f (t^r7))

Theorem 3.5

For a BMPN, .Â/, with exponential firing times and F* non-empty' The

function 91 required to satisfy Corollary 3'3 is of the form,

NP

9r(m) : f h':$) V rn4_ M (3.20)

bgUIog U

(¿' ))(¿'))

i=l

if and only if

r?ønk([B]) : Rank([B I l])'

where tB I l] is the matrix B augmented with the vector I

In this case {h¿, i: P¿ € 2}, satisfies the matrix equation

(3.21)

Blos (h¡)

los (hp)-l (3.22)

Proof

To establish when the form of 91(m) given in equation (3.20) satisfies the

conditions of Corollary 3.3, substitute equation (3.20) into equation (3.18)

to give

I(t¡,1)I

I(tr,2) h',|'i'*') -- l(t¡) v t¡ e Th

Taking logs gives,

h 2

88

(3.23)

3.2 Delayed Firings

I(t¡,L)los(/,t) + ...+ I(tj,,NP)Ios(hNr): los(f (t¡))

This is a system of NT equations in NP unknowns which can be expressed

in matrix form as equation (3.22). Equation (3.22) can be solved if and

only if

nønk([B]) : ,rank([B lU).

conversely if equation (3.22) is satisfied then 91(m) given by (3.20) satisfies

equation (3.18).

Remark

For a BMPN with exponentially distributed firing times we now have two

methods to obtain the equilibrium distribution. We can either construct an

extended net and apply Theorem 3.2, or \rye can apply Corollary 3.3 directly

in conjunction with Theorem 3.5. Note that the product-form obtained in

each case will be equivalent but will appear different as in one case a term

in the product will appear for every place and in the other some terms will

be missing dependent on the S-invariants. This will become clear in the

following example.

The application of Theorem 3.2, Corollary 3.3 and Theorem 3.5 is shown in the fol-

lowing example.

Example 3.2

Consider the BMPN , N in Figure 3.7, and the resulting extended net, .Â/*,

after additional places and transitions are added to account for exponential

firing times, Figure 3.8. The additional places and transitions are enclosed

in dashed circles.

89

3.2 Delayed Firings

P2

P,P1

P4

P5

r1

r3

¡2

t4

A

Figure 3.7: Net without Firing Times

Firstly we will find the equilibrium distribution by using Theorem 3.2 by

considering the extended net as a whole entity, and then we will apply

Corollary 3.3 directly in conjunction with Theorem 3.5 where we can con-

sider ,Â/ as a sub-net of .¡V* which can be analysed independently. The

equivalence of the product-form equilibrium distributions produced by the

two methods will be established. However it will be clear that the product-

form equilibrium distribution is significantly easier to find by application

of Corollary 3.3.

In this example the routing chain of ,l\/* has two communicating classes,

{tr,t",ts,tz} and {ú2, tq,ta,ús}. The functions I e F" satisfy the invariant

measure

l(t')x(t') : /(t')¡(t'¡ : f (tu)y(tu¡ : f (tr)y(tr) : k1

f Ur)xþn) : /(tu)¡(t'¡ : f (t6)v(tu) : f(t")vçt"¡ : ¡x,

where k1 and k2 are non-negative constants.

The incidence matrix A* and the vector C* are given by,

90

3.2 Delayed Firinss

P¡PzPr

P6 P7

P5P

4

Ps P9III

I

t3

IItI\t

5

7

tt

It

't 6

a2

¡4

Itiltltl

,' ta,

'le

IIII

Figure 3.8: Addition of Firing Times

B

0

0

0

I0

0

0

-11

0

0

1

0

0

0

0

1

0

I0

0

0

0

0

0

0

0

1

0

0

j0

0

0

-10

1

0

0

0

0

-10

1

0

0

0

1-1 0

1 -10

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

0

0

1

A1

91

3.2 Delayed Firings

and

C*:

t"n(#)

t"n (ffi)

t"n(ffi)

t'n(#)

t"n (^9ru)

t"n(#)

t"n (#)t"n(#)

respectively.

The extended BMPN, I,/* h.s three minimal support S-invariants. Taking

the inner-product of each with any marking gives,

rn(l) + m(\ + rn(6) + rn(8) : constant

n-L(z) +ræ(3) + m@) +rn(5) + rm(6) + m(7) +rn(8) + rn(9) : constant

and

rn(3) + rn(5) + m(7) + rn (9) : constant

where the constants are non-negative integers determined by the initial

marking.

The augmented matrix [-4. I C.] is row equivalent to the fully row re-

duced matrix

ci+cä+cä+cic;+ci+cä+cä

c;+cici+cä

cä+ci+cici+cä+c;

ciCä

0 0 0 0000000 0 0 000000-1-1 0 100000

1-1 0100000

10 0010001-1 000100

0 000010-l 0000010

-10

1 1

1

92

(3.24)

3.2 Delayed Firings

and

Note that the two rank conditions

ci+cä+ci+ci:o

c;+c;+cä+cJ:o

also result by pre-multiplying C* by the two minimal support T-invariants

(1, 0, 1, 0, 1, 0, 1, 0)

and

(0, 1,0, 1,0,1,0,1)

respectively, and are satisfied for all C.. The rank condition is the only

requirement of Theorem 3.2. Hence solving for the unknown variables y¿

will provide us with a product-form for g(m).

We set log(A) : log(Az) : log(y"): 0 and by inspection write down the

solution for the remaining variables,

and similarly

f (tùf (tz) l(t") x(tt)un: r(h)rþù: /(¿,) :

¡¡¡''

xft2)us:

^ '

xUt)aa: ,ç¿¡'xU2)Y7: xll*)'xUL)ua:

^rtxUz)us: V7¡¡'which from Theorem 3.2 gives 9(m) to be

93

3.2 Delayed Firings

l#] ^"^'

l#] -"'

[#] -'"

lffi)^' l#] -"'

lffi] -'n

(3.25)

The product-form of 9(m) and the functional form of O(*) define the

equilibrium distribution (when it exists) through Theorem 3.1.

Note that the row reduced form of the matrix equation (3.24), is derived

via a particular pivot combination. Each different pivot combination will

produce another product-form for g(m), one for each sufficient place set.

Now consider the product-form result from the application of Corollary 3.3

and Theorem 3.5.

The routing chain of N also has two communicating classes, {Ú1,f3} and

{tr,tn}.The functions / € .F satisfy the invariant measure

f(t1)y(tr) : f(t")y(t") : k'

l(tr)x(tn) : f(tu)y(t") : tc,

where k1 and k2 ate again non-negative constants.

The input bag matrix B and the vector I for the original net Al are given

by,

andtos (f(t1))

tos (f(t2))

tos (f(ts))

tos (f(ta))

":(ill:i)

respectively.

l:

94

3.2 Delayed Firings

The augmented matrix [B I l] is row equivalent to the fully row reduced

matrix

which means that the rank of B is always equal to the rank of [Bll] in this

/ r 0 -l 0 0 lr-lz \lo l 1 o o tz I

ItB B å î ',: )'

case.

As before, we set log(h): 0 and read off the solution for the remaining

variables,

, k*(tz)n': lrrx(t)'

L- k2tú2 - x(t2)'

L- lct,r4 - x(h)'L- kztÚ5 - (tn)'

The functional form of 92(a) is known through corollary 3.3 to be

ez(a) :t#]-'" l#]-"' [#]-'" lå]-"'Theorem 3.5 then gives 91(m) to be

Itcrx1r)l-t" I t, 'l-t" l *, 1-{a) ¡ ¿, 1-{s)gr(m) :L*ffi1 L;øl LilÐl L¡aJ

The function g(m.) is then the product of 91(m) and gr(a). If constants

k1 and k2 aîe removed as factors from g(m*), the power of k1 is simply

the first minimal support S-invariant and the power of k2 is the first minus

the second minimal support S-invariant and hence are constant for all

reachable markings. We can therefore omit k1 and lc2 from the product-

form of g(m.), as any constant factors in the equilibrium distribution will

be absorbed by the normalising constant.

95

3.2 Delayed Firings

Hence we can write

g(m.)

* [x(¿r)]-(u) lx(tu)l*(') [x(¿r)]-(') ¡r1tr¡1-(e) . (3.26)

The product-form invariant measure is then given by Equation (3.19).

Equation (3.26) appears at first glance to be different to Equation (3.25).

However a simple application of the minimal support S-invariants converts

one to the other. To transform (3.26) into (3.25) perform the following

substitutions

*(r) constant - *(4) - -(6) - -(8)

*(2) + constant -*(4)--(5) --(6) -*(7) --(8)--(9)'

As before the constant terms can be discarded as they will appear in the

normalising constant. ¡

We discussed previously why we are interested in systems which possess product-form

equilibrium distributions. In the next chapters we examine algorithms which can be

applied to find performance measures for product-form MPNs through the evaluation

of normalising constants. The algorithms are not just restricted to the BMPNs defined

in this chapter but any net with product-form which satisfies certain requirements.

tffi] -t"

k(r,)l-(') [x(¿,)] *(a)

¡rç¡ n¡1*F)

96

Chapter 4

Cornputational Algorithms

In the previous chapter we established sufficient structural conditions for the existence

of a closed-form equilibrium distribution in MPNs, and necessary and sufficient con-

ditions for that closed-form to be a product over the places. We also discussed in

Chapter 2 the difficulty in analysing large systems, and how the problem is alleviated

to a certain extent in queueing networks by the existence of computationally efficient

algorithms for evaluating performance measures. In the remaining chapters we will

concentrate on evaluating the normalising constant for any product-form Petri net

which satisfies certain conditions. BMPNs, as defined in Section 3.1, are just one class

of nets for which these conditions may hold'

Existing algorithms for single movement queueing networks accomplish the calculation

by sequentially breaking the network down into smaller and smaller sub-networks, with

the normalising constants of each being related (see for example [6], [13]' [17]' [23]).

When the normalising constant of the smallest network can be easily calculated, the

normalising constant of the original network can be found by recursion. Unfortunately

SPNs and batch movement queueing networks are more complicated due to the types

of state transitions and the additional complexity of the reachability graph.

A consideration of the existing algorithms which operate on single movement queueing

networks reveals two essential ingredients,

o a product-form equilibrium distribution, and (4'1)

97

4.O Computational Aleorithms

The nets considered in this chapter are assumed to have a product-form which can be

described by Equations (3.4) and (3.9), thereby satisfying Condition (4.1). Nets for

which the second condition holds are difficult to characterise in general. We define

below a class of nets for which this condition is true, and consider in Section 4.1 some

necessary conditions for (a.2) to hold more generally.

Definition 4.I - S-invariant Reachability

An MPN, (.Â/,ms), with s-inuariant matrir s, is s-invariant reachable if

o

with a reachability set defined by

a state space which can be described by a matrix equation of the

form, Sm: k, where m is a possible state.

Srn: Smo

"(-) : /( fl o¿(-(i)) yT$),

(4.2)

(4.3)

(4.4)

is a necessary and sfficient condition for the reachability of any marleing rn.

Sms gives the initial token distribution of each minimal support S-invariant. Note

that (a.3) is always necessary for the reachability of m, [68], but we are also requiring

sufficiency in the condition. The reachability condition provides a matrix description

of the reachability set thereby satisfying the second condition (4.2).

There are practical SPNs which have product-form and are S-invariant reachable, for

example the set of SPNs which model circuit switched networks [12], and clustering

processes, [71,90] (see Section 3.1). There are also classes of nets, free choicenets and

marked graphs, which are S-invariant reachable under certain conditions [20' 68].

For the remainder of this thesis we restrict ourselves to SPNs which are S-invariant

reachable and have equilibrium distributions of the form

ieP

R(k,P): {m(P) : s(P)m(P) : k}

98

1 S-Inr¡ani ant Reaehat¡ e Pptni l\Iets

where 1l is the normalising constant and Oi(0) : 1.

The function O¿(rn(i)) incorporates limited marking dependent firing rates into the

product-form. An equilibrium distribution of the form given in equation ( .a) appears

in the papers [25], [55] and [57]. Each of the product-form nets of Section 2.3.1, except

for those of Florin and Natkin and Boucherie (Boucherie obtains a product of sub-nets

each of which may not have product-form) can be expressed in this form. For the

product-form closed nets of Section 3.1, Equation (a.a) arises when the function g(m)

has the product (3.9) and O(-) can also be expressed as a product over the places of

the MPN, p

o(-) : l[ o;(rn(i)).t=L

Under the assumption of S-invariant reachability, places which do not appear in any

S-invariant can have no restriction on the number of tokens they can accommodate and

can hence be removed from the normalising constant as geometric sums, as in Example

3 of Henderson and Taylor, [39]. As such, with regard to finding the normalising

constant of S-invariant reachable nets, we assume the reachability set to be finite.

4.L S-Invariant Reachable Petri Nets

Each of the algorithms that we consider in the following chapters assumes S-invariant

reachability. It is an essential condition which provides a matrix description of the

reachability set. In this section we examine S-invariant reachability more closely to

see when it is a valid assumption and what sort of conditions are necessary for it to

be true. We present necessary conditions for S-invariant reachability for SPNs whose

firing and enabling sets are the same for all reachable markings, that is, an enabled

transition can always fire.

Recall that to establish the reachability of some marking m we must do two things,

99

4.1 s- riant Reachable rl Nets

1. Find a non-negative integer solution, ã to Equation (2.7),

AM:m'-m:ATd, and (4.5)

2. Find a LFS for ø.

We require some additional notation. Equation (4.5) is a system of NP equations in

lrr? unknowns. Let the rank of AT be .R¡. We can always find an Rtx Rt sub-matrix

of AT whose determinant is not zero. Call this sub-matrix Â and \et D : ae\L).

The variables of AT corresponding to the columns of Â are basic variables - the pivot

columns in Gauss-Jordan row reduction of (4.5). The remaining lúP - D columns

of AT correspond to free variables. Let the solution vector ã be composed of two

sub-vectors conesponding to the basic and free variables, ã : (õBrãr).

The basic variables can be expressed as linear functions of the remaining N P - D ftee

variables. That is each basic variable {õ(i) i - 1 . . . R,s,} can be written in the form

õ(i): f¿Ge): D "U)ãr(i)

for integers ø(f)

Definition 4.2 - Greatest Common Divisor

Let a1,. . .¡ ún be a set of n integers. The largest integer d that is a cornmon diuisor of

all a¿ is called the greatest common d,iuisor and, is denoted, by

d:(auazr...,an)

The following theorem gives necessary and sufficient conditions for integer solutions

in a single linear equation of the form

crrrt * azrz * '" ! antn -- c a;rc e Z. (4.6)

100

ÕF

flfll

Theorem 4.L - Single Linear Diophantine Equation

An equation of the form (a.6) has integer solutions if and only if d :

(ot,. . ., ún) is a multiple of c.

Proof

This is a standard result which can be found in numerous texts on number

theory such as [1].

tr

Theorem 4.2 - Integer Firing Vector

In an MPN, .Â/, with incidence matrix A, S-invariant matrix S and initial

marking ltts, ân integer solution, d, exists for the system of equations (4.5)'

for some marking rn if and only if each of the following holds

o Sm - Smo,

¡ Each individual equation in the system (4.5) has integer solutions,

and

o There exists a choice of free variables, ã¡, such thaf f¿(õp) is an

integer multiple of D for all i € (1,..., Rt).

Theorem 4.2 is obtained from a discussion in the book by Agnew [1]. The result is

discussed below.

Remarks

The first condition is our definition of S-invariant reachability, Definition 4.1, which

ensures that (a.5) has a real solution.

The second condition is velified by applying Theorem 4.1 to each row in (a.5). That

is, the greatest common divisor of each row must divide its right hand side. Clearly

this is always true in ordinary nets as the greatest common divisor of each row is 1.

101

!-ì í\

1 s-ï Reaehat¡le l\[ef,s

The final condition then ensures that an integer solution exists which satisfies every

equation. A sufficient condition for this to hold is if D : 1, but this is certainly not

necessary.

We now address the issue of negative integer components of firing vectors. Since we

can clearly not fire a transition in reverse how do we cope with negative entries in

firing vectors which arise as solutions to (4.5)? The answer is given by the following

lemma.

Lemma 4.1

In an MPN, .Â,/, the change in marking given by

Am: -At"j

can be obtained by firing the vector

t-"j

for any T-invariant t where rU) > 0

Proof

Consider the change in marking resulting from the firing of the vector

t - ej,Am :AT(t-e¡)

: ATt - At"j

- -AtujSince ATt :0 for all T-invariants t

f1

Lemma 4.1 says that the marking change which would result if transition ú¡ could be

fired in reverse can be achieved by finding a LFS for some other firing count vector.

t02

4 2 Penfonrnance Mensrrnes

The firing count vector being any T-invariant with the jÚh component reduced by

one such that the resulting vector is still non-negative. Recall that in Definition 1.26

we saw that a consistent net is covered by T-invariants. The following corollary is a

consequence of Lemma 4.1 for consistent nets.

Corollary 4.1

In a consistent net any integer solution to (a.5) can be transformed into an

equivalent non-negative integer solution by the application of Lemma 4.1.

As discussed above, for a consistent net any integer solution to (4.5) can be transformed

into an equivalent non-negative integer solution. Theorem 4.2 thus gives necessary and

sufficient conditions for the existence of a non-negative integer firing vector between

any two markings in a consistent net. To establish reachability we must still find a

LFS for that firing vector. We discussed previously that finding LFSs is NP-complete

in the general case.

We have presented here some preliminary results which only solve the problem of

finding integer firing vectors for some change in marking. An investigation into classes

of nets for which LFSs can be found would resolve the second requirement for S-

invariant reachability. An example of such a class is the class of nets which model loss

networks and clustering processes since we know from the literature that loss networks

are S-invariant reachable.


In the following chapters we consider numerous algorithms for evaluating normalising

constants of product-form S-invariant reachable MPNs. In this section we look at how

various performance measures can be defined in terms of these constants. We derive

the normalising constant relationship for the enabling probability and throughput of

a transition and the utilisation of a place.

103

4-2 Performance Measures

Define an auxiliary function G(k,Q) to be

G(k,,Q): t tI a;@þ\ viu)m:Sm=k itPiEQ

and note that the normalising constant from Equation (4.4) is given by

K-r : G(Sms, P).

Theorem 4.3 - Enabling Probability

For an S-invariant reachable MPN with an equilibrium distribution given

by (a.a) the steady state probability that transition ú is enabled is

(4.7)

(4.8)

p(Transition r enabled) : G.(sTo- s'r0)'P'¿) lIv'iþ)G(Smo,2) iep

where

G.(Sm6, P,t) : D lI yl(n)on(,,,(i)+1i(ú))Sm=Sm0 i€P

and /¿(f) is the iÚä element of the input bag for transition ú.

P(Transition ú enabled) D n(m)m:m>I(t)Sm=Sm6

1

(4.e)

Proof

G(Sms, P) ^,t naîoao(*(i))m>I(Ð ieP

Sm=Smg

nrlo t fIy1Ø-"(')on1-1;;¡I

G(Sm¡,2)

With the change of variable m' : m - I(ú) we have,

m,m>11t¡ i€PSm=Sm6

I ilr,'u' D lf vi'Øan(rn'(;)+ri(r))G(k,,P) ieP ¡¡r.¡¡r;,0 ieP

Sm,=Sm0-SI(¿)

Smo - S.I(ú),P,f)G(Sms, P)

ieP

IIv';(').¿eP

P(Transition ú enabled)

G.(

104

Remark

The function g* is the auxiliary function for an MPN with the same net

structure but with the a(m) function shifted by a factor of I(Ú). The con-

volution algorithm for calculating g* is essentially identical to that given

in Theorem 4.1 but with different boundary conditions. For the state inde-

pendent case, where o(m) : 1, only one normalising constant is required

as G*(Sm¡ - S.I(r), P,,t) : G(Sm6 - S.I(r), P).

Throughput

The expression for throughput follows by definition

Throughput of transition f : D zr(m)q(m' f)' (4.10)meTt(.Â/,ms)

Theorem 4.4 - Utilisation Probability

For an S-invariant reachable MPN with an equilibrium distribution given

by ( .4), the steady state probability that place p is not empty is

G(Sms,2 -P(Place p is non emptY) - 1 - G(Sm6, P)

Proof

P(Place p is empty) t zr(m)I[rm(P): 0]

m:Sm=Sm0

(4.11)

G(Sms, P - {p})G(Sms, P)

where 1[.] is the indicator function.

D f[ o¿(-1;¡) yiØ Il*1o¡=o1m:Sm=Smg ¿eP

1

G(Sm6, P)

105

tr

Chapter 5

Convolution Algorithms

In this Chapter we present a number of convolution or recursive type algorithms which

can be used to calculate normalising constants of product-form S-invariant reachable

MPNs.

5.L The State Dependent Case

The following convolution algorithm is derived in the same way as those of Buzen [6]

by partitioning the reachability graph over the possible markings in each place. We

require the following definition.

Definition 5.1. - Marking Sets

The marking set for place P¡ in an S-inuariant reachable Petri net where Sm : k Jor

son'¿e non-negatiue integer uector k, is defi,ned by

M;(k,P) : {m(i) e Z+: Sm : k,}.

The marking sets give the possible sub-markings for each place. For example, if the

condition Sm: k was given by zm(I)+m(2) +3rn(3) :8, then Mr(8,{1,2,3}) :

{0,1,2,,3,4}, M2(8, {1,2,3}) : {0,I,2,9,4,5,6,8} and M"(8,{1,2,3}) : {0, 1,2}.

Notice that '7' is missing from the second marking set.

The first of our convolution algorithms to evaluate the normalising constant is then

106

5.1 The State DePendent Case

given by the following theorem

Theorem 5.1

For the class of MPNs defined in Chapter 4 the following recursive rela-

tionship holds for all non-negative integer k.

G(k,,Q): t A¿u) a" G(k - isp,,Q - {P"}) p: P.p € QieMe(k'8)

where

G(o,Q) __ r,

c(k,{Pr}): t or(Ðal,ieMP(k'{PP})

G(k,Q) :0 V k:þ rn:Sm:k.

Proof

Consider the following partitioning of the reachable markings given by

Srn : k.

G(k, Q) D t il o¿(m(i)) yi@ieMp(k,Q) m:Sm=k i:P;CQ

m(p): j

D a,U) viieMp(k,Q\

Substituting m' : m - jeo gives

t a¿(j) yl

D II o¿(m(i))yiu)m,Sm=k izP¿eQ-{Pp}

^(p): j

t f[ Q¿(^'1;¡) yi'$\m,,Sm,=k-jSp if;eQ-{Pp}

*'(p) : o

G(k, Q)jeMP(k'Q)

107

õ.1 The State Dependent Case

: t oo(r) aic(k- jSp, a - {Pr})ieMP(k'Q)

The boundary conditions are derived from (4.8) as follows. when k : 0,

the only marking which satisfies Sm : k is the zero marking. Substituting

the zero marking into (4.8) gives

G(0, Q) : t.

when only one place p is left in the sub-net, only this place in the marking

affects (4.8), hence

G(k, {Pe}) : t a,u) yi.ieMP(k'{PP})

The last boundary condition arises when k is such that no marking, m,

exists which satisfies Sm : k.

tr

The convolution of Theorem 5.1 was first presented in [11] in a state independent form.

The proof given here is identical except for the inclusion of the O¿(rn(i)) function.

It is a requirement of the above convolution that the marking sets be calculated at

each recursive step. Finding the marking set for each place involves solving linear

Diophantine equations for one unknown at a time, which is a difficult number the-

ory problem. Solving systems of linear Diophantine equations for many variables is

considered in [1] and [1a].

Sereno and Balbo [78] observed the following lemma which when applied to Theorem

5.1 removes the need to calculate the marking sets. Let C¿ be the maximumnumber

of tokens possible in place i of an S-invariant reachable net. We have the following

lemmawhereC:(Cr,...,C¡tp) isthecapacityvectorfortheplacesinthenet' The

notation used in the lemma can be found in Definition 1.21.

108

5.1 The State Dependent Case

Lemma 5.1

For an SPN whose state space is described by Equation 4.2, and for any

place P¿ and n €. Z¡ such that n 1C¿ and n I Mi(k,,P),'

ß(k-nS¿,P-{P¿}):Ø

The lemma can be applied to express Theorem 5.1 as follows'

Theorem 5.2 - Convolution Without Marking Sets

The following recursive relationship holds for all non-negative integer k

CP

c(k,g) : Ðo;(r) yiG(k- jsp,Q- {P,}) p: P,oe Qj=o

where

G(o,Q): t,,

CP

G(k, {&}) : t a,(i) y",j=o

G(k, q¡ : O V k: þ rn: Sm: k.

tr

It is not immediately clear whether Theorem 5.2 is a more efficient method for eval-

uating the normalising constant than Theorem 5.1. While Theorem 5.1 requires the

calculation of marking sets at each recursive step, Theorem 5.2 avoids calculating the

marking sets but has to evaluate more normalising constants, many of which are zeroj

to achieve the final result. We compare the two methods in Example 5.1.

Frosch [26] also extended the work of [11] to a state dependent algorithm in which

O;(rn(i)) takes a special form. Recall that Frosch's nets consist of cyclic state machines

joined together in a certain way by common places (See Section 2.3). His convolution

differs from that of Theorem 5.1 in that the partitioning is over the number of tokens

in each state machine and not each place.

109

5.1 The State Dependent Case

To express Frosch's product-form in a manner consistent with (a'a) we need to label

the places so as to group them into state machines. Let m¡(i) be the marking of place

i in state machine i, Q¿,¡(*¡(e)) corresponds to the element in the product of o(m)

for place i in state machine j and let g¿,¡ be the element of the product for the same

place. If we let ^jU)

a ¿,¡(rn¡(i)) : II lPn,¡(^ ¡(i))l-'t=L

where p¿,j is the state dependent rate out of place i of state machine j' Frosch's

product-form can then be written as

nt M¡

,,(m): 1í ä fÍ [on,¡(-¡(¿))] [y¿,¡(m¡(i))]^it)j=l i=l

where the number of state machines is M and the number of places in state machine

j is M¡.

To present Frosch's convolution we lequire some more notation' Let m¡ be the sub-

marking of state machine j and let /¡(*¡) be the marginal distribution of state machine

j so that we can write

M

n(m) : zr(mr,.. ., mM) : K fl /¡(-¡). (5'1)j=l

Also let Ít¡(Sms) be the reachability set for state machine j and s¡ be the sub-matrix

of S corresponding to the places in state machine j (This is the only place where S¡

will have this meaning)'

As before, we define an auxiliary function,

c1Ë., tut¡ : tM

fI /¡(*¡)j=l

where the normalising constant of Equation (5'1) is given by

K-r - G(Smo, ø).

The convolution is then given by the following theorem which is proved in [26].

merè(.Â/,ms)

110


Theorem 5.3

The following recursive relationship holds for all Ë,

GçË.,u¡ : t f*(^'r)G(Ë- s¡lrn'¡¡,M -r),m'¡¡eÈ¡¡(k\

where, for any integer Mt,

Gço, M'¡ : t,

c1Ë,r¡ : t "f,(*,),m'r €flr(.Â/,mo)

G1Ë.,M'¡ :g v Ë : þ rn: sm: É.

tr

Note that since Frosch's product-form equilibrium distribution is also a standard prod-

uct over the places, Theorems 5.1 and 5.2 can also be applied to his class of nets.


There exist two algorithms specifically for product-form nets with state independent

firing rates. The frrst of these is a convolution algorithm which we present here. The

second is a recursive method which we introduce in Chapter 7.

In the state irídependent case the equilibrium distribution is given by Equation (4.a)

with O(rn(i)) constant for all markings. Note that Theorem 5.2 could also be applied

in this case but would be less efficient than either of the two methods which operate

specifically on state independent nets.

The first of the two algorithms was given by Sereno and Balbo [78] who derived a

relationship between different normalising constants which parallels the work of Buzen

t6]. The convolution can be applied in a tabular manner and only requires limited

memory usage.

111

6.2 The State IndePendent Case

Sereno and Balbo's convolution is also based on their result given in Lemma 5.1.

Theorem õ.4 - Sereno and Balbots Convolution

The auxiliary function given by @.7) when

O(nz(i)) : constant

satisfies the following recursive relationship

G(k,Q): G(k, Q - {Pr}) + y, G(k - Sp' Q)'

Proof

Firstly note that the boundary conditions are identical to those of Theorem

5.2 as the product-form is unchanged except for state independence.

From Theorem 5.2 we have that

G(k,P): g are G(k - iS,P - {Pr}) (5'2)

j=o

cp

G(k,P - {&}) +Da', G(k - iS,P - {&})j=l

CP

G(k,p - {Pr}) -rapDal-t G(k- se - (r - 1)So,P - {Pr})j=1cp-l

G(k,P - {Pr}) + ap Ð yic(k- se - isnP - {Pr}).j=o

The result follows on comparing the last term with (5'2)'

LI2

5.3 Example

P1

P2

rl

P4

t4

12

Figure 5.1: Example 5.1

5.3 Example

Example 5.1 - State Dependent Convolution

The MPN of Figure 5.1, is s-invariant reachable for all initial markings

where at least one transition is enabled. For simplicity we have assumed

Õ(*) : 1 for all markings'

Using the techniques described in previous chapters, the equilibrium dis-

tribution can be written as,

P5

n(m) :,0(,,€)-"' (#)-"' (#)-"'The real number B is the probability that the input bag for transition 2 is

transformed into the input bag for transition 3, indicated by dashed lines

in the figure.

113

5.3 Example

using equations (4.9), (4.10) and (4.11) and with þ:0.5, we applied the

two state dependent convolution algorithms to calculate, for a range of

initial markings, the throughput for each transition and the utilisation of

each place.

We compared the CPU time taken to obtain exact results using the two

convolution algorithms given by Theorem 5'1 and 5.2 respectively with the

time taken by two well known packages, namely SPNP, [9] and GreatSPN,

[g], both of which generate the reachability graph. we obtained perfor-

mance measures for the MPN with numbers of states ranging ftorn 1276

to over 40000. The results are given in Tables 5.1,5.2,5.3 and Figure 5.2'

Figure 5.2 reflects the fact that for this example with a small number

of states, finding marking sets makes the convolution algorithm slower

than both GreatSPN and SPNP but as the number of states increases,

calculation of the marking sets is no longer a liability and the convolution

algorithm becomes progressively more efficient than either package. When

we use Theorem 5.2 we see no improvement in CPU times over Theorem

5.1. This is because the time saved through not calculating the marking

sets is offset by an increase in the number of recursive function calls at

each step in the convolution.

The convolution algorithm of Theorem 5.1 could be improved significantly

by examining more efficient methods for finding the marking sets.

All calculations were performed on a SUN SPARC IPX' E

tt4

5.3 Example

Mo #STATES

PU TAKENSPNP GSPN Theorem 5.1 Theorem 5.2

(b,Ð,Ð,0,Ð L276 l) 6 t4 T2

(7,7,7,7,7) 3235 23 27 43 40

10,10,10,10,10) 8876 130 t37 153 150

(12,72,,12,72,L2) 14975 316 319 290 298

15,15,15,15,15 28551 919 897 67t 70L

L7 ,77 ,r7 ,r7,17 41090 1703 L776 r074 1131

Table 5.1: Time Comparisons

Table 5.2: Performance Measures

Mo: 5,5,5,5 Ð 1276 States

Place Utilisation Transition Throughput1

2

3

4

i)

0.9302330.9999320.953500

0.4767330.465117

1

2

3

4

0.9302330.9302330.4651170.465117

Mo - (7,7,7,7,7) 3235 States


2

3

4

Ð

0.9508200.999997

0.9672t40.4836060.4754t0

1

2

3

4

0.9508200.950820

0.4754100.4754t0

Mo:l 10,10,10,10,10 8876 states


2

3

4

,)

0.9659091.000000

0.9772730.4886360.482955

I2

3

4

0.9659090.9659090.4829550.482955

115

5.3 Example

Mo- 12,r2,L2,r2,r2 14975 States

Place Utilisation Transition Throughput

1

2

3

4

5

0.9716981.000000

0.9811320.4905660.485849

1

2

3

4

0.9716980.971698

0.4858490.485849

Mo: 15,15,15,1 15 28551 States


2

3

4

5

0.9774441.000000

0.9849620.4924810.488722

1

2

3

4

0.9774440.9774440.4887220.488722

Mo: I 17 ,r7 ,17 ,r7 ,,r7) 41090 States

Place Utilisation Transition put

1

2

3

4

5

0.9801331.000000

0.9867550.4933780.490066

1

2

3

4

0.980133

0.980133

0.4900660.490066

Table 5.3: Performance Measures

116

5.3 Example

1800

1600

1400

1200

600

400

200

00 5000 10000 15000 20000 25000 30000 35000 40000

Number of States

Figure 5.2: Graph from Table 5.1

o

tr

t@

+

tr

SPNP OGSPN +

Convolutions E

TT7

Chapter 6

Complementary Convolutions

6.1- Loss Networks

In Section 3.1.1 (page 66) we considered loss networks as an example of closed nets with

product-form equilibrium distributions. Recall that a loss network has the property

that the set of holding places constitute a sufficient place set and only these places

need to appeaÌ in the product-form (3.6). The function o(m) takes the special form

(3.5).

In this chapter we present a convolution algorithm which operates on a difierent region,

the ,,complementary region", to those considered previously and which can evaluate

the normalising constant very efficiently for some MPNs with very large reachability

sets. The existence of the new algorithm is based on a special property which is present

in MpN models of loss networks. Thus we will introduce the algorithm through loss

networks and then consider when a similar convolution can be applied to S-invariant

reachable product-form nets in general.

Consider a loss network with two routes and limited capacity so that too many calls of

one type can restrict the possible numbers of calls of the second type. The state space

for such a system is given in Figure 6.14, where each axis indicates the number of

calls in progress on each route. The dashed lines indicate the independent case where

calls on each of the two routes do not restrict one another. The complementary region

118

6.1 Loss Networks

Route 2 Route 2

Route 1 Route I

A

Figure 6.1: ComplementarY Region

is indicated by the shaded region in Figure 6.18 and is the region within the "box"

but outside the convex region which describes the state space. It is the non-convex

complementary region that we consider in this section.

The state space of a loss network can be described in two ways. We can calculate the

minimal support S-invariants of an MPN model for a loss network and use the fact

that such nets are S-invariant reachable. In a Petri net description, where circuits are

denoted by tokens, each S-invariant ensures that the number of circuits available plus

the number in use remains constant. On the other hand, we can reason the fact that

the state space in loss networks consists of vectors where the number of circuits in

use does not exceed the number available which provides us with a matrix inequality

for the state description. Let P' be the set of holding places for a Petri net model

of a fixed routing circuit switched network. We then have two equivalent expressions

for the reachability set of an MPN model of a loss network each of which provides an

expression for the normalising constant,

B

y*(i)Y-r - G(Smo, P) t il

jzP¡Ep,*(j)lm:Sm=Smo

m(P')zr,tfi,r',=r*, *E''# (6'1)

The existence of two equivalent expressions for the normalising constant is crucial to

119

6.1 s Networks

the existence of the complementary convolution. Equation (6.1) expresses the normal-

ising constant as a sum over a set of markings described by a matrix inequality. It is

this inequality which defines the "regions" shown in Figure 6.1.

In the case of loss networks the following corollary is a consequence of Theorem 5.2

Corollary 6.1 - Convolution for CSNs

For a stationary, MPN, ,'V, which models a loss network, the auxiliary

function given by (a.7) satisfies the following recursive relationship for all

non-negative integer k,cp ",i

G(k,Q): t ? Cru-isp,Q- {P,}) p: P,e Qi=O J'

where

G(o,Q) :1,

G(k, {Pe}) :i,+j=o .t'

tr

The fact that the reachability of a CSN can be defined as a matrix inequality implies

that the set is convex which makes the capacity vector C easy to calculate. We note

again that most existing convolutions operate on a similar convex region. We consider

now a convolution algorithm similar to that given in Corollary 6.1 but defined on the

complementary state space which is generally non convex.

The idea of a complementary state space was used by Mitra [65]. However, instead of

producing a convolution algorithm, Mitra expressed the normalising constant of a CSN

as a power series which, when truncated, gave approximate performance measures.

Consider the following expression for the auxiliary function

^.^(i)G(k, Q) : D fr¡'e¡ee, *w - t

m1e¡'S1o¡m1o¡¡km(o)<c(a)

m(O):m(A)SC(A)

r20

n*r,rr,ffi (6.2)

6-1 f,oss l\Tcf.rx¡orks

The right hand side sums first over a larger but simpler set of markings in which the

number of tokens in each place is constrained only by the capacity of that place and

then subtracts the markings for which the weighted number of tokens in the places,

corresponding to at least one S-invariant, is too large'

We can rewrite (6.2),

G(k, Q) ilj:P¡eQ

ll,,r,rr'ffi.tm(e):s(e)m(e)lk

m(o)<c(o)

U¡C;t

i=O i!

i)fTL

rr lS r#l - G.(k,e). (6.8)

,,I,'rnlk- " r

Clearly, since the first term in (6.3) is easy to calculate, if we can find a method

for calculating G"(k,8) then we have an alternative method for finding the auxiliary

function for the normalising constant. Calculating G"(k, g) will only be of benefit if

the complementary region is small compared with the normal reachability set. This

issue was addressed by Coyle [18] who developed a scaling technique to estimate quickly

the number of states in each of the two regions.

In general, the complementary region will be smaller than the t'forward" region when-

ever the system is close to being independent. In loss networks this means that calls

rarely interfere with each other while in PN terminology it means that behavioural

conflict rarely occurs. The complementary convolution can now be given as follows,

where k. : S(Q)c(a) - k.

Theorem 6.1 - complementary convolution For Loss Networks

For a stationary MPN model of a loss network with normalising constant

given by (6.1), the following relation holds for all non-negative integer k

Cp o,Cr-jc"(k, Q):D,o d:T G"(k + iSp,Q - {Pr}) p: P, € Q

where

G"(k, Q):0 whenever k* ( o,

r2l

6.2 Markovian P rt Nets

cp

c"(k, {Pr}) : Dj=o (co - j)t

if S(i,p)>0 Vi:k.(i) >0otherwise

Yîe-i

and

Õ,: {min

þax; {t"#l}'t,l

where | .l denotes greatest integer less than.

Proof

The proof is given in APPendix A

tr

6.2 Markovian Petri Nets

The complementaly convolution arises because the state space in loss networks can be

expressed as a matrix inequality which defines the various regions described by (6.2)'

The question is then, when can the complementary convolution be applied to Petri

nets more generally and not just to loss networks? This is answered by the following

lemma which gives necessary conditions for the normalising constant of a product-form

S-invariant reachable MPN to have an equivalent matrix inequality description.

Lemma 6.1

Let Al be an S-invariant reachable MPN with S-invariant matrix S' The

two sums given bY

t l[ o¿(rn(i)) yiþ) (6.4)

m(P'):Sm=k ieP'

and

t f[ on(-(;D aiØ (6'5)

m(P'):S(P')m(P')<k ieP'

are identicalfor all k e zls if the set of places P*:P -P',is such that

t22

A r. Mnrkowinn ni l\Tpts

Proof

o S(P.) is the identity matrix, and

o Rank(S(P.)) : Rank(S).

The sets over which each of (6.4) and (6.5) sum are

{*(2') : Sm: k} and, (6.6)

{*(P') : s(P')m(P) I k} (6.7)

respectively.

The possible range of markings for each place of m(P') in (6.6) is given by

m(i) e 0 1k(t) i+j (6.8)

St,¿

where | .l means greatest integer less than. In (6.6) the value for each rn(i)

can range frorn 0 t. L#l with any deficiencies in the equality filled by the

coefficient of one (since S contains the identity matrix). Since equality is

not enforced in (6.7) Equation (6.8) is clearly the solution for each place.

Thus the two sets are equal.

Since the solutions of the sets (6.6) and (6.7) are the same we are left to

show that the cardinality of each set is equal to establish the result.

The presence of the identity matrix in S implies that the matrix is fully row

reduced. The columns in the set P* are then basic variables and those from

the set P' are free variables. Each of the basic variables can be written

uniquely in terms of the free variables. Hence the free variables uniquely

define the entire marking and thus form a sufficient place set.

This means that for every solution ,n(P') in (6.7) there is one and only

one marking m : (m(2'), ttt(P.)) such that Sm : k.

L I

r23

Lemma 6.1 shows when the complementary convolution can be applied to MPNs. Let

p* be a set of places in an MPN satisfying Lemma 6.1. We can then express the

normalising constant given by Equations (a.s) and (a.7) in an equivalent manner

G(Sms, P) : G(Sms, P') : t fl O;(ræ(i)) vTø'm(P'):S(P')m(P')<Sms ieP'

In the same way in which Theorem 6.1 was developed through Equation (6.2) we can

present the following theorem for MPNs in general'

Theorem 6.2 - Complernentary Convolution For MPNs

For a stationary, MPN with a set of places Q satisfying the conditions of

Lemma the following relation holds for all k

cp

c"(k, Q):Ð af,-i Qr(Co- i) G"(k+ jsp, Q- {Prl.) p:Pr€.Qj=o

where

G"(k, Q) :0 whenever k* ( o,

CP

c"(k, {&}) : t al'-i o,(C, - i)j=o

and

mi,, [-u*; {L#;}l\ ,t,] if S(i,p) >0 Vi:k.(i)>0ce otherwise

where I I denotes gleatest integer less than

Proof

The proof is identical to that given in Appendix A with the ;fu terms

replaced by O¡(rn(j))'s.

We have seen two types of convolution algorithms so far. They each have their own

strengths and weaknesses, and can be applied to different network complexities and

topologies. The complementary convolution can be used effectively in some cases

where the more traditional convolution algorithm is intractable.

ce

L24

6.3 Example

6.3 Example

Example 6.1 - Complementary Convolution

I3

t4

I5

Figure 6.2: Simple Switch

This example of a simple switch like model, Figure 6.2, is one of a more

practical nature. Each of the nodes 1,5 and 9 can communicate with nodes

4,8 and 12.

Let y : (At,.. . , g¡r.p). With nine routes and fifteen links the network can

be defined as follows:

V : (2,1.5,3, 1,1,2.3,3, 1,4)

C : (11, 7,11,5,3,4,2,3,2,7,3,2, 13,8,8)

The complementary convolution, Theorem 6.1, calculated the normalising

constants to give blocking on all routes in approximately 120 seconds CPU

time. The results are shown in Table 6.1.

2

3

r25

6.3 Example

RouteLinks Nodes abi EFP

1

2

3

4

5

6

7

8

I

1-4-131-5-141-6-152-7-L32-8-142-9-L53-10-133-11-143-12-15

r-41-8r-125-45-85-I29-49-89-r2

0.03848620.13606210.2066514

0.20018050.06356220.4449Lt70.02319990.06528690.6154803

0.05140.14990.25730.21380.08780.47290.03590.08090.6268

Table 6.1: Blocking for a Simple Switch

The normal or forward state space contains 398,493 states and the comple-

mentary state space has 16,227 states. When compared with the results

given in Table 5.1 where a smaller MPN with 41090 markings required

over 18 minutes cPU time, it is clear that attempting to use a convolution

which operates on a state space with 398,493 markings would be futile'

The power of the complementary convolution is apparent for this example'

The Erlang fixed point algorithm is an approximation technique for eval-

uating blocking probabilities in Loss Networks [51]. It may be suggested

that the Erlang fixed point algorithm would be accurate and efficient in

systems with moderate user interference. To address this issue, in the col-

umn labelled EFP of Table 6.1, we calculated the end to end blocking of

the nine routes using the Erlang frxed point method. For this example, it

can be seen that the Erlang fixed point results do differ, sometimes sig-

nificantly, from the exact results. Moreover, as there is no guarantee of

the accuracy of the Erlang fixed point method, and since the exact results

provide more than just blocking probabilities, in general, it is preferable to

use the exact results if they can be found' tr

126

Chapter 7

Utilisation Recursion

7.L Markovian Petri Nets

We now develop a recursive method for calculating the normalising constant in a state

independent S-invar.iant reachable product-form MPN which is completely different to

any other. An appealing property of this computational method is that its numerical

complexity, or computation time, is dependent only on the structure of the MPN and

not the size of the reachability set. We give an extended definition of utilisation, not of

a single place, but of a group of places. Our analysis is based on a simple relationship

which exists between two such utilisations, leading to a non-trivial result that allows

the probabilities of many states to be added together in the form of geometric sums.

Our result is a generalisation of the work of Koenigsberg [53]. Recall that Koenigs-

berg gave closed form expressions, independent of the number of customers, for the

normalising constants of closed cyclic single movement queueing networks'

Definition 7.L - Utilisation Normalising Constant

The utilisation normalising constant d,ependent on son'Ie uector x e ZÏ is defined to

be

G*(k,P): t IlaTç)'*.9*=¡ i€P

mZx

127

7 -1 Markovian Pcfri l\ef.s

Definition 7.2 - Utilisation

The utilisation of a uector x e ZIP is d,ef,ned by

u* : P(rn(l) ) nt¡...,m(NP) > r")

: t "(m)m:Sm=k

m>x

1- t "(*)m:Sm=k

mzx

(7.1)

Note that if x : e¡, the unit vector with a one in position i and zeros elsewhere, then

U* gives the standard single-server queueing network definition for the utilisation of

place i.

We follow with two theorems which form the basis for our state independent recursive

algorithm for calculating the normalising constant in an S-invariant reachable product-

form MPN.

Theorem 7.! - Utilisation relationship

If x and x' are two vectors of an S-invariant reachable product-form MPN

such that U* ) 0, U*, ) 0 and

then

Sx

128

S*',

7-1 Markovian rt I\ets

Proof

The utilisation of the vector x is

u* ux,

t zr(m)m,Sm=k

m>x

crl,el-,H=n E'T'm>x

1

(7.2)

(7.3)

ux

Ilvi' D llvr'-''¿eP ttt.Stt'r=¡ ieP

m>x

With a change of variable to m' : m - x we have

u*G(k,P)

G(k - Sx,P) flvi'.G(k,P)

D IIYîiSm,=k-Sx i€P

m')0

ieP

Similarly the utilisation of x' is

IIvîi (7.4)¿eP

Since Sx : Sx', we have G(k - Sx, P) : G(k - Sx', P) and division of

(7.3) by (7.4) leads to the result.

t29

7-1 Markovian l\ef s

Theorem 7.2 - Normalising Constant Decomposition

Let x, x' be two utilisation vectors satisfying the hypotheses of Theorem

7.1. The following relationship exists for the normalising constant G(k,P):

G(k,P) : G*(k,P) NP yîi -G*,(k,P)l1l=ryî'nli Ui'- lll=rai'

Proof

Equation (7.5) is derived by substituting from equation (7.1) into equation

(7.2).

The frrst point that we need to address is the existence of the two vectors x and x'.

This is covered by the following Lemma.

Lemma 7.1

If there does not exist two vectors x > 0 and x' ) 0 such that

Sx: Sx' (7.6)

then

G(k,P) : II ,*o(i).itP¡eP

Proof

Equation (7.6) will provide non-trivial solutions for x and x' if and only if

there is a non-trivial solution to the equation

Sx*:0

(7.5)

130

(7.7)

7 1 Markowian ri Nets

where x* : x - x'. The solution for x* can then be split into its positive

and negative components to provide x and x'. If a non-trivial solution does

exist to (7.7) it will always have positive and negative components because

S is a non-negative integer matrix.

Now comp are (7.7) with the definition of the reachability set for S-invariant

reachable nets,7?(*o) :{m:Sm:S*o}

: {m: Sm*: o} (7'8)

where m* : Itt-Iïto. If the only solution to (7.7) is the homogeneous sG-

lution, then clearly from (7.8) the only marking in the reachability set

is the initial marking. The result then follows from the definition of the

normalising constant.

tr

Theorem 7.2 can be used to define a recursion for calculating the normalising constant

in an S-invariant reachable product-form MPN. Observe that the utilisation normal-

ising constants G*(k,P) and, G*,(k,P) arc sums over smaller sets of states than for

G(k,p). A recursion is established by noting that G*(k,P) and. G*,(k,P) can be

expressed in terms of auxiliary functions g with smaller arguments by applying the

well-known inclusion-exclusion principle [24].

Let

n(x,k,P) D TT,Tþ)m.sm=k¿€Pm(i)<t;o¡-L

Ð yi G(k - øSe¡, P - {i}),,a=O

m,¡(x,k,P)

D Iloiþ)m,Sm=k teP

rn(i\<x ¡ ¡n(j)<r ¡

131

V 1 Markowi Pptri ]\Icts

and so on.

Also define

o;-l a¡-L: D D yia'¡G(k - øSe¡ - bSe¡,P - {i',i}),

a=O ä=O

,Sr(x, k,,P) : D rn1*, k,P),ieP

,Sr(x, k,P) : D r,,r'(*, k,P),

,Sr(x, k,P) : pi,i,k(x,k,P),

The general expression for G*(k, P) for some utilisation vector x is then

G*(k,P) :St(*, k,P) -,Sr(x, k,P)

* ^93(x, k,P) -...+ (-l)tP+t,Sr(*,k,P). (7.9)

Note that for a given choice of the vector x, many terms in the above expressron are

zero. For example, consider the two utilisation vectors x : ei * e; and x' = €¡.

G*(k,P)

: ^9r(x, k,P) - S2(x,k,,P)

: pr(x,k,P) * p¡(x, k,P) - p;,¡(x,,k,P)

: G(k,P - {i})+ c(k,P - {j})- G(k,P - {í',i}) (7.10)

i,iePi+i

Dt,j,lcCP

i+j+k+i

etc

L32

7.2 Sinele Movement Queueing Networks

and similarly

G*,(k, P) : G(k,P - {i}). (7.11)

The recursion will end when k and P are sufficiently small such that the normalising

constants describing G*(k,P) and G*,(k,P) arc easily calculated (in many cases as

geometric sums).

As mentioned above, the utilisation vectors x and x' can be calculated by splitting

any integer valued homogeneous solution x* of Sx* : 0 into its positive and negative

components. This provides a number of possibilities. Based on (7.9), the number of

normalising constants that G*(k, P) can be expressed in terms of, for a given vector

x, is

tP Ilfl ('n + t)l - t. (7 't2)L';r J

Equation (7,I2) provides a means for which the utilisation vectors x and x' can be

chosen so as to minimise the number of recursive function calls at each step and thereby

greatly speeding up the recursion. Clearly if there was a choice, x' should be chosen

over x in equations (7.10) and (7.11).

As the size of x and x'increase, the complexity of G*(k,P) and G*,(k,P) increase

exponentially.

We compare this algorithm and that of Sereno and Balbo in ExampleT.l.

7.2 Single Movement Queueittg Networks

In this section we apply the above theory to the special case of single movement

queueing networks. Consider a network of .lú queues, labelled from the set ,Â/, and M

customers with a product form equilibrium distribution of the form

133

7.2 Single Movement Queueing Networks

szr(n) : lG(M,^/)l-' o("s) IIaT'

i=1

where 5 : {1, 2, . . . ,.9} are the labels for the ,9 single server queues in the network, 5 :

{s + t,.. ., ¡\¡} are the set of queues which are not single server, n5 : (n51rr'..,nN)

and n¿ is the number of customers at queue i.

O(.) is an arbitrary function of the state of the non single server queues in the network

which, in the case of Jackson [42] networks, takes the form

(7.13)

The state space for such a queueing network is given by

N

R(M,N) :{"'Dni:M}.i=l

The normalising constant G(M,.Â/) can then be written

s

G(M,A1): t o("s) IIaT'nel¿(M,N) i=1

and y¿ can be interpreted as the relative throughput of queue i

For any two queues i and j, Theorems 7.1 and 7.2 give

s"i: lr",and

M,N - {i})y¡ - G(M,N - {i})Y, (7.r4)

respectively.

G(M,Ìr[): G(

t34

U¿_U¡

7.2 Sinele Movement Queueing Networks

Equation (7.14) can be used repeatedly whenever there are two single server queues'

with different relative throughputs, in the network. The procedure is efficient since it

is only the number of queues in the network which is reduced at each step and not the

number of customers. Consequently the algorithm works independently of the number

of customers in the network.

When equation (7.I4) is used to eliminate all but one of the single server queues the

boundary conditions are

MG(M,,5+{¿}) :D

r=0vi t o("s)

.s'DI+, n;=M-r

(7.15)

When O(ps) takes the form (7.13), Equation (7.15) can be evaluated using Buzen's

algorithm for queues with state dependent service rates.

Equation (7.14) cannot be used directly when yi : yj. Thus the reduction procedure

may reach a point where the queues remaining in the normalising constant consist of

the set 5 along with a set of single server queues, 2, whose relative throughputs are

equal.

In this case the boundary condition becomes

G(M,s + o) : ä(' *r'-; t

) ,"

"",ryf* ¿=M_ro('s) (2.16)

where D : lDl is the number of single server queues with common relative through-

puts.

Equation (7.16) follows by observing that the number of non negative integer vectors

(rr,nr, . . . ,no), that satisfy Ðrn: ? r is (";?;t ) (t". lagl p.a7 Exercise 6)'i€.D

ThecaseS:N

When all the queues in the network are single server queues, Equation (7.14) is valid

for all i and j. Boundary condition (7.15) becomes

135

7.3 ExamPles

G(M,{¡Ð -- aY v i.

and (7.16) reduces to

G(M,D) : M + D _ID -l YM

At first glance it might appear that there could be as many as 2N different normalisa-

tion constants to calculate for a network of N queues'

Implementation of this special case of the utilisation recursion is considered in [15]. It is

shown that, by the judicial use of equation (7.14), the number of recursive calculations

needed is bounded above Ot 4#I. Thus, not only does the algorithm run in a

time independent of the number of customers in the network, but it depends only

quadratically on the number of queues.

Example 2.2 compares Buzen's convolution algorithm with an efficient implementation

of the above algorithm.

7.3 Examples

Example 7.1 - State Independent Firing Rates

We use the product-form MPN of Figure 7.1 taken from Sereno and Balbo's

paper, [78], to compare the time taken by the recursion of Theorem 7 .2 with

that taken by the convolution of Theorem 5.4 by calculating the normalis-

ing constant for a range of initial markings. The constant firing rates given

in [78] lead to the following elements of the product-form:

: 1.0, Az:0.5, ys:113,, Ua: I.2,

: 1.0, ya:817, Uz : I.0

136

Ut

Us

7.3 Examples

t4

t5

Figure 7.1: Example 7.1

s*lStates Normalising

ConstantTime Comparison I seconds

Theorem 5.4 Theorem 7.2

312rl) 28 14.38 0.03 1.07

10,9,8 987 647.60 2.t2 1.10'30,20,10) 35 101 9033.96 1.93 1.10

40,30,20 r22 276 226 024.00 7.35 1.11

50,40,30 300 576 5 358 321.09 17.69 1.11

Table 7.1: Time ComParison

The results, calculated on a SPARC ELC, are given in Table 7.1.

It is particularly important to note that while the convolution proposed

in [78] is polynomial in the number of places and the size of the initial

marking, the utilisation recursion takes a f,ted length of time, independent

of the size of the initial marking.

r37

7.3 Examples

Figure 7.2: Queueing Network

For this particular example the fixed computation time of our method

compares favourably for all initial markings considered. In more complex

examples the relative dimensions of the net and the size of the reachability

set will determine whether this algorithm or that of Sereno and Balbo will

be more efficient.

Example 7.2 - Single Movement Queueing Network

Consider the queueing network in Figure 7.2 consisting of 10 nodes. The

routing probabilities are ] and 1 for solid and dashed arrows respectively'

The relative throughputs for this network are

h : L.0 Az : 0.5 9s : 0.5 A+:0.25 9¡ : 0.5

Ua: 0.25 Uz : !.0 Ua : 0'875 Ug : 0'875 9ro : 1'0'

We evaluated the normalising constants for the network with different num-

bers of customers. The comparitive CPU times for Buzen's method and the

utilisation recursion are given in Table 7.2. Allcalculations were performed

138

7.3 Examples

#Customers

CPU TIME KENBuzen Utilisation

10 1.1 0.10

100 L.I2 0.09

1000 1.13 0.09

10000 1.34 0.10

100000 ó.öô 0.09

200000 7.0 0.10

on a SUN SPARC IPX.

Table 7.2: Time ComParisons

139

Chapter 8

Conclusrons

In this thesis we have concentrated on two aspects of Petri net research. Stochastic

Petri nets which have product-form equilibrium distributions and computational algo-

rithms for evaluating the normalising constants of these nets. In the first case we have

extended some of the existing results and increased the understanding of product-form

in closed nets. In addition we have introduced a number of new computational algo'

rithms for evaluating normalising constants for many types of product-form nets of

which BMPNs are just one example. We have also surveyed the existing work in the

literature.

There is still much work to be done in the areas considered by this thesis. Below we

outline future work which follows from the work contained in the preceding chapters.

We have seen the value of product-form results which hold for any live initial marking.

Balanced nets are just one type of net structure for which this type of product-form

exists. As was the case in queueing theory, there must exist a more general product-

form result which contains each of the existing results as special cases.

In Section 3.2 we added exponential firing times to BMPNs. Considering the structure

of the closed MPN after the inclusion of firing times, and the product-form which

results, it appears that the property of importance is the partitioning of the places,

and transitions into disjoint sets in such a way that the input and output bags of each

140

8.O (lonelrrsions

transition and each place draw places and transitions from only one of the appropriate

sets. It is reasonable to assume that the sets and places may be partitioned into more

than two distinct sets. The obvious conjecture is, what conditions are necessary to

allow the equilibrium di,stribution to be split into independent components based, on a

partition of the places. The work of Section 3.2 splits the equilibrium distribution

into a product of two parts, one for the original places and one for the additional

places. This decomposition like result may have a much wider application which is

not restricted to Balanced nets. The theory could be extended in such a \¡/ay that any

net could be decomposed into sub-nets connected only via transition firing rates.

In Chapter 7 for the first time, we have seen an algorithm which calculates the nor-

malising constant in a fixed length of time for any initial marking. At this stage the

recursion only applies to nets with state independent firing rates. This result needs to

be extended to the state dependent case.

There are many applications of nets in general. We believe that there must also

be many applications of product-form nets waiting to be discovered by imaginative

researchers.

t4L

Appendix A

Proofs

A.L Complementary Convolution Proof

*'3i'"!?äl;ì'-

With the substitution tr : C(8) - m(Q) and for notational convenience letting

S : S(Q) and C : C(Q) we have

g"(k'Q):,,,r.ä"-u *'#rnffi' (A'1)

0(n<C

Because of the not greater than or equal úo condition the set of states over which

the sum is taken is in general non convex. The number of tokens in place 4 in the

complementary system n(i), is bounded above by C¿, but it may be less than this. Let

the maximumvalue of n(i) satisfying n : sn z sc - k and o I n ( c be denoted by

C¿.

Following traditional lines,

cp,Cx-n(k)

Recall the definition of the auxilary function,

g"(k, Q) : t

t tlIn'Sn¿SC-k lc:P¡aê'

0<n<C

"(p): j

ilr,r,rn#.

g'(k,Q)j=o

r42

a(Cx - n(k))!

L.2 Boundarv Conditions for the Comp lementary Convolution

Õe ^.cr- j

\- YP

f^{c,- Ðt

DTIn,Sn¿SC-k lczPxeQ-

Ocn<C

"(p) : j

,Cx-n(k)

so¡(Cn - rz(k))!

Substitute n' : n - jep which gives,

,Cx-n(k)g"(k,Q) DII

n,.Sn,ZSC-k-jSp lc:PxeQ-

nr>0

n'(p) : o

s,¡(C^ - rz(k))!

: f^d+e"(c + isp'Q- {&}) (A'2)

Equation (4.2) provides the basic complementary recursive algorithm linking the nor-

malising constant for one complementary region to that of smaller complementary

regions. As the algorithm proceeds, the complementary regions that it deals with be-

come smaller. It must be noted that the bounds of summation, the Clt ur" functions

of SC - k which itself becomes progressively smaller with each step of the algorithm.

A.2 Boundary conditions for the complemen-tary Convolution

consider Equation (4.1). If SC -k < 0 and ln: sn z sc -k then

NP_NL

D S(i,r)n(j) ( 0 for at least one i.j=l

However, all of the non zero entries in the matrix S : S(2') are positive integers. If

the above equation is to hold, n (j) must be negative for some j, but this contradicts

the condition n I 0. Therefore

9'(k,Q) :0 whenever k > SC

r43

(A.3)

4.3 Findine the Summation Bounds

The second boundary condition,

CP

g'(k,{Pr}) : Dj=o

C"- jUp' (A.4)

(c, - j)t

follows directly from (4.1)

4.3 Finding the Summation Bounds

Theorem 4.1 - summation Bounds of The complementary Region

Let C and S be defined as in section 4.1 and as before let k* : sc - k.

For meZf P-NL constrained by Sm f k. and 0 ( m 3 C,*(p) is bounded

above by C, given by

mi'fm.x, {t#l),t,) if S(i,p) >0 Vi:k.(i) >0otherwise

where I I denotes greatest integer less than.

The not greater than or equal to means that at least one row corresponding

to a positive entry in k* must satisfy a strictly less than condition. Assume

that we wish to calculate the upper bound for place P, customets, cr.

Case (i) - k. has one positive entry

Let i be such that kî > 0, with kf :0 for i + i'we must have

ce

Proof

NP_NL

D S(i, j)m¡ < ki,j=r

in particular,

S(i,p)m, < ki.

If .9(i,p) :0 this inequality is always satisfied and rn(p) is only bounded

above by cT If ^9(i'p) > 0 it must be a positive integer. since we are only

t44

.4'.3 Finding the Summation Bounds

interested in integer values of m@) when calculating its upper bound we

take the closest integer less than "äÐ

We have

rfri t

L A(i,,p) J

cece

if ^9(i,p) > 0

if .9(i,p) : 0

Case (ii) - C. has two or more positive entries

In this case we have

NP_NL

t A(i, j)m¡ < ki for at least one i : k¿* ) 0

j=l

If s(i,p) > 0 v i: ki ) 0 then as for the single positive entry case rn(p)

is bounded above uv Lså)l for each i. since only one of the inequalities

needs to be strictly less than we must take the maximum of these' The

maximum value .f LSäl may exceed Co which is not allowed. Hence, we

ensure that the value we take as a maximum is never greater than cr.

If S(i,p) : 0 for any i: ki ) 0 then rn(p) is unbounded for that i, and

hence is bound above only by Co' Therefore we have

cp: min [max, {t#}l} ,t,) if .9(i,p) >0 Vi:k.(i) >0otherwisece

as required

L45

Appendix B

Basic Notation

Notation Meaning

A Incidence Matrix

A Set of Arcs

AM A Change in Marking

t5(rn) Set of Enabled Transitions in the marking m

"Fs(m) Set of Firable Transitions in the marking m

I(¿¡ ) Input Bag for Transition ú¡

I{ Normalising Constant

m A Marking

*(i) Number of Tokens in Place i

Ills Initial Marking

Net Skeleton

(tr/, *o) Marked Net

NP Number of Places

¡\r^9 Number of Minimal Support S-invariants

NT Number of Transitions

o(¿¡) Output Bag for Transition f¡

P Set of Places

t46

MeaningNotation

Place iP¿

Transition it¿

Reachability Graph for Initial Marking msß(rns)

Reachability Graph for Initial Marking ms of Net 'Al7t(,Â/, ms)

Set of Non-Negative RealsR¡Set of Reals1¿

A Sequence of Transitionso

Firing Count Vector for the Sequence øo

S-invariants

T-invarianttS-invariant MatrixS

Set of TransitionsTWeight Functionw

Set of Non-Negative IntegerszJ+

Set of Integer Vectors of Length lúPZNP

r47

Bibliography

[1] Agnew, J., Explorations in Nurnber Theory, Brooks/Cole Publishing Company,

1972.

[2] Anderson, W.J., Continuous-Time Markou Chains, An Applications Oriented Ap-

proach,, ed: J.Gani, C.C.Hyde, Springer-Verlag, 1991.

[3] Baskett, F., Chandy, K., Muntz, R., Palacios, J., Open, Closed and Mixed Networlcs

of Queues with Different Classes of Customers, J. ACM, YoI. 22, pp.248-260, 1975.

[4] Boucherie, R.J., van Dijk, N.M., Spatial Birth-Death Processes with Multiple

Changes and Applications to Batch Seruice Networks anil Clustering Processes,

Research Report, Free University, Amsterdam, 1988.

[5] Boucherie, R.J., A Characterisation of Independence for Cornpeting Markou Chains

with Apptications to Stochastic Petri .lúeús, Proc. 5úå Int. W'shop., PNPM'93,

Toulouse, France, October 1993.

[6] Buzen, J.P.,Computational Algorithms for Closed Queueing Networles with Erpo-

nential Seruers, Comm. ACM, Vol. 16, No. 9, pp.527-531, 1973.

[7] Chandy, K.M., Herzog, U., Woo, L.5., Parametric Analysis of Queueing Networlcs,

IBM Journal of Research and Development, Vol. 19, pp.43-49, 1975.

[8] Chiola, G., A Software Paclcage for the Analysis of Generalised Stochastic Petri

Net Models, Ploc. Int. Workshop on Timed Petri Nets, IEEDCS Press., Torino,

Italy, July 1985.

148

[g] ciardo, G., Muppala, J., Trivedi, K.s., SPNP: Stochastic Petri Net Package,,Ptoc'

3"d Int. Conf. on Petri Nets and Performance Models, Kyoto, Japan, pp.l42-L5l',

December 1989.

[10] ciardo, G., Trivedi, K.s., A Decomposition Approach for stochastic Petri Net

Mod,els, Proceedings of Petri Nets and Performance Models, Melbourne, pp'74-85,

1991

[11] coleman, J.L., Henderson, w., Taylor, P.G., Product Form Equilibrium Distri-

butions anil an Algorithrn for classes of Batch Mouement Queueing Networlcs and

stochastic Petri l{eús, Technical Report, university of Adelaide, 1992'

[12] Coleman, J.L., Henderson, W., Taylor, P.G., A Conuolution Algorithm for Calcu-

lating Eract Equilibrium Distributions in Resource Allocation Problems with Mod-

erate (Jser Interference, To Appear, IEEE Trans' Comm' '

[13] coleman, J.L., Henderson, w., Taylor, P.G., Product-Form Equilibrium Distribu-

tions and a Conuolution Algorithrn for Stochastic Petri Nets, Submitted' 1993'

[14] Coteman, J.L., Henderson, W., Pearce, C'E'M', Taylor, P'G'' '4 Note on the

Correspond,ence Between Proiluct Forrn Batch Mouement Queueing Networks and'

single Mouement Networks, university of Adelaide, TRC Report, July 1993'

[15] Coleman, J.L., Henderson, W., Taylor, P.G., Efficient Algorithms for Closed Prod-

uct Form Networks containing single seruer Queues, Proc. 8¿h ATRS, Melbourne,

Australia, pp.133-142, December 1993'

[16] conway, 4.8., Georganas, N.D., RECAL - A New Effi'cient Algorithm for the

Eract Analysis of Muttipte-chain closed Queueing Networks, J.ACM, Vol' 33,

pp.768-791, 1986.

[17] Conway, 4., Pinsky, 8., Perforn'ùo,nce Analysis of sharing Policies for Broad'banil

Networles, Proc. 7th Int.ITC Seminar on Broadband Technologies, Morristown,

N.J., 1990.

r49

[18] Coyle A., A Method for Determining Congestion in Resource Access Problems

(Jsing Stochastic Petri Neús, Teletraffic Research Centre, University of Adelaide,

Research Report 16' 1991.

[19] Coyle, 4., Henderson, W., Taylor, P.G., Reduced, Loail Approxi'mations for Loss

Networlcs, To appear.

[20] Desel, J., Esparzâ, J., Reachability in Reuersible Free-Choice Systems, TUM-

19023, Institut Für Informatik, June 1990.

[21] Dijkstra, E.W., Cooperating Sequential Processes, Programming Languages, ed:

F. Genvys, Academic Press, New York, pp'43-112' 1968'

[22] Donatelli, S., Sereno, M.,, On the Product Form Solution for Stochastic Petri

I{eús, Proc. 13th Int. Conf. on Application and Theory of Petri Nets, Sheffield, UK,

pp.154..172, June 1992.

[23] Dziong,2., Roberts, J.W., Congestion Probabilities in a Circuit-Swi,tcheil Inte-

grateil seruices Network, Performance Evaluation, vol. 7, pp.267-284, 1987 '

[24] Feller, W., An Introd,uction to Probability Theory and lts Applicat'í,ons, Third

Edition, Witey and Sons, pp. 98-100, 1950.

[25] Frosch,D., Prod,uct Form Solutions for Closed Synchronized Systerns of Stochastic

Sequential Processes, Proc. 1992 Int. Comp. Sy-p., Taichung, Taiwan, pp'392-402,

December 1992.

[26] Frosch , D., Product Form Closed Synchronized Systems of Stochastic Sequential

Processes: A Conuolution Algorithm, Perconal communication, 1992.

[22] Florin, G., Natkin, S. Matrir Product Form Solution For Closed Synchronized

Queueing Networles, PNPM'89, Kyoto, Japan, pp.29-37, December 1989. Also,

IEEE Trans. Soft. Eng., Vol. 17, No. 2, February 1991.

[28] Gelenbe, E., G-Networks with Triggered Customer Mouement, preprint, 1992'

150

[29] Genrich, J.H., Lautenbach, I(., The Analysis of Distributed Systems by means of

Preilicate Transition-Nets, Lecture Notes Comp. Sci., Vol. 70, pp'123-146, 1979'

[30] Hack, M., Analysis of Production Schemata by Petri.l{eús, Masters Thesis, Dept.

Elec. Eng., MIT, Cambridge, Massachusetts, February 1972'

[31] Haas, P.J., Shedler, G.S., Regeneratiue Simulation of Stochastic Petri Neús, Pro-

ceedings of the International Workshop on Timed Petri Nets, Torino, Italy, pp.12-

23, 1985.

[32] Henderson, W., Queueing Networles with Negatiue Customers and Negatiue Queue

Lengths, J. Appl. Prob., Vol. 30' 1993.

[33] Henderson, W., Lucic, D., Exact Results in the Aggregation and, Disaggregation

of Stochastic Petri Neús., Proc. 4th Int. Workshop on Petri Nets and Performance

Models, Melbourne, Australia, pp.166-175, December 1991'

[34] Henderson, W., Lucic, D. Taylor, P.G.,.A Net Leuel Performance Analysis of

stochastic Petri Nets, J. Aust. Math. soc. series B,31, pp.176-187, 1989.

[35] Henderson, W., Northcote, B.S., Taylor, P.G., Geometric Equilibrium Distribu-

tions for Queues with Interactiue Batch Departures, to appear in Annals of Oper-

ations Research - special Issue on Queueing Networks, 1993.

[36] Henderson, W., Northcote, 8.S., Taylor, P.G., Product Form in Networles of

Queues with Forced Batch Mouement, Submitted.

[37] Henderson, w., Pearce, c.E.M., Taylor, P.G., Van Dijk, N.M., Closed, Queueing

Networks and Batch seruices, Queueing systems, Vol. 16, pp.59-70, 1990.

[38] Henderson, W., Taylor, P.G., Product Form in Networks of Queues with Batch

arriuals and Batch seruices, Queueing systems, vol. 6, pp.71-88, 1990.

[39] Henderson, W., Taylor, P.G., Embedded Processes in Stochastic Petri Neús, IEEE

Trans. Soft. Eng., Vol. 17, No. 2, pp.108-116, 1991.

151

[40] Henderson, w., Taylor, P.G., Sorn e New Results on Queueing Networks with

Batch Mouement, J' Appl' Prob', Vol' 28, pp'409-421' 1991'

[41] Henderson, w., Taylor, P.G., Red,uceil Loail Approximations for Resource Access

problems using Stochastic petri Nets, TRC Report No. 6, University of Adelaide,

1991.

[42] Jackson, J., Networlcs of waiting Lines, operations Research, vol' 15, pp'518-

52t,1957.

[43] Jackson, J., Jobshop-like Queueing Systerns, Management Sc', Vol' 10, pp'131-

t42,1963.

[44] Jensotr, K., High-Leuel Petri Nets, Informatik-Fachberichte, vol' 66, pp'166-180'

1983

[45] Jin, Q., Aoki, K., Stochastic Mod,el anil Behauioural Analysis for a computer

Network system, Proc. Aust. Japan workshop on stochastic models in engineering'

technology and management, Gold coast, Australia,,pp.232-241, July 1993'

[46] Kaufman, J.S., Blocking in a shared, Resource Enuironrnent,IEEE Trans' comm''

Vol. 29, PP.474-481, 1981'

[47] Kartin, s., Taylor, H.M., An Introd,uction to stochastic Modeling, Academic Press'

1984.

[a8] Kelly F.P., /[eúurorks of queues with customers of d,ifferenttypes, J' Appl' Prob''

Vol. 12, pp.542-554, 1975'

[ag] Keily, F.P., Reuersibility and, stochastic Networles, wiley, London, 1979'

[ro] Ketty, F.P., Networks of Quasi-Reuersible Nodes, Applied Probability - computer

science, the Interface: Proc. of the ORSA-TIMS Boca Raton symposium, ed: R'

Disney, Birkhauser, Boston, Cambridge, Mass' 1981'

r52

[51] Kelly, F.P., Blocking Probabilities in Large Circuit Switcheil Networlcs, Adv. Appl.

Prob., Vol. 18, pp.473-505, 1986.

[52] Kelly, F.P., Ioss Networks, Ann. AppI. Prob., vol. 1, pp.319-378' 1991.

[53] Koenigsberg, 8.,, Cyclic Queues, operational Research Quarterly, vol. 9, No. 1,

pp.22-35, 1958.

154) Lazar, 4.4., Robertazzi, T.G., The Geornetry of Lattices for Multiclass Markouian

Queueing Networles, Proc. of the 1984 conf. on Info. sci. and sys., Princeton

University, Princeton, pp.164-168, March 1984'

[55] Lazar, 4.4., Robertazzi, T.G., Marleouian Petri Net Protocols with Product Form

Solution,IEtrE infocom'87, the conference on computer communications, Wash-

ington DC, IEEE computer society press, pp.1054-1062, 1987.

[56] Li, M., Applying Decornposition and Aggregation Theory to the Analysis of

Stochastic Petri Nets anil Queueing Networks, Ph.D Dissertation, Department of

Electrical Engineering, university of ottawa, canada, December 1991.

[57] Li, M., Georganas, N.D., em Pararnetric Analysis of Stochastic Petri Neús, Fifth

International Conference on Modelling and Tools for Computer Performance Eval-

uation, Torino, ItalY, 1991.

[58] Lin, c., Marinescu, D.C., Stochastic High-Leuel Petri Nets and Applications,

IEEE Trans. Comp., Vol. 37, No. 7, pp.815-825, July 1988'

[bg] Marsan, M.4., Chiola, G., On Petri Nets with Deterministic anil Erponential

Transition Firing Times, Proc. 7'h European Workshop on Application and theory

of Petri Nets, Oxford, England, June 1986'

[60] Marsan, M.4., Balbo, G., Conte, G., A Class of Generalized Stochastic Petri Nets

for the Performance Analysis of Multiprocessor Systems, ACM Trans' Comp. Sys''

Vol.2(1), May 1984.

153

[61] Marsan, M.4., Balbo, G', Bobbio, A., Chiola, G', Conte, G', Cumani, A', On

petri Nets with Stochastic Timing, Proc. Int. Workshop on Timed Petri Nets,

Torino, Italy, July 1-3' 1985.

[62] Marsan, M.A., Balbo, G., Bobbio, 4., Chiola, G., conte, G., Cumani, A'., The

Effect of Erecution Policies on the Semantics anil Analysis of Stochastic Petri

.lúeús, IEEE Trans' Soft. Eng., Vol.15, pp.832-845, 1989'

[63] Memmi, G., Roucairol, G., Linear Algebra in Net Theory, Lecture Notes in com-

puter Science, Vol'84, pp.2I3-223, 1980'

[64] Merlin, P.M., A Study of the Recouerability of Computer Systems, PhD' Thesis,

University of California, Irvine, 1974.

[65] Mitra,D., Asymptotic Analysis and Computational Method's for a Class of Sirnple,

circuit-switched, Networlcs with Blocking, Adv. Appl. Prob., vol.19, pp.219-239'

1987

[66] Molloy, M.K., Performance Analysis using stochastic Petri Neús, IEEE. Trans.

on Computers, Vol. 31, No. 9, pp.913-917, September 1982'

[67] Molloy, M.K., Petri Net Modelling, the past, the Present, anil the Future,

PNPM'89, Kyoto, Japan, PP.2-9, December 1989'

[68] Murata, T., Petri Nets: Properties, Analysis and Applications, Proc' IEEE'

Yol.77, No.4, pp.541-580, April 1989'

[6g] Natkitr, S., Les Reseaux d,e Petri Stochastiques et leur Application a l'Eualuation

d,es Systemes Informatiques,, PhD. Thesis, CNAM, Paris, 1980'

[70] Peterson, J.L., Petri Net Theory and the Modeling or systems, Englewood cliffs,

NJ: Prentice-Hall, Inc., 1981.

[21] pollett, P.K., Preseruing Partial Balance in Continuous-Time Marleou Chains,

Adv. Appl. Prob., Vol. 19, pp.431-453, 1987.

154

l72l F[azouk, R.R., The Deriuation of Performance Expressions tor communications

Protocols from Timed, Petri Neús, Proc. ACM SIGCOMM '84, symposium on com-

munications Architecture and Protocols, Montreal, pp.210-217,, June 1984.

[73] Razouk, R.R., Phelps, c.v., Performance Analysis using Timeil, Petri Nets, Proc'

1984 Intl. conf. on Parallel Processing, pp.126-128, August 1984.

[74] Ramchandani, Analysis of AsynchronouE concurrent systems by Petri Nets,,

phD. Dissertation, Dept. Elec. Eng., Massachusetts Institute of Technology, Cam-

bridge, Massachusetts, JulY 1973.

[25] Robertazzi, T.G., Why Most Stochastic Petri Nets are Non-Proiluct Form Net-

works, Technical report, University of New York, 1991'

[76] Roberts, J.W., A Seruice System with Heterogeneous (Iser Requirements, Perfor-

mance of Data communications systems and their Applications, ed: G. Pujolle,

North Holland, PP.423-431, 1981.

[72] Ross, S.M., Introiluction to Probability Models,4th Edition, Academic Press, 1985'

[7g] Sereno, M., Balbo, G., Computational Algorithms for Proiluct Form solution

Stochastic Petri, Nets. Technical Report, University of Torino, Italy, 1993'

[29] Sifaki s, J., (Jse of Petri Nets for Performance Eualuation, \n Measuring, Modelling

and Evaluating computer systems, Ed. H. Beilner, E. Gelenbe, North-Holland,

pp.75-93, 1977.

[s0] Sifakis, J., PerforrÍ¿ance Eualuation of systems using Neús, in Net Theory and

Applications, ed: W. Brauer, pp.307-319, Springer Verlag, 1979'

[g1] Symons, F.J.W. , Introd,uction to Numerical Petri Nets, a General Graphical Model

of concurrent Processing systems, A.T.R., Vol. 14, No. 1, pp.28-33, January 1980.

[82] petri, C.4., I(ommunikation mit Automaten, PhD. Dissertation, University of

Bonn, West Germany, 1962. English Translation, Communication with Automata,

lbÐ

New York: Griffiss Air Force Base. Tech. Rep.RADC-TR-65-377, Vol'l, suppl'l,

1966.

[83] Reisig, W., Petri Nets - An Introduction, EATCS Monographs on Theoretical

Computer Science, Vol. 4, New York: Springer-Verlag, 1985'

[g4] someya, H., Tashiro, T., Murat&, T., Komoda, N., Perfornxance Eualuation of

Job Operation Flows in Computer Systems by Timed Petri NeÚs, Int. W'shop.,

PNPM 89, I(yoto, Japan, pp'104-111, 1989'

[g5] vernotr, M., Zahorjan, J., Lazowska, 8.D., A Comparison of Performance Petri

Nets and, Queuei,ng Networle Models, Research Report 669, Univeristyof Wisconsin,

Madison, 1986.

[g6] walrand, J.,A Discrete-Time Queueing Network, J. Appl. Prob., 20, pp.903-909'

1983.

[87] Watanabe, T., Mizobata, Y., onaga, K., Legal Firing sequence anil Relateil Prob-

lems of Petri.lúeús, PNPM'8$, Kyoto, Japan, pp.277-286, December 1989.

[S8] \ /hittle, P., Equilibrium Distributions for an Open Migration Process, J. Appl.

Prob., No. 5, pp.567-571, 1968.

[Sg] Whittle, P., Nonlinear rnigration procesq Bull. Internat. Inst. Statist., Yol. 42,

pp.642-647, 1969.

[90] whittle, P., systems in stochastic Equilibrium, John wiley, 1986.

[g1] Wong, C.Y.,Dillon, T.S., Forward, K.8., Tirned Places Petri Nets with Stochastic

Representation of Place Times, Proc. Int. Workshop on Timed Petri Nets, Torino,

Italy, pp.96-103, JulY 1985.

[92] ycart ,8., The Philosophers Process: An Ergodic Reuersible Nearest Parti'cle Sys-

tem, Ãrlrlals of Applied Probability, Vol. 3, No' 2, pp'356-363' 1993'

156

[93] Zenie, A., Coloreil Stochastic Petri, Neús, IEEE Comp. Sci. Press, pp.262-27I,

1986

[94] Zuberek, W. M., M-Timed, Petri Nets, Priorities, Preemptions, and Perforrnance

Eualuation of Systems, Advances in Petri Nets (LNCS 222), ed G. Rozenburg,

Springer-Verlag, pp.478-496, July 1985.

t57

stochastic petri nets with product form equilibrium ... · contents list of figures list of tables...

Documents