PERFORMANCE ANALYSIS OF TIMED EVENT GRAPHS WITH

MULTIPLIERS USING (Min, +) ALGEBRA

Samir Hamaci

Jean-Louis Boimond

S

´

ebastien Lahaye

LISA

62 avenue Notre Dame du Lac - Angers, France

Keywords:

Timed event graphs with multipliers, (min,+) algebra, linearization, cycle time.

Abstract:

We are interested in the performance evaluation of timed event graphs with multipliers. The dynamical equa-

tion modelling such graphs are nonlinear in (min,+) algebra. This nonlinearity is due to multipliers and pre-

vents from applying usual performance analysis results. As an alternative, we propose a linearization method

in (min,+) algebra of timed event graphs with multipliers. From the obtained linear model, we deduce the

cycle time of these graphs. Lower and upper linear approximated models are proposed when linearization

condition is not satisﬁed.

1 INTRODUCTION

Timed event graphs (TEG’s) are well adapted to

model synchronization phenomena occuring in dis-

crete event systems (Murata, 1989). Their behav-

ior can be modelled by recurrent linear equations in

(min, +) algebra (Baccelli et al., 1992). When the size

of the model becomes very signiﬁcant, techniques of

analysis developed for these graphs reach their limits.

A possible alternative consists in using timed event

graphs with multipliers, denoted TEGM’s. Indeed,

the use of multipliers associated with arcs is natural

to model a large number of systems, for example,

when the achievement of a speciﬁc task requires sev-

eral units of a same resource, or when an assembly

operation requires several units of a same part.

To our knowledge, few works deal with the per-

formance analysis of TEGM’s. In fact, in the most

of works the proposed solution is to transform the

TEGM into an ordinary TEG, which allows the use

of well-known methods of performances analysis.

In (Munier, 1993) the initial TEGM is the object

of an operation of expansion. Unfortunately, this

expansion can lead to a model of signiﬁcant size,

which does not depend only on the initial structure

of the TEGM, but also on initial marking. With this

method, the system transformation proposed under

single server semantics hypothesis, or in (Nakamura

and Silva, 1999) under inﬁnite server semantics hy-

pothesis, leads to a TEG with |θ| transitions (|θ| is

the 1-norm of the elementary T-semiﬂow of the cor-

responding TEGM).

Another linearization method was proposed in

(Trouillet et al., 2002) when each elementary circuit

of graph contains at least one normalized transition

(i.e., a transition for which its corresponding elemen-

tary T-semiﬂow component is equal to one). This

method increases the number of transitions. Inspired

by this work, a linearization method without increas-

ing the number of transition was proposed in (Hamaci

et al., 2004).

A calculation method of cycle time of a TEGM is

proposed in (Chao et al., 1993) but under restrictive

conditions on initial marking.

The weights on the arcs of a TEGM are nonlinearly

modelled in (min, +) algebra. Based on works given in

(Cohen et al., 1998), we propose a new method of lin-

earization without increasing the number of transition

from the graph. The obtained (min, +) linear model

allows to evaluate the performance of these graphs.

According to initial marking, these performances are

evaluated in an exact or approached way.

This article is organized as follows. Some concepts

on TEGM’s and their functioning are recalled in Sec-

tion 2. The method of linearization is presented in

Section 3. From the equivalent, or approached, TEG

of a TEGM, we deduce the cycle time in the Section

4.

16

Hamaci S., Boimond J. and Lahaye S. (2005).

PERFORMANCE ANALYSIS OF TIMED EVENT GRAPHS WITH MULTIPLIERS USING (Min, +) ALGEBRA.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 16-21

DOI: 10.5220/0001157700160021

Copyright

c

SciTePress

2 RECURRENT EQUATIONS OF

TEGM’s

We assume that the reader is familiar with the

structure, ﬁring rules, and basic properties of Petri

nets, see (Murata, 1989) for more details.

Consider a Petri net deﬁned as a valued bipartite

graph given by a ﬁve-tuple (P, T, M, m, τ ) in which:

• P and T represent the ﬁnite set of places, and

transitions respectively;

• A multiplier M is associated with each arc. Given

q ∈ T and p ∈ P, the multiplier M

pq

(respectively,

M

q p

) speciﬁes the weight (in N) of the arc from

transition q to place p (respectively, from place p to

transition q). A zero value for M codes an absence

of arc;

• With each place are associated an initial marking

(m

p

assigns an initial number of tokens (in N) in

place P) and a holding time (τ

p

gives the minimal

time (in N) a token must spend in place p before

it can contribute to the enabling of its downstream

transitions).

We denote by

•

q (resp., q

•

) the set of places

upstream (resp., downstream) transition q. Similarly,

•

p (resp., p

•

) denotes the set of transitions upstream

(resp., downstream) place p.

An event graph is a Petri net whose each place has

exactly one upstream and one downstream transition.

We denote W the incidence matrix of a Petri net.

A vector θ ∈ N

T

such that θ 6= 0 and W θ = 0 is a

T-semiﬂow. A T-semiﬂow θ has a minimal support

iff there exists no other T-semiﬂow, θ

′

, such that

{q ∈ T | θ

′

(q) > 0} ⊂ {q ∈ T | θ(q) > 0}.

A vector Y ∈ N

P

such that Y 6= 0 et Y

t

W = 0 is

a P-semiﬂow.

In the rest of the paper we assume that TEGM’s are

consistent (i.e., there exists a T-semiﬂow θ covering

all transitions : kθk = T) and are conservative (i.e.,

there exists a P-semiﬂow Y covering all places:

kY k = P ).

Remark 1 We disregard without loss of generality

ﬁring times associated with transitions of a TEG be-

cause they can always be transformed into holding

times on places (Baccelli et al., 1992, §2.5).

With each transition q is associated a counter

variable, denoted n

q

: n

q

is an increasing map from

R to Z ∪ {+∞}, t 7→ n

q

(t) which denotes the

cumulated number of ﬁrings of transition q up to time

t.

In the following, we assume that counter variables

satisfy the earliest ﬁring rule, i.e., a transition q ﬁres

as soon as all its upstream places {p ∈

•

q} contain

enough tokens (M

q p

) having spent at least τ

p

units of

time in place p. When the transition q ﬁres, it con-

sumes M

q p

tokens in each upstream place p and pro-

duces M

p

′

q

tokens in each downstream place p

′

∈ q

•

.

Assertion 1 The counter variable n

q

of a TEGM (un-

der the earliest ﬁring rule) satisﬁes the following tran-

sition to transition equation:

n

q

(t) = min

p∈

•

q , q

′

∈

•

p

⌊M

−1

q p

(m

p

+ M

pq

′

n

q

′

(t − τ

p

))⌋.

(1)

Let us note the presence of inferior integer part to

preserve integrity of Eq. (1). In general, a transition

q may have several upstream transitions ({q

′

∈

••

q})

which implies that its associated counter variable is

given by the min of transition to transition equations

obtained for each upstream transition.

Example 1

n 3

n 4

n 2

n 1

6

3

1

2

3

1

P 1

P 3

P 2

P 4

P 5

1

3

3

3

3

2

2

Figure 1: A TEGM.

The counter variables associated with transitions of

TEGM depicted in the ﬁgure 1 satisfy the next system

equations:

n

1

(t) = ⌊

6+2n

3

(t−1)

3

⌋,

n

2

(t) = min(⌊

3n

1

(t−2)

2

⌋, 3 + 3n

4

(t − 3)),

n

3

(t) = n

2

(t − 1),

n

4

(t) = ⌊

n

3

(t−1)

3

⌋.

In the case of ordinary TEG’s, the transition to

transition equation given in Eq. (1) becomes:

PERFORMANCE ANALYSIS OF TIMED EVENT GRAPHS WITH MULTIPLIERS USING (Min, +) ALGEBRA

17

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

(m

p

+ x

q

′

(t − τ

p

)). (2)

This equation is linear in the algebraic structure

called (min, +) algebra. This structure, denoted Z

min

,

is deﬁned as the set Z ∪ {+∞}, equipped with the

min as additive law (denoted ⊕) and with the usual

addition as multiplicative law (denoted ⊗). The

neutral element of the law ⊕ (resp., ⊗) is denoted

ε = +∞ (resp., e = 0). More generally, the (min, +)

algebra is a dioid (Baccelli et al., 1992).

A dioid (D, ⊕, ⊗) is a semiring in which ⊕ is

idempotent (∀a, a ⊕ a = a). Neutral elements of ⊕

and ⊗ are denoted ε and e respectively.

From the Eq.(2) obtained for each transition, one

can express a TEG as the following recursive matrix

equation:

x(t) = A ⊗ x(t − 1), (3)

where A is a square matrix with coefﬁcient in Z

min

,

and x(t) is the vector of the counter variables associ-

ated with transitions of the graph. See (Baccelli et al.,

1992) for more details on the representation of TEG’s

in the dioid Z

min

.

3 LINEARIZATION OF TEGM’S

A TEGM is linearizable if there exists a change of

variable n

q

(t) = θ

q

x

q

(t) such that x

q

(t) satisﬁes a

(min,+) linear recurrent equation knowing that:

• n

q

(t) is the counter associated with transition q of

TEGM,

• θ

q

is the component of T-semiﬂow associated with

transition q (θ

q

∈ N

∗

).

Proposition 1 A TEGM is linearizable if

∀q ∈ T, ∀p ∈

•

q, ⌊

m

p

M

q p

⌋ ∈ θ

q

N. (4)

Proof: According to assertion (1), we have for

each transition q of a TEGM:

n

q

(t) = min

p∈

•

q , q

′

∈

•

p

⌊M

−1

q p

(m

p

+ M

pq

′

n

q

′

(t − τ

p

))⌋.

Using the change of variable n

q

(t) = θ

q

x

q

(t), and

by distributivity of the multiplication with respect to

the min operator, we have:

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

1

θ

q

⌊(

m

p

M

qp

+

M

pq

′

M

qp

n

q

′

(t − τ

p

))⌋.

From relation

θ

q

M

pq

′

=

θ

q

′

M

qp

, obtained for consistent and conserv-

ative TEGM (see (Munier, 1993)), we have

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

1

θ

q

⌊(

m

p

M

qp

+

θ

q

θ

q

′

n

q

′

(t − τ

p

))⌋,

i.e.,

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

1

θ

q

⌊(

m

p

M

qp

+ θ

q

x

q

′

(t − τ

p

))⌋.

Because θ

q

x

q

′

(t − τ

p

) ∈ N, we ﬁnally obtain

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

(

1

θ

q

⌊

m

p

M

q p

⌋ + x

q

′

(t − τ

p

)), (5)

which corresponds to a (min,+) linear recurrent

equation.

More generally, if the condition (4) is satisﬁed for

each transition, the equation (1) can be expressed as

a (min,+) linear recurrent equation.

Remarks:

• Let us deﬁne an equivalence class of initial mark-

ings for the equivalence relation m

′

≡ m

′′

⇔

∀q ∈ T, ∀p ∈

•

q, ⌊

m

′

p

M

qp

⌋ = ⌊

m

′′

p

M

qp

⌋.

We can notice that all initial markings of a same

equivalence class generate the same ﬁring times

behavior of transitions and give the same (min,+)

model.

• In (Cohen et al., 1998), the authors propose a

linearization method through a similar diagonal

change of counter variables for ﬂuid TEGMs (i.e.,

where initial marking and multipliers can take real

values). Moreover, they state in Prop. IV.6 that the

behavior of a TEGM coincides (in N) with that of

its ﬂuid version if ∀q ∈ T, ∀p ∈

•

q,

m

p

M

qp

∈ θ

q

N.

Thus, under this condition, it is possible to linearize

a TEGM by considering its ﬂuid version . How-

ever, the required condition is more restrictive than

the condition (4).

When the condition (4) is not satisﬁed, we deﬁne

two linear approximated models of the TEGM by con-

sidering a greater (resp., smaller) initial marking.

Deﬁnition 1 The upper (resp., lower) linear model

is obtained by a minimal addition (resp., removal)

of initial tokens in the TEGM, in order to satisfy the

linearization condition (4) for each initial marking.

In other words, in each place p for which

⌊

m

p

M

qp

⌋ /∈ θ

q

N, we add

m

p

(resp., remove m

p

) ini-

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

18

tial tokens until the linearization condition is checked.

We denote x(t) (resp., x(t)) the state vector of the

TEG obtained from the approximate linearization by

addition (resp., removal) of tokens in the TEGM.

We have

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

(

1

θ

q

⌊

(m

p

+

m

p

)

M

q p

⌋ +

x

q

′

(t − τ

p

)),

(6)

where

m

p

is the minimum number of tokens added

in the place p such that ⌊

m

p

+

m

p

M

qp

⌋ ∈ θ

q

N.

and

x

q

(t) = min

p∈

•

q , q

′

∈

•

p

(

1

θ

q

⌊

(m

p

− m

p

)

M

q p

⌋ + x

q

′

(t − τ

p

)),

(7)

where m

p

is the minimum number of tokens

removed in the place p such that ⌊

m

p

−m

p

M

qp

⌋ ∈ θ

q

N.

We have:

∀q, θ

q

x

q

(t) = n

q

(t) ≤ n

q

(t) ≤

n

q

(t) = θ

q

x

q

(t).

Example 2 The TEGM depicted in Fig.1 admits the

elementary T-semiﬂow: θ = (2, 3, 3, 1).

For initial marking M(0)=(6,0,0,3,0), we easily

verify that initial marking of each place satisﬁes the

linearization condition (4), which means that TEGM

is linearizable.

Using the change of variables n

i

(t) = θ

i

x

i

(t)

and thanks to Eq.(5), we obtain the following linear

model:

x

1

(t) = 1 + x

3

(t − 1),

x

2

(t) = min(1 + x

4

(t − 3), x

1

(t − 2)),

x

3

(t) = x

2

(t − 1),

x

4

(t) = x

3

(t − 1).

These equations correspond to the TEG depicted in

ﬁgure 2.

For initial marking M(0)=(6,0,0,4,0), we can note

that the place P4 does not satisfy the linearization

condition.

Thanks to Eqs (6) and (7), we obtain respectively :

x

1

(t) = 1 +

x

3

(t − 1),

x

2

(t) = min(2 + x

4

(t − 3), x

1

(t − 2)),

x

3

(t) = x

2

(t − 1),

x

4

(t) = x

3

(t − 1),

x 3

x 4

x 2

x 1

1

1

1

2

3

1

P 1

P 3

P 2

P 4

P 5

1

Figure 2: TEG obtained by the linearization of the TEGM

of the ﬁgure 1.

and

x

1

(t) = 1 + x

3

(t − 1),

x

2

(t) = min(1 + x

4

(t − 3), x

1

(t − 2)),

x

3

(t) = x

2

(t − 1),

x

4

(t) = x

3

(t − 1).

The evolution of the counter n

2

(t) is depicted in

ﬁgure 3 and is such that n

2

(t) ≤ n

2

(t) ≤

n

2

(t).

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 0

0

4 8 1 2 1 6

2 0

2 4 2 8

T i m e ( t )

E v e n t

n 2

n 2

n 2

Figure 3: The evolution of the counter variables n

2

, n

2

and

n

2

.

PERFORMANCE ANALYSIS OF TIMED EVENT GRAPHS WITH MULTIPLIERS USING (Min, +) ALGEBRA

19

4 PERFORMANCE EVALUATION

4.1 Elements of performance

evaluation for TEG

We recall main results characterizing an ordinary

TEG’s modelled in the dioid Z

min

(Baccelli et al.,

1992), (Gaubert, 1992).

Deﬁnition 2 (Irreducible matrix) A matrix A is

said irreducible if for any pair (i,j), there is an inte-

ger m such that (A

m

)

ij

6= ε.

Theorem 1 Let A be a square matrix with coefﬁcient

in Z

min

. The following assertions are equivalent:

• Matrix A is irreducible,

• The TEG associated with matrix A is strongly con-

nected.

One calls eigenvalue and eigenvector of a matrix A

with coefﬁcients in Z

min

, the scalar λ and the vector

υ such as:

A ⊗ υ = λ ⊗ υ.

When the initial vector x(0) of matrix equation (3) is

equal to an eigenvector of matrix A, the TEG reaches

a periodic regime from the initial state.

Theorem 2 Let A be a square matrix with coefﬁ-

cients in Z

min

. If A is irreducible, or equivalently, if

the associated TEG is strongly connected, then there

is a single eigenvalue denoted λ. The eigenvalue can

be calculated in the following way:

λ =

n

M

j=1

(

n

M

i=1

(A

j

)

ii

)

1

j

. (8)

Regarding the TEG, λ corresponds to the ﬁring

rate identical for each transition. This eigenvalue λ

can be directly deduced from the TEG by

λ = min

c∈ C

M(c)

T (c)

, (9)

where:

• C is the set of elementary circuits of the TEG .

• T(c) is the sum of holding times in circuit c.

• M(c) is the number of tokens in circuit c.

It is possible that several eigenvectors can be associ-

ated with the only eigenvalue of an irreductible ma-

trix.

Deﬁnition 3 Let A be an irreducible matrix of eigen-

value λ. One deﬁnes the matrix denoted A

λ

by

A

λ

= λ

−1

⊗ A.

Theorem 3 (Gondran and Minoux, 1977) Let A be

an irreducible matrix of eigenvalue λ. The j-th col-

umn of the matrix A

+

λ

, denoted (A

+

λ

)

j

, is an eigen-

vector of A if it satisﬁes the following equality:

(A

+

λ

)

j

= A

λ

⊗ (A

+

λ

)

j

. (10)

4.2 Elements of performance

evaluation for TEGM’s

In the case of TEGM’s, the ﬁring rate, denoted λ

m

q

,

is not identical for all transitions. It is deﬁned as fol-

lows:

λ

m

q

=

θ

q

T C

m

, (11)

where θ

q

is the component of the T-semiﬂow associ-

ated with transition q, and T C

m

is average the cycle

time of the TEGM.

The average cycle time of a TEGM can be deﬁned

as follows :

Deﬁnition 4 (Sauer, 2003)

The average cycle time, TC

m

, of a TEGM is the aver-

age time to ﬁre once the T-semiﬂow under the earliest

ﬁring rule (i.e., transitions are ﬁred as soon as possi-

ble) from the initial marking.

The ﬁring rate λ

m

q

of a linearizable TEGM can be

calculated from the (min, +) linear model by:

λ

m

q

= θ

q

λ (12)

where λ is the eigenvalue of the equivalent (min,+)

linear model. This result is a direct consequence of

the linearization proprety.

In the case where we have an approximate lin-

earization, we obtain

λ

m

q

≤ λ

m

q

≤

λ

m

q

,

where

λ

m

q

(resp., λ

m

q

) is the ﬁring rate of the

transition q obtained by using the upper (resp., lower)

approximated linear model of the TEGM.

When components of the eigenvector, associated

with the TEG obtained by linearization, are integer

values, the initial conditions vector of TEGM, de-

noted υ

m

(which allow to reach the periodic regime

from the initial state) can be deduced by:

υ

m

= (θ

1

x

1

(0), ..., θ

n

x

n

(0)) (13)

where x(0) is the eigenvector of the TEG.

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

20

Example 3 One determines the ﬁring rate for each

transition of the TEGM of ﬁgure 1 from the (min, +)

linear model.

For M(0) = (6, 0, 0, 3, 0), the initial marking of

each place veriﬁes the linearization condition. This

TEGM is linearizable.

Thanks to Eq.(8), the production rate of the TEG

obtained after linearization is equal to

1

5

. From

Eq.(12), we deduce the ﬁring rate of each transition:

λ

m

1

=

2

5

, λ

m

2

=

3

5

, λ

m

3

=

3

5

, λ

m

4

=

1

5

.

Thanks to Eq.(11), we deduce that T C

m

= 5.

For initial marking M(0)=(6,0,0,4,0), we have two

linear approximated (min, +) models.

In the case where we add two tokens in the place

P4 in the TEGM, we obtain a TEG with λ is equal to

1

4

.

Thanks to Eq.(12), we deduce the ﬁring rate of

each transition :

λ

m

1

=

3

4

,

λ

m

2

=

2

4

,

λ

m

3

=

2

4

,

λ

m

4

=

1

4

.

Thanks to equation (11), we deduce

T C

m

= 4.

In the case where we remove one token in the place

P4, we obtain a TEG with λ is equal to

1

5

.

Thanks to Eq.(12), we deduce the ﬁring rate of

each transition :

λ

m

1

=

3

5

, λ

m

2

=

2

5

, λ

m

3

=

2

5

, λ

m

4

=

1

5

.

Thanks to Eq.(11), we deduce T C

m

= 5.

Finally, for M(0)=(6,0,0,4,0), we obtain:

4 ≤ T C

m

≤ 5

3

5

≤ λ

m

1

≤

3

4

,

2

5

≤ λ

m

2

≤

2

4

,

2

5

≤ λ

m

3

≤

2

4

,

1

5

≤ λ

m

4

≤

1

4

.

5 CONCLUSION

In order to evaluate the performances of a TEGM

from an equivalent TEG, a technique of linearization

has been proposed in (min,+) algebra. According to

initial marking, a linearization condition was stated.

The performance analysis of a linearizable TEGM,

such as cycle times, is deduced directly from the ob-

tained linear model.

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(1992). Synchronization and Linearity: An Algebra

for Discrete Event Systems. Wiley.

Chao, D., Zhou, M., and Wang, D. (1993). Multiple

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