**2. Petri Net**

2 Will-be-set-by-IN-TECH

diod algebra. The behavior of systems characterized by delays and synchronization can be described by such recurrences [1]. These systems are modeled by Timed Event Graphs (TEG). This latter constitute a subclasses of Timed Petri Nets with each place admits an upstream transition and downstream transition. When the size of model becomes very significant, the techniques of analysis developed for TEG reach their limits. A possible alternative consists in using Timed Event Graphs with Multipliers denoted TEGM. 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 several units of a same resource, or when an assembly

This chapter deals with the performance evaluation of TEGM in dioid algebra. Noting that these models do not admit a linear representation in dioid algebra. This nonlinearity is due to the presence of weights on arcs. To mitigate this problem of nonlinearity and to apply the results used to evaluate the performances of linear systems, we use a linearization method of mathematical model reflecting the behavior of a Timed Event Graphs with Multipliers in

Few works deal with the performance evaluation of TEGM. Moreover, the calculation of cycle time is an open problem for the scientific community. In the case where the system is modeled by a TEGM, 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 evaluation. In [12] 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 TEGM, but also on initial marking. With this method, the system transformation proposed under *single* server semantics hypothesis, or in [14] under *infinite* server semantics hypothesis,

Another linearization method was proposed in [17] when each elementary circuit of graph contains at least one *normalized* transition (*i.e.,* a transition for which its corresponding elementary T-invariant component is equal to one). This method increases the number of transitions. Inspired by this work, a linearization method without increasing the number of transition was proposed in [8]. A calculation method of cycle time of a TEGM is proposed in [2] but under restrictive conditions on initial marking. We use a new method of linearization

This chapter is organized as follows. After recalling in Section 2 some properties of Petri nets, we present in Section 3, modeling the dynamic behavior of TEGM, which are a class of Petri nets, in dioid algebra, precisely in (*min*, +) algebra. In this section we will show that TEGM are nonlinear in this algebraic structure, unlike to TEG. This nonlinearity prevents us to use the spectral theory developed in [5] for evaluate the performances of TEG in (*min*, +) algebra. To mitigate this problem of nonlinearity, we will encode the mathematical equations governing the dynamic evolution of TEGM in a dioid of operators developed in [7], inspired by work presented in [3]. The description of this dioid and the new state model based on operators will be the subject of Section 4. To exploit the mathematical model obtained, a linearization method of this model will be presented in Section 5, in order to obtain a linear model in (*min*, +) algebra and to apply the theory developed for performance evaluation. This latter will be the subject of Section 6. Before concluding, w e give a short example to illustrate this

operation requires several units of a same part.

order to obtain a linear model.

leads to a TEG with |*θ*| transitions.

without increasing the number of transition of TEGM [6].

approach for evaluate the performances of TEGM in dioid algebra.

#### **2.1. Definitions and notations**

Petri Nets (PN) are a graphical and mathematical tool, introduced in 1962 by Carl Adam Petri [15]. They allow the modeling of a large number of Discrete Event Dynamic Systems. They are particularly adapted to the study of complex processes involving properties of synchronization and resource sharing.

The behavior over time of dynamical systems, including evaluation of their performance (cycle time, ...), led to introduce the notion of time in models Petri Net. Several models Petri Net incorporating time have been proposed. These models can be grouped into two classes: deterministic models and stochastic models. The former consider the deterministic values for durations of activity, whereas the latter consider probabilistic values.Among the existing Timed Petri Net include: the Temporal Petri Net [11] associating a time interval to each transition and each place, the T-Timed Petri Net [4] associating a positive constant (called firing time of transition) at each transition and P-Timed Petri Net ; [4], [9] associating a positive constant (called holding time in the place) at each place of graph. It has been shown that P-Timed Petri Net can be reduced to T-Timed Petri Net and vice versa [13]. In the next, for consistency with the literature produced on the dioid algebra, we consider that P-Timed Petri Net.

A *P-Timed Petri Net* is a valued bipartite graph given by a 5-tuple (*P*, *T*, *M*, *m*, *τ*).


**Figure 1.** Example of a P-Timed Petri Net.

More generally, for a Petri Net, we denote *W*<sup>−</sup> = [*Mqp*] (input incidence matrix), *W*<sup>+</sup> = [*Mpq*] (output incidence matrix), *<sup>W</sup>* <sup>=</sup> *<sup>W</sup>*<sup>+</sup> <sup>−</sup> *<sup>W</sup>*<sup>−</sup> (incidence matrix) and considering *<sup>S</sup>* a possible firing sequence from a marking *mi* to the marking *mk*, then a fundamental equation reflecting the dynamic behavior of Petri Net, is obtained:

$$m\_k = m\_i + \mathcal{W} \times \underline{\mathbf{S}}.\tag{1}$$

The graph reaches a periodic regime when there is a firing sequence achievable with *θ* as

A Petri Net is said conservative if all places in the graph form a conservative component.

all places: {*p* ∈ *P*|*Y*(*p*) > 0} = *P*). Such graphs verify the next properties [13]:

• A consistent Petri Net is strongly connected *iff* it is conservative. • A consistent Petri Net has a unique elementary T-invariant.

**3. Dynamic behavior of Timed Petri Nets in dioid algebra**

*a* = *a*). Neutral elements of ⊕ and ⊗ are denoted *ε* and *e* respectively.

*a* � *b* ⇔ *b* = *a* ⊕ *b* (the least upper bound of {a,b} is equal to *a* ⊕ *b*).

*<sup>T</sup>* of a complete dioid <sup>D</sup> is equal to *<sup>x</sup> <sup>x</sup>*∈D

complete dioid always exists and is denoted

The Petri Nets considered here are *consistent* (*i.e.,* there exists a T-invariant *θ* covering all transitions: {*q* ∈ *T*|*θ*(*q*) > 0} = *T*) and *conservative* (*i.e.,* there exists a P-invariant *Y* covering

• A PNPetri Net allows a live and bounded initial marking *m iff* it is consistent and

• The product of multipliers along any circuit of a conservative Petri Net is equal to one.

In the next, 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*.

**Definition 5.** An ordinary *Timed Event Graph* (TEG) is a Timed Petri Net such that each place has exactly one upstream transition and one downstream transition. Weights of arcs are all

These graphs are well adapted to model synchronization phenomena occurring in Discrete Event Dynamic Systems. They admit a linear representation on a particular algebraic structure

**Definition 6.** *A dioid* (D, ⊕, ⊗) is a semiring in which the addition ⊕ is idempotent (∀*a*, *a* ⊕

• A dioid is *commutative* when ⊗ is commutative. The symbol ⊗ is often omitted. Due to idempotency of ⊕, a dioid can be endowed with a natural order relation defined by

• A dioid D is *complete* if every subset *<sup>A</sup>* of D admits a least upper bound denoted

and if ⊗ distributes at left and at right over infinite sums. The greatest element denoted

*x*∈*X x*.

**Example 1.** The set **Z** ∪ {±∞}, endowed with (*min*) as ⊕ and usual addition as ⊗, is a complete dioid denoted **Z**min and usually called (*min*, +) algebra with neutral elements *ε* =

**Example 2.** The set **Z** ∪ {±∞}, endowed with (*max*) as ⊕ and usual addition as ⊗, is a complete dioid denoted **Z**max and usually called (*max*, +) algebra with neutral elements

. The greatest lower bound of every subset *X* of a

Performance Evaluation of Timed Petri Nets in Dioid Algebra 411

*x*∈*A x*,

characteristic vector.

conservative.

called the *dioid* algebra [1].

+∞, *e* = 0 and *T* = −∞.

*ε* = −∞, *e* = 0 and *T* = +∞.

unit.

**Definition 4.** (Conservative Petri Net)

*S* is the characteristic vector of the firing sequence *S*. In Figure 1, the firing sequence *S* = {*n*2}, the characteristic vector is equal to *S<sup>t</sup>* = (0, 1, 0, 0), and from marking *m<sup>t</sup>* <sup>0</sup> = (0, 3, 0, 0, 3, 0), is reached the marking *m<sup>t</sup>* <sup>1</sup> = (0, 0, 0, 3, 0, 0) by firing of the transition *n*2, after a stay of 2 time units of tokens in the places *P*<sup>2</sup> and *P*5.
