**1. Introduction**

26 Will-be-set-by-IN-TECH

multi-functional switches for the operation of certain discrete dynamical systems of common

The authors deeply acknowledge with thanks the valuable suggestions and thought-provoking comments by Dr. B.D. Acharya from time to time while carrying

*Department of Applied Mathematics, Delhi Technological University, Shahbad Daulatpur, Main*

[1] Acharya, B.D (2001). *Set-Indexers of a Graph and Set-Graceful Graphs*, Bull. Allahabad

[2] Best, E. and Thiagarajan, P.S. (1987). Some Classes of Live and Safe Petri Nets,

[4] Jensen, K. (1986). Coloured Petri nets, Lecture Notes in Computer Science, Vol. 254,

[5] Kansal, S., Singh, G.P. and Acharya, M. (2010). On Petri Nets Generating all the Binary

[6] Kansal, S., Singh, G.P. and Acharya, M. (2011). 1-Safe Petri Nets Generating Every Binary *n*−Vectors Exactly Once, Scientiae Mathematicae Japonicae, 74, No. 1, pp. 29-36. [7] Kansal, S., Singh, G.P. and Acharya, M. (2011). A Disconnected 1-Safe Petri Net Whose Reachability Tree is Homomorphic to a Complete Boolean Lattice, proceeding of PACC-2011, IEEE Xplore, Catalog Number: CFP1166N-PRT ISBN: 978-1-61284-762-7. [8] Kansal, S., Acharya, M. and Singh, G.P. (2012). Uniqueness of Minimal 1-Safe Petri Net Generating Every Binary *n*-Vectors as its Marking Vectors Exactly Once, Scientiae

[9] Kansal, S., Acharya, M. and Singh, G.P. (2012). On the problem of characterizing 1-safe Petri nets that generate all the binary *n*-vectors as their marking vectors, Preprint. [10] Singh, G.P., Kansal, S. and Acharya, M. (2012). Embedding an Arbitrary 1-Safe Petri Net in a Boolean Petri Net, Research Report, Deptartment of Applied Mathematics, Delhi

[11] Nauber, W. (2010). *Methods of Petri Net Analysis.* Ch.5 In: Lectures on Design and Analysis With Petri Nets, <http://www.tcs.inf.tu-dresden.de/nauber/dapn.shtml> [12] Peterson, J.L. (1981). Petri Net Theory and the Modeling of Systems, Prentice-Hall, Inc.,

[13] Petri, C.A. (1962). Kommunikation Mit Automaten, Schriften des Institutes fur

[3] Harary, F. (1969).Graph Theory, Addison-Wesley, Reading, Massachusettes.

n-Vectors, Scientiae Mathematicae Japonicae, 71, No. 2, pp. 209-216.

use such as washing machines and teleprinters (e.g., see [1]).

Sangita Kansal, Mukti Acharya and Gajendra Pratap Singh

Concurrency and Nets, Vol. 25, pp. 71-94.

Mathematicae Japonicae, pp., e-2012, 75-78.

Technological University, Delhi, India.

Instrumentelle Mathematik, Bonn.

[14] Reisig, W. (1985). Petri nets, Springer-Verleg, New York.

Englewood Cliffs, NJ.

Springer-Verlag, Berlin, pp. 248-299.

**Acknowledgement**

**Author details**

**12. References**

out the work reported in this chapter.

*Bawana Road, Delhi-110042, India*

Math. Soc. 16, pp. 1-23.

The theory of Discrete Event Dynamic Systems focuses on the analysis and conduct systems. This class essentially contains man-made systems that consist of a finite number of resources (processors or memories, communication channels, machines) shared by several users (jobs, packets, manufactured objects) which all contribute to the achievement of some common goal (a parallel computation, the end-to-end transmission of a set of packets, the assembly of a product in an automated manufacturing line).

Discrete Event Dynamic Systems can be defined as systems in which state variables change under the occurrence of events. They are usually not be described, like the classical continuous systems, by differential equations due to the nature of the phenomenon involved, including the synchronization phenomenon or mutual exclusion. These systems are often represented by state-transition models. For such systems, arise, among others, three problems: Performance evaluation (estimate the production rate of a manufacturing system), resource optimization (minimizing the cost of some resources in order to achieve a given rate of production). To deal with such problems, it is necessary to benefit of models able to take into account all dynamic characteristics of these systems. However, the phenomena involved by Discrete Event Dynamic Systems, and responsible for their dynamics, are much and of diverse natures: sequential or simultaneous, delayed tasks or not, synchronized or rival. From this variety of phenomena results the incapacity to describe all Discrete Event Dynamic Systems by a unique model which is faithful at once to the reality and exploitable mathematically.

The study of Discrete Event Dynamic Systems is made through several theories among which we can remind for example the queuing theory, for the evaluation of performances of timed systems, or the theory of the languages and the automatons, for the control of other systems. The work presented here is in line with theory of linear systems on dioids. This theory involves subclass of Timed Discrete Event Dynamic Systems where the evolution of the state is representable by linear recurrence equations on special algebraic structures called

©2012 Hamaci et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Hamaci et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 operation requires several units of a same part.

**2. Petri Net**

Petri Net.

**2.1. Definitions and notations**

synchronization and resource sharing.

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

Performance Evaluation of Timed Petri Nets in Dioid Algebra 409

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

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

2. *<sup>M</sup>* <sup>∈</sup> **<sup>N</sup>***P*×*<sup>T</sup>* <sup>∪</sup> *<sup>T</sup>*×*P*. Given *<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>* and *<sup>q</sup>* <sup>∈</sup> *<sup>T</sup>*, the multiplier *Mpq*� (resp. *Mqp*) specifies the weight of the arc from transition *nq*� to place *p* (resp. from place *p* to transition *nq*).

4. *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>P</sup>* : *<sup>τ</sup><sup>p</sup>* gives the minimal time a token must spend in place *<sup>p</sup>* before it can contribute

 

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

 

> 

 

1. *P* is the finite set of places, *T* is the finite set of transitions.

3. *<sup>m</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>P</sup>* : *mp* assigns an initial number of tokens to place *<sup>p</sup>*.

to the enabling of its downstream transitions.

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

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 order to obtain a linear model.

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

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

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 approach for evaluate the performances of TEGM in dioid algebra.
