**11. Concluding remarks**

As pointed out, many fundamental issues regarding Boolean Petri nets emerge from the above study. For example, it is found and established that the reachability tree of a 1-safe Petri net can be homomorphically mapped on to the *n*-dimensional complete Boolean lattice, thereby yielding new techniques to represent the dynamics of these Petri nets. One can expect to bring out in the near future some salient features of 1-safe Petri nets in general as a part of a theory that is likely to emerge even in our work.

Following our first discovery of an infinite class of 1-safe star Petri nets that are Boolean, we came across crisp Boolean Petri nets, viz., that generate every binary *n*-vector as marking vector exactly once. This motivated us to move towards a characterization of such 1-safe Petri nets in general. Our work towards this end revealed to our surprise that there can be even such disconnected 1-safe Petri nets. We demonstrated the existence of a disconnected 1-safe Petri net which was obtained by removing the central transition from the star Petri net *Sn*, whose reachability tree can be tactically represented as an *n*-dimensional complete lattice *Ln* [7]. In this disconnected Petri net, the firing of transitions in a particular way (which we may regard as a 'tact' or 'strategy'), gives exactly 2*<sup>n</sup>* marking vectors, repetitions occurring possibly only within a level, that can be arranged as a homomorphic image of the reachability tree of the Petri net, forming the *n*-dimensional complete Boolean lattice *Ln*.

The results of this chapter can perhaps be used gainfully in many purely theoretical areas like mathematics, computer science, universal algebra and order theory, the extent and effectiveness of its utility in solving the practical problem requiring the design of multi-functional switches for the operation of certain discrete dynamical systems of common use such as washing machines and teleprinters (e.g., see [1]).

**Performance Evaluation of Timed**

Samir Hamaci, Karim Labadi and A.Moumen Darcherif

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

**Chapter 18**

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

©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.

Additional information is available at the end of the chapter

product in an automated manufacturing line).

cited.

http://dx.doi.org/10.5772/48498

**1. Introduction**

**Petri Nets in Dioid Algebra**
