State equations of stochastic timed petri nets with informational relations
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STATE EQUATIONS OF STOCHASTIC TIMED PETRI NETS WITH INFORMATIONAL RELATIONS UDC 519.876.5
I. V. Stetsenko
Abstract. The use of timed Petri nets for simulation modeling is analyzed. An equation for transforming states of timed Petri nets and fundamental matrix equations of a stochastic timed Petri net with conflicting and multichannel transitions and informational relations are obtained. Keywords: timed Petri net, simulation modeling, matrix equation. INTRODUCTION Petri nets are universal means for formalizing event-driven processes and are intensively used in various fields. Elements of a Petri net that allow one to describe processes of functioning of complex systems are stochastic time-delayed transitions [1], conflicting and multichannel transitions [2, 3], and informational relations. The latter relations are introduced in [4] for the representation of specific control processes and were used in [5–7] for simulation modeling. Informational relations, as well as multichannel transitions, do not extend the class of Petri nets but make them more convenient for the representation of complex systems. In [8], it is shown that inhibitor Petri nets are equinumerous with priority Petri nets and is proved that both forms of nets are equivalent to a Turing machine. The construction of a Petri net equivalent to a Turing machine is considered in [9]. Timed Petri nets afford the opportunity of simultaneous investigation of both structural and functional properties of a system and its operational characteristics [10]. Stochastic Petri nets are more powerful than Markov processes [11] and, at the same time, their use for simulation modeling is restricted, which is confirmed by the fact that the well-known monograph [12] does not mention Petri nets. The derivation of equations of a stochastic timed Petri net with conflicting and multichannel transitions with informational relations is based on the results of [13, 14]. In [13], fundamental equations of a basic Petri net are described and methods of analytical investigation of its properties are proposed. In [14], the concepts of a state of a timed Petri net and the number of possible transition firings are introduced and a state equation is found for a timed deterministic Petri net with incrementing its simulation time. The main destination of equations obtained in this article is the mathematical description of the rules of functioning of a timed Petri net with a view to constructing (from this description) an algorithm for simulating its model. These equations are represented in matrix form, and it is shown that, in the case of the zero time delay, they are similar to equations from [13], and, in the case of deterministic time delays, they are similar to equations from [14]. CONTROL PROCESSES AND TIMED PETRI NETS WITH INFORMATIONAL RELATIONS Informational relations between a place and a transition make it possible to model relations between control subsystems and a control object that are characterized by transmitting information on a state. For example, Fig. 1 presents a Pe
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