M–timed Petri Nets, Priorities, Preemptions, and Performance Evaluation of Systems

In M–timed Petri nets, firing times are exponentially distributed random variables associated with transitions of a net. Several classes of M–timed Petri nets are discussed in this paper to show increasing “modelling power” of different nets. Conflict–free nets can model M and Ek–type queueing systems. Free—choice nets can also represent Hk–type systems. Systems with several classes of users and with service priorities assigned to user classes require nets with inhibitor arcs. Preemption of service can be represented by extended nets with escape (or generalized inhibitor) arcs. Finally, to provide flexible modelling of scheduling and decision strategies, enhanced Petri nets are introduced with two classes of transitions, immediate and timed ones, and with (exponentially distributed) firing times associated with the timed transitions. It is shown that the behaviour of bounded M–timed Petri nets can be represented by finite “state” graphs which are finite-state continuous-time homogeneous Markov processes. Stationary probabilities of states can thus be obtained by standard techniques used for analysis of Markov chains, and then operational analysis can be applied for performance evaluation. Simple models of interactive systems are used as an illustration of modelling.