Stochastic process reduction for performance evaluation in dtsiPBC

Petri box calculus (PBC) is a well-known algebra of concurrent processes with a Petri net semantics. In the paper, we consider an extension of PBC with discrete stochastic time and immediate multiactions, which is referred to as discrete time stochastic and immediate PBC (dtsiPBC). Performance analysis methods for concurrent and distributed systems with random time delays are investigated in the framework of the new stochastic process algebra. It is demonstrated that the performance evaluation is possible not only via the underlying semi-Markov chains of the algebraic process expressions but also by exploring the reduced discrete time Markov chains, obtained from the semi-Markov chains by eliminating their states with zero residence time (called vanishing states). The latter approach simplifies performance analysis of large systems due to abstraction from many instantaneous activities, such as those used to specify logical conditions, probabilistic branching, as well as urgent or internal (unobservable) work.