A Methodology for Combinining GSPNs and QNs

Generalized Stochastic Petri Nets (GSPNs) are powerful mechanisms to model systems that exhibit instances of parallelism, synchronization, blocking, and simultaneous resource possession. For large systems, however, the solution of GSPNs is computationally very expensive due to the combinatorial growth of the state space. Queuing networks (QNs) provide very efficient solutions for the cases were parallelism, synchronization, blocking, and simultaneous resource possession are not present. This paper presents a methodology by which large GSPNs can be efficiently solved by automatically detecting subnetworks that are equivalent to product-form queuing networks (PFQNs). These subnetworks are replaced in the original GSPN by Flow-Equivalent Service Centers (FESCs). Each FESC is composed of a place/transition pair with marking dependent transition rates. These rates are obtained by solving the QN that corresponds to the subnetwork. The reduced GSPN obtained this way has fewer states than the original GSPN. Performance metrics derived from the reduced GSPN match those obtained from the original network with significant accuracy. This paper presents an algorithm to detect the subnetworks that are equivalent to a PFQN and an algorithm to obtain the service demands of the equivalent PFQN. The computational complexity of the algorithms is a function of the structural properties (number of places, transitions, and arcs) of the GSPN and not of the size of its reachability set. A theorem and two lemmas that serve as a basis for the algorithms are also presented.