Efficient solution of GSPNs using canonical matrix diagrams

Abstract

The solution of a generalized stochastic Petri net (GSPN) is severely restricted by the size of its underlying continuous-time Markov chain. In recent work (G. Ciardo and A.S. Miner, 1999), matrix diagrams built from a Kronecker expression for the transition rate matrix of certain types of GSPNs were shown to allow for more efficient solution; however, the GSPN model requires a special form, so that the transition rate matrix has a Kronecker expression. In this paper, we extend the earlier results to GSPN models with partitioned sets of places. Specifically, we give a more restrictive definition for matrix diagrams and show that the new form is canonical. We then present an algorithm that builds a canonical matrix diagram representation for an arbitrary non-negative matrix, given encodings for the sets of rows and columns. Using this algorithm, a Kronecker expression is not required to construct the matrix diagram. The efficient matrix diagram algorithms for numerical solution presented earlier are still applicable. We apply our technique to several example GSPNs.