On the use of Kronecker operators for the solution of generalized stochastic Petri nets

30 the CTMCs may have more than 10 30 states. A prototype version of the program used to compute the results we presented is available and can be obtained by contacting the second author. The input to the program has a SPNP-style syntax 9]. The eeect of execution policies on the semantics and analyis of Stochastic Petri Nets. IEEE Trans. 29 immediate synchronizing transitions. Preservation of all vanishing markings (Theorem 5.1) is probably never appropriate, since memory usage is the paramount consideration, and this approach increases the already critical size of the probability vectors. In few pathological cases, it could reduce the size of the \local" data structures, but these are not critical. 7 Conclusion We rigorously formalized and implemented an approach based on Kronecker algebra for the solution of the CTMC underlying a GSPN. The results extend previous works by Donatelli, Buchholz, and Kem-per 4, 5, 10, 11, 12], to include immediate synchronizing transitions, quite general marking-dependent behavior, and a reward structure allowing reward impulses associated with immediate transition rings. The restrictions imposed on the GSPN are minimal, thus the approach has obvious practical applications. Furthermore, the structure of the GSPN itself gives strong hints on its decomposition. For example, if an \elimination-based" solution is desired, the GSPN must be decomposed so that immediate transitions are local to a sub-GSPN; if the marking-dependency of a transition does not satisfy our requirements, all the places responsible for this behavior should also be merged in the same sub-GSPN. Memory requirements are still the main limitation to the solution, but these have now been reduced from the size of the transition rate matrix to that of the steady-state probability vector, for a very general class of GSPNs. Even for highly sparse matrices, this corresponds to the ability to solve problems whose state space is one order of magnitude larger than with a traditional solution approach. To further increase the size of models that can be solved, we can use the distributed state-space generation algorithm described in 7], which allows one to partition the memory and execution requirements to generate the state space over a set of workstations, and which exhibits excellent speedups for large problems. The Jacobi method we employed can also be parallelized using a set of workstations, so that the entire solution process is performed in a distributed fashion and uses the available memory. In particular, the \local" matrices occupy …