Hierarchical structuring of superposed GSPNs

Superposed generalized stochastic Petri nets (SGSPNs) and stochastic automata networks (SANs) are formalisms to describe Markovian models as a collection of synchronously communicating components. Both formalisms allow a compact representation of the generator matrix of the Markov chain, which can be exploited for very space efficient analysis techniques. The main drawback of the approaches is that for many models the compositional description introduces a large number of unreachable states, such that the gain due to the compact representation of the generator matrix is completely lost. This paper proposes a new approach to avoid unreachable states without losing the possibility to represent the generator matrix in a compact form. The central idea is to introduce a preprocessing step to generate a hierarchical structure which defines a block structure of the generator matrix, where every block can be represented in a compact form similar to the representation of generator matrices originally proposed for SGSPNs or SANs. The resulting structure includes no unreachable states, needs only slightly more space than the compact representation developed for SANs and can still be exploited in efficient numerical solution techniques. Furthermore, the approach is a very efficient method to generate and represent huge reachability sets and graphs