Memory Efficient Calculation of Path Probabilities in Large Structured Markov Chains

The problem we deal with is the analysis of a class of large structured Markov chains. In particular we assume that the whole state space can be partitioned into disjoint sets (called macro states) in which the process corresponds to the parallel execution of independent jobs. Petri nets and process algebras with phase type (PH) distributed execution times give rise to this kind of model. These models are subject to the phenomenon of state space explosion. It is known that the infinitesimal generator of such models can be handled in a memory efficient way by storing only the "structure \´\´ of the infinitesimal generator as Kronecker expressions or decision diagrams. Less is known instead on how to perform the analysis of the model in a memory efficient manner because in case of most of the available methods the vector of transient or steady state probabilities are stored in an explicit manner. In this paper we consider the calculation of measures connected to the probability that the process passes through a given series of macro states.We show that such measures can be calculated in a memory efficient manner by Laplace transform techniques. The method is illustrated by numerical examples.