This paper presents a new approach for the computation of transient measures in large continuous time Markov chains (CTMCs). The approach combines the randomization approach for transient analysis of CTMCs with a new representation of probability vectors as Kronecker products of small component vectors. This representation is an approximation that allows an extremely space- and time-efficient computation of transient vectors. Usually, the resulting approximation is very good and introduces errors that are comparable to those found with existing approximation techniques for stationary analysis. By increasing the space and time requirements of the approach, we can represent parts of the solution vector in detail and reduce the approximation error, yielding exact solutions in the limiting case.

This paper addresses steady state solution of discrete time stochastic models in which every activity duration is given either by a geometric or a finite support distribution. Finite support distributions can be described by discrete time phase type (DPH) distributions. The behaviour of the whole stochastic model is given by a discrete time Markov chain (DTMC). The DTMC is subject to the so-called state space explosion. We present a technique for obtaining the steady state solution that alleviates this problem. The technique is based on Gaussian elimination combined with an iterative technique. 1

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 macrostates. We show that such measures can be calculated in a memory efficient manner by Laplace transform techniques. The method is illustrated by numerical examples.