Numerical analysis of rational processes beyond Markov chains

Abstract

Matrix exponential distributions and rational arrival processes have been proposed as an extension to pure Markov models. The paper presents an approach where these process types are used to describe the timing behavior in quantitative models like queueing networks, stochastic Petri nets or stochastic automata networks. The resulting stochastic process, which is called a rational process, is defined and it is shown that the matrix governing the behavior of the process has a structured representation which allows one to represent the matrix in a very compact form.The main emphasis of the paper is on numerical techniques to compute the stationary distribution. It is shown which techniques from Markov chain analysis may also be applied for rational processes. Aggregation steps can be used to build an aggregated process which is a Markov chain and allows one to compute certain marginal probabilities for the detailed rational process. Furthermore, a block iteration method is presented which exploits the matrix structure and is an extension of previously published techniques for Markov chains.Finally, we present some examples showing that the extension of Markov models results in models which may have a significantly smaller state space than equivalently or similar behaving Markov chains.