A Numerical Algorithm for the Decomposition of Cooperating Structured Markov Processes

Abstract

Modern computer systems consist of a large number of dynamic hardware and software components that interact according to some specific rules. Quantitative models of such systems are important for performance engineering because they allow for an earlier prediction of the quality of service. The application of stochastic modelling for this purpose is limited by the problem of the explosion of the state space of the model, i.e. the number of states that should be considered for an exact analysis increases exponentially and is thus huge even when few components are considered. In this paper we resort to product-form theory to deal with this problem. We define an iterative algorithm with the following characteristics: a) it deals with models with infinite state space and block regular structure (e.g. quasi-birth&death) without the need of truncation; b) in case of detections of product-form according to RCAT conditions, it computes the exact solution of the model; c) in case of non-product-form, it computes an approximate solution. The very loose assumptions allow us to provide examples of analysis of heterogeneous product-form models (e.g., consisting of queues with catastrophes and/or batch removals) as well as approximating non-product-form models with non-exponential service time distributions and negative customers.