Computational issues in recursive stochastic systems

Summary

Abstract

Estimation of asymptotic quantities in stochastic recursive systems can be performed by simulation or exact analysis. In this paper, we show how to represent a system in order to make computation procedures more efficient. A first part of this paper is devoted to parallel algorithms for the simulation of linear systems over an arbitrary semiring. Starting from a linear recursive system of order m, we construct an equivalent system of order 1 which minimizes the complexity of the computations. A second part discusses the evaluation of general recursive systems using Markovian techniques.

Introduction

Stochastic recursive systems may be used to model many discrete event systems, such as stochastic event graphs [16, 9, 4], PERT networks, timed automata [10] or min-max systems [15]. Qualitative theorems characterizing the asymptotic behavior of the system have been proved recently [2, 22] but efficient quantitative methods are still to be found. We investigate two approaches to estimate the behavior of recursive systems: parallel simulation and exact Markovian analysis. If we consider a linear recursive system of order m, it is essential for both approaches to provide a standard representation of the system that yields a minimum “cost”. A standard representation is a larger system of order 1 which includes the original one, path-wise. The cost is different according to the technique used.

In the first part, we present two algorithms: a space parallel and a time parallel simulation of linear recursive systems. For both of them, we construct an optimal standard representation. This is done by modifying the marking of the associated reduced graph.