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.