On Parametric Steady State Analysis of a Generalized Stochastic Petri Net with a Fork-Join Subnet

Abstract

The performance analysis of parallel systems which are synchronised is an important but challenging task. This is because product form solutions are not available when synchronisation is required. The system of interest is a controlled fork-join network, comprising two parallel processes. Each process is governed by an exponentially distributed delay. The system has a capacity, N, which cannot be exceeded. Thus only N requests for service can be handled concurrently, so that further requests are blocked. The arrival of requests is also governed by an exponential distribution. We model this class of system with a Generalized Stochastic Petri Net (GSPN) which includes a two branch fork-join structure controlled by an environment that enforces the capacity constraint. This GSPN is thus parameterised by the capacity, N. We derive the parametric reduced reachability graph for arbitrary N, and show that it has (N + 1)² markings and one strongly connected component. We also prove that the GSPN is bounded by N, and thus show that one process cannot lead the other by more than N. We obtain the associated family of continuous time Markov chains, and derive the family of global balance equations for arbitrary N. We solve these equations for the steady state probabilities for N = 1 and 2, and then present a theorem for the general form of the solution for arbitrary N > 2 in terms of ratios of polynomials in the transition rates. A scheme of 21 relationships between these polynomials is also obtained. Finally we explore some asymptotic behaviour of the steady state probabilities and relate them to previously obtained closed form approximate solutions.

Supported by Australian Research Council Discovery Project Grant DP0880928.