Analysis of the Fluid Approximation of Stochastic Process Algebra Models

Abstract

In order to avoid the state space explosion problem encountered in the quantitative analysis of large scale PEPA models, a fluid approximation approach has recently been proposed, which results in a set of ordinary differential equations (ODEs) to approximate the underlying continuous time Markov chain (CTMC). This paper presents a mapping semantics from PEPA to ODEs based on a numerical representation scheme, which extends the class of PEPA models that can be subjected to fluid approximation. Furthermore, we have established the fundamental characteristics of the derived ODEs, such as the existence, uniqueness, boundedness and nonnegativeness of the solution. The convergence of the solution as time tends to infinity for several classes of PEPA models, has been proved under some mild conditions. For general PEPA models, the convergence is proved under a particular condition, which has been revealed to relate to some famous constants of Markov chains such as the spectral gap and the Log-Sobolev constant. This thesis has established the consistency between the fluid approximation and the underlying CTMCs for PEPA, i.e. the limit of the solution is consistent with the equilibrium probability distribution corresponding to a family of underlying density dependent CTMCs.