On the Transient Behavior of Ehrenfest and Engset Processes

TL;DR

This paper analyzes the asymptotic distribution of hitting times for the Ehrenfest and Engset processes, using martingale techniques to understand their boundary behaviors as system size grows large.

Contribution

It introduces a novel martingale approach to study the asymptotic hitting time distributions of these classical stochastic processes.

Findings

Asymptotic distributions of hitting times are characterized.

Martingale methods provide new insights into boundary behaviors.

Results apply to large-scale systems with many particles or sources.

Abstract

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies and the Engset process, one of the early (1918) stochastic models of communication networks. This paper investigates the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. The paper relies on martingale methods, a key ingredient is an important family of simple non-negative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.