Martingale proofs of many-server heavy-traffic limits for Markovian queues

TL;DR

This paper reviews the martingale method for proving heavy-traffic limits in many-server queueing models, illustrating techniques with classical and more complex models, and establishing key functional limit theorems.

Contribution

It provides an expository overview of martingale-based proofs for diffusion limits in Markovian queues, including models with finite servers and customer abandonment.

Findings

Martingale representations are constructed via random time changes and thinnings.

Convergence to diffusion limits is proven using the continuous mapping theorem.

Key functional CLT and FWLLN are established with and without martingales.

Abstract

This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.