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/∞, 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.
Highlights
The purpose of this paper is to illustrate how to do martingale proofs of many-server heavy-traffic limit theorems for queueing models, as in Krichagina and Puhalskii [37] and Puhalskii and Reiman [53]
For the more elementary models, we show how key steps in the proof - a functional central limit theorem (FCLT) ((59)) and a functional weak law of large numbers (FWLLN) (Lemma 4.3) - can be done without martingales as well as with martingales
The martingale structure can be used to establish stochastic boundedness of the scaled queueing processes, which we show implies required fluid limits or functional weak laws of large numbers (FWLLN’s) for random-time-change stochastic processes, needed for an application of the continuous mapping theorem with the composition map
Summary
The purpose of this paper is to illustrate how to do martingale proofs of many-server heavy-traffic limit theorems for queueing models, as in Krichagina and Puhalskii [37] and Puhalskii and Reiman [53]. For the more elementary Markovian models considered here, these many-server heavy-traffic limits were originally established by other methods, but new methods of proof have been developed. We want to do more than achieve a quick proof for the simple models, which need not rely so much on martingales; we want to illustrate martingale methods that may prove useful for more complicated models
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