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 Probability Surveys > Vol. 4 (2007) open journal systems 


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

Guodong Pang, Columbia University
Rishi Talreja, Columbia University
Ward Whitt, Columbia University


Abstract
This is an expository review paper illustrating the “martin- gale 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.

AMS 2000 subject classifications: Primary 60F17, 60K25.

Keywords: multiple-server queues,many-server heavy-traffic limits for queues, diffusion approximations, martingales, functional central limit theorems.

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Pang, Guodong, Talreja, Rishi, Whitt, Ward, Martingale proofs of many-server heavy-traffic limits for Markovian queues, Probability Surveys, 4, (2007), 193-267 (electronic). DOI: 10.1214/06-PS091.

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