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


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.


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Guodong Pang. Rishi Talreja. Ward Whitt. "Martingale proofs of many-server heavy-traffic limits for Markovian queues." Probab. Surveys 4 193 - 267, 2007.


Published: 2007
First available in Project Euclid: 16 January 2008

zbMATH: 1189.60067
MathSciNet: MR2368951
Digital Object Identifier: 10.1214/06-PS091

Primary: 60F17 , 60K25

Keywords: diffusion approximations , functional central limit theorems , many-server heavy-traffic limits for queues , Martingales , Multiple-server queues

Rights: Copyright © 2007 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.4 • 2007
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