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

By Guodong Pang, Rishi Talreja, Ward Whitt. 2007

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This is an expository review paper illustrating the "martin- gale method" for proving many-server heavy-traffic stochastic-process lim- its for queueing models, supporting diffusion-process approximations. Care- ful treatment is given to an elementary model - the classical infinite-server model M/M/1, but models with finitely many servers and customer aban- donment 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 stop- ping 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 se- quence 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.

Subfields: Management Information Systems, Management Science and Operations Research, Safety, Risk, Reliability and Quality

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In Management Information Systems

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Citations in the corpus

On many-server queues in heavy traffic (2010, background): “This was first accomplished in the work of Halfin and Whitt [10] for the case of exponential service time distributions and has been continued by many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mande”
Proofs of the martingale FCLT (2007, methodology): “Before applying Theorem VI.3.37 of JS, we note that we have convergence of the finite-dimensional distributions, because we can apply Theorem 11.4.5 of Whitt [22].”
Law of large numbers limits for many-server queues (2007, background): “[17] for a recent survey on the use of martingale methods for establishing heavy-traffic limits of multiserver queues with exponentially distributed service times).”
Heavy-traffic limits for waiting times in many-server queues with abandonment (2008, background): “1 using the technique described in [2].”