Mathematics – Probability
Scientific paper
2007-12-27
Probability Surveys 2007, Vol. 4, 193-267
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/06-PS091 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Scientific paper
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 $M/M/\infty$, 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.
Pang Guodong
Talreja Rishi
Whitt Ward
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