Year: 2007
Type: article
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.
Source: Probability Surveys
DOI: 10.1214/06-ps091
Authors: Guodong Pang, Rishi Talreja, Ward Whitt
Institution: Columbia University
Language: English
Subfield: Management Information Systems
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $675
Funder: National Science Foundation
OpenAlex ID: W1629501955