Probability Surveys

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

Guodong Pang, Rishi Talreja, and Ward Whitt

Full-text: Open access


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.

Article information

Probab. Surveys, Volume 4 (2007), 193-267.

First available in Project Euclid: 16 January 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22]

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


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

Export citation