The Annals of Applied Probability

Heavy-traffic limits for waiting times in many-server queues with abandonment

Rishi Talreja and Ward Whitt

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Abstract

We establish heavy-traffic stochastic-process limits for waiting times in many-server queues with customer abandonment. If the system is asymptotically critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a bounding argument shows that the abandonment does not affect waiting-time processes. If instead the system is overloaded, as in the efficiency-driven (ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh Annual Allerton Conference on Communication, Control and Computing (1999) 1095–1104], we treat customer abandonment by studying the limiting behavior of the queueing models with arrivals turned off at some time t. Then, the waiting time of an infinitely patient customer arriving at time t is the additional time it takes for the queue to empty. To prove stochastic-process limits for virtual waiting times, we establish a two-parameter version of Puhalskii’s invariance principle for first passage times. That, in turn, involves proving that two-parameter versions of the composition and inverse mappings appropriately preserve convergence.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 6 (2009), 2137-2175.

Dates
First available in Project Euclid: 25 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1259158769

Digital Object Identifier
doi:10.1214/09-AAP606

Mathematical Reviews number (MathSciNet)
MR2588242

Zentralblatt MATH identifier
1185.60103

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

Keywords
Multiple-server queues many-server heavy-traffic limits for queues diffusion approximations functional central limit theorems waiting times

Citation

Talreja, Rishi; Whitt, Ward. Heavy-traffic limits for waiting times in many-server queues with abandonment. Ann. Appl. Probab. 19 (2009), no. 6, 2137--2175. doi:10.1214/09-AAP606. https://projecteuclid.org/euclid.aoap/1259158769


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