The Annals of Applied Probability

Asymptotic approximations for stationary distributions of many-server queues with abandonment

Weining Kang and Kavita Ramanan

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A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with N servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as N → ∞, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the N → ∞ and t → ∞ limits cannot always be interchanged.

Article information

Ann. Appl. Probab., Volume 22, Number 2 (2012), 477-521.

First available in Project Euclid: 2 April 2012

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 60F99: None of the above, but in this section

Multi-server queues stationary distribution ergodicity measure-valued processes abandonment reneging interchange of limits mean-field limits call centers


Kang, Weining; Ramanan, Kavita. Asymptotic approximations for stationary distributions of many-server queues with abandonment. Ann. Appl. Probab. 22 (2012), no. 2, 477--521. doi:10.1214/10-AAP738.

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