Asymptotic Approximations for Stationary Distributions of Many-server Queues with Abandonment

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 Y (N) of the system with N servers is represented by a four-component process that consists of the backward 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 first shown that Y (N) is a Feller process, admits a stationary distribution and is ergodic. The main result shows that when the associated fluid limit has a unique invariant state then any sequence {Y /N}N∈N of stationary distributions of the scaled processes converges, as N → ∞, to this 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. The stationary behavior of many-server systems is of interest for performance analysis of computer data systems and call centers.