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

TL;DR

This paper analyzes the long-term behavior of many-server queues with customer abandonment, establishing conditions for the existence of stationary distributions and their convergence to fluid model limits.

Contribution

It introduces a rigorous framework for the stationary distributions of many-server queues with abandonment and proves their convergence to fluid limits under general assumptions.

Findings

The system's state process is a Feller process and admits a stationary distribution.

Scaled stationary distributions are tight and converge to an invariant state of the fluid limit.

Interchanging limits of large number of servers and time may not always be valid.

Abstract

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 $N\rightarrow \infty$, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the $N\rightarrow \infty$ and $t\rightarrow \infty$ limits cannot always be interchanged.