Stationary Distribution Convergence of the Offered Waiting Processes in Heavy Traffic under General Patience Time Scaling

TL;DR

This paper proves that the stationary distributions and moments of offered waiting times in heavy traffic queues with customer abandonment converge to those of a limiting diffusion process, validating diffusion approximations for stationary performance.

Contribution

It establishes the convergence of stationary distributions and moments for GI/GI/1+GI queues under general patience time scaling in heavy traffic, extending previous weak convergence results.

Findings

Stationary distributions converge to the diffusion limit.

Moments of offered waiting times converge to those of the diffusion.

Derived approximation for abandonment probability in stationary state.

Abstract

We study a sequence of single server queues with customer abandonment (GI/GI/1+GI) under heavy traffic. The patience time distributions vary with the sequence, which allows for a wider scope of applications. It is known ([20, 18]) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with non-linear drift, as the traffic intensity approaches one. In this paper, we further show that the sequence of stationary distributions and moments of the offered waiting times, with diffusion scaling, converge to those of the limit diffusion process. This justifies the stationary performance of the diffusion limit as a valid approximation for the stationary performance of the GI/GI/1+GI queue. Consequently, we also derive the approximation for the abandonment probability for the GI/GI/1+GI queue in the stationary state.