Diffusion Approximations for Double-ended Queues with Reneging in Heavy Traffic

TL;DR

This paper analyzes a double-ended queue with customer reneging under heavy traffic, establishing diffusion approximations, convergence results, and stationary distribution for the queue length process.

Contribution

It introduces a diffusion approximation framework for double-ended queues with reneging, linking queue length and waiting time processes under heavy traffic.

Findings

Diffusion-scaled queue length converges to a diffusion process.

Explicit stationary distribution of the limit diffusion process derived.

Provides sufficient conditions for stochastic boundedness of the queue length process.

Abstract

We study a double-ended queue which consists of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system immediately. The matching follows first-come-first-serve principle. If a customer from one class cannot be matched immediately, he/she will stay in a queue and wait for the upcoming arrivals from the other class. Thus there cannot be non-zero numbers of customers from both classes simultaneously in the system. We also assume that each customer can leave the queue without being matched because of impatience. The arrival processes are assumed to be independent renewal processes, and the patience times for both classes are generally distributed. Under suitable heavy traffic conditions, assuming that the diffusion-scaled queue length process is stochastically bounded, we establish a simple asymptotic relationship between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time process, and further show that the diffusion-scaled queue length process converges weakly to a diffusion process. We also provide a sufficient condition for the stochastic boundedness of the diffusion-scaled queue length process. At last, the explicit form of the stationary distribution of the limit diffusion process is derived.