Diffusion Models for Double-ended Queues with Renewal Arrival Processes

TL;DR

This paper models a double-ended queue with renewal arrivals, establishing fluid and diffusion approximations, analyzing heavy traffic limits, and deriving limiting distributions for the queue length process.

Contribution

It introduces a novel diffusion approximation framework for double-ended queues with renewal arrivals and analyzes heavy traffic regimes.

Findings

Diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process.

Steady state and heavy traffic limits interchangeability is established.

Explicit limiting distributions are derived for the queue length process.

Abstract

We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits.