A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation

TL;DR

This paper models a system with catastrophes and repairs as a continuous-time random walk, deriving its probability laws and a jump-diffusion approximation to analyze its behavior under heavy traffic conditions.

Contribution

It introduces a novel jump-diffusion approximation for a system with catastrophes and repairs, providing insights into its transient and steady-state behaviors.

Findings

Derived explicit probability laws for the system.

Established a jump-diffusion approximation for heavy traffic.

Discussed the accuracy of the approximation.

Abstract

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.