Sample Path Moderate Deviation Principle for Queues with Waiting-time Dependent Interarrival and Service Times

TL;DR

This paper establishes a sample-path moderate deviation principle for a queue with waiting-time dependent interarrival and service times, analyzing the queue's behavior under scaling and characterizing the rate functions explicitly.

Contribution

It introduces a novel MDP framework for queues with waiting-time dependent dynamics, including explicit rate functions and analysis of equilibrium points.

Findings

MDP for waiting time process with explicit rate functions

Analysis of stable equilibrium points, zero or positive

Representation of scaled process via linear SDE with reflection

Abstract

We consider a single-server queue where interarrival and service times depend linearly and randomly on customer waiting times, and establish a sample-path moderate deviation principle (MDP) for the waiting time process. The waiting times for the queue can be written as a modified Lindley recursion with a random weight coefficient. Under a natural scaling of the random coefficients, we analyze the fluid behavior of the workload process and derive the stable equilibrium point, which can be zero or a positive value. The moderate-deviation-scaled process is centered around the stable equilibrium point and then represented as a linear stochastic differential equation driven by two random walks together with additional asymptotically negligible error terms and possibly a reflection at zero. The rate functions of MDPs in the two scenarios can be characterized explicitly, and they differ in that the case with zero centering term involves the linearly generalized Skorokhod reflection mapping while the case with positive centering term does not (similar to the corresponding diffusion limits). Our analysis involves the MDP for the associated linearly recursive Markov chains, invoking a perturbation of two independent random walks, and employing martingale techniques to prove the asymptotically exponentially vanishing error terms.