Functional Large Deviations for Cox Processes and $Cox/G/\infty$ Queues, with a Biological Application

TL;DR

This paper establishes a functional large deviations principle for an infinite-server queue driven by a Cox process, motivated by biological gene regulation, providing insights into rare events in such stochastic systems.

Contribution

It introduces a novel large deviations analysis for Cox process-driven queues with biological applications, extending theoretical understanding of rare events in complex stochastic networks.

Findings

Proves a large deviations principle for the equilibrium queue length process.

Models gene regulatory networks as tandem infinite-server queues with modulated arrival rates.

Provides asymptotic estimates for rare fluctuations in biological queueing systems.

Abstract

We consider an infinite-server queue into which customers arrive according to a Cox process and have independent service times with a general distribution. We prove a functional large deviations principle for the equilibrium queue length process. The model is motivated by a linear feed-forward gene regulatory network, in which the rate of protein synthesis is modulated by the number of RNA molecules present in a cell. The system can be modelled as a tandem of infinite-server queues, in which the number of customers present in a queue modulates the arrival rate into the next queue in the tandem. We establish large deviation principles for this queueing system in the asymptotic regime in which the arrival process is sped up, while the service process is not scaled.