Two-parameter Sample Path Large Deviations for Infinite Server Queues

TL;DR

This paper establishes a large deviations principle for the scaled process of an infinite server queue with renewal arrivals and continuous service times, analyzing rare events like ruin and overflow through surface paths.

Contribution

It introduces a two-parameter large deviations principle for infinite server queues with renewal arrivals, extending the understanding of rare event paths in such systems.

Findings

Derived the large deviations principle for the queue process.

Identified most likely paths to ruin and overflow as surfaces.

Applied results to insurance and loss queue scenarios.

Abstract

Let $Q_{\lambda}(t,y) $ be the number of people present at time $t$ with $y$ units of remaining service time in an infinite server system with arrival rate equal to $\lambda>0$. In the presence of a non-lattice renewal arrival process and assuming that the service times have a continuous distribution, we obtain a large deviations principle for $Q_{\lambda}(\cdot) /\lambda$ under the topology of uniform convergence on $[0,T]\times\lbrack0,\infty)$. We illustrate our results by obtaining the most likely path, represented as a surface, to ruin in life insurance portfolios, and also we obtain the most likely surfaces to overflow in the setting of loss queues.