Two-parameter sample path large deviations for infinite-server queues

Abstract

Let $Q_{\lambda}(t,y)$ be the number of people present at time $t$ with at least $y$ 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 $[0,T]\times[0,\infty)$. We illustrate our results by obtaining the most likely paths, represented as surfaces, to overflow in the setting of loss queues, and also to ruin in life insurance portfolios.