Two-parameter Sample Path Large Deviations for Infinite Server Queues

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 $Q_{\lambda }(\cdot )\mathrm{/}\lambda$ under the topology of uniform convergence on $[0,T]\times [0,\mathrm{\infty })$. 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.