Two-parameter Sample Path Large Deviations for Infinite-Server Queues

Let Q_λ(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 λ > 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_λ(·)/λ under the topology of uniform convergence on [0, T] × [0, ∞). 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.