  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.
