It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lorden's inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

In this study we assume an inventory system for perishable commodities in which the lifetimes of the items stored are i.i.d. random variable with finite mean.

We utilize the analogy between this inventory system and a queueing system with impatient customers, to study the process of the lost demand, the death of the unused items and the number of items in the system.

This paper presents a simple and computationally tractable method which recursively computes the stationary probabilities of the queue size in an M/G/1 queueing system with variable service rate. For each service two possible service types are available and the service rule is characterized by two switch-over levels. The computational approach discussed in this paper can be applied to a variety of queueing problems.