The idle period of the finite G/M/1 queue with an interpretation in risk theory

We consider a G/M/1 queue with restricted accessibility in the sense that the maximal workload is bounded by 1. If the current workload Vt of the queue plus the service time of an arriving customer exceeds 1, only 1 − Vt of the service requirement is accepted. We are interested in the distribution of the idle period, which can be interpreted as the deficit at ruin for a risk reserve process Rt in the compound Poisson risk model. For this risk process a special dividend strategy applies, where the insurance company pays out all the income whenever Rt reaches level 1. In the queueing context we further introduce a set-up time a ∈ [0,1]. At the end of every idle period, an arriving customer has to wait for a time units until the server is ready to serve it.