The idle period of the finite \(G/M/1\) queue with an interpretation in risk theory (Q967287)

The paper deals with the \(G/M/1\) queue where the workload (virtual waiting time) \(V_t\) is bounded by \(1\): If \(V_t\) plus the service time of an arriving customer exceeds \(1\), only \(1-V_t\) of the service requirement is accepted. The paper focus on the idle period \(I\), i.e. the duration between the time when the queue becomes empty and the next customer arrives, of this finite queue, because it can be interpreted as the deficit at ruin for a risk reserve process \(R_t\) in the compound Poisson risk model with a constant behavior strategy: when the risk reserve process reaches level \(1\) dividends are paid out with constant rate \(1\), such that \(R_t\) is constant until the next claim occurs. In the paper expressions for the LST and the distribution of \(I\) are given. Two methods are used, which base on sample path analysis: the first method bases on collecting subsequent overshoots over level \(1\) in the original \(G/M/1\) workload process and the second method on the observation that the idle period can be seen as the overshoot of the workload process in a dual \(M/G/1\) queue. The second method is used to construct a modified \(M/G/1\) process and to define a formula for the distribution of the idle period in the finite \(G/M/1\) queue with set-up time \(a\in[0,1]\), where after each busy period an arriving customer has to wait \(a\) time units until the server is ready to serve it.