Cost optimization of queueing systems with interruptions

We consider a queueing system $M|G|1$ with possible vacations in server operations for principal customers (for example, if a server is leased). A cost optimization problem is solved. As control parameters, we use the probability $\alpha$ of the vacation and its duration. Under fairly general assumptions about the system behavior during vacations, we show that the optimal value of the probability $\alpha$ is either 0 or 1. We also give necessary and sufficient conditions for a vacation to be carried out, i.e., $\alpha=1$. With constant vacation durations, we find conditions such that $\alpha=1$, and the vacation duration is optimal. Two examples are considered. In the first example, the revenue from the vacation is a linear function of its duration, and, in the second example, the revenue is a quadratic function.