A vacation queue $\mathrm{M}|\mathrm{G}|1$ with close-down times

We consider a single-channel system with service vacations, a Poisson input flow, and an arbitrarily distributed service time. Interruptions in service can mean either a complete shutdown of the server for a random period of time or a transition to a different (nonstandard) regime—they can occur either at the end of busy periods when the system is operating in the standard regime, or at the end of vacations at which the system contains no customers. We assume that there is a random timeout before a possible vacation and that the vacation occurs at the end of this timeout if no customers were received by the system during the timeout. Otherwise, the vacation is canceled and the system resumes standard operations. We consider three regimes with different conditions regarding the presence of timeouts and the rules for resuming the standard regime. Under fairly general assumptions concerning distributions of timeout times, we obtain durations of vacations, and processes describing the performance of the system during interruptions, formulas for the distribution, and expectation of the number of customers in the system in the stationary regime. Corresponding examples are given. For a number of special cases our results coincide with those available in the literature.