Queueing systems with vacations, interruptions, and delays in device operation

Single-channel systems where service vacation is triggered when the system is free from customers are studied. Vacations in service can mean either a complete shutdown of the server or a transition to a different regime. If, during a vacation, the number of customers in the system reaches a certain fixed threshold, the vacation terminates and the unit resumes standard operations. A delay begins if the system is free from customers after a vacation; the delay interrupts at the moment of arrival of the first customer. Then a new vacation begins if no customers have arrived. The input of the queue outside vacations is a Poisson flow, and the remaining variables, which control the performance of the system (the service time, the durations of vacations and delays) have arbitrary distributions. A stationary distribution for the number of customers in the system is determined under the assumption that the process controlling the number of customers in the system during the vacation and the planned duration of the vacation are independent. We also study the asymptotic behavior for the number of completed vacations.