This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization.

This research analyzes a multi-server retrial queue with two types of calls: working vacation and working breakdown. The incoming call may enter the retrial queue and attempt to seize a server after a random delay if all the servers are unavailable upon arrival. In its idle time, the server makes outgoing calls. All the servers take a synchronous working vacation when the system empties after regular service. The system may fail at any time due to disasters, forcing all the calls within the service area to leave the system and causing all the main servers to fail. When the main servers fail, the repair process begins immediately. The standby servers serve arriving customers at a lower level of service during the working breakdown or working vacation. For this model, we derive an explicit expression for the stationary distribution with the help of the quasi-birth-and-death process and the matrix geometric method. Further, the formulas of various system performance indices are developed. An application example is given and several numerical experiments are performed to verify the analytical results. We also perform the comparative analysis of retrial queues with/without two-way communication and two-way communication retrial queues with/without disasters. The results reveal that the proper consideration of outgoing calls to the server can reduce the average time spent in the buffer. Furthermore, a more reliable server reduces the server idle rate.