Stochastic Study of a Retrial Queue with Impatience under a Multi-Level Environment Incorporating Sensor-Enabled Shock Detection

This study analyzes an M/M/1 retrial queue with impatient customers under a multi-level stochastic environment, utilizing a sensor as a notification mechanism for detecting random shocks. Customer arrivals occur according to a random process, with arrival rates depending on the environmental state, and each arriving customer is attached with a sensor tag for continuous monitoring. If a customer arrives when the server is idle, service is started immediately. Otherwise, customers are forwarded to an orbit of infinite capacity, and an orbiting customer makes reattempts after random time intervals under a first-in-first-out (FIFO) discipline. Because of impatience, customers waiting in the orbit may abandon the system if their waiting times are too long. The system is also subject to random shocks that instantaneously disrupt the server. Upon the occurrence of a shock, the server immediately enters a repair phase, and interruption information is broadcast via the sensor to all tagged customers, resulting in the removal of both customers in service and those waiting in the orbit. After the completion of repair, the server resumes service at any environmental state level with probability. Such system has wide applications in real world scenarios, including wireless sensor networks and modern telecommunication systems such as 4G/5G networks. The probability generating function method is used to obtain the steady-state probabilities, and important performance metrics are derived. Numerical results are presented to bring out the effect of some parameters on system performance.