A Retrial Queueing Model with Optional Service, Server Breakdowns, and Customer Impatience

This paper studies a single-server retrial queue with optional service, server breakdowns, and customer impatience. Customers arrive according to a Poisson process. An arriving customer who finds the server idle immediately receives mandatory service. Upon completion of the mandatory service, the customer may proceed to an optional service with probability p , where the mandatory and optional service times are exponentially distributed. If the server is busy upon arrival, the customer joins an orbit and retries for service after an exponentially distributed retrial time. Both mandatory and optional services are provided by the same server. During the busy period, including both mandatory and optional services, server breakdowns occur according to a Poisson process. Upon breakdown, the server immediately enters repair, where the repair times are exponentially distributed and the customer receiving service at the instant of breakdown is lost. During the repair period, arriving customers continue to join the orbit, while orbiting customers may abandon the system due to impatience before receiving service, where the impatience times are exponentially distributed. After completion of the repair process, the server returns to the normal operating state and resumes providing mandatory service. The steady-state behavior of the system is analyzed using probability generating functions (PGFs), leading to explicit expressions for several important performance measures. Numerical experiments are presented to illustrate and validate the analytical results.