Unreliable Server Retrial Queue with Optional Service and Multi-Phase Repair

In this paper, the retrial unreliable server queue with batch arrivals is considered. The arrival rates of the units are different and dependent upon the joining probabilities according to the server status. On arrival, if unit finds the busy server, he may retry for the service after a random duration of time. The server facilitates the essential service and optional service, if opted after essential service. Moreover, the server is unreliable and subject to the breakdown while rendering essential/optional service. The failed server may immediately undergo for the compulsory multiphase repair or may wait to start the repair due to any technical reasons. The server can also avail the optional vacation under the Bernoulli schedule after finish the service of each unit or may continue to serve the next unit. The variables corresponding to elapsed times of general distributed service process, retrial process, repair process and vacation duration, as supplementary variables and used to frame the governing equations. By using the probability generating functions of joint distributions of the units at different states of the server, the performance characteristics of the system are derived. To validate the results, the sensitivity analysis has been performed by taking the numerical illustration