Optimal repairable Mx/G/1 queue with multi-optional services and Bernoulli vacation

This article deals with the modified Bernoulli vacation schedule for the unreliable server batch arrival queueing system with essential and multi‐optional services under N‐policy. The server is subject to breakdown while rendering service to the customers. At the completion of service of the customer, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. The server follows N‐policy, i.e. the server will not initiate the service until the queue size reaches or exceeds N. All customers demand for the first ‘essential’ service whereas only some of them demand for other ‘optional’ services. After getting first essential service, the customer may demand for the second optional phase service with some probability; if the customer completes its second optional phase service, he may demand for the next optional phase services with some other probabilities and so on. We derive the system size distribution at random points and at departure points and other performance indices by employing generating function and supplementary variable techniques. By taking numerical illustration, the sensitivity analysis is also conducted to determine the effects of the system parameters on various performance measures.