Finite M/Ek/1 queue with server breakdowns, start-up, second optional service and reneging

The current study is about a M/M/1 queueing system of finite capacity with vacation, server failures, second optional service and customer impatience in transient mode. Each customer will be given k number of mandatory phases wise service. Vacation is allowed for customer if the system has no clients to be served and there -by it returns to normalcy once N-customers are pooled and initiates start-up. Further, we assumed two types of repair facilities with an optional probability. The server first provides essential service to all customers and second optional service will be provided with a probability of “p”. The customer may renege in first phase of service. Runge-Kutta Method is applied to find Transient state probabilities and computed various performance indices like expected length of system, the mean waiting time etc. and also used sensitivity analysis to check the effect of various parameters.