Analysis of Markovian queueing system with server failures, N-policy and second optional service

The current work details the behaviour of a finite Markovian queueing system with a vacation in which the server may face problems of breakdowns while in service. The repair process does start immediately after a breakdown which immediately resumes the service. During this period any new customer is allowed to join the system. Whenever the server finds nobody, the server goes on vacation and resumes service after N customers are accumulated. Meanwhile, it triggers pre-service called start-up. Further, we considered two types of repair facilities for the broken-down server with an optional probability. The server first provides essential service to all customers and the second optional service will be provided with a probability of "p". The customer may renege in the first phase of service. We adopted Runge-Kutta Method to find Transient state probabilities and computed various performance indices like the expected length of the system, the mean waiting time etc. We then performed the sensitivity analysis to explore the effect of different parameters.