Transient solution of /G/1 with Second Optional Service Subject to Reneging during Vacation and Breakdown time

In this paper, we present a batch arrival non- Markovian queuing model with second optional service. Batches arrive in Poisson stream with mean arrival rate λ, such that all customers demand the first essential service, whereas only some of them demand the second ‘optional’ service. We consider reneging to occur when the server is unavailable during the system breakdown or vacations periods. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.