Time Dependent Solution Of M [ X ] / G / 1 Queueing Model With second optional service , Bernoulli k - optional Vacation

This paper investigate a queueing model where in customers arrive in batches according to compound Poisson process with rate λ and are served one by one in FIFO basis. All incoming customers demand the first ’essential’ service wherein only some of them demand the second ’optional’ service. The service times of both essential and optional services follow arbitrary(general)distribution with different vacation policies. In addition to this, due to the annoyance of seeing long queue in the system the customer may decide not to join the queue, called balking. Such a customer behavior is considered in both busy time and vacation time of the system. For this model, We obtain the time dependent solution and the corresponding steady state solutions. Also, we derive the performance measures, the mean queue size and the average waiting time explicitly.