Computation of queue length probabilities for bulk service queues with vacation and feed back

A M/G A,B /1 queue with infinite waiting space is considered. There is a single server who can serve at a time, a batch of minimum size 'A' and maximum size 'B'. The server is free to take vacation and the customer has a feedback facility. For such a system the LST and pgf are formulated from which the queue length probabilities and some other performance measures are derived. A special case of this queue with distrihutions of service and vacation time as exponential is solved by both analytical approach (i.e. root-finding method) and by the method of characteristic equations. The numerical results obtained by both methods are found to agree well. It is also found that, the queue length has a direct trend (increase or decrease) with feed back probability as well as the arrival rate when all other parameters are kept same.