A discrete time bulk service queue with accessible batch: Geo/ NB (L,K) /1

This paper considers a discrete-time queue of packets awaiting movement according to the bulk service rule (L, K). Time is assumed to be divided into equal intervals called slots. The analysis has been carried out under the assumption that (i) service times of batches are independent of the number of packets in any batch and (ii) there is no simultaneous arrival of packets and departure of batch in a single slot. The packets arrive one by one and their inter arrival times follow geometric distribution. The arriving packets are queued in FIFO order. One server transports packets in batches of minimum number L and maximum K and service times follow negative binomial distribution. The server accesses new arrivals even after service has started on any batch of initial number 'j' (L= j < K). This operation continues till the random service time of the ongoing batch is completed or the maximum capacity of the batch being served attains 'K' whichever occurs first. The distribution of system occupancy just before and after the departure epochs is obtained using discrete-time analysis. The primary focus is on the various performance measures of the steady state distribution of the batch server at departure instants and also on numerical illustrations.