The present paper is an attempt to explore a discrete – time queueing models which is suitable for the performance evaluation of Asynchronous Transfer Mode (ATM) multiplexers and switches. In these models, the time axis is divided into fixed-length slots and the service of a customer must start and end of slot boundaries. Most analytical studies of discrete-time queues assume constant service times equal to one slot, an infinite buffer capacity and/or uncorrelated arrival process. A discrete time bulk service rule (L,K)the inter arrival time are assumed to be independent and identically distributed random variable follows negative binomial distribution. The time arrival between the consecutive arrivals of customers is described in terms of a discrete probability mass function (p.m.f) (S(k) is the probability of having an interval of an integer number of x time units between the arrival customer number n and customer number n+1. The service time of customers n is given in terms of a discrete p.m.f S(k)(follows geometric distribution. The arrival and service are independent and identically distributed (i.i.d). The arrival of customer in First In First Out (FIFO) order. One server transports packets in batches of size minimum L packets and maximum K. The steady state analysis for the considered queuing system is presented and the generating functions of the number of customers in the system are obtained. We also obtained the closed form of expressions for some performance measures of the system.
