In this paper the authors consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of type i of batch size bi, i=1,...,r with probability αi or is a departure of a single customer with probability γ or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1-γ-Σiαi or 1-Σiαi according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space the authors will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.
