Transient analysis of queues with heterogeneous arrivals

In this paper we 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 typei of batch sizeb _i, i=1,...,r with probability _i or is a depature of a single customer with probability 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––_{i i} or 1–_{i i} according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we 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.