A discrete-time priority queueing system is studied, in which two different classes of fixed-length packet-trains arrive according to independent batch geometric streams. The packets in each packet-train arrive at the rate of one packet per slot (train arrivals), resulting in a correlated arrival stream. The service time of one packet is deterministic of one slot. The motivation for the work comes from ATM networks with diverse traffic sources and correlated packet arrival stream. Using the probability generating function method, the joint distribution of queue lengths and the waiting time distribution are obtained for each class. Numerical results are presented. Comparision is made with the case of ‘batch arrivals’, where all packets of a packet-train arrive simultaneously at the buffer.