Abstract: This paper analyses an N×N nonblocking ATM input and ouput switch queueing with multiple priorities at input ports.The arrival process of each priority to each input port is Bernoulli with probability p1,p2,…,pR,and each class has a dedicated buffer with infinite size. The speed-up factor is S (B ≤ S ≤ N),and the output buffer size is B with backpressure.Each cell has an equal probability (1/N) being sent to output ports.The average delay time of the considered approaches is derived by theoretical analysis and computer simulations.