We propose a new model for the performance analysis of an ATM switching fabric based on a single-buffered Banyan network. In this model, we use a three-state, i.e., "empty", "new" and "blocked" Markov chain model to describe the behavior of the buffer within a switching element. In addition to traditional statistical analysis including throughput and delay, we also examine the delay variation. Performance results show that the proposed model is more accurate in describing the switch behavior under a uniform traffic environment in comparison with the "two-state" Markov chain model developed by Jenq (1983) and Yoon (1990). Based on the "three-state" model, we study a packet priority scheme which gives the blocked packet higher priority to be routed forward during contention. For a fully loaded six-stage switch, the analytical model predicts that the delay variation can be reduced by 35%, while the simulation shows a 22% reduction.