This paper is concerned with the queuing process which arises when one imposes nonzero crossover times on a nonpreemptive priority system with two classes of customers. The objective is to derive the Laplace-Stieltjes transform of the waiting-time distribution function and the queue-size generating function for each class of customers. The results are obtained by first analyzing a simple single queue model with walking time and then extending this model to fit the two queue problem by an appropriate choice of variables.