The thesis deals with three priority queues. Chapters I and II treat a queueing model with two service units in tandem and a single server alternating between them. Chapter III deals with two independent service units with a single server serving alternately between them and Chapter IV treats a single server M/G/1 queue with a priority rule based on the ranking of the service times. In Chapter I the server serves the two service units alternately with a non-zero switching rule in unit 1 and a zero switching rule in unit 2. The case of zero switching rule for unit 1 is dealt in Chapter II. In both cases the distributions of busy period, virtual waiting time and queue length and their ergodic properties are studied in terms of Laplace transforms. In Chapter III we consider the alternating priority queues with a non-zero switching in each unit. Distributions of busy period and queue length are discussed. In Chapter IV we study the virtual waiting time process of an M/G/1 queue under this priority rule: within each generation customers are served in the order of shortest (or longest) service times. Here we also study the limiting behavior of the virtual waiting time, and compare the means of the limiting distributions with those of first come, first served discipline. Applications of the different priority models are discussed.