We shall be concerned with an irreducible Markov chain, which we shall call "the system." For simplicity we shall assume that the system is aperiodic, but this is not essential. The reader is referred to [1] for explanations of the terminology used. We first state some general theorems which provide criteria for determining whether the system is transient, recurrent-null or ergodic (recurrent-nonnull). These are then applied to the Markov chains associated with certain queuing processes recently studied by D. G. Kendall [4], [5]; most of the results have already been obtained by Kendall using direct methods, and the main purpose of the present paper is to illustrate the application of general theorems to this type of problem.