Many physical systems are well-modeled by queueing systems in which time is slotted, the distribution of the number of entities that arrive during a slot is dependent upon the evolution of a discrete time, discrete state Markov chain, and the number of entities that may be served during a slot is limited to some number, say R. Techniques for analyzing systems in this, or closely related, class have appeared in the literature, but distributions have been presented in only rare instances, limited to the case R = 1. Yet, distributions are very important, not only in performance evaluation, but in design, especially for sizing buffers in integrated (BISDN) communications systems and intermediate storage space designs in manufacturing systems. In this dissertation, a numerically stable methodology based on eigenanalysis and probability generating function technique has been developed for producing both occupancy and delay moments and distributions for the equilibrium process described above. Feasibility of the methodology is demonstrated through numerical results for two examples of an important subclass. Special attention is paid to obtaining accurate numerical values; and wherever available, numerical values are compared to those previously obtained in the literature. Furthermore, additional important models amenable to analysis by the same methodology are discussed and numerically feasible approaches for obtaining important performance measures are suggested. / Ph. D.