This paper gives results on various traffic processes and on the sojourn time distribution for a class of models which operate as Markov processes on finite state spaces. The arrival and the service time processes are assumed to be independent renewal processes with interval distributions of phase-type. The queue capacity is finite. A general class of queue disciplines are considered. The primary models studied are from the M/E_k/Φ/L class. The input, output, departure and overflow processes are analyzed. Furthermore, the sojourn time distribution is determined. Markov renewal theory provides the main analytical tools. It is shown that this work unifies many previously known results and offers some new results. Various extensions, including a balking model, are studied. / Ph. D.