This dissertation deals with the problem of analyzing feed-forward priority queueing systems. In this type of system incoming units enter one of n priority queues if the service facility is busy. Units in the highest priority queue are served on a first-come-first-served basis while units in the lower priority queues will only be serviced if there are no higher priority units. A predefined delay time is employed for each queue so that a unit waiting in a lower priority queue is able to transfer to the next higher queue. The waiting unit will transfer to a higher priority queue if its waiting time becomes equal to the predefined delay time. Otherwise, the unit enters the service facility. Units receive service until completion, and at that time, leave the queueing system. The initial approach in this dissertation to analyze such a system is to develop the steady state mathematical expressions for a feed-forward system with two priority queues (FF₂). This represents the simplest case for feed-forward queueing systems (FF_n), and development of the mathematical expressions for this model indicates the tractability of higher order models. Numerical results are obtained from the FF₂ mathematical model by defining input parameters for the delay process, interarrival distribution, and service distribution. To aid in validating the cumulative waiting time results, W_m(t), produced by the mathematical model, an FF₂ simulation model is developed. Results from the two procedures compare favorably which aids in substantiating the notion that the FF₂ model is correct as well as the notion that the simulation model is operating properly. As is often the case, when systems become too complex to be analyzed mathematically, simulation is warranted. Such is the case in this research. Thus, a general simulation model is developed for analyzing higher order systems. This model is capable of representing any FF_n queueing system by simple manipulation. A modified Runge-Kutta procedure is developed for an FF₇ system to ascertain the behavior of the system under transient conditions. Transient results are also obtained from the simulation model. A comparison of the results from the Runge-Kutta and simulation procedures indicate that they agree quite favorably. This agreement aids in substantiating that both methods are properly formulated. The major goal of this research, to develop a model for analyzing FF_n queueing systems, has been accomplished. Various mathematical expressions for analyzing FF₂ systems have been developed. Because higher order models are mathematically intractable, a general simulation model has been developed and substantiated for both steady state and transient conditions. A modified Runge-Kutta procedure has also been developed to aid in analyzing the transient case. / Ph. D.