This paper presents a numerical technique for the solution of a class of queueing problems having fixed priority assignment, synchronous service, constant service time, and finite number of sources. The analysis yields waiting-time distributions, namely, the probabilities of waiting exactly nW service cycles or longer. It applies to scanners, multiplexer channels, loop configurations, etc. It is especially useful for worst case design in that it permits the calculation of overflow probabilities. Results are given in the form of curves and are compared with those for exponential service times. A priority scheme is outlined that allows partial elimination of the geometrical priorities inherent to loop multiplexers. © 1972, IEEE. All rights reserved.