Delay Analysis of a Two-Queue, Nonuniform Message Channel

A Message Channel is dejined as a tandem connection of single server queues in which the successive service times experienced by any particular customer are scaled versions of the same random variable, and thus it serves as a model for sparsely connected store-and-forward data communications networks (or network segments) where messages typically preserve their lengths as they traverse the system. A particular instance of such a nonstandard queueing model is analyzed in this paper. The system consists of two single server queues in tandem subject to a Poisson arrival process (at the first queue) and providing service according to scaled versions of a sequence of two-level, discrete random variables. A set of recursive equations that can be used to solve the model for any given scaling factor at the second queue (normalized with respect to the jirst queue service) is explicitly derived. In addition, complete solutions are displayed for several cases of interest, and the equilibrium mean cumulative waiting times for these instances are compared as a method of indicating the impact of the scaling factor on the operation of the system. The extension of several results to systems with more general service time processes is discussed.