Message Delays in Repeated-Service Tandem Connections

One of the inherent complexities associated with queueing models for store and forward data communications networks arises from the fact that messages typically preserve their length as they traverse the system. The interarrival and service sequences at queues internal to the network are thus dependent, making standard methods of analysis realistically inappropriate (except in an approximate sense). In an effort to find methods of dealing with such nonstandard queueing systems, a model for sparsely-connected networks (or network segments) called a message channel has been studied. A message channel is defined 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. A number of recent results concerning the delay properties of such systems are presented. These include: ordering relations for the successive waiting times in the channel; characterizations of waiting time properties under extreme conditions (simultaneous arrival of all customers): and, simple bounds on performance parameters for systems with uniformly bounded service processes. A general integral equation for the distribution of the equilibrium cumulative waiting time in queueing systems (which are instances of uniform message channels) is also derived. The solution of this equation for systems with discrete service time distributions and Poisson arrivals is indicated and explicitly exhibited for the case of two discrete levels (corresponding to a message switching system with two packet classes).