Deterministic Analysis of Queueing Systems with Heterogeneous Servers

Using deterministic (sample-path) analysis, we generalize and extend fundamental properties of systems with “stationary deterministic flows” as introduced by Gelenbe (1983) and Gelenbe and Finkel (1987). Primarily, we provide conditions for stability and instability for general queueing models, and focus attention on multichannel queueing systems with servers that work at different rates. Stability analysis is important in computer applications and usually precedes any further investigation of the system in question. Our results complement and extend those of Gelenbe and Finkel by making weaker assumptions, allowing multichannel facilities with heterogeneous servers, and including more general queueing disciplines such as processor sharing and LCFS-PR. The key to our stability analysis is a deterministic version of the renewal-reward theorem which we call Y=i,X, and a relationship that shows the “operational analysis” definition of average service times, when considered as the observation period t --t cc, coincides with the standard definition of average service times for all stable queueing systems. Our analysis is completely deterministic and avoids any stochastic assumptions about the system under investigation; thus, it provides the practitioner with a method that often leads to a better and deeper understanding of the system under consideration. It also gives a powerful tool to determine which properties of the system are independent of the usually needed probabilistic assumptions. As an illustration, a sample-path relationship that gives the long-run average busy period (cycle) for a general queueing model is given and utilized to derive several well-known results under weaker conditions.