Classification of Discrete-Time Queues

Abstract

In this article we classify discrete-time queues based on scheduling rules and observation epochs combinations. This classification leads to {\em coherent}, {\em sub-coherent}, and {\em super-coherent} systems when {\em observed} waiting times are, respectively equal to, less than, or larger than {actual} waiting times. We then explore the consequences of this classification. Specifically, we discuss invariant properties of {\em coherent} systems including queue-lengths, waiting times, servers' busy times, busy periods, Pollaczek-Khinchine formula, and other common characteristics. An important consequence is that a performance characteristic of a system with specific scheduling rule and observation epoch combination extends to the entire class. An unresolved issue in the literature is the assertion that Little's law does not apply for discrete-time queues that incorporate certain scheduling rules. Using this classification, we reconcile the generality of Little's law and its applicability to all discrete-time queues regardless of scheduling rules.