The Distributional Little's Law and Its Applications

This paper discusses the distributional Little's law and examines its applications in a variety of queueing systems. The distributional law relates the steady-state distributions of the number in the system (or in the queue) and the time spent in the system (or in the queue) in a queueing system under FIFO. We provide a new proof of the distributional law and in the process we generalize a well known theorem of Burke on the equality of pre-arrival and postdeparture probabilities. More importantly, we demonstrate that the distributional law has important algorithmic and structural applications and can be used to derive various performance characteristics of several queueing systems which admit distributional laws. As a result, we believe that the distributional law is a powerful tool for the derivation of performance measures in queueing systems and can lead to a certain unification of queueing theory.