A rate balance principle and its application to queueing models

TL;DR

This paper introduces a rate balance principle applicable to general stochastic processes, especially those with birth and death transitions, and demonstrates its usefulness in deriving results for specific queueing models.

Contribution

It presents a new rate balance principle for non-Markovian processes and applies it to derive results in queueing theory models.

Findings

Derived a recursion for residual service times in Mn/Gn/1 queues.

Established a recursion for residual inter-arrival times in G/Mn/1 queues.

Unified approach for analyzing non-Markovian queueing systems.

Abstract

We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth and death like transitions, for which it is shown that for any state $i$, the rate of two consecutive transitions from $i-1$ to $i+1$, coincides with the corresponding rate from $i+1$ to $i-1$. This observation appears to be useful in deriving well-known, as well as new, results for the Mn/Gn/1 and G/Mn/1 queueing systems, such as a recursion on the conditional distributions of the residual service times (in the former model) and of the residual inter-arrival times (in the latter one), given the queue length.