Sojourn time in a single server queue with threshold service rate control

TL;DR

This paper analyzes the sojourn time in a single-server queue with threshold-based service rate control, providing Laplace transforms, moments, and new matrix generating functions for complex inspection scenarios.

Contribution

It introduces a novel analytical approach for threshold-based service rate control in queues, including matrix generating functions for generalizations.

Findings

Laplace transform of stationary sojourn time derived

Procedure to compute all moments established

Extended analysis to random inspection times using new matrix tools

Abstract

We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times. This extension requires the development of a new methodological tool, that is "matrix generating functions". The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.