An M/M/c queue with queueing-time dependent service rates

TL;DR

This paper introduces a novel analysis method for an M/M/c queue where service rates depend on individual queueing delays, modeling real-world scenarios like healthcare and call centers, and demonstrates the impact of variable service rates on performance.

Contribution

It presents a new Markov chain model and recursive solution for analyzing queueing systems with delay-dependent service rates, a novel approach in queueing theory.

Findings

Service rate differences significantly affect queueing delay performance.

The model provides a recursive method to compute the stationary distribution.

Numerical experiments illustrate the impact of delay-dependent service rates.

Abstract

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on queueing time performance.