Steady-state analysis of shortest expected delay routing

TL;DR

This paper analyzes a queueing system with two non-identical exponential servers using a novel series representation of the equilibrium distribution, providing insights into its asymptotic behavior and enabling efficient numerical computation.

Contribution

It extends the compensation approach to inhomogeneous random walks in the quadrant, deriving a series of product-forms for the equilibrium distribution in a complex queueing system.

Findings

Series expression for equilibrium distribution derived

Asymptotic behavior of probabilities analyzed

Numerical method for equilibrium probabilities developed

Abstract

We consider a queueing system consisting of two non-identical exponential servers, where each server has its own dedicated queue and serves the customers in that queue FCFS. Customers arrive according to a Poisson process and join the queue promising the shortest expected delay, which is a natural and near-optimal policy for systems with non-identical servers. This system can be modeled as an inhomogeneous random walk in the quadrant. By stretching the boundaries of the compensation approach we prove that the equilibrium distribution of this random walk can be expressed as a series of product-forms that can be determined recursively. The resulting series expression is directly amenable for numerical calculations and it also provides insight in the asymptotic behavior of the equilibrium probabilities as one of the state coordinates tends to infinity.