Asymptotic Analysis of Spectral Properties of Finite Capacity Processor Shared Queues

TL;DR

This paper analyzes the spectral properties of finite capacity processor-shared queues, deriving asymptotic expressions for eigenvalues that determine customer response time distributions, with numerical validation.

Contribution

It introduces an asymptotic approach to analyze eigenvalues of large finite capacity queue models, simplifying the eigenvalue problem to standard differential equations.

Findings

Asymptotic eigenvalue approximations match numerical results

Tail distribution of sojourn times characterized by dominant eigenvalues

Reduction to Airy equation simplifies spectral analysis

Abstract

We consider sojourn (or response) times in processor-shared queues that have a finite customer capacity. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite capacity models where the system can only hold a large number $K$ of customers. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Airy equation. The dominant eigenvalue leads to the tail of a customer's sojourn time distribution. Some numerical results are given to assess the accuracy of the asymptotic results.