On Spectral Properties of Finite Population Processor Shared Queues

TL;DR

This paper analyzes the spectral properties of finite population processor-shared queues, deriving asymptotic solutions for response times by studying eigenvalues and eigenvectors of large linear systems, linking them to differential equations.

Contribution

It introduces an asymptotic approach to analyze eigenvalues of large finite population queue models, connecting spectral analysis to differential equations like Hermite's.

Findings

Eigenvalues approximate solutions to differential equations

Dominant eigenvalue determines tail behavior of response times

Asymptotic methods simplify complex eigenvalue problems

Abstract

We consider sojourn or response times in processor-shared queues that have a finite population of potential users. 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 population models where the total population is $N\gg 1$. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the tail of a customer's sojourn time distribution.