A Many-Server Functional Strong Law For A Non-Stationary Loss Model

TL;DR

This paper establishes a strong law of large numbers for the fraction of occupied servers in a non-stationary loss model, simplifying analysis by using a semimartingale representation without tracking individual job ages.

Contribution

It introduces a novel approach to derive a many-server FSLLN in a non-Markovian, non-stationary setting without detailed state tracking, via a semimartingale framework.

Findings

Fluid limit characterized by a Volterra integral equation

Simplifies analysis by avoiding age or residual service time tracking

Proves uniqueness of the fluid limit solution

Abstract

The purpose of this note is to show that it is possible to establish a many-server functional strong law of large numbers (FSLLN) for the fraction of occupied servers (i.e., the scaled number-in-system) without explicitly tracking either the age or the residual service times of the jobs in a non-Markovian, non-stationary loss model. This considerable analytical simplification is achieved by exploiting a semimartingale representation. The fluid limit is shown to be the unique solution of a Volterra integral equation.