The M/M/Infinity Service System with Ranked Servers in Heavy Traffic

TL;DR

This paper analyzes an infinite-server queueing system with ranked servers under heavy traffic, deriving asymptotic behavior of the server index distribution as the arrival rate grows large.

Contribution

It provides the first two terms of asymptotic expansions for the moments of the server index in an M/M/Infinity system under heavy traffic conditions.

Findings

L/lambda tends to a uniform distribution on [0,1]

Asymptotic moments of server index are explicitly characterized

Heavy traffic limit results for ranked server systems

Abstract

We consider an M/M/Infinity service system in which an arriving customer is served by the first idle server in an infinite sequence S_1, S_2, ... of servers. We determine the first two terms in the asymptotic expansions of the moments of L as lambda tends to infinity, where L is the index of the server S_L serving a newly arriving customer in equilibrium, and lambda is the ratio of the arrival rate to the service rate. The leading terms of the moments show that L/lambda tends to a uniform distribution on [0,1].