Large-scale Join-Idle-Queue system with general service times

TL;DR

This paper analyzes a large-scale parallel server system with general service times under the Join-Idle-Queue routing, proving that the system's stationary distribution concentrates at equilibrium, ensuring minimal waiting in the limit.

Contribution

It establishes the asymptotic behavior of the Join-Idle-Queue system with arbitrary service time distributions as the number of servers grows large.

Findings

Stationary distribution concentrates at equilibrium as n→∞

Waiting probability for arriving customers vanishes in the limit

System remains stable under the condition λ/μ<1/2

Abstract

A parallel server system with $n$ identical servers is considered. The service time distribution has a finite mean $1/\mu$, but otherwise is arbitrary. Arriving customers are be routed to one of the servers immediately upon arrival. Join-Idle-Queue routing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where $n\to\infty$ and the customer input flow rate is $\lambda n$. Under the condition $\lambda/\mu<1/2$, we prove that, as $n\to\infty$, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant equal $\lambda/\mu$. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.