Asymptotic distributions and chaos for the supermarket model

TL;DR

This paper analyzes the asymptotic behavior and chaos in the supermarket model with many queues, providing explicit convergence rates to the limiting distribution and quantifying the independence (chaos) among queues as the system size grows.

Contribution

It offers precise rates of convergence to equilibrium and demonstrates the chaotic independence of queues in large systems, extending understanding of the model's asymptotic properties.

Findings

Total variation distance between equilibrium and limiting distribution is of order n^{-1}.

System starting from general initial conditions also converges at rate n^{-1}.

Queues exhibit chaotic behavior with joint distributions close to product laws at rate n^{-1}.

Abstract

In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \lambda n, where 0<\lambda <1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}.