@article{EJP391,
author = {Malwina Luczak and Colin McDiarmid},
title = {Asymptotic distributions and chaos for the supermarket model},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {12},
year = {2007},
keywords = {Supermarket model, join the shortest queue, random choices, power of two choices, load balancing, equilibrium, concentration of measure, law of large numbers, chaos},
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 $d \geq 2$ 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 $n\to\infty$. 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 $1/n$ 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 $1/n$.},
pages = {no. 3, 75-99},
issn = {1083-6489},
doi = {10.1214/EJP.v12-391},
url = {http://ejp.ejpecp.org/article/view/391}}