There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where 0<λ<1. Upon arrival each customer selects d≥2 servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as n→∞ the maximum queue length takes at most two values, which are lnlnn/lnd+O(1).
Publié le : 2006-03-14
Classification:
Supermarket model,
join the shortest queue,
random choices,
power of two choices,
maximum queue length,
load balancing,
equilibrium,
concentration of measure,
60C05,
68R05,
90B22,
60K25,
60K30,
68M20
@article{1147179980,
author = {Luczak, Malwina J. and McDiarmid, Colin},
title = {On the maximum queue length in the supermarket model},
journal = {Ann. Probab.},
volume = {34},
number = {1},
year = {2006},
pages = { 493-527},
language = {en},
url = {http://dml.mathdoc.fr/item/1147179980}
}
Luczak, Malwina J.; McDiarmid, Colin. On the maximum queue length in the supermarket model. Ann. Probab., Tome 34 (2006) no. 1, pp. 493-527. http://gdmltest.u-ga.fr/item/1147179980/