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Asymptotic distributions and chaos for the supermarket model

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Asymptotic distributions and chaos for the supermarket model
2.	Creator	Author's name, affiliation, country	Malwina J Luczak; London School of Economics; United Kingdom
2.	Creator	Author's name, affiliation, country	Colin McDiarmid; University of Oxford
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Supermarket model, join the shortest queue, random choices, power of two choices, load balancing, equilibrium, concentration of measure, law of large numbers, chaos
3.	Subject	Subject classification	Primary 60C05; secondary 68R05, 90B22, 60K25,60K30, 68M20.
4.	Description	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 $1/n$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2007-01-24
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/391
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-391
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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