Open Access
2007 Asymptotic distributions and chaos for the supermarket model
Malwina Luczak, Colin McDiarmid
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Electron. J. Probab. 12: 75-99 (2007). DOI: 10.1214/EJP.v12-391


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 $1/n$.


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Malwina Luczak. Colin McDiarmid. "Asymptotic distributions and chaos for the supermarket model." Electron. J. Probab. 12 75 - 99, 2007.


Accepted: 24 January 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1131.60005
MathSciNet: MR2280259
Digital Object Identifier: 10.1214/EJP.v12-391

Primary: 60C05
Secondary: 60K25 , 60K30 , 68M20 , 68R05 , 90B22

Keywords: chaos , concentration of measure , Equilibrium , Join the shortest queue , Law of Large Numbers , load balancing , power of two choices , random choices , Supermarket model

Vol.12 • 2007
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