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March 2006 On the maximum queue length in the supermarket model
Malwina J. Luczak, Colin McDiarmid
Ann. Probab. 34(2): 493-527 (March 2006). DOI: 10.1214/00911790500000710


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).


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Malwina J. Luczak. Colin McDiarmid. "On the maximum queue length in the supermarket model." Ann. Probab. 34 (2) 493 - 527, March 2006.


Published: March 2006
First available in Project Euclid: 9 May 2006

zbMATH: 1102.60083
MathSciNet: MR2223949
Digital Object Identifier: 10.1214/00911790500000710

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

Keywords: concentration of measure , Equilibrium , Join the shortest queue , load balancing , maximum queue length , power of two choices , random choices , Supermarket model

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 2 • March 2006
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