The Annals of Probability

On the maximum queue length in the supermarket model

Malwina J. Luczak and Colin McDiarmid

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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).

Article information

Ann. Probab., Volume 34, Number 2 (2006), 493-527.

First available in Project Euclid: 9 May 2006

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 68R05: Combinatorics 90B22: Queues and service [See also 60K25, 68M20] 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Supermarket model join the shortest queue random choices power of two choices maximum queue length load balancing equilibrium concentration of measure


Luczak, Malwina J.; McDiarmid, Colin. On the maximum queue length in the supermarket model. Ann. Probab. 34 (2006), no. 2, 493--527. doi:10.1214/00911790500000710.

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