The Annals of Applied Probability

On the power of two choices: Balls and bins in continuous time

Malwina J. Luczak and Colin McDiarmid

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Abstract

Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, where λ>0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when d≥2, there is an integer-valued function md(n)=ln ln n/ln d+O(1) such that, in the equilibrium distribution, the maximum load of a bin is concentrated on the two values md(n) and md(n)−1, with probability tending to 1, as n→∞. We show also that the maximum load usually does not vary by more than a constant amount from ln ln n/ln d, even over quite long periods of time.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 3 (2005), 1733-1764.

Dates
First available in Project Euclid: 15 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1121433767

Digital Object Identifier
doi:10.1214/105051605000000205

Mathematical Reviews number (MathSciNet)
MR2152243

Zentralblatt MATH identifier
1079.60016

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 68R05: Combinatorics 90B80: Discrete location and assignment [See also 90C10] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Keywords
Balls and bins random choices power of two choices maximum load load balancing immigration–death equilibrium

Citation

Luczak, Malwina J.; McDiarmid, Colin. On the power of two choices: Balls and bins in continuous time. Ann. Appl. Probab. 15 (2005), no. 3, 1733--1764. doi:10.1214/105051605000000205. https://projecteuclid.org/euclid.aoap/1121433767


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