## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 25 (2020), paper no. 1, 13 pp.

### Load balancing under $d$-thinning

Ohad Noy Feldheim and Jiange Li

#### Abstract

In the classical balls-and-bins model, $m$ balls are allocated into $n$ bins one by one uniformly at random. In this note, we consider the $d$-thinning variant of this model, in which the process is regulated in an on-line fashion as follows. For each ball, after a random bin has been selected, an overseer may decide, based on all previous history, whether to accept this bin or not. However, one of every $d$ consecutive suggested bins must be accepted. The *maximum load* of this setting is the number of balls in the most loaded bin. We show that after $\Theta (n)$ balls have been allocated, the least maximum load achievable with high probability is $(d+o(1))\sqrt [d]{\frac{d\log n} {\log \log n}}$. This should be compared with the related $d$-choice setting, in which the optimal maximum load achievable with high probability is $\frac{\log \log n} {\log d}+O(1)$.

#### Article information

**Source**

Electron. Commun. Probab., Volume 25 (2020), paper no. 1, 13 pp.

**Dates**

Received: 27 August 2019

Accepted: 9 December 2019

First available in Project Euclid: 3 January 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1578020668

**Digital Object Identifier**

doi:10.1214/19-ECP282

**Zentralblatt MATH identifier**

07149364

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

**Keywords**

balls and bins load balancing two-choice thinning

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Feldheim, Ohad Noy; Li, Jiange. Load balancing under $d$-thinning. Electron. Commun. Probab. 25 (2020), paper no. 1, 13 pp. doi:10.1214/19-ECP282. https://projecteuclid.org/euclid.ecp/1578020668