## Abstract

In the two-thinning balls-and-bins model, an overseer is provided with uniform random allocation of m balls into n bins in an on-line fashion. The overseer may reject the allocation of each ball, in which case it is placed into a new bin, drawn independently, uniformly at random. The purpose of the overseer is to reduce the maximum load, that is, the difference between the maximum number of balls in a single bin and the average number of balls among all bins. We provide tight estimates for three quantities: the lowest achievable maximum load at a given time m, the lowest achievable maximum load uniformly over the entire time interval [m]:= {1, 2, . . ., m} and the lowest achievable typical maximum load over the interval [m], that is, a load which upper-bounds 1 − o(1) portion of the times in [m]. In particular, for m polynomial in n and sufficiently large, we provide an explicit strategy, which achieves a typical maximum load of (log n)^{1/2}+o(1) , asymptotically the same as that can be achieved at a single time m. In contrast, we show that no strategy can achieve better than Θ(_{log}^{log}_{log}^{n}_{n} ) maximum load for all times up to time m.

Original language | American English |
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Pages (from-to) | 795-850 |

Number of pages | 56 |

Journal | Annals of Applied Probability |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2024 |

### Bibliographical note

Publisher Copyright:© Institute of Mathematical Statistics, 2024.

## Keywords

- Balls-and-bins
- load balancing
- two-choice
- two-thinning