Load balancing under d-thinning

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 Θ(n) balls have been √ allocated, the least maximum load achievable with high probability is (Forumala Prsrnted). This should be compared log log n with the related d-choice setting, in which the optimal maximum load achievable with high probability is (Forumala Prsrnted).