The Bottleneck Birthday Problem
This is a fun problem with limited practical impact; it is an incremental contribution to combinatorial probability.
The paper introduces the Bottleneck Birthday Problem (BBP), a variant of the classic birthday problem, and provides recurrence relations for exact computation of probabilities and number of people, using restricted Stirling numbers of the second kind in a novel way. Numerical results and complexity comparisons are presented.
We introduce a fun problem that can be considered as a variant of the classic birthday problem, the Bottleneck Birthday Problem (BBP). It is stated as: what is the maximum number of people we have to choose so that no day of the year has more than r >= 1 birthdays incident on it with probability at least 1/2? We provide a survey of techniques used in the literature on occupancy and load balancing problems to derive recurrence relations for exact computation of the probability, and the number of people keeping probability fixed at a threshold. Further, we show that restricted Stirling numbers of the second kind can be used to derive an additional recurrence, in a novel way. We provide complexity comparisons and numerical results from an implementation of the recurrences.