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.

翻译：我们引入一个有趣的问题，可视为经典生日问题的一个变体——瓶颈生日问题（BBP）。该问题表述为：至少需要选取多少人，才能使得一年中任何一天发生生日事件的人数不超过r≥1的概率不低于1/2？我们综述了文献中用于占据和负载均衡问题的技术，以推导精确计算概率的递推关系，以及保持概率固定于阈值时的人数。进一步，我们以新颖方式证明，限制性第二类斯特林数可用于推导另一个递推关系。我们提供了递推关系实现的复杂度比较与数值结果。