Given two independent Poisson processes 𝒜 and ℬ with rates respectively α and β (α < β) we match each point of 𝒜 with the closest point of ℬ that has not already been matched. The points of 𝒜 are taken in random order. It is shown that the point process of unmatched points of ℬ is a renewal process with the same interval distribution as the busy period of an M/M/1 queue. The distribution and moments of the distance between a typical point of 𝒜 and the corresponding matched point of ℬ are obtained. Variants of the matching process in which the assigned point of ℬ must lie to the right of the point of 𝒜, and in which the matching distance must be less than a fixed tolerance are studied. The use of matched samples to control for bias in observational studies is discussed.
