This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.
The author is grateful Alex Marynych for a number of illuminating discussions. Thanks go to an anonymous referee for a pointer to the literature and contructive comments that lead to improvement of the presentattion.
"On sequential selection and a first passage problem for the Poisson process." Electron. Commun. Probab. 26 1 - 13, 2021. https://doi.org/10.1214/21-ECP377