Some issues of the secretary problem with random lifetimes of scanned objects

O. K. Zakusylo
Theory of Stochastic Processes
Vol.29 (45), no.1, 2025, pp.118-128
We consider a variation of the secretary problem, in which n objects have random lifetimes after their scan. After scanning each object in turn, an observer either continues the process or chooses one of the scanned objects. In the latter case, the selection process ends, although the selected object may already disappear. We assume that its lifetime has a geometric distribution with parameter α. Unlike all previous formulations of the secretary problem, in which the scanning started with the first object, we investigate the optimal strategy of a belated observer who joined the selection process with a delay. In the final part of the article, we investigate the transient regime when α = α (n) → 1 as n → ∞. It turns out that the behavior of the threshold level L = L(n) can be arbitrary, as long as lim_{n → ∞} L/n ≥ 1/e.
DOI: https://doi.org/10.3842/tsp-2495432112-85
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