On sequential selection and a first passage problem for the Poisson process

Abstract

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.

Diffusion approximations in the online increasing subsequence problem

“…with initial values X (0) = x 0 and L(0) = 0 will have the same distribution as (X, L). By virtue of the additive realisation through Π * , the online increasing subsequence problem is transformed into an online bin packing problem [10]; details of this equivalence are found in [14]. Here, the generic item of some size x observed at time t (an atom of Π * ) can be either packed or dismissed.…”

“…The above relations did not use the feasibility constraint. For feasible control we have p(1) < 1, hence from (8) follows the benchmark bound [1,5,[13][14][15] q(1) < 1, i.e. EL(t) <…”

“…The strategy with ψ(t, x) = √ 2/ν will be called stationary. It can be shown by a variational argument [14] that the stationary strategy achieves the maximum expected length over the class of strategies that satisfy the mean-value constraint EX (1) ≤ 1. This implies the benchmark bound EL(1) < √ 2ν for feasible strategies since they meet the mean-value constraint.…”