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.
Highlights
When a shuttle carrying a large number of hotel guests arrives at the hotel, the passengers start queuing and pass the exit door at times of a Poisson process
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
Summary
When a shuttle carrying a large number of hotel guests arrives at the hotel, the passengers start queuing and pass the exit door at times of a Poisson process. Since the marks sorted into increasing order themselves comprise a homogeneous Poisson process, zooming in the marks scale with factor t and changing the metaphore, it is seen that the number of items packed under the smallest first policy coincides with the exit count N (t) from the shuttle problem we started with. The formula seem to require substantial analytic work to extract the desired second term of the asymptotic expansion We circumvent this by resorting to elementary probabilistic tools, with the core of our approach being the observation that N (t), for each fixed t, has the same distribution as the entrance count M (t) appearing in the following dual shuttle entrance problem. Throughout we shall be using the notation ν(t) := EN (t), σ2(t) := VarN (t)
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