  We revisit the problem of maximising the expected length of increasing
subsequence that can be selected from a marked Poisson process by an online
strategy. Resorting to a natural size variable, the problem is represented in
terms of a controlled partially deterministic Markov process with decreasing
paths. Refining known estimates we obtain fairly complete asymptotic expansions
for the moments, and using a renewal approximation give a novel proof of the
central limit theorem for the length of selected subsequence under the optimal
strategy.
