Generalizing the batch-to-M/G/1 translation to partial information

Develop a proof that extends the batch‑relaxation argument translating optimality of weighted discounted policies to steady‑state tail optimality in the M/G/1 to the partial‑information setting, resolving the technical obstacle arising from dependencies among job sizes within busy periods so that a partial‑information analogue can be established.

Background

The full‑information proof leverages a reduction to finite batches (busy periods) and optimality of a weighted discounted single‑machine policy, then translates it back to the M/G/1 to obtain strong tail optimality.

In the partial‑information setting, the authors identify that instances arising from busy periods induce subtle dependencies among job sizes that break the standard batch optimality‑to‑M/G/1 translation, and explicitly note their inability to generalize this part of the argument.

References

Above, we have focused on the full-information case, and for good reason: we have not been able to generalize part of this argument to the partial-information case. The issue has to do with a subtle difference between the traditional stochastic batch setting \citep[Section~10.1]{pinedo_scheduling_2016}, which assumes independent job sizes, and the instances that arise from busy periods, which can have subtle dependencies between jobs' sizes (\cref{sec:reduction}).

— Strongly Tail-Optimal Scheduling in the Light-Tailed M/G/1 (2404.08826 - Yu et al., 12 Apr 2024) in Section 1.7 (Technical challenge: translating from the batch relaxation to the M/G/1)