MMP.8.SD:5.1 - Reserve a resource for an observed later request

A portable power unit has enough energy to serve one task in either of two slots, with no recharge. Serving the current task in slot 1 yields 4 units on an already selected gain scale. In slot 2 a different task arrives with probability 0.6 and yields 10 if served. Arrival is observed before the slot-2 decision. Unused energy has terminal value zero; the current task cannot be deferred. Serving consumes the whole unit.

Let b be remaining energy, either 0 or 1, and d indicate the later arrival. At slot 2,

V_2(b,d) = 10*b*d.

Use the retained state (time, remaining energy, current request). The slot-1 comparison is

serve now: 4 + 0 = 4
reserve:   0 + 0.6*10 + 0.4*0 = 6.

The resulting policy reserves energy, then serves if the request arrives. It has larger expected gain than immediate service, although it yields zero when no request arrives. If the requirement is a gain of at least 4 in every admitted case, serving now meets it and reservation does not.

Two histories can show the same later request but differ in whether the energy was already used. Merging them would make the slot-2 service appear available after both histories. Retaining b prevents that error.

Suppose the arrival probability changes to 0.3, with the other premises unchanged. Reservation now gives 3 and immediate service gives 4, so the first action changes. If instead the only available probability statement is 0.55≤p≤0.65, reservation gives expected gain between 5.5 and 6.5 and exceeds 4 throughout. Refining p is unnecessary for that comparison.