A reserved instance is paid for whether it runs or not, so it pays off exactly when it runs more than reserved price ÷ on-demand price of the time. Applied layer by layer to a load profile, that threshold picks the level to reserve: the floor the profile spends most of its day above, with everything on top of it bought on demand.
Core ideas
The threshold is a single division. The work is in applying it to a layer of the fleet rather than to the fleet as a whole.
Reserved costs r an hour for every hour of the term whether or not it serves anything; on demand costs d an hour only when it runs. Over any period, reserving is cheaper exactly when the instance runs more than u* = r/d of it. At 7¢ and 12¢ that is 7/12, or 58.3% of the hours — and nothing else about the profile enters the threshold.
The decision is taken one instance at a time, from the bottom of the profile upward. The first instance runs all twenty-four hours and is an easy reservation; the thirty-sixth runs for one hour and is an obvious mistake. Somewhere between them the share of hours above the level falls through u*, and that level is the optimum.
Raising the reserved level by one costs 24r a day and saves d for every hour the profile is above the new level. So raise while hours above × d > 24r, which is exactly hours above / 24 > u*. Costing every level from zero to the peak gives the same answer — 12 here — and the two agreeing is the check.
The material
Where u* comes from, the marginal argument that turns it into a level, the two ways of getting that level wrong, and an interruption model for spot.
Let r be the price of a reserved instance-hour, charged for every hour of the term, and d the price of an on-demand instance-hour, charged only when the instance runs. Over a term of H hours an instance that runs uH of them costs rH reserved and duH on demand, so reserving is cheaper exactly when
u > u* = r/d.
The term length H cancels, so u* is a property of the two prices alone.
7¢ against 12¢ gives u* = 7/12 = 58.3%. Read it as a duty cycle: an instance that will be up for more than fourteen hours of every twenty-four should be reserved, and one that will not should not be. Every other question on this page is which instances those are.
Stack the profile as layers. Layer k is the k-th instance, and it runs during exactly those hours when demand is at or above k. Its utilisation is therefore the share of hours the profile spends above k, which falls as k rises — so there is one crossing, and it is the level to reserve up to.
The marginal rule is worth preferring because it never evaluates a total bill. It compares one day of reservation against one day of the hours that reservation covers, and it is the same argument that picks an order quantity or a staffing level: raise while the margin is positive, stop when it turns.
This profile uses 432 instance-hours a day and peaks at thirty-six instances. All on demand: 432 × $0.12 = $51.84. Reserved to 12: 288 reserved hours at $0.07 plus 144 on-demand at $0.12, which is $20.16 + $17.28 = $37.44, saving $14.40 a day — 27.8%. Reserved to the peak of thirty-six: 864 × $0.07 = $60.48, which is $8.64 worse than reserving nothing at all.
The peak hour’s instances run for one hour in twenty-four, a utilisation of 4.2% against a threshold of 58.3%. Reserving them buys twenty-three idle hours each, at full price, every day of the term. The instinct behind it is a good one — the peak is the part that hurts — but the peak is precisely the part with the worst duty cycle, and a commitment is a bet on hours rather than on importance.
The opposite error is quieter. Reserving nothing leaves the bottom twelve instances, which run every hour of every day at 100% utilisation, paying the on-demand premium for capacity that is never not needed. That is the $14.40 a day the optimum recovers.
A spot instance-hour at s can be reclaimed part-way through, and the work still has to be done, so an interrupted hour is re-bought on demand. With a reclaim probability q the expected delivered cost is s + q·d, which at s = $0.04, d = $0.12 and q = 15% is $0.058 — genuinely below $0.12, but nowhere near the headline $0.04.
So spot is a third arm on the same comparison rather than a free lunch: it wins while interruptions are rarer than two in three, and it is the burst layer — the one with the utilisation too low to reserve — that it belongs on.
Everything above assumes the profile is known. When it is a forecast with a distribution around it, the quantity to commit to is a quantile of that distribution rather than a level of a known shape, which is the newsvendor problem — solved in “The Newsvendor Problem” in the Operations Research course “Inventory Models”, with the under-commitment and over-commitment costs in the shortage and overage roles.
Interactive
Reserve the floor, buy the spikes, and price spot's interruptions
Method
The threshold first, because it is one division and it decides how the profile is read.
u* = r/d. That is the duty cycle a commitment needs to pay for itself, and it is independent of the term, the profile and the size of the fleet. At 7¢ and 12¢ it is 58.3%, or just over fourteen hours a day.
For each level k, count the hours the profile is at or above k. That count divided by the period is the k-th instance’s utilisation, and it is what the threshold is compared against.
Stop at the first level where it does not. Here that is 12, with 45.8% of hours above it against a threshold of 58.3%. No total has been computed at any point in this step.
The optimum, reserving nothing, and reserving the peak: $37.44, $51.84 and $60.48 a day. The third being worse than the second is the number that makes the argument, and it is invisible if only the recommendation is priced.
s + q·d is what a delivered spot hour costs, and q < (d − s)/d is the condition for it to beat on demand at all. Put it on the burst layer, which is the layer whose utilisation was too low to reserve.
The profile uses 432 instance-hours a day and peaks at thirty-six instances. The threshold is computed once and then applied upward.
The +$8.64 line is the one to remember. Reserving the peak is not a cautious version of reserving the optimum — it is worse than making no commitment at all, because the instances it commits to are the ones that run for an hour a day.
The spot block shows why an interruption model is not optional. The sticker price is a third of on demand; the delivered price is under half of it; and at a reclaim rate of two in three the two are equal and spot stops being cheap at all. Quoting $0.04 without q is quoting a price for something you may not receive.
For a faded attempt, raise the reserved price to 10¢ and leave on demand at 12¢. Compute the new u* first and predict from it alone whether the optimal level rises or falls, then find the level and the three daily bills in the lab and say how much of the original saving survived.
You should be able to compute u* = r/d, turn a profile into hours-above-a-level, pick the reserved level by the marginal rule and confirm it by scanning every level, and price both the reserve-nothing and reserve-the-peak plans beside the one you recommend.
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