Resource time: preserve the unit before multiplying a rate

Resource-time counts simultaneous allocations rather than elapsed job duration alone. Four devices held for one hour contribute four device-hours even if their kernels run for only half that hour. Include setup, loading, evaluation, and idle residence when the chosen accounting boundary holds those devices throughout. An autoscaled job needs the integral of its changing resource count, or a sum over intervals with constant counts. Keep CPU core-hours, device-hours, memory-capacity time, and storage-capacity time separate until an explicit conversion or rate justifies combining them.

Use event timestamps from the allocation lifecycle and document whether intervals include their start and exclude their end. Overlapping worker logs must not count one shared allocation repeatedly. Conversely, summing job wall times can miss simultaneously allocated devices. Kernel utilization helps explain why allocated capacity produced little useful work, but it does not replace the allocation inventory. Retain both measurements with their identities and clocks so an apparent optimization can be traced to shorter residence, fewer devices, more useful execution, or merely a changed boundary.

A hypothetical service can charge each allocation with a minimum duration and a subsequent billing quantum. Apply that rule to each eligible lifecycle before aggregation. Rounding total fleet time can understate a bill when many short allocations each incur a minimum. Conversely, rounding every task independently would overstate it if tasks legally share one continuing billed allocation. Cancellation, suspension, and restart may define new lifecycles under a real contract. Read those rules rather than infer them from a smooth utilization graph or a convenient arithmetic simplification.

The formula here is one deliberately specified policy: a positive quantum, a minimum duration, and upward rounding of the larger duration for each allocation. It is not a universal cloud-billing equation. Convert seconds to hours only after using consistent units inside the rounding operation. Different resource classes can have different units and policies, so aggregate their computed costs rather than their raw quantities. A smaller number of requests does not establish a smaller bill unless it changes a quantity or rule that actually carries a charge.

A billing export can distinguish raw usage amount from amount in pricing units, a usage interval from an invoice period, and a resource identifier from an optional or missing field. Fee, usage, credit, refund, and adjustment records may follow different semantics. Multiplying every row by an apparent rate can duplicate an already calculated cost or mishandle a fee with no usage quantity. Preserve charge type, rate version, currency or accounting unit, and record identity; reconcile one documented basis instead of making every column look interchangeable.

The function lab provides a continuous resource-count analogy. Reset, explicitly set a=0.75 and x=2, predict the integral, then Play or Step. Repeat from a fresh Reset with a=1.5 and x=2 explicitly set. For the normalized curve n(t)=a t² from zero to x, displayed areas are two and four. They are normalized count-time units, not device-hours inferred from a real cluster. The instrument has no allocations, billing minima, invoice records, or rounding. Its useful lesson is why an area and an endpoint count are different quantities.