OPS.15.1:4 - Solution

Define the quantity from the operating question, reconstruct the events and their subjects, derive the intervals or count changes, and then aggregate. Keep incomplete observation visible in the result. Return the number with enough meaning to use or recompute it.

OPS.15.1:4.1 - Choose what the decision needs to measure

Name the subject and the consequence being compared. A customer’s time to an accepted result, a machine’s occupied time and the number of service visits answer different questions even when they concern the same order.

Define membership of the counted population. State when a subject enters and leaves that population. For unfinished orders, choose the arrival and completion events for the service being considered. For a ready queue, include the conditions that make the next operation possible.

Separate unique cases from visits. A return for correction may create another visit to a station while the same customer case remains unfinished. If the case is closed and later reopened, decide whether the receiving question concerns separate episodes or one extended service. Use its operating meaning; do not decide from the number of log rows.

OPS.15 supplies the decision-specific account, and OPS.3 helps identify its subjects. A short statement of the chosen quantity is enough when it removes the ambiguity.

OPS.15.1:4.2 - Relate the records to the operating events

For the events used in the calculation, recover the subject, event meaning and occurrence time as far as the available information supports them. Distinguish occurrence time from recording time when their difference affects the result. Reconcile time zones, clock offsets and timestamp precision before comparing events from different systems.

Recover the relationships needed by the question. A dispatch can concern several order lines; one machine cycle can process several parts; one invoice can cover charges for several deliveries. Keep the common event identifiable while following each related subject. A row per event-subject relation is a useful representation, but its row count is not automatically an event count.

Check what the recorded label establishes. “Completed” may mean that a worker reported completion, that inspection accepted the result or that the customer received it. If the receiving comparison needs a later event, either obtain it or report the result under the available boundary.

Use MMP.7 when selection, censoring or the recording procedure affects an inference. Missing records do not by themselves show that the corresponding work did not occur.

OPS.15.1:4.3 - Construct the intervals and count changes

For each relevant subject or visit, pair the boundary events belonging to that episode. Its elapsed residence is end time minus start time. If only a range for an endpoint is known, propagate that range instead of selecting an unsupported point.

For resource use, reconstruct the intervals during which the resource is occupied under the chosen meaning. Occupied machine time, operator attention and reservation can differ. A machine can run unattended while its operator serves another machine.

Combine intervals according to the quantity. For a single resource’s occupied fraction, take the duration of the union of its occupied intervals; overlapping descriptions of the same occupation are counted once. To measure total occupied resource-hours across distinct resources, sum those resource-specific durations. To measure part residence during batch processing, retain each part’s interval.

Calendar time and available working time are also different quantities. For working time, intersect the interval with the relevant open-calendar intervals before adding durations. A calendar conversion does not remove a customer’s overnight wait from elapsed service time.

For counts, follow entries and exits of the chosen population. With fixed case identity and no splitting or merging, ending population equals starting population plus entries minus exits. Include cancellations or other exits under their own meanings. If the subject is split or combined, recover the changed counting unit before using that balance.

OPS.15.1:4.4 - Respect the observation window

Choose a positive-duration observation window for a time-average quantity. Include the subjects already present at its beginning and those still present at its end.

For a subject entering at a and leaving at d, its contribution within the window [u,v] is:

overlap = max(0, min(d,v) - max(a,u)).

For a subject known to remain present through v, use v as the overlap endpoint without asserting completion at v. An unknown arrival before u similarly contributes from u while leaving the full residence unresolved.

Add these overlaps to obtain the population-time area. Divide by v-u to obtain the mean population during that window. This is the area under the population count: each present subject contributes one unit for each unit of time it remains present.

For a fixed counting unit without splitting or merging, an empty-to-empty interval gives a useful relation. Its population-time area equals the sum of residence times of all episodes that leave. With at least one departure, mean population equals departure rate multiplied by mean residence of those same episodes. The departure rate is the successful completion rate only when every departure meets the chosen completion condition. If no episode occurs, the area and mean population are zero; a mean residence over zero departures is undefined.

For example, three cases enter at zero. One is cancelled at hour one; the others complete successfully at hours two and three. Area is 1+2+3=6 case-hours over three hours, so mean population is two. All three departures contribute to that relation. The two successful completions remain a separate result count.

On an arbitrary cut window, the departed-episode mean and departure rate alone generally omit the partial intervals. Calculate each present episode’s overlap with the window, including episodes still unfinished at its end.

This finite observation identity does not estimate an unseen tail, establish a long-run regime or explain what caused waiting. OPS.10 uses the appropriate capacity or queueing model when those further questions matter.

OPS.15.1:4.5 - Aggregate only compatible observations

Keep the population, event pair, clock and weighting rule consistent within the reported aggregate. Combine totals before dividing when group sizes differ. An unweighted mean of departmental means gives each department equal weight, not each case equal weight.

Choose the weighting from the question. A case-weighted mean describes a typical counted case. Resource-demand weighting describes a different burden. Retain classes when the proposed change affects them differently or an aggregate conceals a service commitment.

To compare policies, use the same recipient boundary unless changing that boundary is the point of the decision. If internal admission moves later, report upstream waiting together with internal residence before concluding that customers receive results sooner.

For uncertain or incomplete observations, derive the consequence still supported. A residence known to be between four and ten hours cannot decide an eight-hour criterion. It can already satisfy an upper limit of twelve hours under the same assumptions. Choose further observation through C.11.DUA; a complete history is not a prerequisite for every comparison.

OPS.15.1:4.6 - Return the quantity to the operating decision

Explain the result in terms the recipient uses: for example, “the machine was occupied for forty of the sixty available minutes” or “the shorter internal time excludes four hours of waiting before admission”.

Retain the definition and observations needed to reproduce a consequential quantity, using the existing account when it suffices. Distinguish a direct calculation from a model-based estimate or forecast. MMP.13 supplies inference when the result must generalize beyond the observed population.

When a check fails, localize the repair: a subject was counted twice, a boundary event was misidentified, a clock was shifted, an unfinished interval was excluded, or the aggregation rule answered another question. Correct that construction and the decisions that used it. A changed route, acceptance event or recording procedure reopens the affected quantity.