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.