Connection window: the next sailing sets the price of a miss

The formula, in words

• Everything is measured backwards from the departure. The feeder's arrival is hours before it, the cutoff is hours before it, and the transfer is a duration in between. The box reaches the stack at the arrival minus the transfer, it clears when that is at or before the cutoff, and the number the page calls slack is the arrival minus the transfer minus the cutoff. Three quantities owned by three different parties collapse into one, which is why the page prints it: only you know which of the three you can move.
  $$\mu=A−T−C$$
  where $\mu$ is the mean slack, $A$ the mean arrival hours before departure, $T$ the mean transfer hours, and $C$ the cutoff hours before departure.
• The spreads add in squares. Two independent spreads combine as the square root of the sum of their squares, so 6 and 8 give 10 rather than 14. The consequence is that the wider leg dominates, and it dominates more than its share: taking the 6 to zero buys 2 hours of combined spread, and taking the 8 to zero buys 4. This is the same arithmetic the lesson on variance runs on a supplier's lead time, arriving here from the other end.
  $$\sigma=\sqrt{\sigma_A^2+\sigma_T^2}$$
  where $\sigma$ is the combined spread, $\sigma_A$ the arrival spread, and $\sigma_T$ the transfer spread, in hours.
• The share that clears is a property of your numbers, not of anybody's service. It is the standard normal function of the slack divided by the combined spread. On the numbers this page opens with that is $\Phi$ of 1.00, which is 84.1%. The function is evaluated with a rational approximation accurate to better than one part in ten million, so the third digit is real. What is not real is any claim about a connection you have not measured.
  $$p=\Phi\left(\frac{\mu}{\sigma}\right)$$
  where $p$ is the share that clears and $\Phi$ the standard normal distribution function.
• A miss is priced at one interval, and that is an assumption. A box that misses waits for the next sailing, so the expected added delay is the share that misses multiplied by the days to that sailing. It says nothing about a second miss, a full sailing or a cancelled one. It is a straight line through the origin, which means the interval is a multiplier on everything else and it is the only number here that nobody at the terminal controls.
  $$D=(1−p)I$$
  where $D$ is the expected added delay per box and $I$ the sailing interval, in days.

A worked example

The inputs. The feeder arrives 36 hours before departure on average, varying by 6. The terminal takes 14 hours, varying by 8. The cutoff is 12 hours before departure. The next sailing is 7 days later. Two hundred boxes.

Ten hours of slack, and ten hours of spread. Six and eight came to ten, not fourteen: the two spreads together are four hours less than their sum, and they always are, because the only way two independent spreads combine is through their squares.

1.1 days added, averaged across every box on the connection, including the 84.1% that never waited a minute. That is the number in the finding, and it is the number that is worth carrying because it is the one that changes when nothing at the terminal changes at all.

The interval is the multiplier, and nobody at the terminal owns it. Hold everything and move only the interval:

The same connection, twenty one times worse. On a daily service, sixteen boxes in a hundred miss and the whole cargo is four hours later on average. At three weeks between sailings the same sixteen boxes cost more than three days across everything. Nothing about the feeder changed, nothing about the terminal changed, and the difference is a timetable.

The cutoff is the slope. Now hold the interval at seven days and move only the cutoff:

Six hours at the cutoff bought 0.7 days a box. Six hours the other way cost 1.3. The curve is not symmetric about the visitor's position, and it steepens on the wrong side, which is what a normal always does once the slack falls inside one spread of zero.

Moving the feeder's mean six hours earlier does exactly the same thing, because the slack is one number made of three and it does not care which of them moved. The page prints the slack for that reason.

The check that makes it stick, and it is the whole page in one move. A perfect feeder arrives at exactly 36 hours before departure every single time.

Removing six hours of feeder spread took two hours out of the combined ten. Clearance went from 84.1% to 89.4%, and better than ten boxes in a hundred still miss with a feeder that is never once late.

Removing eight hours took four hours out of the ten, and clearance went to 95.2%.

In the worked example, removing feeder spread raises clearance by 5.3 percentage points; removing terminal spread raises it by 11.1 points.

The two numbers worth carrying

1. 1.1 days a box, of which none is anybody's transit time. It is the price of a share that misses, and the price is set by an interval nobody at the terminal publishes.
2. The combined spread is 10 hours from a 6 and an 8. The sum is a number that does not appear anywhere in the arithmetic and is the number most people would have used.

Check it yourself

A feeder that is never once late still leaves better than ten boxes in a hundred missing. The same perfection on two legs is worth twice as much on one of them, and which one is decided by nothing but which number was bigger before you started.

What this does not model

Correlation between the two legs, which is common and always makes the combined spread wider than this page draws; a second miss, and therefore any capacity on the next sailing; a cancelled or blank sailing; a box given priority over another rolled box; the difference between operational, transactional and storage time inside the transfer figure, which have different causes and different remedies and are collapsed here into one duration; any distinction between what a hub does and what a gateway does, the second involving a change of mode with its own constraints; congestion, which makes the transfer spread depend on how many boxes are already there rather than staying a fixed number; the shape of a real transfer time, which has a floor and a long tail where a normal has neither; the direct routing this connection was created instead of, and its own frequency and waiting; what a day costs, in any currency; detention, demurrage, storage and every other contract clock, all of which start on different events and none of which is a transit time; and the documentary and release processes that can hold a box after every physical movement has finished.