In short: A sixty day lead time and a lead time averaging sixty days across a forty-five to ninety-five day spread go into the same field, and most replenishment systems cannot tell them apart. What decides whether you stock out is demand over lead time, meaning the total that accumulates during the replenishment window rather than either quantity on its own. Ocean delays are bounded below by the physical transit and unbounded above, so the distribution is skewed and a normal approximation understates the tail that matters. Everything here depends on your system retaining the original promised date through a reschedule, which is the first field to check.
A sixty day lead time is a planning parameter. A lead time that averages sixty days and lands anywhere between forty-five and ninety-five is a different problem, and most replenishment systems cannot tell the two apart because both are entered into the same field.
For anyone importing across an ocean, that spread is usually the largest single source of risk in the plan. The demand forecast can be excellent and the service level still fails, because the container that was supposed to arrive in week 34 arrived in week 39 and the buffer was sized for an average that never happens.
What the variance actually does
The quantity that determines whether you stock out is demand over lead time, meaning the total demand that accumulates during the replenishment window rather than either quantity on its own.
When lead time is fixed, that is a sum of a known number of periods and the variance is simply lead time times demand variance. When lead time is itself random, a second term appears:
Variance of demand over lead time equals lead time mean times demand variance, plus demand mean squared times lead time variance.
Two things about that second term. It scales with the square of average demand, so it grows fast on high volume items. And it is entirely absent from the standard safety stock formula, which means every system using that formula is systematically under-buffering exactly the items where the exposure is largest.
Run the numbers on a typical imported line. Weekly demand of 2,000 units with a standard deviation of 400. Lead time averaging 10 weeks with a standard deviation of 2.5 weeks. First term: 10 times 160,000, or 1.6 million. Second term: 4 million times 6.25, or 25 million. The term being ignored is fifteen times the term being used, and the standard deviation of demand over lead time is about 5,150 units rather than the 1,265 the simple formula implies.
That is a missing variable rather than a tuning error, and no adjustment to the service level target corrects for it in a stable way.
The decomposition itself is old and settled. Hadley and Whitin gave it in Analysis of Inventory Systems in 1963, and Eppen and Martin returned to the question in Management Science in 1988 to show that plugging an average lead time into a normal safety stock formula produces errors that vary systematically with the item's demand level, so the mistake is not distributed evenly across the catalogue.
Getting the distribution rather than the two moments
The closed-form variance is a good improvement and it still assumes normality, which is shaky when lead times have a long right tail, as ocean freight reliably does. Delays are bounded below by the physical transit time and unbounded above, so the distribution is skewed and a normal approximation understates the tail you care about.
The exact approach is a convolution. If demand per period follows a probability mass function, demand over exactly L periods is that function convolved with itself L times. When lead time is random, take the mixture: weight each L-fold convolution by the probability that lead time equals L, and sum. The result is the true distribution of demand over lead time, with the skew intact.
This is computationally cheap on any realistic grid, and it removes the normality assumption entirely. Where demand does not have a clean parametric form, a Monte Carlo version does the same job by sampling a lead time, then sampling that many periods of demand, and repeating enough times to build the distribution empirically.
Either way, the output is a full distribution rather than a mean and a standard deviation, and that is what the ordering decision needs.
Turning the distribution into a date
Here is where the topic stops being statistical and becomes operational, because the actual question a planner faces is when to place the order.
Given a quantity that has to be on the shelf by a particular week, every candidate order date has an expected cost with three components.
Order too early and you pay for holding stock you did not need yet, plus the capital tied up in it, plus in some categories the shelf life you consumed while it sat there.
Order too late and you pay the expected cost of arriving after the requirement, which is the probability of a late arrival multiplied by whatever a shortage costs. That probability comes directly from the lead time distribution, and it is the reason the distribution matters rather than the average: the probability of being late when you order at the average lead time is roughly fifty percent, which is almost never the intended risk posture.
Between those, there is an expected total cost curve across order dates, and it has a minimum. That minimum is the recommendation, and the difference between it and the neighbouring options tells you how much the timing decision is worth. Sometimes the curve is flat and the decision does not matter much, which is useful to know because it frees attention for decisions that do.
Worth running the numbers on one line to see the shape of it.
The requirement is 5,000 units on the shelf in week 40. The item costs 20, so the lot is 100,000 of stock, and at a 25% annual carrying rate holding it costs 481 a week. Arriving late costs 4,000 a week, made up of expedited replacement and the margin on sales missed while short.
The lead time distribution measured on this lane, in weeks: 8 with probability 0.10, 9 with 0.20, 10 with 0.25, 11 with 0.20, 12 with 0.15, and 13 with 0.10. That averages 10.4 weeks.
Price each candidate order date, expressed as weeks before the requirement.
Ten weeks ahead, roughly the average: 0.40 expected weeks early and 0.80 expected weeks late, so 192 plus 3,200, or 3,392.
Eleven weeks ahead: 0.95 early and 0.35 late, so 457 plus 1,400, or 1,857.
Twelve weeks ahead: 1.70 early and 0.10 late, so 818 plus 400, or 1,218.
Thirteen weeks ahead: 2.60 early and nothing late, so 1,251.
Fourteen weeks ahead: 3.60 early, so 1,732.
The minimum sits at twelve weeks, with thirteen inside 3% of it, so the curve is genuinely flat across two weeks and the planner has a fortnight of latitude to spend on other constraints. Ordering at the average lead time costs 3,392 against an optimum of 1,218, nearly three times as much, and the whole of that difference comes from where the average sits relative to the lateness risk. Forty-five percent of orders placed ten weeks ahead on this lane arrive late.
The output is a date, a quantity, and the cost of the alternatives. A planner can act on that. A service level percentage requires them to do the translation themselves, usually under time pressure, usually by adding a week for safety.
Getting the lead time history
All of this needs a lead time distribution per supplier and per lane, and this is where most attempts stop, because the data is frequently missing.
You need the promised date and the actual receipt date for every purchase order, retained historically. Systems commonly overwrite the promised date when a supplier reschedules, which destroys the record of the original commitment and makes the reliability measurement impossible. If that is happening, fixing it is a small change with a large payoff, and it needs to happen before anything else here becomes possible.
Some practical notes on the measurement once you have it.
Segment before pooling. Lead times differ by supplier, by lane, by mode and by season, and a single pooled distribution mixing air and ocean is meaningless. Chinese New Year and the pre-holiday peak shift ocean transit times materially, so a seasonal split is usually warranted.
Distinguish supplier delay from transit delay. A supplier who ships late and a carrier who sails slowly both show up as a late arrival, and they call for entirely different responses. If you have the ship date, split the total into production lead time and transit lead time and model them separately.
Use enough history to see the tail, since the whole point is the right side of the distribution and thirty orders will not reveal a delay that happens once in twenty. Two years is a reasonable minimum, longer if order frequency is low.
Watch for censoring here too. Orders still in transit at the end of your data window have no receipt date and dropping them biases the sample toward the fast ones, which is exactly the wrong direction.
One more measurement trap sits inside the definition of the start date. Lead time recorded from purchase order creation includes however long the order waited for supplier confirmation, which is your own procurement process rather than the supplier's performance. The symptom is a distribution with a fat left shoulder, a long low bump of orders that ran slow for reasons the supplier never saw, and a supplier who disputes your scorecard with some justification.
The check is a scatter of lead time against the gap between order creation and supplier confirmation. Where they correlate, you are measuring two processes stacked on each other, and the fix is to record both timestamps and model them separately. It matters for the ordering decision, because the confirmation delay is yours to shorten and the transit spread is not.
What tends to come out of it
Three findings recur when businesses measure this for the first time.
Reliability varies far more between suppliers than anyone expected, and the variation does not correlate well with the supplier scorecard, because scorecards usually measure average on-time performance rather than spread. A supplier who is always four days late is easy to plan around. A supplier who averages on time by being three weeks early half the time and three weeks late the other half is expensive, and a conventional scorecard rates them highly.
The cheapest service improvement available is often reducing variance rather than reducing the mean. Cutting average lead time by a week is a commercial negotiation. Getting the supplier to be consistent is frequently a scheduling conversation, and the buffer reduction from halving the spread usually exceeds the reduction from shortening the average by the same amount.
And a portion of what gets recorded as demand variability turns out to be lead time variability wearing a different label, because late arrivals produce stockouts which produce lost sales which look like demand volatility.
Where this runs out
Lead time distributions are estimated from the past and the past includes disruptions that will not repeat and excludes disruptions that will. A distribution fitted through a period containing a canal closure or a port strike carries that event in its tail as though it were a recurring property of the lane. Judgement is required about whether to include those observations, and the honest answer is that both choices are defensible and you should state which you made.
The cost of a shortage also has to come from somewhere. Lost margin on the missed unit is the defensible floor. Anything beyond that is a judgement about customer relationships, and I would rather make that judgement explicit in a parameter than bury it in a service target. Businesses that refuse to estimate it are still making the estimate, just implicitly and without review.
Finally, none of this helps if the constraint is upstream capacity rather than transit. When a supplier is allocating scarce output, your lead time is a function of your priority in their queue, and the useful response is a commercial one rather than a statistical one.
Start with whether your system retains original promised dates. Everything in this piece depends on that one field surviving a reschedule.