A warehouse stops
for 30 minutes.
Why does it take 10 hours to catch up?
Follow a hypothetical warehouse through a normal morning, a thirty-minute stop and the hours it takes to catch up.
A normal hour.
This warehouse can process 1,000 orders an hour. Usually, 950 arrive. That leaves room for 50 more.
Those 50 spare processing slots are what the warehouse can use to catch up when a disruption creates extra work.
At 9:00 a.m., processing stops.
Orders keep arriving. Nothing is processed for thirty minutes. By 9:30 a.m. there are 475 extra orders waiting.
Half an hour of incoming orders at 950 an hour adds 475 orders to the queue.
950 ORDERS/HOUR ARRIVING · 0 PROCESSED · 100% THROUGH THE STOP
The warehouse restarts. The backlog barely moves.
At 9:30 a.m. processing returns to 1,000 orders an hour. New orders still arrive at 950 an hour. There is only room to reduce the backlog by 50 orders an hour.
The backlog shrinks by the difference between what the warehouse processes and what keeps arriving: 1,000 minus 950, or just 50 orders an hour.
475 EXTRA ORDERS ÷ 50 PER HOUR = 9 HOURS 30 MINUTES AFTER THE RESTART
- 9:00 A.M.Processing stops
Orders continue to arrive at 950 an hour.
- 9:30 A.M.Processing restarts
475 extra orders are waiting. Spare room: 50 an hour.
- 7:00 P.M.Backlog cleared
Ten hours after the stop began, the extra orders are gone.
Recovery is complete when the extra backlog returns to zero.
Half an hour of lost work takes the rest of the day to absorb.
Almost all of the warehouse's capacity is already needed to keep up with incoming orders.
What changes with a little more capacity?
Keep arrivals at 950 orders an hour. Give the warehouse a little more processing capacity after the restart.
INCOMING DEMAND FIXED AT 950 ORDERS/HOUR · STOP FIXED AT 30 MINUTES
With 1,000 orders/hour of processing and 950 still arriving, there is room to reduce the backlog by 50 orders an hour. The same 475 extra orders take 9h 30m to clear after the restart — 10h 00m from the moment the stop began, finishing at 7:00 p.m.
Fifty extra processing slots an hour doubles the room available to reduce the backlog, from 50 orders an hour to 100. At the same demand, a 5% capacity increase nearly halves the total recovery time in this simplified example.
The room left over decides the day.
The room left over in a normal hour determines how quickly this warehouse can catch up after a disruption.
Capacity planning has to account for both normal demand and the time needed to recover from interruptions. Real decisions also depend on costs, changing demand and differences in the work each order requires.
Model, assumptions and source
The model
μ₀ = 100 work units/hour (base capacity)
λ = baseline utilization % → units/hour
μ = 100 + added capacity %
D = stop duration in minutes
Peak backlog Q = λ × D / 60
Drain = Q / (μ − λ) × 60 minutes
Total recovery = D + drain
Queue(t) = λt/60 for t ≤ D, else max(0, Q − (μ−λ)(t−D)/60)
Total recovery is measured from the start of the interruption until the extra backlog returns to zero. It is not the waiting time of an individual shipment.
The split between keeping up with new arrivals and reducing the backlog describes net flow only. The model says nothing about the order in which individual items are picked.
Assumptions
- One bottleneck with constant incoming demand, zero initial backlog and a continuous-flow approximation.
- Processing stops completely during the interruption. Arrivals continue.
- Full effective capacity resumes immediately when the interruption ends.
- Extra capacity increases processing capacity, while incoming demand remains fixed at the baseline rate.
- No further outages, demand changes, cancellations, finite-buffer blocking or downstream constraints.
Provenance
Illustrative capacity model. Modeled scenario outputs — not observed terminal data, not reporting, and not a proprietary market dataset. Costs and real operating constraints are not modeled, so no economically optimal capacity level or financial saving is claimed.
Mathematical background: Cathy Wu, MIT 1.041/1.200, Graphical analysis II: Cumulative diagrams, Spring 2026. The scenario values and the recovery formula here are Gettier's own derivation using flow conservation and a deterministic queue; MIT neither supplies nor endorses this dataset.
Units in this story
The model is normalized: base capacity is 100 work units per hour. This story shows the same model at ten times that scale, so 100 units/hour reads as 1,000 orders/hour and 95% demand reads as 950 orders/hour. Percentages and all times are unchanged.