Explore the warehouse model.
Change the assumptions behind the story, then inspect or download the calculated scenarios. This is an illustrative model — not observed operational data.
Model controls and result
DEMAND IS HELD FIXED WHEN CAPACITY IS ADDED
At 95% baseline utilization, a 30-minute complete stop accumulates 47.5 work units. Processing resumes at 100 units/h while arrivals continue at 95 units/h, leaving 5 units/h to clear the backlog. It is gone 10h 00m after the stop began.
SCENARIO U95-D30-C0 · EFFECTIVE UTILIZATION 95%
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
The model is normalized to a base capacity of 100 work units per hour. In the guided story that same capacity is shown as 1,000 orders per hour, so one model unit represents ten orders. Recovery times and percentages are identical either way.
Set the recovery time
you can live with.
Every scenario this model produces: 39 baseline utilization values (60–98%) × 4 stop durations × 4 capacity uplifts. Filter it, sort it, inspect any single record with its assumptions, load a row into the controls above, or download the whole thing as JSON or CSV.
| Record | |||||
|---|---|---|---|---|---|
| 60% | +20% | 50% | 1h 00m (60 min) | U60-D30-C20 | |
| 61% | +20% | 50.83% | 1h 01m (61 min) | U61-D30-C20 | |
| 62% | +20% | 51.67% | 1h 02m (62.1 min) | U62-D30-C20 | |
| 63% | +20% | 52.5% | 1h 03m (63.2 min) | U63-D30-C20 | |
| 64% | +20% | 53.33% | 1h 04m (64.3 min) | U64-D30-C20 | |
| 65% | +20% | 54.17% | 1h 05m (65.5 min) | U65-D30-C20 | |
| 60% | +10% | 54.55% | 1h 06m (66 min) | U60-D30-C10 | |
| 66% | +20% | 55% | 1h 07m (66.7 min) | U66-D30-C20 | |
| 61% | +10% | 55.45% | 1h 07m (67.3 min) | U61-D30-C10 | |
| 67% | +20% | 55.83% | 1h 08m (67.9 min) | U67-D30-C20 |
Data dictionary and units
- base_utilization_pct
- Incoming demand as a percentage of base capacity (100 work units/hour). 60–98.
- interruption_minutes
- Length of the complete stop, in minutes. 15, 30, 45 or 60.
- extra_capacity_pct
- Added processing capacity after restart. Demand is unchanged. 0, 5, 10 or 20.
- arrival_units_per_hour / effective_capacity_units_per_hour
- λ and μ in work units per hour.
- effective_utilization_pct
- λ ÷ μ × 100 — demand against capacity after the uplift.
- peak_backlog_units
- Work units waiting at the moment processing restarts.
- drain_minutes_after_restart
- Minutes from restart until the extra backlog reaches zero.
- total_recovery_minutes
- Minutes from the start of a complete interruption until the extra backlog returns to zero, including the interruption. Not individual shipment delay.