Industry: Government and Public Sector Product: ModelRisk Application: Improving Healthcare Access through Probabilistic Modeling
A regional primary-care network is held to an access standard: a routine appointment within 18 days. With 42 provider full-time equivalents (FTE) seeing roughly 3,700 referrals a week, the planning spreadsheet looks healthy — average utilisation around 0.95, an expected wait of well under a day. On the deterministic numbers the region "meets the standard."
The ModelRisk rebuild modelled access as what it actually is — a queue — and found the average hides the problem. The median week does clear almost instantly, but the mean wait is 9.7 days, the 90th percentile is 32.9 days, and the 99th is 75 days. The probability that a patient waits longer than the 18-day standard is 20.4%. The network breaches its own access target roughly one week in five, and the spreadsheet's sub-day average gave no warning.
Queues are non-linear. When demand sits at 95% of capacity, a 5% bad week does not produce a 5% longer wait — it produces a backlog that takes weeks to clear, because the spare capacity available to absorb the surge is exactly the sliver between utilisation and 1.0. Feed mean demand and mean capacity into the model and you get the wait at the mean operating point, which is comfortable. But appointment waits are driven by the weeks where demand spikes and provider availability dips at the same time, and a point estimate averages those weeks away. The honest output is a right-skewed wait distribution, not a single number.
The headline distribution shows the wait a patient experiences across 60,000 simulated weeks. The mean (green line) is 9.7 days; the 90th percentile (VaR 90%) is 32.9 days and the 99th is 75 days. The black dotted line is the 18-day standard — 20.4% of the distribution sits beyond it. The mass near zero is the comfortable median week; the long right tail is the congestion the network is actually judged on.
The model is a backlog simulation, which is how primary-care access behaves in practice: a standing queue accumulates when a week's demand exceeds capacity and drains when it does not, and the wait a patient sees is the season-average backlog divided by weekly throughput — Little's Law, W = L / λ, expressed in days. Each trial runs a 26-week season.
W = L / λ
The inputs:
The resulting utilisation runs from 0.65 at P10 to 1.30 at P90 (median 0.92). Critically, demand exceeds capacity outright in 37.8% of weeks — and in those weeks the backlog grows rather than clears.
The staffing sweep re-runs the same demand and availability draws across provider counts from 38 to 58 FTE and reads off the breach probability. At the current 42 FTE the probability of exceeding the 18-day standard is 20.4%. To hold that breach probability at or below 10% the network needs about 48 FTE; to reach 5% it needs roughly 53 FTE. The curve is the business case: six additional FTE halve the breach rate, and eleven bring it to one week in twenty. Because the sweep reuses the actual simulated weeks, the recommended establishment is robust to the same demand surges that break the current one.
This chart is the reason the point estimate is dangerous. Each grey dot is a simulated week; the red curve is mean wait against utilisation. Below about 0.85, waits are near zero — there is slack to absorb noise. As utilisation climbs toward 1.0 (amber line, demand equal to capacity) the wait turns sharply upward, and in the 37.8% of weeks where utilisation exceeds 1 the mean wait is 25.1 days — past the standard. Wait and utilisation correlate at 0.90. A network planned to sit at 0.95 utilisation looks efficient and is one bad week from the steep part of the curve; the deterministic plan parks the region exactly where the cliff begins.
Swinging each input across its P10–P90 range, appointment-length throughput moves the mean wait by about ±10.5 days, weekly referral volume by ±7.7 days, and provider availability by ±2.5 days. The ranking surprised the network: shaving variability off consultation length — protecting delivered throughput — moves the wait more than managing referral demand does, because at high utilisation every minute of lost throughput pushes the system further up the congestion cliff.
The average wait answers "is the region usually fine?" Access policy needs "how often, and how badly, is it not?" Monte Carlo simulation in ModelRisk turns one expected wait into a breach probability and a staffing curve — and at 95% utilisation, the breach probability is the only number that matters.