Medical Service Wait Time Estimator
ANA›Life Services Authority›National Calculator Authority›Medical Service Wait Time Estimator
.calc-container { max-width: 640px; margin: 2rem 0; padding: 1.5rem; background: #fff; border: 1px solid #ddd; border-radius: 8px; box-shadow: 0 1px 3px rgba(0,0,0,0.06); font-family: system-ui, -apple-system, sans-serif; } .calc-container h3 { font-family: Georgia, serif; font-size: 1.15rem; color: #1a1a1a; margin-bottom: 1rem; padding-bottom: 0.5rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-row { display: flex; align-items: center; gap: 0.75rem; margin-bottom: 0.75rem; flex-wrap: wrap; } .calc-row label { min-width: 160px; font-size: 0.9rem; color: #333; font-weight: 500; } .calc-row input[type="number"], .calc-row select { flex: 1; min-width: 120px; max-width: 200px; padding: 0.5rem 0.6rem; border: 1px solid #ccc; border-radius: 4px; font-size: 0.9rem; font-family: system-ui, sans-serif; color: #1a1a1a; background: #fafaf8; } .calc-row input:focus, .calc-row select:focus { outline: none; border-color: var(--ac, #3d5a80); box-shadow: 0 0 0 2px rgba(26,74,138,0.12); } .calc-row .unit { font-size: 0.82rem; color: #888; min-width: 30px; } .calc-btn { display: inline-block; margin-top: 0.5rem; padding: 0.55rem 1.5rem; background: var(--ac, #3d5a80); color: #fff; border: none; border-radius: 4px; font-size: 0.9rem; font-weight: 600; cursor: pointer; font-family: system-ui, sans-serif; } .calc-btn:hover { opacity: 0.9; } .calc-result { margin-top: 1.25rem; padding: 1rem 1.25rem; background: #f0f6fc; border-left: 3px solid var(--ac, #3d5a80); border-radius: 0 6px 6px 0; display: none; } .calc-result.visible { display: block; } .calc-result-label { font-size: 0.78rem; text-transform: uppercase; letter-spacing: 0.06em; color: #666; margin-bottom: 0.25rem; } .calc-result-value { font-size: 1.6rem; font-weight: 700; color: var(--ac, #3d5a80); } .calc-result-detail { font-size: 0.85rem; color: #555; margin-top: 0.5rem; line-height: 1.5; } .calc-note { margin-top: 1rem; font-size: 0.8rem; color: #888; font-style: italic; } .calc-grid { display: grid; grid-template-columns: 1fr 1fr; gap: 0.75rem; margin-top: 0.75rem; } .calc-grid-item { padding: 0.6rem 0.8rem; background: #f8f9fa; border-radius: 4px; border: 1px solid #eee; } .calc-grid-item .label { font-size: 0.75rem; color: #888; text-transform: uppercase; letter-spacing: 0.04em; } .calc-grid-item .value { font-size: 1.1rem; font-weight: 600; color: #1a1a1a; } @media (max-width: 720px) { .calc-row { flex-direction: column; align-items: flex-start; gap: 0.3rem; } .calc-row label { min-width: auto; } .calc-row input[type="number"], .calc-row select { max-width: 100%; width: 100%; } .calc-grid { grid-template-columns: 1fr; } } .calc-chart { margin: 1rem 0; text-align: center; } .calc-chart svg { max-width: 100%; height: auto; } .calc-chart-legend { display: flex; flex-wrap: wrap; justify-content: center; gap: 0.6rem 1.2rem; margin-top: 0.6rem; font-size: 0.8rem; color: #555; } .calc-chart-legend span { display: inline-flex; align-items: center; gap: 0.3rem; } .calc-chart-legend i { display: inline-block; width: 10px; height: 10px; border-radius: 2px; font-style: normal; } .calc-related { max-width: 640px; margin: 2rem 0 1rem; padding: 1.25rem 1.5rem; background: #f8f9fa; border: 1px solid #e8e8e8; border-radius: 8px; } .calc-related h3 { font-family: Georgia, serif; font-size: 1rem; color: #1a1a1a; margin: 0 0 0.75rem; padding-bottom: 0.4rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-related-list { list-style: none; padding: 0; margin: 0 0 0.75rem; display: grid; grid-template-columns: 1fr 1fr; gap: 0.4rem 1.5rem; } .calc-related-list li a { font-size: 0.88rem; color: var(--ac, #3d5a80); text-decoration: none; } .calc-related-list li a:hover { text-decoration: underline; } .calc-browse-all { margin: 0.5rem 0 0; font-size: 0.9rem; font-weight: 600; } .calc-browse-all a { color: var(--ac, #3d5a80); text-decoration: none; } .calc-browse-all a:hover { text-decoration: underline; } @media (max-width: 720px) { .calc-related-list { grid-template-columns: 1fr; } }
Medical Service Wait Time Estimator
Estimates expected patient wait time using the M/M/c Erlang-C queuing model, which models random patient arrivals and random service times across multiple providers.
Patient Arrival Rate (patients/hour)
Average number of patients arriving per hour (λ)
Service Rate per Provider (patients/hour)
Average number of patients one provider can serve per hour (μ)
Number of Providers (doctors/nurses)
Total number of parallel service providers (c)
Average Service Time (minutes)
Average time to serve one patient in minutes (overrides service rate if changed)
Estimate Wait Time
Fill in the fields above and click Estimate Wait Time.
function medCalc() { const resultDiv = document.getElementById("med-result");
// --- Read inputs --- const lambda = parseFloat(document.getElementById("med-arrival-rate").value); let mu = parseFloat(document.getElementById("med-service-rate").value); const c = parseInt(document.getElementById("med-num-providers").value); const avgMin = parseFloat(document.getElementById("med-avg-service-min").value);
// --- Sync service rate from average service time (minutes → patients/hour) --- if (!isNaN(avgMin) && avgMin > 0) { mu = 60.0 / avgMin; document.getElementById("med-service-rate").value = mu.toFixed(4); }
// --- Input validation --- if (isNaN(lambda) || lambda ⚠ Arrival rate must be a positive number."; return; } if (isNaN(mu) || mu ⚠ Service rate must be a positive number."; return; } if (isNaN(c) || c ⚠ Number of providers must be at least 1."; return; }
// --- Traffic intensity (server utilization) --- // rho = lambda / (c * mu) — must be = 1.0) { resultDiv.innerHTML = "⚠ System is overloaded (ρ = " + rho.toFixed(3) + " ≥ 1). The queue will grow indefinitely. Add more providers or reduce arrival rate."; return; }
// --- Erlang-C formula --- // P0 = probability that all servers are idle // C(c, λ/μ) = Erlang-C = probability a patient must wait
const a = lambda / mu; // offered load (Erlangs)
// Compute sum_{n=0}^{c-1} a^n / n! let sumTerms = 0; let factorial = 1; for (let n = 0; n 0) factorial *= n; sumTerms += Math.pow(a, n) / factorial; }
// c! term let cFactorial = 1; for (let i = 1; i " + "Expected Wait in Queue (Wq)" + "" + Wq_minutes.toFixed(1) + " minutes" + "Expected Total Time in System (W)" + "" + W_minutes.toFixed(1) + " minutes" + "Probability a Patient Must Wait (Erlang-C)" + "" + (erlangC * 100).toFixed(1) + "%" + "Average Queue Length (Lq)" + "" + Lq.toFixed(2) + " patients" + "Average Patients in System (L)" + "" + L.toFixed(2) + " patients" + "Server Utilization (ρ)" + "" + utilPct + "% per provider" + "Offered Load (a = λ/μ)" + "" + a.toFixed(2) + " Erlangs" + "" + "" + urgency + "
"; }
#### Formula — M/M/c Erlang-C Queuing Model
Traffic intensity (utilization per server):
ρ = λ / (c · μ) (must be < 1 for a stable queue)
Offered load:
a = λ / μ (Erlangs)
Erlang-C probability (chance a patient waits):
C(c, a) = [ (aᶜ / c!) · 1/(1−ρ) ] ÷ [ Σₙ₌₀ᶜ⁻¹ (aⁿ/n!) + (aᶜ/c!) · 1/(1−ρ) ]
Expected wait time in queue:
Wq = C(c, a) / (c·μ − λ) [hours]
Expected total time in system (Little's Law):
W = Wq + 1/μ
Average queue length:
Lq = λ · Wq (Little's Law)
Where: λ = arrival rate (patients/hr), μ = service rate per provider (patients/hr), c = number of providers
#### Assumptions & References
- Patient arrivals follow a Poisson process (random, memoryless) — standard assumption for walk-in clinics and EDs.
- Service times follow an exponential distribution (M/M/c model).
- All providers are identical in service rate.
- Queue discipline is FIFO (first-in, first-out); no priority triage modeled.
- System must be stable: ρ = λ/(c·μ) < 1; otherwise the queue grows without bound.
- Erlang-C formula: Erlang, A.K. (1917). Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. Post Office Electrical Engineers' Journal.
- Green, L.V. (2006). Queueing Analysis in Healthcare. In: Patient Flow: Reducing Delay in Healthcare Delivery. Springer.
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