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.
Average number of patients arriving per hour (λ)
Average number of patients one provider can serve per hour (μ)
Total number of parallel service providers (c)
Average time to serve one patient in minutes (overrides service rate if changed)
Fill in the fields above and click Estimate Wait Time.
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.
- No patient balking, reneging, or appointment scheduling is modeled.
- 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.
- Recommended utilization for medical services: ρ ≤ 0.80 to avoid excessive wait times (Green, 2006).
- For triage-based systems (e.g., CTAS/ESI), use priority queuing (M/M/c/K) models instead.