Estimate average queue delay when ambulance calls compete for a limited number of staffed units. This calculator uses an Erlang C multi-server queue, treating each ambulance as a server and adjusting its service rate for operational availability. It reports utilization, the probability an arriving call must wait, average queue delay, and total time through the modeled service cycle. This simplified model supports scenario comparison, not dispatch decisions.
Effective service rate per unit μ = availability × 60 ÷ cycle minutes
Utilization ρ = arrival rate ÷ (units × μ)
Average queue wait Wq = Erlang-C wait probability ÷ (units × μ − arrival rate)
Average queue wait is the long-run delay before a unit becomes available under steady, independent arrivals and exponentially distributed service times. Real emergency systems often have priority, travel geography, relocations, and time-varying demand.
Do not interpret this queue delay as a promised response time. Dispatch, triage, travel-to-scene time, coverage policy, and emergency standards are outside the model.
With 10 staffed units, 5 calls per hour, a 75-minute call cycle, and 90% availability, effective service is 0.72 calls per unit-hour. Fleet utilization is 69.4%. The Erlang C model estimates an average queue wait of about 1.9 minutes and a 22.2% probability that an arrival waits.
At or above full long-run utilization, arrivals meet or exceed service capacity and a steady-state queue has no finite average.
No. It estimates time until a unit becomes available; travel-to-scene time must be added separately.
It is the modeled share of arriving calls that find all entered ambulances busy.
It provides a transparent multi-unit queue estimate, but its steady and memoryless assumptions simplify real dispatch operations.
Not by itself. Real planning must consider priority, geography, response standards, peaks, mutual aid, and qualified operational review.
| Item | Meaning |
|---|---|
| Symbol | Definition |
| λ | Average call arrivals per hour |
| μ | Effective calls served per unit-hour |
| c | Available staffed ambulances |
| ρ | λ ÷ (c × μ); must be below 1 |
| Wq | Average time waiting for a free unit |