Queue Waiting Time Calculator (M/M/1)
Average wait, queue length and time in the system for a single counter, checkout or server with random arrivals, from the arrival and service rates.
Results
The dashed line marks the value you entered.
What this tool does
The most important lesson of queueing theory is in the chart: waiting time does not grow steadily with how busy the server is — it explodes as utilisation approaches 100 %. A counter busy 83 % of the time makes people wait about 25 minutes on average; at 95 % the same service gives waits of over an hour and a half. That is why a little spare capacity buys so much. The M/M/1 model assumes random (Poisson) arrivals, one server, first come first served and service times that vary randomly; real queues with steadier service do somewhat better.
Formula
ρ = λ ÷ μ · L_q = ρ² ÷ (1 − ρ) · W_q = L_q ÷ λ · W = 1 ÷ (μ − λ) (M/M/1)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
la | Arrivals per hour | /h |
mu | Customers one server can handle per hour | /h |
WQ | Average wait before being served | min |
RH | Utilisation | % |
LQ | Average people waiting in line | — |
L | Average people in the system | — |
W | Average total time (wait + service) | min |
ST | Queue | — |
Worked example
- Arrivals per hour10 /h
- Customers one server can handle per hour12 /h
- Average wait before being served25.00 min
- Utilisation83.3 %
- Average people waiting in line4.167
- Average people in the system5.000
- Average total time (wait + service)30.00 min
- QueueStable — it empties from time to time
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.