How to use it
- Choose M/M/1 for a single server or M/M/c for several servers sharing one queue.
- Enter the arrival rate λ and the service rate μ of one server in the same time unit, and pick that unit.
- For M/M/c enter the number of servers.
- Read the utilization and the average wait first; the rest of the table shows queue length, time in the system and the chance of an empty system.
Formulas
With a = λ/μ and ρ = a ÷ c (must be below 1):
- P₀ = 1 ÷ (Σₖ₌₀^{c−1} aᵏ/k! + aᶜ ÷ (c!(1 − ρ)));
- probability of waiting (Erlang C) C = aᶜ ÷ (c!(1 − ρ)) × P₀;
- Lq = C × ρ ÷ (1 − ρ), Wq = Lq ÷ λ;
- W = Wq + 1/μ, L = λ × W.
For one server these reduce to ρ = λ/μ, L = ρ ÷ (1 − ρ), W = 1 ÷ (μ − λ).
Worked example
A single counter where 4 customers arrive per hour and the clerk serves 5 per hour:
- utilization ρ = 4 ÷ 5 = 80 %, and a newcomer has to wait with probability 80 %;
- the counter is empty 20 % of the time;
- on average 3.2 people wait, 4 are in the system;
- the average wait in line is Wq = 0.8 ÷ (5 − 4) = 0.8 h, the total time W = 1 h.
Add a second clerk serving 3 per hour each (M/M/2, λ = 4, μ = 3): utilization 66.67 %, chance of waiting 53.33 %, Wq = 0.2667 h, about 16 minutes.
Tips
- Keep utilization well below 100 % when waits matter: the last few percent cost the most time.
- One shared queue beats separate queues with the same total capacity; that is what M/M/c models.
- Real arrivals may not be random (appointments, batches); then the model is only a rough guide.