Queueing Theory Calculator (M/M/1, M/M/c, M/G/1)

Queueing theory predicts how long people, packets or jobs wait when they arrive at random and a limited number of servers handle them: checkout lanes, support desks, web servers, database connections. From the arrival and service rates this calculator gives the utilisation, the average queue length, wait and time in the system, the chance of waiting at all and how many are likely to be present.

How to use this tool

1. Choose a model: one server (M/M/1), several servers sharing a queue (M/M/c), or one server with any service-time spread (M/G/1).
2. Enter the arrival rate λ and the service rate μ of one server in the same time unit.
3. For M/M/c enter the number of servers; for M/G/1 the variation of the service time.
4. Optionally enter a wait threshold to see the chance of waiting longer.

Formula

ρ = λ ÷ (cμ); Lq = C(c, a) × ρ ÷ (1 − ρ); Wq = Lq ÷ λ; W = Wq + 1 ÷ μ; L = λ × W

a = λ ÷ μ is the offered load and C(c, a) the Erlang C probability that all c servers are busy (for one server it equals ρ). For M/G/1, Lq = (λ²σ² + ρ²) ÷ (2(1 − ρ)) with σ the standard deviation of the service time (Pollaczek–Khinchine). L = λW is Little's law, which holds for any stable queue. The queue is stable only when ρ < 1.

Example

An M/M/1 queue with λ = 2 arrivals and μ = 3 services per minute has ρ = 2 ÷ 3 = 0.667. On average L = 2 customers are in the system and W = 1 minute is spent there; the queue holds Lq = 1.333 and the wait before service is Wq = 0.667 minutes. Little's law checks out: 2 × 1 = 2. With four arrivals per minute and μ = 2 the queue would be unstable.