Math Calculators

Queueing Theory Calculator (M/M/1)

Calculate the average wait time, queue length and utilization of a simple single-server (M/M/1) queue.

Utilization (ρ) —

Avg number in system (L) —

Avg wait in system (W) —

Avg number in queue (Lq) —

Avg wait in queue (Wq) —

About the Queueing Theory Calculator (M/M/1)

An M/M/1 queue models a single server with random (Poisson) arrivals at rate λ and random (exponential) service times at rate μ. It's the simplest queueing model, but it captures the core trade-off: as arrivals approach the service rate, waits grow rapidly.

Formula: ρ = λ/μ · L = λ/(μ−λ) · W = 1/(μ−λ) · Lq = λ²/(μ(μ−λ)) · Wq = λ/(μ(μ−λ))

The server must process faster than customers arrive (μ > λ) or the queue grows without bound. This model is used in call centers, checkout lines and network traffic analysis.