What are the assumptions of the M/M/1 queue?

Arrivals follow a Poisson process (rate λ), service times are exponentially distributed (rate μ), there is a single server, FIFO discipline, and the system is stable when λ < μ.

What is traffic intensity (ρ)?

Traffic intensity ρ is defined as λ/μ. It represents server utilization. The system is stable only if ρ < 1.

How do I get average number in the system (L)?

For M/M/1: L = ρ / (1 - ρ). Once you enter λ and μ in the calculator, it will compute L for you.

How do I compute average waiting time in queue (Wq)?

For M/M/1: Wq = λ / (μ × (μ − λ)) = ρ / (μ − λ).

Queuing Theory Calculator (M/M/1)

This tool calculates the main steady-state performance measures of an M/M/1 queue: server utilization (ρ), average number in the system (L), average number in queue (Lq), average time in system (W), average waiting time in queue (Wq), and key probabilities. Just enter the arrival rate (λ) and the service rate (μ).

Arrival rate (λ)

Units: customers per time unit (e.g. per hour)

Service rate (μ)

Must be > λ for a stable system

Probability of n customers in system optional

We will compute P(n) = (1 − ρ)ρⁿ

Utilization (ρ)

—

ρ = λ / μ

Avg # in system (L)

—

L = ρ / (1 − ρ)

Avg # in queue (Lq)

—

Lq = ρ² / (1 − ρ)

Avg time in system (W)

—

W = 1 / (μ − λ)

Avg waiting time (Wq)

—

Wq = λ / (μ(μ − λ))

P0 (empty system)

—

P0 = 1 − ρ

P(n)

—

P(n) = (1 − ρ)ρⁿ

Formulas used (M/M/1)

This calculator assumes:

Arrivals: Poisson with rate λ
Service: Exponential with rate μ
1 server, first-come-first-served, infinite buffer
Steady-state exists only if λ < μ

Traffic intensity: \( \rho = \frac{\lambda}{\mu} \)
Probability of zero customers: \( P_0 = 1 - \rho \)
Average number in system: \( L = \frac{\rho}{1 - \rho} \)
Average number in queue: \( L_q = \frac{\rho^2}{1 - \rho} \)
Average time in system: \( W = \frac{1}{\mu - \lambda} \)
Average waiting time in queue: \( W_q = \frac{\lambda}{\mu(\mu - \lambda)} = \frac{\rho}{\mu - \lambda} \)
Probability of n customers: \( P(n) = (1 - \rho)\rho^n \)

Example

Suppose a help desk receives on average λ = 5 calls/hour and can serve μ = 8 calls/hour. Then ρ = 5/8 = 0.625, so the system is stable. The calculator will show L, Lq, W, Wq automatically.

Frequently asked questions

What if λ ≥ μ?

Then the queue grows without bound and the standard steady-state M/M/1 formulas do not apply. You need to increase service capacity (raise μ) or reduce arrivals (lower λ).

Can I change the time unit?

Yes. If you enter both λ and μ in the same time unit (per minute, per hour, per day), the formulas still work. The waiting times (W, Wq) will be in that same time unit.

Related concepts

Little’s Law (L = λW) still holds and can be used to cross-check the results. See the Little’s Law calculator for a quick validation. :contentReference[oaicite:0]{index=0}