Queuing Theory Calculator (M/M/1)
Free M/M/1 queuing theory calculator. Enter arrival rate (λ) and service rate (μ) to get utilization, L, Lq, W, Wq, probability system is empty, and probability of n customers in the system. Assumptions clearly explained.
Full original guide (expanded)
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).
Units: customers per time unit (e.g. per hour)
Must be > λ for a stable system
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
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}
Formula (LaTeX) + variables + units
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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: \( P(n) = (1 - \rho)\rho^n \)
- No variables provided in audit spec.
- Little’s Law calculator — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/littles-law
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
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