M/M/1 Queue Calculator
In an M/M/1 queue L = ρ/(1 − ρ): with 8 arrivals and 10 services per hour, ρ = 0.8 and the average number in the system is 4.
4
- Utilisation ρ = λ/μ
- 0.8
- Probability the system is empty P₀ = 1 − ρ
- 0.2
- Average queue length Lq = ρ²/(1 − ρ)
- 3.2
- Average time in the system W = 1/(μ − λ)
- 0.5 h
- Average time in the system W
- 30 min
- Average waiting time in the queue Wq = ρ/(μ − λ)
- 0.4 h
- Average waiting time in the queue Wq
- 24 min
Step-by-step solution
- ρ = λ/μ = 8 ÷ 10 = 0.8
- L = ρ/(1 − ρ) = 4; Lq = ρ²/(1 − ρ) = 3.2
- W = L/λ = 0.5 h (Little’s law)
M/M/1 Queue Formula
L = ρ/(1 − ρ), W = 1/(μ − λ)
ρ = λ/μ, Lq = ρ²/(1 − ρ), Wq = ρ/(μ − λ)
Worked Example
Customers arriving at 8 per hour to a server handling 10 per hour:
- ρ = λ/μ = 8 ÷ 10 = 0.8
- L = ρ/(1 − ρ) = 4; Lq = ρ²/(1 − ρ) = 3.2
- W = L/λ = 0.5 h (Little’s law)
Probability the system is empty P₀ = 1 − ρ: 0.2 · Average queue length Lq = ρ²/(1 − ρ): 3.2 · Average time in the system W = 1/(μ − λ): 0.5 h · Average time in the system W: 30 min · Average waiting time in the queue Wq = ρ/(μ − λ): 0.4 h · Average waiting time in the queue Wq: 24 min
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Frequently asked questions
What is Little’s law?
L = λW and Lq = λWq: average number = arrival rate × average time. It holds for any stable queue.
Why does waiting time explode as ρ → 1?
W = 1/(μ − λ): as arrivals approach the service rate, the denominator goes to zero.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). M/M/1 Queue Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/mm1-queue-calculator/
IEEE
Scientific Converters, "M/M/1 Queue Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/mm1-queue-calculator/