Queueing Theory Calculator
Result
M/M/1 queue calculator for arrival rate λ and service rate μ. Requires μ > λ and returns the expected queue length Lq.
Formula
- Queueing Theory Calculator models a stable M/M/1 queue from the arrival rate λ and service rate μ. The model requires μ > λ. It returns Lq, the expected number of customers waiting in the queue, using Lq = λ²/[μ(μ-λ)]; the expected number in the system is L = λ/(μ-λ).
- The main inputs are Arrival rate λ, Service rate μ. Enter them in the units and format shown in the calculator.
- Keep comparable quantities in compatible units whenever the formula combines or compares them.
- Changing any input above recalculates the result with the same calculation engine used by this page.
Example: Queueing Theory
- Arrival rate λ: 2
- Service rate μ: 3
If λ = 2 arrivals per time unit and μ = 3 services per time unit, L = 2/(3-2) = 2 and Lq = 2²/[3(3-2)] = 4/3 ≈ 1.3333.
Sample result: Calculating…
Frequently asked questions
What condition is required for an M/M/1 queue?
The service rate μ must be greater than the arrival rate λ; otherwise the queue is not stable.
What does Lq mean?
Lq is the expected number of customers waiting in the queue, excluding any customer currently in service.
What does L mean?
L is the expected number of customers in the system, including both the queue and the customer in service.
What does the Queueing Theory calculate?
M/M/1 queue calculator for arrival rate λ and service rate μ. Requires μ > λ and returns the expected queue length Lq.
Which values should I enter?
Enter Arrival rate λ, Service rate μ using the units and format shown beside each field.
What formula does this calculator use?
The calculator uses the following relationship: L = λ/(μ-λ); Lq = λ²/[μ(μ-λ)].