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M/M/1 Queueing Calculator — result sheet
Calculate utilization, idle probability, average queue lengths, and waiting times for a stable single-server M/M/1 queue.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: ρ = λ/μ; P₀ = 1−ρ; L = ρ/(1−ρ); Lq = ρ²/(1−ρ); W = 1/(μ−λ); Wq = ρ/(μ−λ).
The M/M/1 model describes Poisson arrivals, exponentially distributed service times, one server, an unlimited waiting area, and steady-state operation. The stability condition λ < μ is checked before any result is returned.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Arrival rate λ — customers per time unit; minimum 0; maximum 1000000
- Service rate μ — customers per time unit; minimum 1.0E-6; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Arrival rate λ | 6 |
| Service rate μ | 10 |
Utilization = 60%; average system size L = 1.5; average queue size Lq = 0.9; mean system time W = 0.25 time units; mean queue wait Wq = 0.15 time units.
Assumptions and limits
- Arrivals follow a Poisson process and service times are exponentially distributed with one server.
- The queue is first-in, first-out with unlimited capacity and the rates use one consistent time unit.
- The formulas describe long-run steady-state averages; they do not predict a particular customer's wait or model finite capacity, priorities, balking, or multiple servers.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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