Calculator
Example data table
These sample scenarios show how utilization changes results.
| Scenario | λ (per hour) | μ (per hour) | ρ | W (hours) | Wq (hours) |
|---|---|---|---|---|---|
| A | 6 | 10 | 0.6 | 0.25 | 0.15 |
| B | 8 | 10 | 0.8 | 0.50 | 0.40 |
| C | 9 | 10 | 0.9 | 1.00 | 0.90 |
Note: values are illustrative, based on M/M/1 assumptions.
Formula used
- ρ = λ / μ (utilization)
- P0 = 1 − ρ (empty system probability)
- L = ρ / (1 − ρ) (mean in system)
- Lq = ρ² / (1 − ρ) (mean in queue)
- W = 1 / (μ − λ) (mean time in system)
- Wq = λ / (μ(μ − λ)) (mean waiting time)
- Pn = (1 − ρ)ρⁿ for integer n ≥ 0
- Little’s Law checks: L = λW and Lq = λWq
How to use this calculator
- Choose rate mode or mean-time mode, based on your data.
- Enter λ and μ, or enter mean interarrival and service time.
- Optional: enter n to compute the probability of n customers.
- Click Calculate to see results above the form.
- Use CSV or PDF to share the computed summary.
- If you see instability, reduce λ or increase μ.
FAQs
1) What does “single server queue” mean?
It models one service channel handling arrivals one-by-one. Examples include one cashier, one helpdesk agent, or one machine processing jobs.
2) What assumptions does the M/M/1 model use?
Arrivals follow a Poisson process, service times are exponential, and there is one server with infinite waiting space and first-come first-served order.
3) Why must λ be less than μ?
If arrivals come as fast as, or faster than, service capacity, the queue grows without bound. A steady-state average is only defined when λ < μ.
4) What is utilization ρ and how should I interpret it?
ρ is the fraction of time the server is busy. As ρ approaches 1, waiting times rise sharply, even if capacity is only slightly exceeded.
5) What is the difference between W and Wq?
W is total time in the system: waiting plus service. Wq counts only the waiting portion before service starts.
6) What does Pn represent?
Pn is the probability that exactly n customers are in the system at steady state. It helps estimate crowding or buffer requirements.
7) Can I use mean times instead of rates?
Yes. The tool converts mean interarrival time to λ and mean service time to μ using reciprocals, then applies the same M/M/1 equations.
8) When should I avoid using this model?
Avoid it when arrivals are strongly scheduled, service times are not memoryless, priorities exist, or multiple servers share work. Consider other queue models then.