Calculator inputs
Choose a queue model, enter demand and service data, then calculate the steady-state measures.
Formula used
ρ = λ / μP₀ = 1 − ρPₙ = (1 − ρ)ρⁿL = λ / (μ − λ)Lq = λ² / [μ(μ − λ)]W = 1 / (μ − λ)Wq = λ / [μ(μ − λ)]
The standard formulas apply directly to a stable M/M/1 queue. Finite capacity uses truncated state probabilities and effective throughput. General service uses the Pollaczek–Khinchine mean waiting formula.
M/G/1: Wq = λE[S²] / [2(1 − ρ)]M/M/1/K: P₀ = (1 − ρ) / (1 − ρᴷ⁺¹)Finite source: Pₙ ∝ N!/(N−n)! × (λ/μ)ⁿ
How to use
- Select the queue model matching your system assumptions.
- Choose rates or average times as the input method.
- Enter arrival demand and one-server service capacity.
- Add capacity, population, or variability data when required.
- Set probability, planning, precision, and comparison options.
- Submit the form and review stability before other results.
- Export the results or test a practical scenario.
Example data
| Example | Model | Arrival input | Service input | Extra setting |
|---|---|---|---|---|
| Bank service counter | M/M/1 | 8 per hour | 12 per hour | n = 3 |
| Website request processor | M/M/1/K | 45 per minute | 60 per minute | K = 25 |
| Repair technician | Finite population | 0.15 per hour per machine | 2 per hour | N = 12 |
| Variable service desk | M/G/1 | 6 per hour | 10 per hour | SD = 4 minutes |
Queue assumptions and interpretation
The default model assumes independent Poisson arrivals and exponential service times. It also assumes one server, unlimited waiting space, and first-come service. Real systems may include priorities, abandonment, interruptions, or changing demand.
Utilisation below fifty percent normally provides a strong buffer. Values above eighty-five percent often create sensitive waiting times. These bands are guidance rather than universal operating rules.
Frequently asked questions
What is a single server queue?
It is a waiting system with one service channel. Customers arrive, wait when necessary, and leave after service. Examples include one cashier, processor, technician, or reception desk.
Why must λ be below μ?
A stable unlimited queue needs service capacity above average demand. Otherwise, unfinished work accumulates without a finite long-run average. Temporary operation can still occur, but steady-state formulas fail.
What does utilisation mean?
Utilisation estimates the server’s busy share over time. Higher utilisation uses capacity efficiently but increases congestion risk. Near-full utilisation can create very large average delays.
What is the difference between L and Lq?
L counts everyone inside the queueing system. Lq counts only customers waiting before service begins. Their difference is the average number receiving service.
What is the difference between W and Wq?
W is total time from arrival until departure. Wq includes only the time spent waiting. Average service time equals the difference between them.
When should I use M/M/1/K?
Use it when the total system capacity is limited. Arrivals finding K customers are blocked or rejected. The model reports effective throughput and blocking probability.
When should I use M/G/1?
Use it when service times are not exponential. You must provide service-time variance or equivalent variability. More variability usually increases queue waiting and congestion.
What is a finite population queue?
It limits arrivals to a known source population. Busy or failed sources cannot generate another request immediately. Machine repair and closed-user systems often use it.
Are probability approximations always exact?
M/M/1 occupancy probabilities are exact under its assumptions. M/G/1 tail and occupancy outputs need distribution details for exact values. This calculator labels those outputs as practical approximations.