Calculator Inputs
Example Data Table
| Model | λ | μ | c | σs | ρ | Wq | W | Scenario note |
|---|---|---|---|---|---|---|---|---|
| M/M/1 | 6 | 10 | 1 | — | 0.6000 | 0.1500 min | 0.2500 min | Stable single-server line |
| M/M/c | 18 | 12 | 2 | — | 0.7500 | 0.1071 min | 0.1905 min | Two equal servers |
| M/D/1 | 12 | 20 | 1 | 0 | 0.6000 | 0.0375 min | 0.0875 min | Fixed service time |
| M/G/1 | 8 | 12 | 1 | 0.05 | 0.6667 | 0.1133 min | 0.1967 min | Variable service time |
These rows show how different queue structures change waiting time under similar traffic conditions.
Formula Used
Queueing delay depends on arrival intensity, service capacity, server count, and service-time variability.
Common utilization: ρ = λ / (cμ) for multi-server systems, or ρ = λ / μ for single-server systems.
M/M/1: Wq = λ / [μ(μ − λ)], W = Wq + 1/μ, Lq = λWq, L = λW.
M/M/c: P0 = 1 / [Σ(a^n / n!) + a^c / (c!(1−ρ))], where a = λ / μ. Then Pw = [a^c / (c!(1−ρ))]P0 and Wq = Pw / (cμ − λ).
M/D/1: Wq = ρ / [2μ(1 − ρ)], W = Wq + 1/μ.
M/G/1: Wq = λE[S²] / [2(1 − ρ)], where E[S²] = σs² + (1/μ)².
All reported queue lengths use Little’s Law: Lq = λWq and L = λW.
How to Use This Calculator
- Select the queue model that matches your system design.
- Enter arrival rate and service rate in the same time basis.
- Set the number of servers for shared-service systems.
- Add service-time standard deviation when using the M/G/1 model.
- Choose an optional target time and warning threshold.
- Press calculate to see delay, backlog, utilization, and capacity guidance.
Frequently Asked Questions
1. What does queueing delay mean?
Queueing delay is the waiting time before service begins. It excludes actual service time, but total system time includes both waiting and service.
2. When should I use M/M/1?
Use M/M/1 for one server when arrivals and service times are random and memoryless. It is a common baseline for engineering capacity analysis.
3. When is M/M/c better than M/M/1?
Choose M/M/c when several identical servers share the same waiting line. It estimates delay reduction, waiting probability, and multi-server utilization.
4. Why does utilization matter so much?
As utilization approaches one, little spare capacity remains. Waiting time can rise sharply, even if the system still appears technically stable.
5. What is the benefit of M/D/1?
M/D/1 assumes fixed service time. Lower service variation usually reduces waiting, making it useful for paced machines, repeatable tasks, and deterministic processing lines.
6. Why does M/G/1 need standard deviation?
M/G/1 accounts for service-time variability. Standard deviation helps estimate the second moment of service time, which directly influences queueing delay.
7. What happens if arrival rate exceeds capacity?
The queue becomes unstable. In practice, backlog grows without bound, and calculated steady-state delay formulas no longer represent a workable operating condition.
8. Can I compare design alternatives here?
Yes. Change models, service rates, server counts, or variability assumptions to compare how each design affects delay, queue length, and target compliance.