Mean Queue Length Calculator

• Utilization: for single server, ρ = λ/μ. For multi server, ρ = λ/(cμ). The calculator requires ρ < 1.
• M/M/1: Lq = ρ²/(1−ρ), then Wq = Lq/λ.
• M/D/1: Lq = ρ²/(2(1−ρ)).
• M/G/1: Pollaczek–Khinchine using service variation Cs: Lq = λ²(1+Cs²)/(2μ²(1−ρ)).
• M/M/c: Erlang C waiting probability P(wait), then Wq = P(wait)/(cμ−λ) and Lq = λWq.
• G/G/1: Kingman approximation: Wq ≈ (ρ/(1−ρ))((Ca²+Cs²)/2)(1/μ), and Lq = λWq.
• System totals: L = Lq + λ/μ and W = L/λ (Little’s Law).

1. Select the queue model that matches your process assumptions.
2. Enter arrival rate (λ) and service rate (μ) using the same time unit.
3. If using multi server, enter the number of servers (c).
4. If using variability models, set Ca and Cs (0 means perfectly regular).
5. Click Calculate. Results appear above the form under the header.
6. Use CSV/PDF buttons to export a shareable report.

Utilization drives nonlinear growth

Mean queue length rises slowly at low ρ, then accelerates sharply near 1.0. In a single server case, ρ=λ/μ. With μ=12 per hour, moving λ from 6 to 9 raises ρ from 0.50 to 0.75, while Lq increases from 0.50 to 2.25 customers. At λ=11.4, ρ=0.95 and Lq becomes 18.05, showing why high utilization is risky even when service looks adequate.

M/M/1 versus M/D/1 at equal load

Holding λ=8 and μ=12 gives ρ=0.667. M/M/1 yields Lq=1.333 and Wq=Lq/λ=0.167 time units. M/D/1 yields Lq=0.667 and Wq=0.083 because deterministic service removes half of the waiting variability. In both models, the mean number in system is L=Lq+λ/μ, so L is 2.000 for M/M/1 and 1.333 for M/D/1.

Service variability penalty in M/G/1

The Pollaczek–Khinchine result shows Lq scales with (1+Cs²). At λ=8, μ=12, and Cs=1.6, (1+Cs²)=3.56, producing Lq≈2.373 and Wq≈0.297. If process improvements reduce Cs to 1.0, (1+Cs²)=2.00 and Lq falls to ≈1.333. This is a practical lever: reduce rework, batching, and interruptions to cut service dispersion.

Multi server relief under M/M/c

Parallel servers reduce congestion when offered load spreads. With c=3, μ=10, and λ=24, utilization is ρ=0.80. Erlang C gives a waiting probability near 0.65, Wq≈0.087, and Lq≈2.09. Compare that to a single server with μ=30 and λ=24: ρ=0.80 but Lq=3.20, so pooling helps, yet waiting still rises quickly as ρ approaches 1.

G/G/1 for real world arrivals

When arrivals bunch and service varies, Kingman’s approximation uses (Ca²+Cs²)/2. For λ=10, μ=15, Ca=1.3, Cs=1.1, ρ=0.667 and the factor equals 1.45, giving Wq≈0.193 and Lq≈1.93. If Ca drops to 1.0 through appointment smoothing, Wq falls to about 0.147 and Lq to about 1.47.

Planning targets and sensitivity checks

Many operations target ρ between 0.70 and 0.85 to balance cost and delay. Use the graph to sweep λ, verify stability. Stress test with λ up 10% and μ down 10%; if ρ crosses 1.0, the model predicts unbounded growth. Keep units consistent, export CSV/PDF for documentation, and compare models to justify staffing, scheduling, or automation changes.

What does Lq represent in this calculator?

Lq is the average number of customers waiting, excluding those being served. It measures congestion in the queue only, not the whole system.

What is the difference between Lq and L?

Lq counts only waiting customers. L counts everyone in the system: waiting plus in service. The calculator uses L = Lq + λ/μ, so service capacity directly adds to system size.

Why must utilization ρ stay below 1?

If ρ ≥ 1, arrivals meet or exceed service capacity over time. The expected queue grows without bound, so steady state averages like Lq and Wq are not meaningful.

Which queue model should I select?

Use M/M/1 for random arrivals and exponential service, M/D/1 for fixed service times, M/G/1 when only service variability is known, M/M/c for multiple identical servers, and G/G/1 when both arrival and service variability matter.

How do Ca and Cs affect waiting?

Ca and Cs are coefficients of variation. Larger values mean more clustering or uneven service, which increases Wq and Lq. In G/G/1, waiting scales roughly with (Ca²+Cs²)/2, so reducing either variability improves performance.

How can I reduce mean queue length in practice?

Lower λ through smoothing or throttling, raise μ with faster handling or automation, add servers c when work can be parallelized, and reduce variability by standardizing steps, limiting batching, and removing rework. Small changes near ρ=1 can have large benefits.