What is the Webster formula used here?

The HCM Webster formula estimates average queue length at a signalized intersection: Q = (v * C) / (2 * (1 - (v/s))) + (v^2) / (2 * s * (1 - (v/s))), where v is arrival flow rate (veh/h), C is cycle length (s), and s is saturation flow rate (veh/h). This tool uses the simplified uniform delay form adapted for queue estimation.

What is the M/D/1 model?

M/D/1 is a queuing model with Poisson (random) arrivals, deterministic (constant) service time, and a single server. The average queue length is Lq = ρ² / (2 * (1 - ρ)), where ρ = λ / μ is the utilization (traffic intensity). It is commonly used for toll plazas and service counters with fixed service times.

What does degree of saturation (v/c) mean?

Degree of saturation is the ratio of demand flow rate (v) to capacity (c). Values below 0.85 are generally acceptable. Above 0.90, queues grow rapidly. At or above 1.0, the system is over-capacity and queues are theoretically unbounded.

Why does queue length increase so sharply near v/c = 1.0?

Queue length is a nonlinear function of the v/c ratio. As demand approaches capacity, even small increases in flow cause disproportionately large queue growth. This is the fundamental instability of queueing systems operating near their capacity limit.

For the HCM Webster model: flow rate in vehicles per hour (veh/h), saturation flow in veh/h, cycle length in seconds, and effective green time in seconds. For M/D/1: arrival rate (λ) and service rate (μ) in the same time units (e.g., customers per minute).

What is a typical saturation flow rate?

A standard saturation flow rate for a through lane at a signalized intersection is 1,800 to 1,900 vehicles per hour of green (veh/hg). Adjustments are applied for lane width, heavy vehicles, grade, and turning movements per the HCM.

What is the 95th percentile queue length?

The 95th percentile queue is the length exceeded only 5% of the time, used to size turn bays and storage so they rarely overflow. It is larger than the average queue — traffic software often estimates it as roughly 1.5 to 2 times the average for typical v/c ratios.

Queue Length Calculator

Name: Queue Length Calculator
Availability: InStock
Author: Nham Vu

Calculate average queue length at a traffic signal or service point using the HCM Webster formula and M/D/1 queuing theory.

Queue Parameters

Queuing Model

HCM Webster

Traffic signals

M/D/1

Service queues

Arrival Flow Rate (veh/h)

Saturation Flow Rate (veh/h)

Cycle Length (s)

Effective Green (s)

Results

Enter parameters and click Calculate

Formula Reference

HCM Webster Uniform Queue (simplified):

Q = (v² × C) / (2 × s × (g/C) × (1 − v/c))

where v = arrival flow (veh/h), s = saturation flow (veh/h),
C = cycle (s), g = effective green (s), c = capacity = s × (g/C)

Summary

Calculate average queue length at a traffic signal or service point using the HCM Webster formula and M/D/1 queuing theory.

How it works

Select the queuing model: HCM Webster (traffic signals) or M/D/1 (deterministic service queues).
Enter the arrival flow rate, capacity (or service rate), and cycle/timing parameters.
The calculator applies the selected formula to compute the average queue length.
Results include average queue (vehicles or customers), degree of saturation (v/c ratio), and overflow indicator.
Adjust inputs and see results update immediately to explore different scenarios.

Use cases

Estimate vehicle queues at signalized intersections for traffic engineering studies.
Size turn pockets and storage bays at signalized intersections.
Analyze customer wait lines at toll booths, check-ins, or single-window service counters.
Evaluate the impact of signal timing changes on queue length.
Compare degree of saturation before and after capacity improvements.
Determine whether a queue will overflow into upstream intersections.
Support Level of Service (LOS) assessments for traffic impact analyses.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu