What formula does the signal model use?

It uses the HCM uniform delay form: d1 = 0.5 * C * (1 - g/C)^2 / (1 - (g/C) * (v/c)), then average queue = d1 * v / 3600, where C is cycle length (s), g is effective green time (s), v is arrival flow (veh/h), and c is capacity. The original Webster (1958) delay equation is the historical reference, but it does not use effective green time and is not what this tool computes.

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