What formula is used for the traffic signal model?

The signal model uses the FHWA Traffic Signal Timing Manual (Chapter 3, Equation 3-5) planning approximation: Qavg = v * (C - g) / 3600 vehicles per lane for undersaturated conditions (X = v/c < 1.0). This represents the average number of vehicles arriving during the red interval.

What is the Q95 screening estimate for signals?

Per FHWA TSTM Equation 3-6, the 95th-percentile queue is approximated as roughly 2 * Qavg. This is strictly a planning screening heuristic. Complex arrivals, coordination, actuation, and critical turn bay storage require full HCM back-of-queue or microsimulation analysis.

What is the difference between M/M/1 and M/D/1?

Both assume Poisson (random) arrivals at rate λ and a single server. M/M/1 assumes exponential (variable) service times, yielding Lq = ρ² / (1 - ρ). M/D/1 assumes constant (deterministic) service times, cutting the queue length in half: Lq = ρ² / (2 * (1 - ρ)).

What happens when demand exceeds capacity (v/c ≥ 1 or ρ ≥ 1)?

When arrival rate equals or exceeds service capacity, steady-state queuing formulas no longer hold. The queue grows over time without a finite steady-state bound. Sizing queues in oversaturated conditions requires specifying an analysis duration (peak period length) and arrival profile.

Can I enter an arrival rate of zero?

Yes. With no arrivals, utilization, average queue length, and queue wait are zero. The displayed system time is 1 / μ: the expected service time for a hypothetical arriving customer.

Queue Length Calculator

Estimate average queue length and waiting times using FHWA signal timing planning formulas, M/M/1 exponential service, or M/D/1 deterministic service models.

Queue Parameters

Operational Model

Signal FHWA TSTM M/M/1 Random Svc M/D/1 Constant Svc

Arrival Flow per Lane (v) (veh/h)

Saturation Flow Rate per Lane (s) (veh/hg)

Cycle Length (C) (s)

Effective Green (g) (s)

Calculation Results

Enter parameters and click Calculate

Model Relationships & Governing Equations

FHWA Signal Timing Manual (Chapter 3) Planning Expressions:

c = s × (g / C)
X = v / c (degree of saturation)
Q_avg = v × (C − g) / 3600 [veh/lane]
Q₉₅ ≈ 2 × Q_avg (screening approximation)

Valid for undersaturated flow (X < 1.0) with Poisson/random arrivals. Back-of-queue and incremental delay require full HCM procedures.

Summary

This calculator provides planning-level queue estimates for both transportation and operations engineering. For traffic signals, it computes undersaturated average red-phase queue per lane using the FHWA Traffic Signal Timing Manual formula (Equation 3-5) and displays an approximate Q95 screening estimate. For analytical service points, it computes utilization (ρ), queue length (Lq), total in system (L), queue waiting time (Wq), and total time (W) under M/M/1 (exponential service) or M/D/1 (deterministic service) queuing models.

How it works

Select the operational model: FHWA Traffic Signal (planning level), M/M/1 (exponential service), or M/D/1 (deterministic service).
For traffic signals, enter hourly flow per lane, saturation flow rate, cycle length, and effective green time.
For service queues, enter arrival rate (λ) and service rate (μ), selecting per-minute or per-hour time units.
The calculator verifies stability limits (v/c < 1.0 or ρ < 1.0) and evaluates closed-form queue length and delay expressions.
Results display average queue length, wait times, capacity utilization, and model-specific planning cautions.

Use cases

Perform planning-level screening of signalized intersection queue lengths per lane.
Estimate a 95th-percentile screening queue for initial signal timing reviews.
Model single-server customer lines with random service times (e.g., tech support, retail checkouts).
Model single-server queues with constant service duration (e.g., automated car wash, automated toll booths).
Compare wait times and line lengths between exponential and deterministic service distributions.

Frequently Asked Questions

Last updated: 2026-09-17 · Reviewed by Nham Vu