ITE Journal - March 2020 - 35

GHM's Solution
The foundation of GHM's solution is a minimum safe and
comfortable DISTANCE to STOP, defined as the "critical distance"
(xC), which is composed of an allocated perception-reaction
distance (xPR) plus a minimum braking distance (xBr). It is expressed
mathematically as:
v20	
xC = xPR+ xBr= v0* tPR+ _____
2amax

(1)

Where:

xC= Critical distance - the minimum safe and comfortable stopping
distance, (feet [ft.] or meters [m])
v0= Maximum uniform (constant) initial/approach velocity, (foot
per second [ft./s] or meter per second [m/s])
tPR= Maximum allocated driver-vehicle perception-reaction time, (s)
amax= Maximum uniform (constant) safe and comfortable deceleration, (ft./s2 or m/s2)
GHM's GO solution is the minimum TIME needed for a vehicle
to travel across the critical distance (xC) and is thus the minimum
yellow change interval (Ymin) required to eliminate the dilemma
zone. The solution is calculated by dividing the critical distance by
the vehicle's maximum constant velocity across that distance. For
driver-vehicles that maintain their initial velocity (v0) across the
critical distance, this is expressed mathematically as:
v20	
_____
2a
x
v
tPR _____
max
_0___
c
Ymin = __
(2)
v0 = v0 + v0
Which reduces to the well-known kinematic equation:
v0	
Ymin = tPR+ _____
2amax

(3)

Since restrictive yellow laws (drivers must not enter the intersection on yellow) prevailed in their jurisdiction, GHM's original
yellow time solution also included the minimum clearance interval
(tCl) to allow a vehicle with length (L) to travel straight through and
exit an intersection with a width (w), expressed as:
tCl =

w+L
v0

_____	

(4)

Internationally, "permissive" yellow change laws (driver-vehicles
may enter the intersection during the entire yellow interval) are
most common and the clearance interval function is often handled
by employing a separate "all-red" interval.
Figure 1 illustrates the above concepts for both restrictive (YR)
and permissive (YP ) yellow timing policies.
This article promotes the most common permissive yellow
change interval timing policy, but practitioners should note that
where restrictive yellow laws prevail, the yellow interval must also
handle the clearing function.

Figure 1. GHM's minimum STOP and GO equations plotted and
referenced to a signalized intersection.

Limitations of GHM's Kinematic Equation
An essential concept to be recognized is that GHM's Kinematic
Equation can only be derived if both the initial velocity (v0) which is
used to calculate the minimum stopping distance and the vehicle's
velocity while traversing the minimum stopping distance are the
same. Where a vehicle must slow down for any reason, such as to
negotiate a turn, the initial velocity (v0) and the vehicle's velocity
while traversing the critical distance are NOT the same and GHM's
Kinematic Equation cannot be used. This point has been reiterated
in correspondence by Dr. Alexei A. Maradudin, the sole surviving
author of the original GHM paper:5
"This formula which we derived, cannot be applied to turning lanes
or to any situation where the driver must decelerate within the critical
distance. The formula can only be applied to vehicles which start at the
maximum allowable speed measured at the critical stopping distance
and which proceed at a constant speed into the intersection."
Järlström has devised a new protocol to extend the kinematic
equation for situations where a vehicle must slow down within the
minimum stopping distance based on GHM's logic.

GHM's Logic Extended to Turning Movements
A central axiom of traffic signal timing is that, at the onset of the
yellow indication, a "reasonable" driver farther from the intersection than their minimum stopping distance (critical distance) has
sufficient distance to stop comfortably and should do so. Likewise,
w w w .i t e.or g

M a rch 2020

35


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ITE Journal - March 2020

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