Addis Ababa Institute of Technology (AAiT), 2011

1 Traffic Engineering– CENG 6302

Chapter 5.1
Basic Principles of
Intersection Signalization

Instructor: Dr. Bikila Teklu

bikilatek@yahoo.com
 Basic Principles of Intersection
Signalization
2

Discharge headways, saturation flow rates,

and lost times

Allocation of time and the critical lane
concept

The concept of left-turn equivalency

Delay as a measure of service quality

 Terms and Definitions
3

Components of a signal cycle

Cycle (one complete rotation through all the indications)

Cycle length (C)

Interval

Change interval (yi)

Clearance interval (ari)

Green interval (Gi)

Red interval (Ri)

Phase (Gi + yi + ari)

 Terms and Definitions
4

Types of signal operation

Pretimed operation

Semi-actuated operation (detectors on minor road)

Full-actuated operation (detectors on every approach)

Computer control (computer coordinates large number of signals)

Treatment of Left Turns

Note: Protected and permitted may

Permitted left turns also apply to right turns, where the

Protected left turns conflicts are between the right-turn
vehicular movement and the pedestrian

Compound left turns movement in the conflicting cross walk
 Discharge Headways
5

When the light turns GREEN, there is a queue of vehicles waiting to

be discharged, that were stopped during the preceding RED
phase.

As the queue moves, headway measurements are taken as follows :

First headway – b/n start of GREEN and front wheel of first
vehicle crossing the stop line

Second headway – time b/n 1st and 2nd vehicles’ front wheels
crossing the stop line

Subsequent headways measured similarly

Only headways through the last vehicle in queue (at the start
of GREEN light) are considered to be operating under
saturated conditions
 Discharge Headways
6

Note: First headway is

relatively long
(perception-reaction
sequence, move foot
from brake to
accelerator and
accelerate)

Eventually, headways
tend to level out (occurs
when queued vehicles
have fully accelerated by
the time they cross the
stop line)
 Saturation Headway and
Saturation Flow Rate
7

Saturation Headway (h) – the constant headway achieved

when the average headways level out (often after 4th or 5th
headway position)

Saturation flow rate (s) – the number of vehicles entering
the intersection per hour if every vehicle consumes “h”
seconds of green time and if the signal were always green

s = 3600/h
where s = saturation flow rate, vehicles per hour of green
per lane (veh/hg/ln)
h = saturation headway, seconds/vehicle (s/veh)
 Start-Up Lost Time
8

The average headway per vehicle is actually greater than

“h” seconds.

The first three or four headways involve additional time as
drivers react to the GREEN signal and accelerate.

The sum of these additional times is the Start-Up Lost time

l1 = ∑ ∆ i
i
where l1 = start-up lost time, s/phase
∆i = incremental headway (above “h” seconds)
for vehicle i, s
 Start-Up Lost Time
9

It is possible to model the amount of GREEN time required

to discharge a queue of “n” vehicles as,
Tn = l1 + nh
where Tn = GREEN time required to move a queue of “n”
vehicles through a signalized intersection, s
l1 = start-up lost time, s/phase
n = number of vehicles in queue
h = saturation headway, s/veh
(Note: this model is not of great use, but illustrates the basic
concepts of saturation headway and start-up lost times.)
 Clearance Lost Time & Total Lost Time
10

Associated with stopping the queue at the end of the

GREEN signal

Difficult to observe in the field, as it requires that the
standing queue of vehicles be large enough to consume all
of the GREEN time.

In such a situation, the clearance lost time, l2, is defined as
the time interval between the last vehicle’s front wheels
crossing the stop line, and the initiation of the GREEN for the
next phase.

Total Lost Time per phase (s/phase) = l1 + l2

 Effective GREEN Time
11

The actual signal goes through a series of intervals for

each signal phase: Green, Yellow, All-red, Red

The Yellow and All-red intervals are a transition between
the GREEN and RED.

In terms of modeling, there are only two time periods of
interest:

Effective green time – amount of time that vehicles are
moving (at a rate of one vehicle every h seconds)

Effective red time – amount of time that the vehicles are
not moving
 Effective GREEN Time
12

gi = Gi + Yi – tLi
where gi = effective green time for movement(s) i, s
Gi = actual green time for movement(s) i, s
Yi = sum of yellow and all-red intervals for
movement(s) i, s (Yi = yi + ari)
yi = yellow interval for movement(s) i, s
ari = all-red interval for movement(s) i, s
tLi = total lost time for movement(s) i, s
This model results in an effective green time that may be fully
utilized by vehicles at the saturation flow rate (i.e. at an
average headway of “h” s/veh)
 Capacity of an Intersection Lane or
Lane Group
13

The capacity of an intersection lane or lane group may be

computed as:

ci = si ⋅ 
gi 
C 
 
where: ci = capacity of lane or lane group i, veh/hr
si = saturation flow rate for lane or lane group i, veh/hg
gi = effective green time for lane or lane group i, s
C = signal cycle length, s

Sample Problem
 Critical-Lane and Time-Budget
14

Time-Budget – in its simplest form, is the allocation

of time to various vehicular and pedestrian
movements at an intersection through signal control

The Critical-Lane concept involves the

identification of specific lane movements that will
control the timing of a given signal phase.
 Critical-Lane and Time-Budget
15

Assume a simple two-phase signal:

During each phase four lanes of traffic moving
simultaneously (Demand not evenly distributed among all;
lanes 1 & 2 have the most “intense” demand)

The signal must be timed
to accommodate traffic
in these lanes.

Note: most intense
demand does not
necessarily imply highest
volume (too many left-
turning vehicles may be
critical as well)
 Critical-Lane and Time-Budget
16

In general, the following rules apply to the

identification of critical lanes:

There is a critical lane and critical-lane flow for
each discrete signal phase provided

Except for lost times (when no vehicles move)
there must be one and only one critical lane
moving during every second of effective green
time in the signal cycle
 Maximum Sum of Critical-Lane Volumes
17

Can be considered as a general measure of the

“capacity” of the intersection.

To determine the Maximum Sum of Critical-Lane
Volumes:

Determine the total lost times in the hour

Divide the remaining time (total effective
green time) by the saturation headway
 Maximum Sum of Critical-Lane Volumes
18

Assuming the total lost time per phase (tL) to be constant for
all phases, the total lost time per signal cycle is:
Where L = lost time per cycle, s/cycle
L = N ⋅ tL tL = total lost time per phase (l1 + l2), s/phase
N = number of phases in the cycle

The total lost time in an hour depends upon the number of
cycles occurring in the hour:

(
LH = L ⋅ 3600
C
) Where LH = lost time per hour, s/hr
L = lost time per cycle, s/cycle
C = cycle length, s
 Maximum Sum of Critical-Lane Volumes
19

The remaining time within the hour is devoted to effective

green time for critical lane movements,

TG = 3600 − LH where TG = total effective green time

in the hour, s

This time may be used at a rate of one vehicle every “h”

seconds, where h is the saturation headway:

Where Vc = maximum sum of critical


Vc = 
TG 
h  lane volumes, veh/hr
  h = saturation headway, s/veh
 Maximum Sum of Critical-Lane Volumes
20

Merging the previous equations,

1  3600 
Vc = 3600 − Nt L  
h  C 

For the intersection on slide 15:

Two-phase signal, cycle length of 60-sec, total lost times
of 4s/phase, and a saturation headway of 2.5s/veh

1   3600 
Vc = 3600 − 2 * 4 *   = 1,248 veh / hr
2.5   60 
 Maximum Sum of Critical-Lane Volumes
21

As cycle length increases,

“capacity” increases
(Because lost times are
constant per cycle)

Capacity decreases, as
the number of phases
increases (Because for
each phase, there is one
full set of lost times in the
cycle)

Having a cycle length that is considerably longer than what is

desirable causes increases in delay.
 Finding an appropriate cycle length
22

Desirable cycle length (Cdes), s:

Nt L
Cdes =
 Vc 
1−  
 (3600 / h ) * PHF * ( v / c ) 

PHF (Peak Hour Factor) – to account for the flow rate in the
worst 15-minute period of the hour

v/c (volume to capacity ratio) – to provide some excess capacity
to avoid failure of individual cycles or peak periods on a
specific day
 Finding an appropriate cycle length
23

Consider the case of three-phase signal, with tL=4 s/phase,

saturation headway of 2.2 s/veh, PHF of 0.9 and Vc=1200 veh/hr,

For v/c = 1.0, Cdes = 65 s,

For v/c = 0.95, Cdes = 85 s,
The practical limit

For v/c = 0.90, Cdes = 130 s, for pretimed signals
is 120 s.

For v/c = 0.85, Cdes = 290 s, and

For v/c = 0.8, Cdes = -648.6 s

The negative cycle length signifies that there is not enough time
within the hour to accommodate the demand with the required
green time plus the 12 s of lost time per cycle.
Sample Problem
 The Concept of Left-Turn Equivalency
24

The left turn is the most difficult process to model at a

signalized intersection.

May be shared-lane operation or exclusive-lane operation

A left-turning vehicle will consume more effective green time
traversing the intersection than will a similar through vehicle

The most complex case is that of a permitted left turn made
across an opposing vehicular flow from a shared lane

Through vehicle equivalents – how many through vehicles
would consume the same amount of effective green time
traversing the stop line as one left-turning vehicle?
 The Concept of Left-Turn Equivalency
25

“In the same amount of time, the left lane discharges five
through vehicles and two left-turning vehicles, while the right
lane discharges eleven through vehicles”

i.e. eleven through vehicles are equivalent to five through vehicles

plus two left-turning vehicles
11 = 5 + 2ELT ; ELT = (11-5)/2 = 3.0
 The Concept of Left-Turn Equivalency
26

 The left-turn equivalent depends upon a number of factors:

 How left turns are made (protected, permitted, compound)
 The opposing traffic flow
 The number of opposing lanes

 ELT increases as the opposing

flow increases
 For any given opposing flow,
however, ELT decreases as the
number of opposing lanes
increases
 The Concept of Left-Turn Equivalency
27

 A number of models, including the HCM, calibrate a multiplicative

adjustment factor that converts an ideal (or through) saturation flow
rate to a saturation flow rate for prevailing conditions:

s prev = sideal * f LT
s prev (3600 / h prev ) hideal
f LT = = =
sideal (3600 / hideal ) h prev

sprev = saturation flow rate under prevailing conditions, veh/hg/ln;

sideal = saturation flow rate under prevailing conditions, veh/hg/ln;
fLT=left-turn adjustment factor; hideal=saturation headway under
ideal conditions, s/veh; hprev= saturation headway under prevailing
conditions, s/veh
 The Concept of Left-Turn Equivalency
28

 The prevailing headway, hprev, is computed as,

h prev = ( PLT ELT hideal ) + [(1 − PLT )hideal ]

hideal
f LT =
h prev
hideal
f LT =
( PLT ELT hideal ) + [(1 − PLT )hideal ]
1 1
f LT = =
PLT ELT + (1 − PLT ) 1 + PLT ( ELT − 1)
Where PLT is the proportion (fraction) of left-turning traffic.
 The Concept of Left-Turn Equivalency
29

EXAMPLE:
 An approach to a signalized intersection has two
lanes, permitted left-turn phasing, 10% left-turning
vehicles, and a left-turn equivalent of 5.0. The
saturation headway for through vehicles is 2.0
s/veh. Determine the equivalent saturation flow rate
and headway for all vehicles on this approach.
 Delay as a Measure of Effectiveness
30

 Commonly used measures to characterize operational

quality of signalized intersections:
 Delay: amount of time consumed in traversing the

intersection (difference between arrival time and

departure time)
 Queuing: number of vehicles forced to queue

behind the stop-line during a RED signal phase

 Stops: percentage or number of vehicles that must

stop at the signal

 Types of Delay
31

 Commonly used forms of delay are:

 Stopped-time delay: the time a vehicle is stopped in queue while
waiting to pass through the intersection
 Approach delay: includes stopped-time delay but adds the time
loss due to deceleration to stop and reacceleration back to the
desired speed
 Time-in-queue delay: time from a vehicle joining an intersection
queue to its discharge across the STOP line on departure
 Travel time delay: difference between the driver’s expected
travel time through the intersection and the actual time taken (a
more conceptual value)
 Control delay: delay caused by a control device (traffic signal
or a STOP sign). Roughly equal to time-in-queue delay plus
acceleration-deceleration delay.
 Types of Delay
32

 Delay measures can be for:

 A single vehicle [s/veh]

 Average for all vehicles

over a given time period

[s/veh]
 An aggregate total value

for all vehicles over a

given time period [veh-
secs, veh-mins, veh-hours]
 Basic Theoretical Models of Delay
33

 Cumulative vehicles arriving and departing vs. time at

a given signal location
 Vehicles are assumed
to arrive at a uniform
rate of flow of v
vehicles per unit time
 For example: If the
arrival flow rate, v, is
1800 vehs/h, then
one vehicle arrives
every 3600/1800 =
2 s.
 Basic Theoretical Models of Delay
34

 Three critical parameters can be estimated from the

previous illustration:
 The total time that any vehicle i spends waiting in the
queue, W(i): horizontal time-scale difference between
the time of arrival and departure
 The total number of vehicles queued at any time t, Q(t):
vertical vehicle-scale difference between the number
of vehicles that have arrived and departed
 The aggregate delay for all vehicles passing through the
signal is the area between the arrival and departure
curves (vehicles x time).
 Basic Theoretical Models of Delay
35

 This model makes two major simplifications:

 Assumption of uniform arrival rate (actual
arrivals would be random & for coordinated
systems arrivals are in platoons)
 It is assumed that queue is forming at a point
location (as if vehicles were stacked on top of
one another) i.e. that backward growth of the
queue in space is ignored.
 Basic Theoretical Models of Delay
Three Delay Scenarios
36

Overall Stable
Stable flow
(some phases fail)
throughout

Overall failure
(v/c > 1.0)
 Components of Delay
37

 Uniform delay (uniform

arrivals and stable flow)
 Random delay (Due to
randomly distributed
nature of delays)
 Overflow delay (when
demand exceeds
capacity)
 Webster’s Uniform Delay Model
38

 Basic delay model (Assumption of stable flow and

simple uniform arrival function)
 Webster’s Uniform Delay Model
39

 Aggregate delay can be estimated as the area

between the arrival and departure curves
1 Where UDa = aggregate uniform delay, veh-secs
UDa = RV
2 R = length of the RED phase, s
V = total vehicles in queue, vehs

 Since traditionally traffic signals are developed in

terms of GREEN time,
 
R = C 1 −  g   Where C = cycle length, s

  C  g = effective green time, s
 Webster’s Uniform Delay Model
40

 The height of the triangle, V, is the total number of

vehicles in the queue [Includes vehicles arriving
during RED, R, plus those that join the end of the
queue while it is moving out of the intersection
during time tc (see figure)]
 Thus, it is important to determine the time it takes
for the queue to clear, tc
 This is done by equating the number of vehicles
arriving during the R + tc equal to the number of
vehicles departing during the period tc
 Webster’s Uniform Delay Model
41

ν ( R + tc ) = stc
s
R + tc =  tc
v
s 
R = tc  − 1
v 
R
tc =
s 
 − 1
v 
 Webster’s Uniform Delay Model
42

Substituting for tc  
 R   vs 
V = v( R + tc ) = v  R +  = R 
  s    s − v 
 − 1
  v 
And for R
  g   vs 
V = C 1 −    
   
C s − v 
Aggregate delay (in veh-secs for one signal cycle)
2
1 1 2  g   vs 
UDa = RV = C 1 −  
2 2  C   s − v 
 Webster’s Uniform Delay Model
43

Average uniform delay per vehicle (dividing UDa by

the number of vehicles arriving during the cycle , vC:
2
 g
1− 
1  C
UD = C
2  v
1 − s 
Where UD = average uniform delay per vehicles, s/veh
C = cycle length, s
g = effective green time, s
v = arrival flow rate, veh/h
 Webster’s Uniform Delay Model
44

 Alternatively, the capacity, c, can be used rather than

the saturation flow rate, s [Note that s = c/(g/C)

2
 g  g 
2
1 −
 C  0.50 C 1 −
1  C 
UD = C =
2 1 −  g  v 
  C  c  ( ) 1 − 

g
 C
 X

Where X = v/c ratio, or degree of saturation

(Note that the maximum value of X is 1.0 as the
uniform flow model assumes no overflow)
 Modeling Overflow Delay
45

 “Oversaturation” is used to describe extended time periods

during which arriving vehicles exceed the capacity of the
intersection approach to discharge vehicles.
 During oversaturation, delay consists of both uniform delay and
overflow delay
 Modeling Overflow Delay
46

 The formula for the uniform delay component may be

simplified by setting X to 1.
2 2
0.50C 1 − g  0.50C 1 − g 
 C   C 
UDo = = = 0.50C 1 − g 
 g   g   C 
1−   X 1−  1.00
 C  C
 To this, the overflow delay may be added:
1 T2
ODa = T (vT − cT ) = (v − c) Aggregate overflow delay,
2 2 veh-secs
T
OD = [ X − 1] Average overflow delay per vehicle, s/veh
2
Note: OD is obtained by dividing ODa by cT
 Delay Models in the HCM 2000
47

 The HCM 2000 delay model includes:

 the uniform delay model,
 an overflow delay model, and
 a term covering delay from an existing or
residual queue at the beginning of the
analysis period.
 Delay Models in the HCM 2000
48

d = d1 PF + d 2 + d 3 d = control delay, s/veh

d1 = uniform delay component, s/veh
 1 − g 
2
 PF = progression adjustment factor
 C    c  d2 = overflow delay component, s/veh
d1 =   *  
  1 − min(1, X ) *
2 g  d3 = delay due to pre-existing queue,
  C  

s/veh
T = analysis period, h
  8kIX  X = v/c ratio
d 2 = 900T ( X − 1) + ( X − 1) 2 +   C = cycle length, s
  cT  
k = incremental delay factor for
actuated controller settings; 0.50
for all pre-timed controllers
Note that the PF factor I = upstream filtering/metering
accounts for the effect of adjustment factor; 1.00 for all
individual intersection analyses
platooned arrival patterns c = capacity, veh/h
 Examples
49