Bandwidth Optimization

1

• Topic 4
• Bandwidth Optimization

2
Bandwidth Optimization Methodology

• Half-Integer Algorithms by Brooks and Little
• Based on minimizing interferences
• Providing equal bandwidth solution when speeds
  are the same for both directions
• Calculating interference for each individual
  intersection based on a referencing intersection,
  m
• The referencing intersection m is the one that
  has the minimum green

3
Half-Integer Offset (Center Red)
m
j
i
Simultaneous
Alternate
Time
Gm
Gj
Ri
Rm
Rj
Gi
Distance
4
Simultaneous Offset - Forward (Lower
Interference)
m
j
IL
Time
Gm
Gj
0
Rm
Rj
Distance
5
Simultaneous Offset - Forward (Upper
Interference)
m
j
0
IU
Time
Gm
Gj
Rm
Rj
Distance
6
Simultaneous Offset - Forward (No Interference
Slack Time)
m
j
IS
Time
Gm
Gj
0
Rm
Rj
Distance
7
Simultaneous Offset - Forward (No Interference
Slack Time)
m
j
IS
IS
Time
Gm
Gj
Rm
Rj
Distance
8
Alternate Offset - Forward (Lower Interference)
m
j
IL
Time
Gm
Rj
0
Rm
Gj
Distance
9
Alternate Offset - Forward (Upper Interference)
m
j
IU
0
Time
Rj
Gm
Rm
Gj
Distance
10
Alternate Offset - Forward (No Interference)
m
j
IS
Time
Rj
Gm
Rm
Gj
Distance
11
Simultaneous Offset - Backward (Lower
Interference)
j
m
Time
Gm
Gj
IL
0
Rj
Rm
Distance
12
Summary of Equations
KL, KU, and K are integers K is even -
simultaneous K is odd - alternate
13
Summary of Equations
KL, KU, and K are integers K is even -
simultaneous) K is odd - alternate
14
Summary of Brooks Algorithm
a. Find intersection m with smallest green b.
Travel times to right of m (forward) are
positive and to the left (backward) are
negative c. Calculate least allowable IL and IU
for each intersection d. Perform total
interference minimization e. Identify the
optimal progression band and offsets f.
Construct the time-space diagram g. Adjust split
of directional bandwidth if desired
15
Graphical Illustration
1
2
m
3
IL,3
IU,2
Time
IU,1
Distance
16
Example
Speed is 40 mph
Int. Gj Rj Dmj -0.5(Rm-Rj) Tmj 0.5(TmjTjm) -IS Imax KU IU KL IL ?mj
1 45 35 1500 -2.5 -25.56 -25.56 -5 40
2 40 40 - - - - - - - - - - -
3 45 35 330 -2.5 5.62 5.62 -5 0 1 -1 3.120 -36.88 43.12 0 1 -8.12 31.87
4 50 30 1830 -5.0 31.19 31.19
5 45 35 4830 -2.5 82.31 82.31 -5
17
Illustration of Intersection 2 and 4
(Simultaneous Offset)
m2
4
Rj
26.19
Gj
Time
Gm
Rm
Distance
18
Illustration of Intersection 2 and 4
(Alternate Offset)
m2
4
3.81
Time
Rj
Gm
Gj
Rm
Distance
19
Illustration of Intersection 2 and 5
(Simultaneous Slack Times)
m2
5
0.19
4.81
Rj
Gj
Time
Gm
Rm
Distance
20
Example (continued)
Int. Gj IU IL
1 45 11.94 23.06
2 40 0 0
3 45 3.12 31.87
4 50 26.19 3.81
5 45 -0.19 -4.81
Rank Int. IU IL
1 4 26.19 3.81
2 1 11.94 23.06
3 3 3.12 31.87
4 2m 0 0
5 5 -0.19 -4.81
21
Example (continued)
Intersection by Rank Intersection by Rank Intersection by Rank Intersection by Rank Intersection by Rank ITotal
1 2 3 4 5 ITotal
U U U U U 26.19
L U U U U 11.94 3.81 15.75
L L U U U
L L L U U
L L L L U
L L L L L
Interference Rank Int. K-value Offset ?mj (Reference Start of Green)
L 1 4 1 Alternate
U 2 1 1 Alternate
U 3 3 0 Simultaneous
U 4 2m - -
U 5 5 2 Simultaneous
22
Offset to Start of Green (Alternate)
m
Rj
?G,mj
Time
Gm
?R,mj
Rm
Distance
23
Offset to Start of Green (Simultaneous)
m
Rj
?G,mj
Time
Gm
?R,mj 0
Rm
Distance
24
Bandwidth with LT Phases
Time
Intersection 1
Intersection 2
Space
25
Major Terms
Phase B
OB left turn
Cycle
OB through
IB through
IB left turn
Link length in-between
26
Bandwidth Maximization
Outbound
Inbound Bandwidth
Outbound Bandwidth
27
Bandwidth Maximization
Inbound Bandwidth
Outbound Bandwidth
28
Phasing Sequence
Leading
Lead-Lag
Lag-Lead
Lagging
16 Combinations
4 types of left turn sequence for each
intersection
29
Methodology
Maximum Bandwidth Bo Bi Bo lt Gomin Bi lt
Gimin
Bmax Go,min Gi,min Ii,min
j
X
30
Methodology
Bmax Go,min Gi,min Ii,min
Interference
Bmax Constant
Exception Bmax Gi,min
31
Methodology
Bmax Go,min Gi,min Ii,min
Interference
Bmax Constant
Exception Bmax Gi,min
32
Upper Interference
Iup
X Intersection that has the smallest inbound
green
33
Iujp Gix - (-Rxn Txj Rjp Gij Tjx ) mod C
GixnCL
Iujp
(-Rxn Txj Rjp Gij Tjx )
-Rxn Txj Rjp Gij
Gix
-Rxn Txj
-Rjp
-Rxn Txj Rjp
0
Rxn
-Rxn
Intersection X
Intersection j
34
Iujp Gix - (-Rxn Sox Txj Rjp Gij Tjx )
mod C
GixnCL
Iujp
(-Rxn SoxTxj Rjp Gij Tjx )
-Rxn SoxTxj Rjp Gij
Gix
-Rxn SoxTxj
-Rjp
-Rxn SoxTxj Rjp
0
Rxn
-RxnSox
-Rxn
Intersection X
Intersection j
35
No valid upper interference
(-Rxn SoxTxj Rjp Gij Tjx )
Iujp
If Gix - (-Rxn SoxTxj Rjp Gij Tjx
) lt-Sij, no valid upper interference.
-Rjp
Intersection X
Intersection j
36
Iujp Gix - (-Rxn Txj Rjp Gij Tjx ) mod C
T.T.34 sec
16
Iu 55-4114
41
3034226134181 181-14041
55
30342261
30
38
61
3045-34
45
303422
22
16
3034
41
j
30
55
-30
0
x
37
Iujp Gix - (-Rxn Sox Txj Rjp Gij -Tjx )
mod C
T.T.34 sec
20
41
-(-34)34
55
34-341861
34
34
0
61
Sox
45
34-3418
18
34-34
20
j
41
Intersection j has the smallest outbound green
34-341861-3445 Iu 55-4510
55
34
x
38
Iujp Gix - (-Rxn Sox Txj Rjp Gij -Tjx )
mod C
3045
T.T.34 sec
20
45
304
55
Sox
-(-30)30
304-341861
30
38
0
61
41
304-3418
18
304-34
20
j
45
Intersection j has the smallest outbound green
304-341861-3445 Iu 55-4510
55
30
x
39
ILjp (-Rxn Txj Sj Rjp Tjx) mod C
ILjp
Go,min
Slack Time, Sj Sj Goj Go,min
0
Rxn
j
x
40
ILjp (-Rxn Txj Sj Rjp Tjx) mod C
T.T.50 sec
24
41
55
3850-41550149
38
30
IL 9
60
45
3850-415
3850
15
3850-4
24
41
j
38
55
-38
0
x
41
No valid lower interference
T.T.34 sec
16
Lower interference calculation causes upper
interference occurring. Intersection j must move
up by 4 sec to reduce upper interference. Inbound
band 55-1441
41
Iu 2455-6118
55
3034-42261
30
38
61
3034-42234116
24
140-11624gtgt6
45
3034-422
3034-4
22
3034
16
j
41
30
55
-30
0
x
42
Lower Interference
Gij Gix
Gij
C
ILjp
C - ILjp lt (Gij Gix)Sj
ILjp gt C Sj
43
ILjp (-Rxn Txj Soj Rjp Tjx) mod C
2-sec slack
T.T.95 sec
30
55
0
41
140-1328lt10 No lower interference
-1695-4-3895132
16
18
x
45
-1695
65
-1695-4
30
38
-1695-4-38
55
j
41
0
16
-16
x
44
Example

• There are three coordinated intersections A, B,
  and C with a cycle length of 140 sec. Travel
  times and phasing sequence and splits are shown
  below. Determine the maximum bandwidth and
  offsets.

T.T.34 sec
T.T.56 sec
A
B
C
38
18
16
57
61
50
41
64
64
30
25
18
45
Example
T.T.34 sec
T.T.56 sec
A
B
C
38
18
16
57
61
50
41
64
64
30
25
18
B Iu 61-(-1834-306434)61-84-23, not valid
(23gt3) IL (-1834-9-3034)11 C Iu
61-(-1890-256490)61-610, IL
(-1890-16-2590)121, not valid
(140-12119gt3) Therefore, it has a 11 lower
interference caused by B and zero interference by
C. Bandwidth 41(61-11)91
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Homework (C60)
T.T.36 sec
T.T.51 sec
T.T.27 sec
T.T.41 sec
20
10
10
30
25
25
30
26
30
20
25
15
30
25
12
10
15
13
10
15
48
Homework

• There are three coordinated intersections A, B,
  and C with a cycle length of 140 sec. Travel
  times and phasing sequence and splits are shown
  below. Determine the maximum bandwidth and
  offsets.

T.T.34 sec
T.T.56 sec
38
25
20
61
64
45
57
41
55
30
18
18
49
Homework

• There are two intersections A and B, running
  coordinated control for the major street, with
  the cycle length of 60 sec.
• For intersection A, The outbound through time is
  15 sec, and the inbound through time is 20 sec
  the outbound left-turn interval is 10 sec, and
  the inbound left-turn interval is 15 sec.
• For intersection B, The outbound through time is
  20 sec, and the inbound through time is 25 sec
  the outbound left-turn interval is 15 sec, and
  the inbound left-turn interval is 20 sec.
• The distance between the two intersections is
  1320 feet.
• Questions
• Assuming intersection A is running leading left
  turns, and Intersection B is running leading left
  turn for outbound direction and lagging left turn
  for inbound direction. At the speed limit of 30
  mph for both directions, what is the maximum
  total bandwidth (of both directions) between the
  two intersections?
• If the speed limit is 20 mph, what is the maximum
  bandwidth when intersection A is running leading
  left turn, and intersection B is running lagging
  left turn?
• All red and yellow time can be ignored for the
  calculation.
• Please list all your calculations and adjustments
  if any (Tips use Synchro to optimize bandwidth
  manually to check your results)

50
Brooks Algorithm A Special Case
Speed is 40 mph
Int. Gj Rj Dmj -0.5(Rm-Rj) Tmj 0.5(TmjTjm) -IS Imax KU IU KL IL ?mj
1 45 35 1500 -2.5 -25.56 -25.56 -5 40
2 40 40 - - - - - - - - - - -
3 45 35 330 -2.5 5.62 5.62 -5 0 1 -1 3.120 -36.88 43.12 0 1 -8.12 31.87
4 50 30 1830 -5.0 31.19 31.19 0 -1 26.19 -13.81 0 1 -36.19 3.81
5 45 35 4830 -2.5 82.31 82.31 -5 0 1 2 79.81 35.81 -0.19 1 2 -44.81 -4.81
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