1.225J
1.225J (ESD 225) Transportation Flow Systems
Lecture 10
Control of Isolated Traffic Signals
Profs. Ismail Chabini and Amedeo R. Odoni
Lecture 10 Outline
‰ Isolated saturated intersections
‰ Definitions: Saturation flow rate, effective green, and lost time
‰ Notation for an intersection approach variable
‰ Two assumptions for delay models
‰ Average delay per vehicle: deterministic term Wq,A
‰ Average delay per vehicle: stochastic term Wq,B
‰ Webster optimal green time settings: two approaches intersection and
numerical example
‰ Webster cycle time optimization procedure
‰ Mid-day and evening-peak examples
‰ Lecture summary
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Lecture 10, Page 2
1
Isolated Saturated Intersections
Average departure rate
from an approach
qN
P2
Saturation regime
for P2
qE
qW
P1
Saturation regime
for P1
qS
q
Average arrival rate
at an approach
‰ An implication of saturation regime: need to efficiently allocate green
times (gN, gS) and (gE, gW)
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Lecture 10, Page 3
Saturation Flow, Effective Green, and Lost Time
Saturation flow
s
Rate of discharge
of queue in
fully-saturated
green periods
Lost time (l1)
Red
Time
Lost time (l2)
Effective green time (g)
Green (k)
Amber (a)
Red
• Total lost time l = l1 + l2 (typically 2 sec)
• Green (k) + Amber (a) = Effective green time (g) + Total lost time (l) Ÿ l = k + a - g
• Effective green time (g) u Saturation flow (s) = Total vehicles discharged during (k + a)
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Lecture 10, Page 4
2
Notations for An Intersection Approach
‰ Webster
x s : saturation flow rate
x g : effective green time
x c : cycle time
g
: fraction of effective green in cycle time
c
xO
‰ Webster
‰ Meaning
q
O
arrival rate (veh/unit of time)
Os
average flow rate at exit of an approach
q
Os
x
‰y
‰ Queueing Theory
degree of saturation
U
P
O
P
q
s
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Lecture 10, Page 5
Two Assumptions for Delay Models
‰ Assumption (A):
• The interarrival times are constant (view arrivals as evenly spaced
at rate q)
• Service time is constant during effective green and zero in the rest
of the cycle
• Average waiting per vehicle is denoted by Wq , A
‰ Assumption (B):
• The interarrival times are exponentially distributed with rate q
• Service time is constant with service rate Os
• Average waiting per vehicle is denoted by Wq , B
‰ Webster formula for total waiting time per vehicle:
d
Wq , A Wq , B correction factor obtained by simulation
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Lecture 10, Page 6
3
Average Delay per Vehicle: Term Wq,A
Cumulative
vehicles in queue
q (c g )
(s q)
q·(c-g)
s-q
q
Time
c-g
g
c
‰ Total waiting during c per approach:
1
q (c g )
q(c g )[(c g ) ]
2
sq
q (c g ) 2 s
˜
2
sq
q (c g ) 2
1
˜
2
(1 Ox )
‰ Total arrivals during cycle c: qc
­ q (c g ) 2
1 ½ 1
˜
¾˜
2
(1 Ox) ¿ qc
¯
‰ Wq, A ®
c (1 g c) 2
˜
2 (1 Ox)
c (1 O ) 2
2(1 Ox)
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Lecture 10, Page 7
Average Delay per Vehicle: Term Wq,B
‰ Interarrival times are exponentially distributed with rate q, and
service times are deterministic with rate Os
‰ Average waiting time for M/D/1 queueing system:
‰ Wq , B
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1 (q Os ) 2
˜
2 q (1 q Os)
1
U2
˜
2 O (1 U )
1
x2
˜
2 q(1 x)
Lecture 10, Page 8
4
Webster’
Webster’s Average Delay Per Vehicle Model
‰ Average delay per vehicle: d Wq , A Wq , B correction term
1
‰d
§ c ·3
c(1 O ) 2
x2
0.65¨¨ 2 ¸¸ x ( 25O )
2(1 Ox) 2q(1 x)
©q ¹
‰ Wq , A dominates for very small values of x
‰ Wq , B dominates for large values of x (xo 1)
‰ Small value of x is not an important case from an optimization
standpoint
‰ Optimal green time setting problem: Find OE, OW, ON, and OS such that
the total delay is minimum
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Lecture 10, Page 9
Observed Delay vs.
vs. Webster’
Webster’s Model
Relative delay per vehicle =
Average delay per vehicle / Cycle time
Semi-empirical curve
to fit results
Terms Wq,A + Wq,B
Term Wq,A
Degree of saturation, x
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Lecture 10, Page 10
5
“Optimal Settings”
Settings”: A Two Approaches Intersection
‰ x1
q1
, x2
O1s1
q2
O2 s2
‰ Note: • (q1, s1) and (q2, s2) are given
• (O1+ O2 )c= c
• if x1n, then x2 p and vice versa
‰ Total delay | Wq(,1B) ˜ q1 Wq(, B2 ) ˜ q2
xi2
1 2
˜ qi
¦
2 i 1 qi (1 xi )
1 2 xi2
¦
2 i 1 (1 xi )
‰ Minimum total delay: Total delays are about the same on both
approaches
x12
x22
x
1 x1
x x1
x
1 x2
x2
O2 § g 2 c
¨
O1 ¨© g1 c
g2 ·
¸
g1 ¸¹
q2 s 2 §
¨
q1 s1 ¨©
y2 ·
¸
y1 ¸¹
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Lecture 10, Page 11
Numerical Example 1
x Saturation flow rate s 1800 veh/hr for all arms (approaches)
Lost time L 10 sec
Cycle length c 60 sec
x qN
qS
600 veh/hr ; q E
x yN
yS
600
1800
x y N S
x
g N S
g E W
1
; y E W
3
13
16
x g N S g E W
x 2 g E W g E W
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1
; yE
3
qW
yW
300 veh/hr
300
1800
1
6
1
6
2
60 10 50 sec
3g E W
50 sec Ÿ g E W
50 3 | 17 sec; g N S | 33 sec
Lecture 10, Page 12
6
Cycle Time Optimization
‰ " Optimal" cycle : co
n
‰y
¦y,
i
1.5 L 5
1 y
n number of phases (typically n 2)
i 1
­l average time lost per phase (l | 2 sec)
®
¯ R all - red time ( R | 6 sec)
‰ L n l R,
‰ Typically L | 10 sec
‰ Two phases :
Phase A
all red
all red
k
Phase B
c
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Lecture 10, Page 13
Numerical Example 2: MidMid-Day
x s 1600 veh/hr in each direction ( N o S ; S o N ; E o W ; W o E )
2 phases; all reds 6 sec/cycle; lost time 2 sec/phase
x qN
x
x
x
x
x
600 veh/hr ; qW 400 veh/hr; q E 300 veh/hr
300
3
400 2
600 3
y N yS
; yE
; yW
1600 16
1600 8
1600 8
3 2 5
2
3
yN S
; L 2 ˜ 2 6 10 sec
; y
; yE W
8 8 8
8
8
1.5 u10 5
53 sec; optimal cycle
co
1 5 8
2
3
g N S # (53 10) | 26 sec; g E W # (53 10) 17 sec
5
5
xN xS 0.764; xW 0.779; xE 0.585
qS
x Wq , N
Wq ,S # 18.4 sec; Wq ,W # 28.6 sec; Wq , E # 20.0 sec
x Lq , N
Lq , S # 3.07 veh; Lq ,W # 3.18 veh; Lq , E # 1.67 veh
x Total delay/hr # 11 hours
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Lecture 10, Page 14
7
Numerical Example 2(cont.): EveningEvening-Peak
x s 1600 veh/hr in each direction ( N o S ; S o N ; E o W ; W o E )
2 phases; all reds 6 sec/cycle; lost time 2 sec/phase
x qN
x
x
x
x
x
qS
800 veh/hr ; qW qE 600 veh/hr
800 1
600 3
y N yS
; yW yE
1600 2
1600 8
1
3
1 3 7
yN S
; yE W
; y
; L 2 ˜ 2 6 10 sec
2
8
2 8 8
1.5 u10 5
160 sec; optimal cycle
co
1 7 8
4
3
g N S # (160 10) | 86 sec; g E W # (160 10) 64 sec
7
7
xN xS 0.93; xW xE 0.9375
x Wq , N
Wq ,S # 62.0 sec; Wq ,W
Wq , E # 88.3 sec
x Lq , N
Lq , S # 13.8 veh; Lq ,W
Lq , E # 14.7 veh
x Total delay/hr # 57 hours
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Lecture 10, Page 15
Lecture 10 Summary
‰ Isolated saturated intersections
‰ Definitions: Saturation flow rate, Effective green, and Lost time
‰ Notations for an intersection approach variable
‰ Two assumptions for delay models
‰ Average delay per vehicle: deterministic term Wq,A
‰ Average delay per vehicle: stochastic term Wq,B
‰ Webster optimal green time settings: Two approaches intersection
and numerical example
‰ Webster cycle time optimization procedure
‰ Mid-day and evening-peak examples
1.225, 12/3/02
Lecture 10, Page 16
8