Introduction to Transportation
Systems
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PART III:
TRAVELER
TRANSPORTATION
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Chapter 27:
Deterministic Queuing
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Deterministic Queuing
Applied to Traffic Lights
‹ Here
we introduce the concept of
deterministic queuing at an
introductory level and then apply
this concept to setting of traffic
lights.
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Deterministic Queuing
Deterministic Queuing
In the first situation, we consider λ(t), the arrival
rate, and µ(t), the departure rate, as
deterministic.
Deterministic Arrival and Departure Rates
λ(t)
µ(t)
Figure 27.1
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Deterministic Queuing
Deterministic Arrival and Departure Rates
(continued)
λ(t)
vehicles/hr
2000
1000
0
1
2
3
4
time
µ(t)
vehicles/hr
1500
time
Figure 27.1
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Queuing Diagram
Cumulative
Arrivals
4000
available
capacity for
departure
maximum queue
= 500
3000
2000
arrivals
1000
1
Figure 27.2
available capacity
for departure
2
3
4
5
Time
(hours)
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Another Case
‹ Now,
the numbers were selected
to make this simple; at the end of
four hours the system is empty.
The queue dissipated exactly at
the end of four hours. But for
example, suppose vehicles arrive
at the rate of 1,250/hour from t=3
to t=4.
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Another Queuing Diagram
Cumulative
Arrivals
4250
4000
available
capacity
for
departure
3000
2000
arrivals
1000
1
2
available
capacity
for
departure
3
4 4hrs 5
10min
‹
‹
Time
(hours)
CLASS DISCUSSION
What is the longest queue in this system?
What is the longest individual waiting time?
Figure 27.3
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Computing Total Delay
Area Between Input and Output Curves
Cumulative
(4,4000)
4000
(3,3000)
A i l
3000
2000
1000
(2,1000)
1
Figure 27.4
2
3
4
Time (Hours)
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Choosing Capacity
µ (t) = 2000
µ (t) = 1500
µ (t) = 500
CLASS DISCUSSION
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A Traffic Light as a
Deterministic Queue
Service Rate and Arrival Rate at Traffic Light
Service
Rate µ(t)
µ
Time
R
G
R
G
Arrival
Rate
λ
Time
Figure 27.5
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Queuing Diagram per Traffic
Light
Cumulative
Arrivals
Arrival at Rateλ
Service at
Rate µ
to
Time
R
Figure 27.6
G
R
G
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Queue Stability
All the traffic must be dissipated during the
green cycle.
If R + G = C (the cycle time),
then λ(R + t0) = µt0.
Rearranging t0 =
λR
µ-λ
If we define λ = ρ (the “traffic intensity”),
µ
Then t0 = ρR
1-ρ
For stability t0 ≤ G = C - R .
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Delay at a Traffic Signal --
Considering One Direction
D=
λR2
2(1 - ρ)
The total delay per cycle is d
d= D =
R2
λC
2C(1 - ρ)
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Two Direction Analysis of
Traffic Light
Flows in East-West and North-South Directions
λ4
λ2
λ1
λ3
Figure 27.7
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D1 = λ1R12
2(1 - ρ1)
where ρ1 = λ1
µ
We can write similar expressions for
D2, D3 , D4. We want to minimize
DT , the total delay, where
DT = D1 + D2 + D3 + D4
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Choosing an Optimum
Remembering that
R2 = R1
R4 = R3 = (C - R1)
we want to minimize DT where
DT =
λ1R12
λ2R12
λ3(C-R1)2 λ4(C-R1)2
+ 2(1 - ρ ) + 2(1 - ρ ) + 2(1 - ρ )
2(1 - ρ1)
2
3
4
To obtain the optimal R1, we differentiate the
expression for total delay with respect to R1 (the
only unknown) and set that equal to zero.
dDT
dR1
λ1R1
λ2R1 λ3(C-R1) λ4(C-R1)
= 1 - ρ + 1 - ρ - 1 - ρ -
1 - ρ
=0
1
2
3
4
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Try a Special Case
λ1 = λ2 = λ3 = λ4
Therefore, ρ1 = ρ2 = ρ3 = ρ4.
The result, then, is
R1 = C , R3 = C
2
2
This makes sense. If the flows are equal, we would
expect the optimal design choice is to split the
cycle in half in the two directions.
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‹ The
text goes through some further mathematical derivations of other cases for the interested student.
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