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Civil Engineering Department College of Engineering
__________________________________
Transportation Engineering I
CIV 367
Lecture 2B_ Fundamentals of Traffic Flow
Kwasi Agyeman – Boakye ( [email protected])
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Shockwaves in Traffic Stream
Shockwaves describe the phenomenon of backups and queuing on a highway due to a sudden
reduction of the capacity of the highway. Traffic Waves.wmv . What other situations lead to a
reduction in capacity?
The sudden reduction in capacity could be due to accidents, reduction in the number of lanes,
restricted bridges sizes, work zones, a signal turning red. Shockwaves often occur as part of
Interrupted Traffic flow.
At boundary between two traffic states a shock wave exists,
moving along the road at speed Csw.
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Shockwaves Forward ,Backward and Stationary
C12 > 0, Forward moving shockwave;
positive shockwave speed moving in direction of traffic
C13 > 0, Backward moving shockwave;
negative shockwave speed moving in opposite direction.
“Backward”: discontinuity moves in opposite
direction of the moving traffic;
“forward”: discontinuity moves in the same direction
of the moving traffic;
“forming”: increase of congested portion over time;
“recovery”: decrease of congested portion over
time;
“frontal”: shock wave is at the downstream end of
the congested region;
“rear”: shock wave at the upstream end of the
congested region;
“stationary ”: shock wave remains at the same
position in space
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Shockwave Diagrams
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Calculation of Shockwaves
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Calculation of Shockwaves
Answer
Qmax = 2400veh/h/lane Um = 75km/h
(a) Greenshield Equation
Question
A two-lane single direction section of motorway has
a capacity of 2400 veh/h/lane at a speed of 75 km/h.
(a) Derive the form of the Green shields speed -
density curve and calculate all the relevant
parameters for the curve.
(b) During a period when traffic is flowing at 1800
veh/h/lane, a vehicle breaks down in one lane
reducing the road to single lane moving at full
capacity. It takes 20 minutes before the broken-down
vehicle is cleared. Afterwards, conditions reverts to
full capacity.
(i) What is the backwards propagation speed of the
shock wave caused by the breakdown?
(Hint, to work on a 2-lane flow-density diagram)
(ii) What is the maximum distance upstream from the
breakdown point where the effects will be felt?
(iii) At what time will the motorway be back to normal
at the point of breakdown?
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Calculation of Shockwaves
Construct a fundamental q – k diagram for 2 lanes
State at capacity, qm = 4,800 km/h,
Km = 64veh/km
Before the breakdown flow is 3,600 veh/h for the 2
lanes - State 1
During the breakdown flow reduced to 1 lane full
capacity 2,400 veh/h – State 2
State 1, q1 = 4,800 veh/h
State 2, q2= 2,400 veh/h
The densities at state 1 and 2 are;
Uf = 2Um = 2x75 = 150km/h
qm = UmKm and Kj = 2Km
Kj = 2x qm = 2x2400 = 64 veh/km/lane
Um 75
a = Uf = 150 b= Uf/Kj = 2.34
Hence Greenshield Equation
U = 150 – 2.34K
(b)
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Calculation of Shockwaves
ii) After the breakdown is removed the traffic
returns to state 3. Wave speed moves from 2 to 3.
Csw23 = q2 – q3 = 2400 – 4800 = -53 km/h
k2 – k3 109.3 - 64
And then moves from 3 to 1 again.
Csw13 = q3 – q1 = 4800 – 3600 = 37.5 km/hr
k3 – k1 64 - 32
dmax = Csw12x(20 + t) = Csw23xt
Shockwave from State 1 to State 2
Csw12 = q1 – q2 = 3600 – 2400 = -15.5 km/hr
k1 – k2 32 - 109.3
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Calculation of Shockwaves
Question
A road consists of 4 lanes, 2 in each direction. The
maximum capacity of 2 lanes in one direction is
2000 veh/h. When vehicles are stationary in a
jamming condition, the average length occupied by
a vehicle is 6.25m. During a period of observation,
the actual volume of traffic in one direction is
steady at the rate of 1200 veh/h. This flow is
brought to a halt when a traffic signal turns red and
a queue forms.
Find the time in seconds which elapses from the
moment the signal turns red until the stationary
queue reaches another intersection 75m from the
signal. Assume a linear relationship between
speed and concentration. Ans 58.7 sec
15.5 (1/3+t) = 53t
t = 0.137hr = 8.3min
Hence Maximum Distance,
dmax = Csw23xt
= 53x0.137
= 7.26km
ii) Motorway returns to normal when the shock wave
from 3 to 1 has travelled forward to meet the start of
the breakdown at (1), i.e. time when return to normal
is:
= 20min + 8.3min + dmax/Csw31x60
= 20 + 8.3 + 7.3x60/37.5 = 20+8.3+11.73 = 40min.
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Queue Theory
Queues have an effect on traffic flow and capacity. Their study is very important to the traffic
engineer , especially in congestion situations.
Queue theory deals with the use of mathematical algorithms to describe the processes that
result in the formation of queues, so that a detailed analysis of the effects of queues can be
undertaken.
A queue is formed when arrivals wait for a service or an opportunity, such as the arrival of an
accepted gap in a main traffic stream, the collection of tolls at a toll booth or of parking
fees at a parking garage, signalised intersections, bottlenecks etc
For proper analysis the following characteristics have to be considered;
•Arrival Distribution
•Service Method
•Characteristics of the Queue Length
•Service Distribution
•Number of channels
•Oversaturated and Undersaturated Queues
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Queue Characteristics
Arrival Distribution
Arrivals can be described as either a deterministic distribution or a random distribution. Poisson
distribution which typifies a combination of both is often used to describe light-to-medium traffic.
It is generally used in queuing theory.
0.00
0.05
0.10
0.15
0.20
0.25
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Arrivals in 15 minutes
Prob
abili
ty o
f Occ
uran
ce
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Queue Characteristics
Service Method
• First come – first served ( first in – first out)/ FIFO eg. Toll booth, signal
•Last in – first served ( Last in – first out)/ LIFO, boarding and exiting a bus,
•A system of priority eg. Giving priority to buses.
•No regular system of priority eg. A telephone operator
Characteristics of the Queue Length
• Finite Queue: Maximum number of units in the queue is specified eg. Where the waiting area
is limited. Between short distance signalised stops.
•Infinite Queue; maximum number of units in the queue is limitless.
Service Distribution
Can be considered as random. The poisson and negative exponential distribution is often used.
Number of Channels
The number of waiting lines. Could be a single channel or a multiple channel.
Oversaturated and Undersaturated Queues
Oversaturated has arrival rate greater than service rate. Length of queue does not reach a study
state but continues to increase.
Undersaturated has arrival less than the service rate. Also length of queue may vary but will
reach a steady state with the arrival of units.
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Single-Channel, Undersaturated,
Infinite Queues – A typical queue
Assume the rate of arrival is q veh/h and the service rate is Q veh/h. Also assume that both the
rate of arrivals and the rate of service are random, the following relationships can be developed.
1. Traffic Intensity, ρ
2. Probability of n units in the system , P(n)
Where n is the number of units in the system, including the unit being served
Rate of arrival
Queue Service
area
q Q
System
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Single-Channel, Undersaturated,
Infinite Queues – A typical queue
3. The expected number of units in the system, E(n)
4. The expected number of units waiting to be served (thus, the mean queue length) in the
system, E(m)
Note that E(m) is not exactly equal to E(n) -1, the reason being that there is a definite probability
of zero units being in the system, P(0).
5. Average waiting time in the queue, E(w)
6. Average waiting time on arrival, including queue and service, E(v)
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Single-Channel, Undersaturated,
Infinite Queues – Calculations
Probability that zero units are in the system P(0);
P(0) = 1 – q = 1 – 425 = 0.32
Q 625
Hence the operator will be free 32 percent of the
time.
b) Average number of vehicles in the system
E(n) = 425 = 2
625 – 425
c) Average waiting time for the vehicles that wait;
E(v) = 1 = 0.005hr
625 – 425
Qn. On a given day, 425 veh/h arrive at a tollbooth
located at the end of an off-ramp of a rural
expressway. If the vehicles can be serviced by only a
single channel at the service rate of 625veh/h
determine
a) The percentage of time the operator of the
tollbooth will be free
b) The average number of vehicles in the system
c) The average waiting time for the vehicles that wait
(Assume Poisson arrival and negative exponential
service rate)
Solution
q=435veh/h Q = 625veh/h
a) For the operator to be free, the number of vehicles
in the system must be zero;
Hence using the following equation
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Single-Channel, Undersaturated,
Infinite Queues – Calculations
The toll booth at Pokuase in Accra can handle 120 veh/h, the time to process a vehicle being
expontentially distributed. The flow is 90veh/h with a Poissonian arrival pattern. Determine:
i) The average number of vehicles in the system
ii) The length of the queue
iii) The average time spent by vehicles in the system
iv) The average time spent by vehicles in the queue
Ans. i)3 ii) 2.25 iii)120sec iv)90sec