2B_Fundamentals of Traffic Flow

KAAF UNIVERSITY COLLEGE

Civil Engineering Department College of Engineering

__________________________________

Transportation Engineering I

CIV 367

Lecture 2B_ Fundamentals of Traffic Flow

Kwasi Agyeman – Boakye ( [email protected])


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.

Traffic Waves.wmv

Traffic Waves.wmv

Traffic Waves.wmv

Traffic Waves.wmv

Traffic Waves.wmv


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


Shockwave Diagrams


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?



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)



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



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.


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


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


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.


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



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)



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



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

2B_Fundamentals of Traffic Flow

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