Traffic Light Sequences Essay, Research Paper
MEI Mechanics-1 Maths course-work
An probe into automatic Traffic Lights
Daniel Baird
The Edinburgh Academy
Fall 1996
Table of contents
Table of contents 2
Introduction 3
Question 3
Purposes 3
Simplifications and Premises 4
Road 4
Traffic Lights 4
Vehicles 4
General premises. 4
Sequence OF THE TRAFFIC LIGHT 5
MODEL 1 5
Maximal distance travelled in order for a vehicle to unclutter the system 5
Time period of one rhythm 6
Time period of green visible radiation 6
Dead Time 6
Time period of ruddy visible radiation 7
MODEL 2 8
Data aggregation 9
Real Situation 9
List of other factors 9
Decision 10
Introduction
In this brief study, my purpose is to try to find the factors that affect the form of operation of impermanent traffic visible radiations in operation while roadworks or another obstructor is barricading one lane of a two-way, individual carriageway route. Once this has been accomplished, I shall try to make a simple mathematical theoretical account and effort to use it to existent life.
Question
When major roadworks take topographic point on a bipartisan individual carriageway route, contractors often regulate a one-way flow of traffic in jumping waies by usage of automatically controlled traffic visible radiations. What considerations affect the timings of such visible radiations?
Purposes
1. Analyse job.
2. Propose a expression.
3. suggest mistakes involved.
4. expression into ways where timings from expression could be modified other than simplest instance.
Simplifications and Premises
+ Road
n The route is bipartisan, individual carriageway.
n The route is directly, there are no obstructors.
n Road surface is changeless, clash is changeless.
+ Traffic Lights
N Drivers have infinite forbearance and ever obey traffic visible radiations absolutely.
n All sets of visible radiations are absolutely coordinated, and their form invariable.
+ Vehicles
n Vehicle size is changeless.
n Vehicle acceleration/deceleration is changeless.
n Distance between vehicles is changeless.
n All vehicles are perfect.
n Distance travelled by autos between traffic visible radiations is changeless.
General premises.
N Weather is changeless, clear and dry.
n The clip that the gold visible radiation is displayed for is quasi-instantaneous and therefore can be ignored.
N.B. These premises have been made in order to let my theoretical accounts to work decently. The theoretical values that the theoretical accounts formulate are to be taken with a pinch of salt and therefore to take history of other variables, mistake bounds should be applied to cut down these disagreements.
Sequence OF THE TRAFFIC LIGHT
1. Red
2. Red/Amber
3. Green
4. Amber
5. Red
MODEL 1
Distance between traffic signals. 34m
Mean velocity of traffic. 5ms-1
Time for traffic to unclutter the system ( 34m ) at 5ms-1 6.8sec
Mean length of one vehicle. 4m
Average inter-vehicle spread ( dynamic ) . 5m
Average inter-vehicle spread ( inactive ) . 0.5m
Number of vehicles per flow. 8
Time hold between autos get downing at first set of signals. 1sec.
Duration of gold visible radiation. 3sec s.
In this theoretical account, it is assumed that every rhythm of the visible radiations, no mre or less than eight vehicles pass through the system in one way. The minimal distance a auto has to go to unclutter the sytem is 34m. On the visible radiations turning green, the first auto immediately reaches a velocity of 5ms-1 one second subsequently the 2nd auto does the same and so on.
Maximal distance travelled in order for a vehicle to unclutter the system
34+ ( 7 & # 215 ; 0.5 ) + ( 8 & # 215 ; 4 ) =69.5m
The last vehicle in the flow has
to go the 34m between the traffic visible radiations plus the distance between itself and the first set of visible radiations. This latter distance is the length of all eight autos plus the distance of seven inactive inter-car spreads.
Time period of one rhythm
Time = Distance / velocity
= 69.5m / 5ms-1
= 13.9 sec s
This is the clip taken for the first vehicle to unclutter the system, nevertheless the last vehicle has to wait 7 seconds before it can get down to travel.
13.9+7=20.9 sec s
The clip from the first auto get downing until the last auto uncluttering the last set of visible radiations is hence 20.9 seconds.
Time period of green visible radiation
= 35.5m / 5m/s
= 7.1 sec s
This is the clip for the last vehicle to unclutter the first set of visible radiations, nevertheless, we have to see the clip hold between autos get downing.
7.1+7=14.1 sec s
Therefore the minimal clip the green visible radiation must be illuminated is 14.1 seconds.
Dead Time
Time = period of one rhythm & # 8211 ; period of green visible radiation.
= 20.9s & # 8211 ; 14.1s
= 6.8s
Therefore, the system can non be clear until 6.8 seconds after the green visible radiation has been extinguished.
Time period of ruddy visible radiation
To happen the continuance of the ruddy visible radiation, I shall map out the two sets of visible radiations and fill in the spaces!
Light one Period/seconds Light two
Green 14.1 Red
Amber 3 Red
Red 6.8 Red
Red 3 Red/Amber
Red 14.1 Green
Red 3 Amber
Red 6.8 Red
Red/Amber 3 Red
Therefore the period of the ruddy visible radiation is ( 6.8+3+14.1+3+6.8 ) =33.7 seconds
MODEL 2
In order to bring forth an equation for these traffic visible radiations, I shall present some terra incognitas.
Variable Symbol
Distance between traffic signals. Dsig
Mean velocity of traffic. V
Mean length of one vehicle. L
Average inter-vehicle spread ( inactive ) . Gs
Number of vehicles per flow. n
Time hold between autos get downing at first set of signals. T
Duration of gold visible radiation. A
Maximal distance. Dmax
Maximal clip. Tmax
Dead Time. Tdead
Time period of green visible radiation. Tgreen
Time period of ruddy visible radiation. Tred
Dmax = Dsig+ [ ( n-1 ) Gs ] +nL
Dmax = Dsig+nGs+nL-Gs
Tmax = ( Dmax/v ) + ( n-1 ) T
Tmax = Dmaxv-1+nt-t
Tgreen = ( Dmax & # 8211 ; Dsig ) /v + T ( n-1 )
Tgreen = Dmaxv-1 & # 8211 ; Dsigv-1 + tn & # 8211 ; T
Tdead = Tmax & # 8211 ; Tgreen
Tred = 2Tdead +Tgreen + 2A
Data aggregation
The information I have used was collected in Broughton, Edinburgh. A building site near the route is barricading one lane of this two-way, individual carriageway route. The information was collected by the pupils in the Mechanics A-level maths set. Stop tickers were used to clip the traffic signals, and a wheel swayer to mensurate the dimensions of the route. The conditions was clear and blowy.
Real Situation
In a existent state of affairs it is even more complicated as non merely would we hold to bear in head the acceleration and slowing of the vehicles, but even these would non be changeless.
Many other factors besides consequence the consequences, such as if the traffic is already traveling at the first set of signals, besides factors such as conditions conditions consequence the manner of drive.
Factor Error bounds
Car Length x0.75m
Inter-car spread 2.6mx0.2m
Time when system is empty 5secx2.5sec
Speed of flow 4.26ms-1 & # 215 ; 0.75ms-1
List of other factors
+ Weather
+ Time of twenty-four hours
+ Presence of particular vehicles, e.g. Articulated lorries, bikes, Equus caballuss etc.
+ Reckless drivers leaping visible radiations.
Decision
This undertaking has given me an interesting penetration into the externally simple maths of mundane systems. I would see these theoretical accounts to be anything but perfect as there are far to many random factors involved and this makes it excessively hard to pattern accurately without pandemonium theory.
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