A report on application of probability to control the flow of traffic through a highway system

A report on application of probability to control the flow of traffic through a highway system

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  PROBABILITY AND STATISTICS
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  TOPIC- A REPORTON APPLICATION OF PROBABILITY TO CONTROLTHE FLOWOF TRAFFIC THROUGH A HIGHWAY SYSTEM PROJECT- 1 NAME- ABHIJIT PATRA
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  CONTENT PROBABILITY AND ITSTERMINOLOGY CALCULATION EXAMPLE APPLICATION OF PROBABILITY TO CONTROL TRAFFIC FLOW
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  PROBABILITYANDRELATEDTERMINOLOGY  Probability meansthe mathematical chance that something might happen, is used in a numerous day-to-day applications, including in weather forecasts, sports strategies, insurance options, games and recreational activities, making business.  In other word probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how like the proposition is true.  Probability of an event is a number between 0 and 1 where 0 indicates impossible event and 1 indicates certain events.  Event: – an event is a set of outcomes of an experiment to which probability is assigned. Event is the subset of sample space  Sample space: – sample space of an experiment is the set of all possible outcomes or results of that experiment. A sample space is usually denoted as a set notation and every outcomes are listed as elements of the set.
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  Types of events 1.Mutually exclusive events: The events are said to be mutually exclusive or disjoint when they can not occur at the same time Example- rolling a die once. The outcome a) odd number b) even number Here the two events a and b are mutually exclusive event. 2. Independent event: The event are said to be independent if the occurrence of one does not affect the probability of occurrence of other . Example- if we flip a coin in the air and get the outcome as head ,then again if we flip the coin but this time we get the outcomes as tail. In both cases the occurrence of both the events is independent of each other.
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  CALCULATION Probability of anevent = sample space = S Event occurred = E Probability of a is denoted as P(E) P(E) = NUMBER OF GIVEN OUTCOME NUMBER OF ALL POSSIBLE OUTCOME |E| |S|
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  EXAMPLE Let consider twodice rolled once . The event is sum of the outcome is even. Find the probability of the event. Ans- If two dice rolled once the sample space s= {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5), (2,6),(3,1),(3,2),(3,3),(3,4),(3,5), (3,6),(4,1),(4,2),(4,3),(4,4) (4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3) ,(6,4)(6,5)(6,6)} |S|=𝟔𝟐 Event E= the sum of the outcome is even = {(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5), (4,2),(4,4),(4,6) (5,1),(5,3),(5,5),(6,2),(6,4),(6,6)} |E|=18 P(E) = |𝑬| |𝑺| = 18 36
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  APPLICATIONOF PROBABILITY TOCONTROL THE TRAFFIC FLOW The aim of this model is to provide a simple yet accurate representation of lane changing. We observe the probability a specific car will switch at a given time, which is accurate from the macroscopic point of view. This approach is reasonable because there is no definite rule to account for drivers’ decisions to switch lanes: even if switching lanes is objectively correct in a given situation, the driver may not make the best decisions. Our model follows the ‘keep right except to pass’ (KREP) rule, widely used in traffic systems around the world. Cars stay in the rightmost lane unless the car in front of them is driving below their preferred speed.
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   In thatcase, they could switch to the next lane on the left until they are able to drive at their preferred speed, and continue driving on that lane until they overtake the slower car and have enough empty distance to return to the right lane  If a car in any lane (except the leftmost) is traveling at a lower speed than its desired speed because it is impeded by the car in front, then there is a chance the driver will decide to switch to the left lane.  First, however, the driver must take into account safety concerns — it must be separated enough from neighboring cars to be able to switch lanes safely.  The “two-second rule”, a common practice among drivers for car-following, forbids changing lanes if a car is less than two seconds behind the car in front of it or behind it. CONTINUED….
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  THANK YOU