DynaTraffic – Models and mathematical prognosis

Simulation of the distribution of trafficwith the help of Markov chains

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What is this about?

• Models of traffic situations• Graphs:

– Edges, Vertices– Matrix representation– Vector representation

• Markov chains– States, transition probabilities– Special states: periodic, absorbing, or transient– Steady-state distribution

• Matrix vector multiplication

DynaTraffic help to understand and learn these concepts

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The goal: analysis of a traffic system

We are interested in this question:„How many cars are thereat a certain time on a lane?”

In order to be able to make statements about the development of a system, we need a model.I.e., first we build a model and then we control and observe this model.

Mathematical prognosis step 1

Build a model of an everyday situation

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Photo from a side perspective

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Photo of the layout

© Google Imagery 2007

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Model of the layout, with cars

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Elements for the model without cars

Nodes

Arrows

What does your model look like?

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Model of the layout, without cars

Stop points nodesLanes edges

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Representation in DynaTraffic

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- Characters to label lanes- Colored arrows and slightly different arrangement

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Models

• Why does one build models?– To better understand systems– Models are a useful tool to examine systems

• Definition of a model: A simplified representation used to explain the workings of a real world system or event.(Source: http://en.wiktionary.org/wiki/model)

• Mathematical Models try to capture the relevant parameters of natural phenomena and to use these parameters for predictions in the observed system.

Mathematical prognosis step 2

Transformation of the model

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Why a transformation method?

We are concerned with traffic on single lanes and analyze the traffic with the help of a Markov model.

For that lanes must be verticesTransformation of the situation graph!

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Transformation recipe

Transformation of situation graph to line graph:• Each edge become a vertex.• There is an edge between two vertices, if one

can change from one lane to the other in the traffic situation.

• Each vertex has an edge to itself.

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Transformation step a)

Each edge becomes a vertex.

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Transformation step b)

There is an edge between two vertices, if one can change from one lane to the other in the traffic situation.

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Transformation step c)

Each vertex has an edge to itself.I.e.: a car can remain on a lane!

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The good news concerningthis transformation ☺

We do not need to do this transformation, it is already done in DynaTraffic. But we should understand it…

Situation graph Line graph to the situation graph

Mathematical prognosis step 3

Define assumptions

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Define the process

• Every 10 seconds, each car takes a decision with a certain probability (so-called transition probability):– „I change to another lane“– „I remain on this lane“

• The realization of decisions is called a transition: cars change their state, if necessary.

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The transition graph

The transition probabilities areentered in the line graph

Transition graph (= Markov chain)

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Our Markov chain

The vertices represent possible states, i.e., lanes on which a car can be.The edges show to which other lanes a car can change from each lane.

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Meaning ofthe transition probability?

„If there is a car onlane A now, it will inthe next transitionchange to lane Bwith a probabilityof 83%.

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Alternative representationof the transition graph

Transition graph Transition matrix

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How to read the transition matrix

From

To

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Empty entries inthe transition matrix?

If an edge does not exist, there is a 0 in the transition matrix at the corresponding entry.

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Summary:our traffic model

Photo Model with cars

Model without cars

Transition graphTransition matrix

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Summary:our traffic model

Photo Model with cars

Model without cars

Transition graphTransition matrix

Step 1: build a model of an everyday situation

Step 2

: tran

sform

ation

of th

e mod

el

Step 3: define assumptions

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Demo DynaTraffic

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Our Markov chain again

The vertices represent possible states, i.e., lanes on which a car be.The edges show on which other vertices a car can change from one vertex, and with which probability this happens per transition.

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„Do a transition“?

To calculate how many cars there are going to be on a certain lane, one needs:

– The number of cars on the individual lanes.– The probabilities leading to the certain lane.

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How many cars are on lane Aafter the next transition?

Cars on individual lanes:A: 3 carsB: 4 carsC: 7 cars

Probabilities leading to lane A:A A : 0.17C A : 0.83

Calculation:3 * 0.17 + 7 * 0.83 = 6.32 cars

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Probabilities for transitions

The required probabilities can directly be read from the transition matrix!

3 * 0.17 + 7 * 0.83 = 6.32 cars

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State vector

The number of cars per lane in the state vector notation

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Calculate transitions

As seen: Multiplication of the first row of the transition matrix with the current number of cars on the first lane (= second entry of the state vector) gives the new number of cars on the first lane.

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Calculate transitions compactly

With a matrix vector multiplication a transition can be calculated at once for all lanes!

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Matrix vector multiplication

+ + + + =

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Properties of transition graphs

Based on the transition probabilities, certain states of a transition graph can be classified. States can be absorbing, periodic, or transient.There are further steady-state distributions and irreducible transition graphs.

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Absorbing state

A state which has no out-going transitionwith positive probability

Over time all cars conglomerate there!

Where does this happen with real traffic?- Junkyard- dead-end one-way street ;-)

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Periodic states

States take periodically the same values

Traffic oscillates between certain states.

Where are such streets in everyday life?- e.g. between work and home

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Transient state

A state to which a care cannever return.

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Irreducible transition graph

Each state is reachable from every other state.

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Is the following graph irreducible?

No! (State D is not reachable from every other state!)

Which transition probabilites couldbe changed in order to make this graph ireducible?

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Properties of the transition matrix

Column sum = 1: stochastic matrix

Column sum ≠ 1:

Column sum < 1:total number of cars goes toward 0

Column sum > 1:total number of cars grow infinetly

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Steady-state distribution

If a transition graph is irreducible and does not have periodic states, then the system swings into a steady-state distribution, independently of the initial state.

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Notation of transition probabilities

„The probability to change fromvertex A to vertex B is 10%”

P(A, B) = 0.1

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Summary

Properties of states:– Periodic: Cars move to and fro.– Absorbing: All cars are finally there.– Transient: A car never returns there.

Transitions graphs can be irreducible: Cars can change from every state to every other state.

Distributions can be steady-state: the system has swung into.

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Models and their limitations

Lanes can hold a infinitely large number of cars in our model. This is not realistic!

Therefore:– Simulation stops, of > 2000 cars on a lane.– In the upper-limit mode a individual upper limit

(< 2000) can be defined per lane.

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The upper-limit mode

Possible application: Different parking areas. Cars should fill the parking areas C, D, and E in this order.

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Layout of parking areas in DynaTraffic

Upper limits of lanes are displayed

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Process upper-limit:lane C reached its capacity

Set all edges incident to C to 0.No more cars should arrive.

This is not a stochastic matrix any more!Columns must be normalized.

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Process upper-limit:lane C is unlocked again

Original row of the lane is reestablishedOnly outgoing edges are reestablished:This is ok for all vertices.Normalize column sum.

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What is this about?

• Models of traffic situations• Graphs:

– Edges, Vertices– Matrix representation– Vector representation

• Markov chains– States, transition probabilities– Special states: periodic, absorbing, or transient– Steady-state distribution

• Matrix vector multiplication

DynaTraffic help to understand and learn these concepts

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Summary

We model and analyze a traffic system with the help of Markov chains.– How does the traffic distribution evolve?– Does the system swing into?

Like this we can make predictions about the system based on our Markov model!

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DynaTraffic

Situationgraph

Transitiongraph

Statestatistics

Transition matrix& state vector

Control of transitions

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Demo DynaTraffic