1. DynaTraffic – Models and mathematical prognosis
Simulation of the distribution of traffic with the help of Markov chains

2. What is this about? Models of traffic situations Graphs: Markov chains
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
Graphentheorie

3. The goal: analysis of a traffic system
We are interested in this question: „How many cars are there at 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.

4. Mathematical prognosis step 1
Build a model of an everyday situation

5. Photo from a side perspective

6. Photo of the layout
© Google Imagery 2007

7. Model of the layout, with cars

8. Elements for the model without cars
Nodes
Arrows
 What does your model look like?

9. Model of the layout, without cars
Stop points  nodes
Lanes  edges 4 3 7

10. Representation in DynaTraffic4 3 7
Characters to label lanes
Colored arrows and slightly different arrangement

11. 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:
Mathematical Models try to capture the relevant parameters of natural phenomena and to use these parameters for predictions in the observed system.

12. Mathematical prognosis step 2
Transformation of the model

13. 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 vertices
 Transformation of the situation graph!

14. 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.

15. Transformation step a)
Each edge becomes a vertex.

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

17. Transformation step c)
Each vertex has an edge to itself. I.e.: a car can remain on a lane!

18. The good news concerning this 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

19. Mathematical prognosis step 3
Define assumptions

20. 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.

21. The transition graph
The transition probabilities are entered in the line graph
 Transition graph (= Markov chain)

22. 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.

23. Meaning of the transition probability?
„If there is a car on lane A now, it will in the next transition change to lane B with a probability of 83%.

24. Alternative representation of the transition graph
Transition graph Transition matrix

25. How to read the transition matrix
From To

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

27. Summary: our traffic model
Model with cars
Model without cars
Photo
Transition matrix
Transition graph

28. Summary: our traffic model
Step 1: build a model of an everyday situation
Model with cars
Step 2: transformation of the model
Model without cars
Photo
Step 3: define assumptions
Transition matrix
Transition graph

29. Demo DynaTraffic

30. 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.

31. „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.

32. How many cars are on lane A after the next transition?
Cars on individual lanes:
A: 3 cars B: 4 cars C: 7 cars
Probabilities leading to lane A: A  A : 0.17
C  A : 0.83
Calculation: 3 * * 0.83 = 6.32 cars

33. Probabilities for transitions
The required probabilities can directly be read from the transition matrix!
3 * * 0.83 = 6.32 cars

34. State vector
The number of cars per lane in the state vector notation

35. 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.

36. Calculate transitions compactly
With a matrix vector multiplication a transition can be calculated at once for all lanes!

37. Matrix vector multiplication=

38. 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.

39. Absorbing state
A state which has no out-going transition with positive probability
 Over time all cars conglomerate there!
Where does this happen with real traffic?
- Junkyard
- dead-end one-way street ;-)

40. 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

41. Transient state A state to which a care can never return.
Ein Zustand der nicht transient ist, ist rekurrent

42. Irreducible transition graph
Each state is reachable from every other state.

43. Is the following graph irreducible?
No! (State D is not reachable from every other state!)
Which transition probabilites could be changed in order to make this graph ireducible?

44. 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

45. 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.

46. Notation of transition probabilities
„The probability to change from vertex A to vertex B is 10%”
P(A, B) = 0.1

47. 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.

48. 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.

49. The upper-limit mode
Possible application: Different parking areas. Cars should fill the parking areas C, D, and E in this order.

50. Layout of parking areas in DynaTraffic
Upper limits of lanes are displayed

51. 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.

52. Process upper-limit: lane C is unlocked again
Original row of the lane is reestablished
Only outgoing edges are reestablished: This is ok for all vertices.
Normalize column sum.

53. What is this about? Models of traffic situations Graphs: Markov chains
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
Graphentheorie

54. 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!
Markov und Leiterspiel:
Markov und Monopoly:

55. DynaTraffic Situation graph Transition graph State statistics
Control of transitions
State statistics
Transition matrix & state vector

56. Demo DynaTraffic