Understanding Traffic Flow Theories: From Queuing to Cellular Automata

2. Introduction - Fatmah Ebrahim

3. There are 500,000 traffic jams a year. That’s 10,000 a week. Or 200-300 a day. Traffic congestion costs the economy of England £22bn a year1 [1] Eddington Transport Study, Rod Eddington (2006) Traffic Jam Facts

4. Queuing Theory Macroscopic Flow Theory Kinetic Theory Cellular Automata Three Phase Theory Vehicle Following Model Bifurcations Computer Model Mechanical Model Data Analysis Conclusions Overview

5. Queuing Theory - Roger Hackett

6. Parameters of Queuing Theory Flow rate: q Capacity: Q Intensity: x=q/Q

7. Pollaczek-Khintchine Formula

11. Macroscopic Flow Theory - Peter Edmunds

12. Macroscopic Flow Theory We need to define three variables: Spatial density, K: the number of vehicles per unit length of a given traffic system. Flow, Q: the number of vehicles per unit time Speed, v: the time rate of progression These lead to the fundamental equation of traffic flow: Q=Kv

13. We can obtain an equation that resembles the equation of continuity for fluid flow: This is based on the assumption that no vehicles enter or leave the road. It can be adapted for n traffic lanes and for inflows or outflows of gΔxΔt. Modeling traffic flow as a fluid

14. To solve this, we assume that the flow q is a function of the density k. We obtain the equation: This is solved by the method of characteristics. Eventually, after introducing a parameter s along the characteristic curves, the final equation can be derived:

15. An analytic solution of this equation is usually impossible and so what is done in practice is to draw the graph of q=kf(k) against k.

16. Kinetic Traffic Flow Theory - Joshua Mann

17. Kinetic Theory • Developed to explain the macroscopic properties of • gases. • Pressure, temperature and volume are modelled by • considering the motion and molecular composition of • the particles. • Original theory was static repulsion.

18. Primitive Speed Equation • Convection term: change of the average speed V due to a spatial speed gradient carried with the flow V. • Pressure term: change of average speed V as a result of individual vehicles that travel at v < V and v > V. • Smooth acceleration: change of average speed V due to smooth individual accelerations. • Discrete acceleration 1: change of the average speed V due to events that cause a discrete change in the number of vehicles with expected speed v. • Discrete acceleration 2: change of the average speed V due to a discrete change in the total number of vehicles.

19. Modified Speed Equation • Where W is a drivers desired speed. • is the relaxation time.

20. Advantages and Disadvantages • Advantages: • It provides a realistic representation of multiclass traffic. • It reproduces phenomena observed in congested traffic. • It helps to relate traffic flow models to the behaviour of the driver. • Disadvantages: • The individual behaviour of drivers is still not fully accounted for. • The model cannot fully describe complex traffic flows in towns.

21. Cellular Automata Traffic Model - Joshua Mann

22. Cellular Automata • An idealization of a physical system. • Physical quantities take a finite set of values and • space and time are discrete. • Traffic flow is modelled using the road traffic rule.

23. Road Traffic Rule Model • Vehicles modelled as point particles moving along a line • of sites. • A vehicle can only move if its destination cell is free. • If the destination cell is freed at the same time as motion • the vehicle does not move until after the cell is vacated • as it cannot observe the other vehicles motion.

24. 1 0 0 1 Applications • Traffic light situation. • Numbers in grid are turn flags and indicate priority. • Condition allowed is right turn on red light.

25. Advantages and Disadvantages • Advantages: • It enables the study of traffic flow in towns and cities. • It allows the implication of certain road regulations to be modelled. • Disadvantages: • It does not account in any way for the behaviour of the driver. • The individual speed of vehicles is not accounted for. • The differing sizes of vehicles are not accounted for.

26. Three Phase Theory - Eóin Davies

27. The Three Phase Theory of Traffic Flow Classical Theory (Two Phases): Free Flow Congested Three Phase (Congested phase split into two): Free Flow Synchronized flow Wide-moving jam

28. Fundamental Hypothesis of Three Phase Traffic Theory

29. Transitions • Free Flow -> Synchronised Flow • Synchronised Flow -> Wide-moving Jam

30. Conclusions • It is qualitative theory. • It is a description of traffic patterns not an explanation. • Not widely accepted. • Based on data from German freeways - there is no reason that the results would match other roads in other countries.

31. Vehicle Following Model - Steven Kinghorn

32. Vehicle Following Model (VFM) • VFM studies the relationship between two successive vehicles. • Each following vehicle responds to the vehicle directly in front. Following vehicle Leading vehicle Velocity of the leading vehicle Velocity of the following vehicle Separation distance between two vehicles

33. Response = Sensitivity Stimulus General form of model • Response – Braking or accelerating • Sensitivity – Driver reaction time • Stimulus – Change in relative speed • One example of a VFM equation: - • Other VFM’s have different variations in sensitivity. For example, a VFM developed by Gazis, Herman & Potts (1959) has a greater sensitivity for smaller spacing between vehicles: - (1) Speed of following vehicle Speed of leading vehicle (2) (3)

34. VFM in Computer simulation • Limitations – following vehicles only react to the vehicles directly in front. However, majority of drivers would look further ahead to gauge traffic conditions. • Computer simulations can be created to introduce many different types of traffic systems (Traffic lights, lanes closer etc) • By applying a vehicle following model, we can study how congestion might be caused and develop ways to reduce it.

35. Bifurcations - Roger Hackett

36. Bifurcations This is the reaction time delay vehicle following model.

37. The Computer Model - Alex Travis

38. Intelligent Driver Model v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distance s0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicle b: comfortable braking deceleration δ is set to 4 as convention s: distance of vehicle ahead v: velocity of vehicle ∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.

39. Acceleration on free road Deceleration due to car ahead v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distance s0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicle b: comfortable braking deceleration δ is set to 4 as convention s: distance of vehicle ahead v: velocity of vehicle ∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.

40. Graphs Produced for Single Lane Model

41. The Mechanical Model - Eóin Davies

42. Mechanical Model Q=k.v Q=Flow k=density v=velocity Want to confirm this relation. Need to measure these variables.

43.  Mechanical Model Release balls at a fixed rate. Density and speed of balls varies when angle of ramp changes x Figure 1

44. Method • 1. Set value of flow by releasing bearings at fixed intervals. • 2. Measure speed of balls at certain angle of ramp. • 3. Measure Density at different flow rates. • 4. Use Q=k.v to calculate flow.

45. Results • Comparing set flow and flow calculated using K.v

46. Data Analysis - Peter Edmunds

47. Data Analysis We needed to analyze data to investigate which of the theories already mentioned is the most appropriate for traffic flow. On the 28th of January our group attempted to take data from the M1. This was a failure. Professor Heydecker from the Transport Department at UCL very kindly allowed us to use his data, taken in conjunction with the Highways Agency.