1. Traffic Flow Fundamentals
CE331 Transportation Engineering

2. Objectives
Understand the fundamental relationships among traffic parameters
Estimating traffic parameters using the fundamental relationship
Queuing Models

3. Some Terms Speed (u) Density (k) Volume (V) Flow (q)
Rate of motion (mph)
Density (k)
Rate of traffic over distance (vpm)
Volume (V)
Amount of traffic (vph)
Flow (q)
Rate of traffic (vph); equivalent hourly rate

4. Basic Relationships q = k u Low volumes Highest speeds High volumes
Lower speeds
Highest volumes
Medium density
Maximum density
No speed or flow
q = k u

5. Basic Relationships

6. Flow-Density Example
If the spacing between vehicles is 500 feet what is the density?
d = 1/k k = 1/d = 1 veh/500 feet
= vehicles/foot = veh/mile

7. Speed-Density Relationship
Max speed
0 density uf
Max density
0 speed
Speed kj
Density

8. Speed-Density Relationship

9. Flow-Density Relationship
qcap
Optimum density
kcap kj
Density

10. Flow-Density Relationship

11. Flow – density (and speed)
Slope of these lines is the space mean speed at this density B KB
Do the dimensional analysis

12. Flow-Density Example
If the space mean speed is 45.6 mph, what is the flow rate?
q = kus = (10.6 veh/mile)(45.6 mph) = veh/hr

13. Speed-Flow Relationshipuf
Speed
“Optimal” speed for flow maximization
ucap
qcap= kcapucap
qcap
Flow

14. Speed-Flow Relationship

15. Highway capacity manual
Source 1985 highway capacity manual

16. 2000 Highway Capacity Manual

17. Speed density relationship
Capacity Drop
Source: Maze, Schrock, and VanDerHorst, “Traffic Management Strategies for Merge Areas in Rural Interstate Work Zones

18. Speed-Flow-Density Relationship (Greenshield’s Linear Model)uf
ucap
qcap
qcap
kcap kj

19. Special Case Greenshield’s Model
Linear
(Only) When Greenshield’s Model holds,

20. Greenshield Linear Model

21. Example 1
A highway section has an average spacing of 25ft under jam conditions and a free-flow speed of 55mph. Assuming that the relationship between speed and density is linear, determine the jam density, the maximum flow, the density at maximum flow, and the speed at maximum flow.

22. Example 2
Traffic observations along a freeway lane showed the flow rate of 1200vph occurred with an average speed of 50mph. The same study also showed that the free-flow speed is 60mph and the speed-density relationship follows the Greenshield’s model. What is the capacity of this lane?

23. Example 3
A section of highway is known to have a free-flow speed of 55mph and a capacity of 3300vph. In a given hour, 2100 vehicles were counted at a point along the road. If Greenshield’s model applies, what would be the space mean speed of these 2100 vehicles?

24. Queuing Theory
The theoretical study of waiting lines, expressed in mathematical terms
input
output
server
queue
Delay= queue time +service time

25. Queuing theory Common ways to represent (model) queues
Deterministic queuing
Steady state queuing
Shock waves

26. (all the information necessary to completely describe the system)
Common Assumptions
The queue is FCFS (FIFO).
We look at steady state: after the system has started up and things have settled down.
(all the information necessary to completely describe the system)

27. Typical cases where queuing is important
Bottle necks – capacity reductions
Lane closures for work zones on multi-lane facilities
Toll booths
Where else would we experience a line in traffic?

28. Deterministic queuing model
Treats the bottle neck like a funnel
Assumptions
Constant headways through out analysis period – what does this imply about density?
Capacity does not vary with traffic flow variable (speed, density or flow)

29. Deterministic queuing treats the waiting line as if it has no length

30. Deterministic queuing example
Example – Suppose we have a lane closure on a freeway and the single lane capacity at the closure is 1,400 vehicle per hour. Assume that during the first hour we expect the traffic volume of 2,000 vehicles per hour for one hour and then the volume to reduce to 800 vehicle per hour. At the peak, how long will the queue be and how long will it take to dissipate?
Draw a queuing diagram (cumulative arrivals on Y axis, time on X axis)

31. Deterministic queuing model example

32. Deterministic queuing traffic signal case

33. Diagram for traffic signal

35. Example λ t0 R

36. Example

37. Calculating Delay
= Average Delay

38. Intersection delay in both directions

39. Multiple directions

40. Minimizing delay

41. Steady State Queuing Arrival Distribution Service Method
Described by a Poisson distribution
Service Method
Usually first come first serve
Service Distribution
Follows a negative exponential distribution
Number of channels
Assumption of steady state
Average arrival rate is less than average service rate

42. The M/M/1 System
Poisson Process
Exponential server
output
queue

43. Steady state queuing equation variables
q = Average arrival rate
Q = Average service rate
n = Number of entities in
the system
w = Time waiting in the
queue
v = Time in the system

44. Arrivals follow a Poisson process
Readily amenable for analysis
Reasonable for a wide variety of situations
a(t) = # of arrivals in time interval [0,t]
q = mean arrival rate
0 = small time interval
Pr(exactly 1 arrival in [t,t+]) = q
Pr(no arrivals in [t,t+]) = 1-q
Pr(more than 1 arrival in [t,t+]) = 0
Pr(a(t) = n) = e-q t (q t)n/n!

45. Model for Interarrivals and Service times
Customers arrive at times t0 < t1 < Poisson distributed
The differences between consecutive arrivals are the interarrival times : n = tn - t n-1
n in Poisson process with mean arrival rate q, are exponentially distributed,
Pr(n  t) = 1 - e-q t
Service times are exponentially distributed, with mean service rate Q:
Pr(Sn  s) = 1 - e-Qs

46. System Features  This is a Markovian System
arrival times are independent
service times are independent of the arrivals
Both inter-arrival and service times are memoryless
Pr(Tn > t0+t | Tn> t0) = Pr(Tn  t)
future events depend only on the present state
 This is a Markovian System

47. Example of where assumptions are violated
4-way
Stop
Arrivals are not random
2-way
Stop

48. Common one channel equations
Example – suppose that cars take an average of 5 seconds at a stop sign. If 9 cars per minute arrive at the sign what is the probability of having 5 in the system what is the probability of having five or fewer.
P(5) = (9/12)5(1-9/12) = 0.06
P(4) = 0.08
P(3) = 0.11
P(2) = 0.14
P(1) = 0.19
P(0) = 0.25
0.83

49. Common queuing equations
Expected number of vehicles Expected number of Vehicles
in the system in the queue

50. Common queuing equations
Average wait in the queue Average wait in the system

51. Common queuing equations
Probability of spending less than
time t in the system
Probability of spending less than
time t in the queue

52. Common queuing equations
Probability of having more than n vehicles in the system

53. Shock wave theory Recognizes that density is a variable
Recognizes the dynamics of the traffic flow
Considers non-steady state condition
More realistically represents traffic flow

54. Example shock wave

55. Flow – density and speed
Slope of these lines is the space mean speed at this density B KB

56. Back Ward Moving Shock wave
Speed of the Shock wave

57. Example
Suppose you have 2,000 vehicle per hour approaching a lane closure at and average speed of 65 mph. The capacity of the lane closure is 1,400 vehicles per hour and at the maximum capacity move at 20 mph. Assuming the approaching vehicles are evenly distributed between the two lanes. How fast is the shock wave traveling backwards?
Q1 = 2,000 vehicles per hour
K1 = 1,000 per lane per hour/65 mph = vehicles per mile
Q2 = 1,400 vehicles per hour
K2 = 1,400 per lane per hour/20 mph = 70 vehicles per mile
How fast would the shock wave move backwards if all vehicle approach in a single lane?

58. Forward moving shock wave