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Traffic Flow Fundamentals

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Title: Traffic Flow Fundamentals


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)
  • 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
Highest speeds
Low volumes
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
  • 0.002 vehicles/foot 10.6
    veh/mile

7
Speed-Density Relationship
0 density
Max speed
0 speed
Max density
Speed
Density
8
Speed-Density Relationship
9
Flow-Density Relationship
Flow
qcap
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) 481.5
    veh/hr

13
Speed-Flow Relationship
Speed
Optimal speed for flow maximization
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 (Greenshields
Linear Model)
uf
ucap
qcap
qcap
kj
kcap
19
Special Case
  • Greenshields Model
  • Linear
  • (Only) When Greenshields 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
Greenshields 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 Greenshields 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
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
34
(No Transcript)
35
Example
?
µ
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
  • 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
output
Exponential server
queue
43
Steady state queuing equation variables
  • 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 lt t1 lt .... -
    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
  • arrival times are independent
  • service times are independent of the arrivals
  • Both inter-arrival and service times are
    memoryless
  • Pr(Tn gt t0t Tngt 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
15.38 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
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