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Traffic Flow Fundamentals

- CE331 Transportation Engineering

Objectives

- Understand the fundamental relationships among

traffic parameters - Estimating traffic parameters using the

fundamental relationship - Queuing Models

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

Basic Relationships

Highest speeds

Low volumes

High volumes

Lower speeds

Highest volumes

Medium density

Maximum density

No speed or flow

q k u

Basic Relationships

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

Speed-Density Relationship

0 density

Max speed

0 speed

Max density

Speed

Density

Speed-Density Relationship

Flow-Density Relationship

Flow

qcap

Density

Flow-Density Relationship

Flow density (and speed)

Slope of these lines is the space mean speed at

this density

B

KB

Do the dimensional analysis

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

Speed-Flow Relationship

Speed

Optimal speed for flow maximization

qcap kcapucap

qcap

Flow

Speed-Flow Relationship

Highway capacity manual

Source 1985 highway capacity manual

2000 Highway Capacity Manual

Speed density relationship

Capacity Drop

Source Maze, Schrock, and VanDerHorst, Traffic

Management Strategies for Merge Areas in Rural

Interstate Work Zones

Speed-Flow-Density Relationship (Greenshields

Linear Model)

uf

ucap

qcap

qcap

kj

kcap

Special Case

- Greenshields Model
- Linear
- (Only) When Greenshields Model holds,

Greenshield Linear Model

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.

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?

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?

Queuing Theory

- The theoretical study of waiting lines, expressed

in mathematical terms

input

output

server

queue

Delay queue time service time

Queuing theory

- Common ways to represent (model) queues
- Deterministic queuing
- Steady state queuing
- Shock waves

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)

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?

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)

Deterministic queuing treats the waiting line as

if it has no length

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)

Deterministic queuing model example

Deterministic queuing traffic signal case

Diagram for traffic signal

(No Transcript)

Example

?

µ

t0

R

Example

Calculating Delay

Average Delay

Intersection delay in both directions

Multiple directions

Minimizing delay

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

The M/M/1 System

Poisson Process

output

Exponential server

queue

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

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!

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

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

Example of where assumptions are violated

4-way Stop

Arrivals are not random

2-way Stop

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

Common queuing equations

Expected number of vehicles

Expected number of Vehicles in

the system in

the queue

Common queuing equations

Average wait in the queue Average wait

in the system

Common queuing equations

Probability of spending less than time t

in the system

Probability of spending less than time

t in the queue

Common queuing equations

Probability of having more than n vehicles in the

system

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

Example shock wave

Flow density and speed

Slope of these lines is the space mean speed at

this density

B

KB

Back Ward Moving Shock wave

Speed of the Shock wave

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?

Forward moving shock wave