Models of Traffic Flow

1
Chapter 5

• Models of Traffic Flow

2
Introduction

• Macroscopic relationships and analyses are very
  valuable, but
• A considerable amount of traffic analysis occurs
  at the microscopic level
• In particular, we often are interested in the
  elapsed time between the arrival of successive
  vehicles (i.e., time headway)

3
Introduction

• The simplest approach to modeling vehicle
  arrivals is to assume a uniform spacing
• This results in a deterministic, uniform arrival
  patternin other words, there is a constant time
  headway between all vehicles
• However, this assumption is usually unrealistic,
  as vehicle arrivals typically follow a random
  process
• Thus, a model that represents a random arrival
  process is usually needed

4
Introduction

• First, to clarify what is meant by random
• For a sequence of events to be considered truly
  random, two conditions must be met
• Any point in time is as likely as any other for
  an event to occur (e.g., vehicle arrival)
• The occurrence of an event does not affect the
  probability of the occurrence of another event
  (e.g., the arrival of one vehicle at a point in
  time does not make the arrival of the next
  vehicle within a certain time period any more or
  less likely)

5
Introduction

• One such model that fits this description is the
  Poisson distribution
• The Poisson distribution
• Is a discrete (as opposed to continuous)
  distribution
• Is commonly referred to as a counting
  distribution
• Represents the count distribution of random events

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Poisson Distribution
P(n) probability of having n vehicles arrive in
time t ? average vehicle arrival rate in
vehicles per unit time t duration of time
interval over which vehicles are counted e base
of the natural logarithm
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Example Application

• Given an average arrival rate of 360 veh/hr or
  0.1 vehicles per second with t20 seconds
  determine the probability that exactly 0, 1, 2,
  3, and 4 vehicles will arrive.

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Poisson Example

• Example
• Consider a 1-hour traffic volume of 120 vehicles,
  during which the analyst is interested in
  obtaining the distribution of 1-minute volume
  counts

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Poisson Example

• What is the probability of more than 6 cars
  arriving (in 1-min interval)?

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Poisson Example

• What is the probability of between 1 and 3 cars
  arriving (in 1-min interval)?

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Poisson distribution

• The assumption of Poisson distributed vehicle
  arrivals also implies a distribution of the time
  intervals between the arrivals of successive
  vehicles (i.e., time headway)

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Negative Exponential

• To demonstrate this, let the average arrival
  rate, ?, be in units of vehicles per second, so
  that

• Substituting into Poisson equation yields

(Eq. 5.25)
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Negative Exponential

• Note that the probability of having no vehicles
  arrive in a time interval of length t i.e., P
  (0) is equivalent to the probability of a
  vehicle headway, h, being greater than or equal
  to the time interval t.

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Negative Exponential

• So from Eq. 5.25,

(Eq. 5.26)
Note
This distribution of vehicle headways is known as
the negative exponential distribution.
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Negative Exponential Example

• Assume vehicle arrivals are Poisson distributed
  with an hourly traffic flow of 360
  veh/h.Determine the probability that the
  headway between successive vehicles will be less
  than 8 seconds.Determine the probability that
  the headway between successive vehicles will be
  between 8 and 11 seconds.

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Negative Exponential Example

• By definition,

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Negative Exponential Example
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Negative Exponential
For q 360 veh/hr
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Negative Exponential
20
Queuing Systems

• Queue waiting line
• Queuing models mathematical descriptions of
  queuing systems
• Examples airplanes awaiting clearance for
  takeoff or landing, computer print jobs, patients
  scheduled for hospitals operating rooms

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Characteristics of Queuing Systems

• Arrival patterns the way in which items or
  customers arrive to be served in a system
  (following a Poisson Distribution, Uniform
  Distribution, etc.)
• Service facility single or multi-server
• Service pattern the rate at which customers are
  serviced
• Queuing discipline FIFO, LIFO

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D/D/1 Queuing Models

• Deterministic arrivals
• Deterministic departures
• 1 service location (departure channel)
• Best examples maybe factory assembly lines

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Example

• Vehicles arrive at a park which has one entry
  points (and all vehicles must stop). Park opens
  at 8am vehicles arrive at a rate of 480 veh/hr.
  After 20 min the flow rate decreases to 120
  veh/hr and continues at that rate for the
  remainder of the day. It takes 15 seconds to
  distribute the brochure. Describe the queuing
  model.

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M/D/1 Queuing Model

• M stands for exponentially distributed times
  between arrivals of successive vehicles (Poisson
  arrivals)
• Traffic intensity term is used to define the
  ratio of average arrival to departure rates

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M/D/1 Equations

• When traffic intensity term lt 1 and constant
  steady state average arrival and departure rates

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M/M/1 Queuing Models

• Exponentially distributed arrival and departure
  times and one departure channel
• When traffic intensity term lt 1

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M/M/N Queuing Models

• Exponentially distributed arrival and departure
  times and multiple departure channels (toll
  plazas for example)
• In this case, the restriction to apply these
  equations is that the utilization factor must be
  less than 1.

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M/M/N Models