arrival pattern

28
Chapter 5 Models of Traffic Flow

Upload: jacqueline-herrera-cruz-fabio

Post on 26-Mar-2015

404 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: arrival pattern

Chapter 5

Models of Traffic Flow

Page 2: arrival pattern

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)

Page 3: arrival pattern

Introduction• The simplest approach to modeling

vehicle arrivals is to assume a uniform spacing

• This results in a deterministic, uniform arrival pattern—in 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

Page 4: arrival pattern

Introduction

• First, to clarify what is meant by ‘random’:

• For a sequence of events to be considered truly random, two conditions must be met:1. Any point in time is as likely as any other

for an event to occur (e.g., vehicle arrival)2. 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)

Page 5: arrival pattern

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

Page 6: arrival pattern

Poisson Distribution

!

)()(

n

etnP

tn

P(n) = probability of having n vehicles arrive in time tλ = average vehicle arrival rate in vehicles per unit timet= duration of time interval over which vehicles are countede= base of the natural logarithm

Page 7: arrival pattern

Example Application

Given an average arrival rate of 360 veh/hr or 0.1 vehicles per second; with t=20 seconds; determine the probability that exactly 0, 1, 2, 3, and 4 vehicles will arrive.

Page 8: arrival pattern

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

Page 9: arrival pattern

Poisson Example What is the probability of more

than 6 cars arriving (in 1-min interval)?

6

0

1

616

i

inP

nPnP

(0.5%)or 005.0

995.01

)012.0036.0090.0180.0271.0271.0135.0(16

nP

Page 10: arrival pattern

Poisson Example What is the probability of between

1 and 3 cars arriving (in 1-min interval)? 32131 nPnPnPnP

%2.72

%0.18%1.27%1.2731

nP

Page 11: arrival pattern

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)

Page 12: arrival pattern

Negative Exponential• To demonstrate this, let the average

arrival rate, , be in units of vehicles per second, so that

3600

q

Substituting into Poisson equation yields

!3600

)(

3600

n

eqt

nP

qtn

(Eq. 5.25)

sec

veh

hsec

hveh

!

)()(

n

etnP

tn

Page 13: arrival pattern

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.

Page 14: arrival pattern

Negative Exponential

• So from Eq. 5.25,)()0( thPP

36003600

1

1 qtqt

ee

This distribution of vehicle headways is known as the negative exponential distribution.

(Eq. 5.26)

1 !0

10

x

Note:

Page 15: arrival pattern

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.

Page 16: arrival pattern

Negative Exponential Example

• By definition,

thPthP 1

818 hPhP

551.0

4493.01

1

18

3600)8(360

3600

e

ehPqt

Page 17: arrival pattern

Negative Exponential Example

1161.0

551.03329.01

551.01

8111

811118

3600)11(360

e

hPhP

hPhPhP

Page 18: arrival pattern

Negative Exponential

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35

Time (sec)

Pro

b (

h >

= t

)

e^(-qt/3600)

For q = 360 veh/hr

Page 19: arrival pattern

Negative Exponential

c.d.f.

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35

Time (sec)

Pro

ba

bili

ty (

h <

t)

1 - e^(-qt/3600)

8

0.551

Page 20: arrival pattern

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 hospital’s operating rooms

Page 21: arrival pattern

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

Page 22: arrival pattern

D/D/1 Queuing Models

• Deterministic arrivals• Deterministic departures• 1 service location (departure

channel)• Best examples maybe factory

assembly lines

Page 23: arrival pattern

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.

Page 24: arrival pattern

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:

Page 25: arrival pattern

M/D/1 Equations

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

)1(2

2

)1(2

)1(2

2

t

w

Q

Page 26: arrival pattern

M/M/1 Queuing Models

• Exponentially distributed arrival and departure times and one departure channel

When traffic intensity term < 1

1

)(

1

2

t

w

Q

Page 27: arrival pattern

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.

0.1N

Page 28: arrival pattern

M/M/N Models

Qt

Qw

NNN

PQ

NNn

P

N

N

n

N

c

n

c

c

1

)/1(

1

!

)/1(!!

1

2

10

1

0

0