Chapter 5
Models of Traffic Flow
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)
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
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)
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
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
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.
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
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
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
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)
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
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.
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:
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.
Negative Exponential Example
• By definition,
thPthP 1
818 hPhP
551.0
4493.01
1
18
3600)8(360
3600
e
ehPqt
Negative Exponential Example
1161.0
551.03329.01
551.01
8111
811118
3600)11(360
e
hPhP
hPhPhP
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
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
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
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
D/D/1 Queuing Models
• Deterministic arrivals• Deterministic departures• 1 service location (departure
channel)• Best examples maybe factory
assembly lines
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.
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:
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
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
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
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