Chapter
p III
TRANSPORTATION SYSTEM
ANALYSIS
Tewodros N.
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Lecture Overview
 Traffic engineering studies
 Spot speed studies
 Volume studies
 Travel time and delay studies
 Parking studies
 Fundamental
F d
l principles
i i l off traffic
ffi fl
flow
 Traffic flow elements
 Flow-density relationships




Fundamental diagram
g
of traffic flow
Mathematical relationships describing traffic flow
Shock waves in traffic streams
Gap and gap acceptance
 Queuing Analysis
 Queuing Patterns
 Q
Queuingg models
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Queuing Analysis
Delay = actual travel - ideal travel time
what is the ideal travel time?
1. Travel time under free flow conditions and
22. Travel
T
l time
i at capacity.
i
 Queuing delay:- delay that results when demand exceeds its capacity
Queuing
Q
i
Discipline
Input Source
(Customers)
Queue
Arrival Rate
Served
Customer
Service Facility
Queuing System
Transport Engineering
Tewodros N.
Service Rate
School of Civil and
Environmental Engineering
Input parameters
Mean arrival rate (λ):- is rate at which customers arrive
at a service facility.
=3600/
Mean service rate (μ):(μ): is the rate at which customers
(vehicles depart from a transportation facility.
=3600//
The number off servers (N)
( )
Queue discipline
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Queue Disciplines
 First in first out (FIFO):- first-come, first- served (FCFS) service discipline.
Example:- Prepaid taxi queue at airports
 First
Fi t in
i last
l t outt (FILO):(FILO) the
h customers are serviced
i d iin the
h reverse order
d
of their entry.
Example:- the people who join an elevator
fi t are the
first
th last
l t ones to
t leave
l
it.
it
 Served in random order (SIRO):- every customer in the queue is equally
likely to be selected.
 Priority scheduling:- customers are grouped in priority classes on the
basis of some attributes
Example:- Treatment of VIPs in p
preference to other p
patients in a
hospital
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Queuing Patterns
Constant
arrival
and
constant
service
rates
Varying
arrival
rate and
constant
service
rate
Constant
arrival
rate and
varying
service
rate
Varying
arrival
and
service
rates
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Queuing models
 Notation for describing queue is given by
X / Y/ N
Where:- X the arrival distribution type should be used
used,
Y the service distribution type should be used,
N represents the number of servers.
 M/M/1,
 M/M/N,
 Multiple single servers’
servers
 D/D/N
Where:- D stands for deterministic:- the arrival and service times of each
vehicle are known
M stands Markovian:- exact arrival and/or service time of each vehicle
is unknown
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
M/M/1 model
Arrival times and service rates follow markovian
distribution or exponential distribution which are
probabilistic
b b
ddistributions.
b
Only one server.
Assumptions
Customers are assumed to be patient.
p
System is assumed to have unlimited capacity.
Users arrive from an unlimited source.
The queue discipline is assumed to be first in first out.
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
M/M/1 model Cont…
 Percentage of X number of customers are in the system.
݂(‫(ݔݎ=)ݔ=ܺ(ܲ=)ݔ‬1−r)
Where: ܷ‫ = ݎ݋ݐ݂ܿܽ ݊݋݅ݐܽݖ݈݅݅ݐ‬r = /
Where: The average number of customers at any time in the system
 The average number of customers in the queue at any time
is
 Expected time a customer spends in the system
 Expected time a customer spends in the queue
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Example 1
 The Vehicles arrive at a toll booth at an average rate of
300 per hour. Average waiting time at the toll booth is
10s per vehicle.
vehicle If both arrivals and departures are
exponentially distributed, what is the average number
of vehicles in the system, average queue length, the
average delay
d l per vehicle,
hi l the
h average time
i a vehicle
hi l iis
in the system?
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
M/M/N model
Multi -server model with N number of servers
 Here
H
i the
is
h average service
i rate ffor N identical
id i l service
i
counters in parallel. For x=0
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
M/M/N model Cont…
The average number of customers in the system is
The average queue length
The expected time in the system
The expected
p
time in the qqueue
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Example 1
Consider the Example 1 as a multi-server
problem with two servers in parallel.
parallel
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Multiple single servers' model
N numbers of identical independent parallel
servers which receive customers from a same
source but in different parallel queues each one
receiving customers at a rate of / .
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Example 3
Consider the problem 1 as a multiple single
server's model with two servers which work
independently with each one receiving half the
arrival rate that is 150 veh/hr.
veh/hr
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
D/D/N model
The arrival and service rates are deterministic
tthat
at iss the
t e arrival
a va and
a d service
se v ce times
t es of
o each
eac
vehicle are known.
A
Assumptions
i
Customers are assumed to be patient.
System is assumed to have unlimited capacity.
Users arrive from an unlimited source.
The queue discipline is assumed to be first in first out.
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Example 4
Morning peak traffic upstream of a toll booth is given in the
table below. The toll plaza consists of three booths, each of
which can handle an average
g of one vehicle everyy 8 seconds.
Determine the maximum queue, the longest delay to an
individual vehicle.
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Solution
 Service rate is given as 8 seconds per vehicle.
 This implies for 10 min, 75 vehicles can be served by each server.
 Hence
H
225 vehicles
hi l can be
b servedd by
b 3 servers in
i 10 min.
i
Transport Engineering
Tewodros N.
School of Civil and
Environmental Engineering
Tha k Y
Thank
You!!