chapter iii transportation system analysis · 2013. 3. 6. · chapter iii transportation system...

Chapter III pTRANSPORTATION SYSTEM

ANALYSISANALYSIS

Tewodros N.www.tnigatu.wordpress.com

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Lecture Overview Traffic engineering studies

Spot speed studies Volume studies Travel time and delay studies Parking studies

F d l i i l f ffi fl Fundamental principles of traffic flow Traffic flow elements Flow-density relationships Fundamental diagram of traffic flow g Mathematical relationships describing traffic flow Shock waves in traffic streams Gap and gap acceptance

Queuing Analysis Queuing Patterns Queuing models Q g

Transport Engineering School of Civil andTewodros N. Environmental Engineering

Queuing Analysis Delay = actual travel - ideal travel time what is the ideal travel time?what is the ideal travel time?

1. Travel time under free flow conditions and 2 T l i i2. Travel time at capacity.

Queuing delay:- delay that results when demand exceeds its capacity Q i

Input Source Queue Service Facility Served

Queuing Discipline

Queuing System

(Customers) Queue Service Facility Customer

Arrival Rate Service Rate


Input parameters Mean arrival rate (λ):- is rate at which customers arrive

at a service facility.

=3600/Mean service rate (μ): is the rate at which customersMean service rate (μ):- is the rate at which customers

(vehicles depart from a transportation facility.

/=3600/The number of servers (N) f ( )Queue discipline


Queue Disciplines First in first out (FIFO):- first-come, first- served (FCFS) service discipline.

Example:- Prepaid taxi queue at airports Fi t i l t t (FILO) h i d i h d First in last out (FILO):- the customers are serviced in the reverse order

of their entry. Example:- the people who join an elevator

fi t th l t t l itfirst are the last ones to leave 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 attributesExample:- Treatment of VIPs in preference to other patients in a p p

hospital


Queuing Patterns Constant

arrival and

constant

Varying arrival

rate and constantconstant

service rates

constant service

rate

Constant arrival

rate and varying service

Varying arrival

and serviceservice

rate

service rates


Queuing models Notation for describing queue is given by

X / Y/ N Where:- X the arrival distribution type should be usedWhere:- X the arrival distribution type should be used,

Y the service distribution type should be used, N represents the number of servers.

M/M/1,M/M/N, Multiple single servers’Multiple single 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


M/M/1 modelArrival times and service rates follow markovian

distribution or exponential distribution which are b b d bprobabilistic distributions.

Only one server.Assumptions

Customers are assumed to be patient. pSystem is assumed to have unlimited capacity. Users arrive from an unlimited source. The queue discipline is assumed to be first in first out.


M/M/1 model Cont… Percentage of X number of customers are in the system.

( )= ( = )= (1−r)Where:- = r = /Where: r /

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


Example 1The 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 If both arrivals and departures are10s per vehicle. If both arrivals and departures are exponentially distributed, what is the average number of vehicles in the system, average queue length, the

d l hi l h i hi l iaverage delay per vehicle, the average time a vehicle is in the system?


M/M/N model Multi -server model with N number of servers

H i h i f N id i l iHere is the average service rate for N identical service counters in parallel. For x=0


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 time in the queue p q


Example 1Consider the Example 1 as a multi-server

problem with two servers in parallelproblem with two servers in parallel.


Multiple single servers' modelN numbers of identical independent parallel

servers which receive customers from a same source but in different parallel queues each onesource but in different parallel queues each one receiving customers at a rate of / .


Example 3Consider 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/hrarrival rate that is 150 veh/hr.


D/D/N modelThe arrival and service rates are deterministic

that is the arrival and service times of each t at s t e a va a d se v ce t es o eacvehicle are known. A iAssumptions

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.


Example 4Morning 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 of one vehicle every 8 seconds. g yDetermine the maximum queue, the longest delay to an individual vehicle.


Solution Service rate is given as 8 seconds per vehicle. This implies for 10 min, 75 vehicles can be served by each server. H 225 hi l b d b 3 i 10 i Hence 225 vehicles can be served by 3 servers in 10 min.


Tha k Y !Thank You!Y

chapter iii transportation system analysis · 2013. 3. 6. · chapter iii transportation system...

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