EP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN

Size: px
Start display at page:

Download "EP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN"

Transcription

1 EP2200 Queueg theory ad teletraffc systems Queueg etworks Vktora Fodor

2 Ope ad closed queug etworks Queug etwork: etwork of queug systems E.g. data packets traversg the etwork from router to router Ope ad closed etworks Ope queug etwork: customers arrve ad leave the etwork (typcal applcato: data commucato) Closed queueg etworks: ad out flows are mssg costat umber of customers crculate the etwork (applcato: computer systems) γ p q q 2 q 3 p 2 p 23 γ 2 q 4 p 34 2

3 Ope queug etworks- A tadem system q q 2 Posso() Exp(µ ) Exp(µ 2 ) The most smple ope queug etwork Assume a Posso arrval process ad depedet expoetally dstrbuted servce tmes What s the departure process from queue? Iterdeparture tme: Customer leaves queue behd: tme of servce of ext customer Customer leaves empty system behd: tme to ext arrval + tme of servce L( f τ µ µ ( t)) ρ + ( ρ) s + µ s + s + µ 2 2 ρµ ( s + ) + µ ρµ s + + µ ( s + )( s + µ ) ( s + )( s + µ ) s + Departure process: Posso ()! Same for //m but ot for systems wth losses ad ot for /G/m systems 3

4 A tadem system q q 2 Posso() Exp(µ ) Exp(µ 2 ) Tadem system Queue s a // queue Departure process from Queue s Posso Thus Queue 2 s also a // queue State of the tadem queue: N( 2 ) p()p( 2 ) Jackso theorem: the etwork behaves as f set of depedet queues that s: p( 2 ) p( )p( 2 ) Proof: see Vrtamo otes 4

5 odelg commucato etworks - ote o the depedece assumpto terarrval tme packet sze correlato! The product form p( 2 ) p p 2 apples oly f the arrval ad servce processes are depedet For two trasmsso lks seres queue 2 s ot a //-queue Correlato betwee servce tmes of a customer the two queues determed by the packet legth ad the lk trasmsso rate Correlato betwee arrval ad servce tmes For two cosecutve packets the terarrval tme at the secod queue ca ot be smaller tha the servce (that s trasmsso) tme of the frst packet at the frst queue E.g. there wll ot be ay queug queue 2 f the trasmsso rate at queue 2 s larger Product form soluto does ot apply 5

6 odelg commucato etworks - ote o the depdece assumpto Klerock s assumpto o depedece Traffc to a queue comes from several upstream queues Superposto of Posso processes gve a Posso process Traffc from a queue s spread radomly to several dowstream queues Partal processes are Posso wth testy p (Σ p ) It s assumed to create depedet arrval ad servce processes Product form soluto apples E.g. etwork of large routers 6

7 Ope Jackso s queug etworks where the product form works Ope queug etwork arrvals to the etwork from all arrval pot a departure pot s reachable queues wth fte storage ad m expoetal servers Eve fte storage f last queue the etworks Customers from outsde of the etwork arrve to ode as a Posso process wth testy γ 0 The servce tmes are depedet of the arrval process (ad servce tmes other queues) A customer comes from ode to ode after servce wth the probablty p or leaves the etwork wth the probablty p 0 - p. Note t allows feedback (e.g p ). The arrval process ot Posso aymore but the queue behaves as f the arrval would be Possoa. Network s stable f all the queues are stable. γ p q q 2 q p p p 34 γ 2 q 4 7

8 Ope Jackso s queug etworks Flow coservato: arrval testy to ode s γ + p Jacksos theorem: The dstrbuto of umber of customers the etwork has product form queues behave as depedet //m queues! (we do ot prove same as for tadem queues) p ( ) p ( ) p ( ) 2 γ p q q 2 q p p p 34 γ 4 q 4 8

9 Ope Jackso s queug etworks Flow coservato: arrval testy to ode : γ + Example : sgle feedback queue p p γ µ Performace measures as f t would be // Though the arrval process s ot Posso Stablty: / µ < γ + p γ p ρ µ p( k) k ( ρ) ρ 9

10 Ope Jackso s queug etworks Arrval testy ad state probablty + γ p ( ) ( ) ( ) P 2 For the // case: P P ( ) ( ρ ) ρ ad ρ / < P µ Example 2 calculate arrval testes gve the stablty rego the possble arrval rates whe the etwork s stable calculate the probablty that the etwork s empty calculate the probablty that there s oe customer the etwork 0

11 Lttle s theorem apples to the etre etwork! Good because T s hard to calculate f there are feedback loops. The mea umber of customers the etwork ad the average tme spet the etwork are (e.g. // case) The mea umber of odes a customer vsts before leavg: {Sum arrval testy to the queues} / {arrval testy to the etwork} N T N N / γ ρ ρ + p V / γ γ Ope Jackso s queug etworks ea performace measures

12 Closed Jackso s queug etworks Not exam materal ths year Closed queug etwork queues wth fte storage ad m expoetal servers K customers crculatg the etwork o arrvals ad departures The servce tmes are depedet of the arrval process (ad servce tmes other queues) A customer comes from ode to ode after servce wth the probablty p Queues ca ot be depedet sce there s a fxed umber of customers p q q 2 q p p p 34 p 42 p 4 q 4 2

13 3 Flow coservato: arrval testy to ode the problem s that oe of the -s are kow: Lmted set of states sce the sum of the customers s costat K: C based soluto: state: vector of umber of customers per queue - complex Algorthmc soluto e.g. // (*) gves a set of depedet equatos wth soluto of e.g.: we have to select the oe that gves sum of etwork state probabltes equal to oe Gordo-Newell: state probabltes wthout calculatg arrval testes (wthout proof) ) ( p Closed Jackso s queug etworks ( ) } 0 { 2 K S }... { }... { e e e e α S K K e G e G P ) ( µ µ

14 Summary Queug etworks: set of queug systems customers move from queue to queue Appled to etworkg problems: depedece of queues have to be esured Ope queug etworks Burke: Output process of a //m queue s Possoa Jackso theorem: etwork state probablty has product form f //m queues Closed queug etworks ot exam materal Number of customers costat State of queues s depedet Gordo-Newell ormalzato 4

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.-Ig. Georg Carle

More information

IS 709/809: Computational Methods in IS Research. Simple Markovian Queueing Model

IS 709/809: Computational Methods in IS Research. Simple Markovian Queueing Model IS 79/89: Comutatoal Methods IS Research Smle Marova Queueg Model Nrmalya Roy Deartmet of Iformato Systems Uversty of Marylad Baltmore Couty www.umbc.edu Queueg Theory Software QtsPlus software The software

More information

Outline. Basic Components of a Queue. Queueing Notation. EEC 686/785 Modeling & Performance Evaluation of Computer Systems.

Outline. Basic Components of a Queue. Queueing Notation. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. EEC 686/785 Modelg & Performace Evaluato of Computer Systems Lecture 5 Departmet of Electrcal ad Computer Egeerg Clevelad State Uversty webg@eee.org (based o Dr. Raj Ja s lecture otes) Outle Homework #5

More information

Network of Markovian Queues. Lecture

Network of Markovian Queues. Lecture etwork of Markovan Queues etwork of Markovan Queues ETW09 20 etwork queue ed, G E ETW09 20 λ If the frst queue was not empty Then the tme tll the next arrval to the second queue wll be equal to the servce

More information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

The Erlang Model with non-poisson Call Arrivals

The Erlang Model with non-poisson Call Arrivals The Erlag Model wth o-posso Call Arrvals Thomas Boald Frace Telecom R&D 38-40 rue du geeral Leclerc 92794 Issy-les-Mouleaux, Frace thomasboald@fracetelecomcom ABSTRACT The Erlag formula s kow to be sestve

More information

Queueing Networks. γ 3

Queueing Networks. γ 3 Queueg Networks Systes odeled by queueg etworks ca roughly be grouped to four categores. Ope etworks Custoers arrve fro outsde the syste are served ad the depart. Exaple: acket swtched data etwork. γ µ

More information

PTAS for Bin-Packing

PTAS for Bin-Packing CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,

More information

Random Variables and Probability Distributions

Random Variables and Probability Distributions Radom Varables ad Probablty Dstrbutos * If X : S R s a dscrete radom varable wth rage {x, x, x 3,. } the r = P (X = xr ) = * Let X : S R be a dscrete radom varable wth rage {x, x, x 3,.}.If x r P(X = x

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution: Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

Lecture 2 - What are component and system reliability and how it can be improved?

Lecture 2 - What are component and system reliability and how it can be improved? Lecture 2 - What are compoet ad system relablty ad how t ca be mproved? Relablty s a measure of the qualty of the product over the log ru. The cocept of relablty s a exteded tme perod over whch the expected

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos

More information

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

The expected value of a sum of random variables,, is the sum of the expected values:

The expected value of a sum of random variables,, is the sum of the expected values: Sums of Radom Varables xpected Values ad Varaces of Sums ad Averages of Radom Varables The expected value of a sum of radom varables, say S, s the sum of the expected values: ( ) ( ) S Ths s always true

More information

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?

More information

Lecture 9: Tolerant Testing

Lecture 9: Tolerant Testing Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have

More information

Naïve Bayes MIT Course Notes Cynthia Rudin

Naïve Bayes MIT Course Notes Cynthia Rudin Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.

More information

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty:

More information

Open and Closed Networks of M/M/m Type Queues (Jackson s Theorem for Open and Closed Networks) Copyright 2015, Sanjay K. Bose 1

Open and Closed Networks of M/M/m Type Queues (Jackson s Theorem for Open and Closed Networks) Copyright 2015, Sanjay K. Bose 1 Ope ad Closed Networks of //m Type Qees Jackso s Theorem for Ope ad Closed Networks Copyrght 05, Saay. Bose p osso Rate λp osso rocess Average Rate λ p osso Rate λp N p p N osso Rate λp N Splttg a osso

More information

Rare Events Prediction Using Importance Sampling in a Tandem Network

Rare Events Prediction Using Importance Sampling in a Tandem Network Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No http://www.jcss.com ISSN 345-3397 Rare Evets Predcto Usg Importace Samplg a Tadem Network A. Behrouz Safaezadeh Departmet

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

Class 13,14 June 17, 19, 2015

Class 13,14 June 17, 19, 2015 Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral

More information

Steady-state Behavior of a Multi-phase M/M/1 Queue in Random Evolution subject to Catastrophe failure

Steady-state Behavior of a Multi-phase M/M/1 Queue in Random Evolution subject to Catastrophe failure Advaces Theoretcal ad Appled Mathematcs ISSN 973-4554 Volume, Number 3 (26), pp. 23-22 Research Ida Publcatos http://www.rpublcato.com Steady-state Behavor of a Mult-phase M/M/ Queue Radom Evoluto subect

More information

APPENDIX A: ELEMENTS OF QUEUEING THEORY

APPENDIX A: ELEMENTS OF QUEUEING THEORY APPENDIX A: ELEMENTS OF QUEUEING THEORY I a pacet rado etwor, pacets/messages are forwarded from ode to ode through the etwor by eterg a buffer (queue) of a certa legth each ode ad watg for ther tur to

More information

Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations

Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 ISS 5-353 Optmal Strategy Aalyss of a -polcy M/E / Queueg System wth Server Breadows ad Multple Vacatos.Jayachtra*, Dr.A.James

More information

Lecture Notes Forecasting the process of estimating or predicting unknown situations

Lecture Notes Forecasting the process of estimating or predicting unknown situations Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal

More information

CHAPTER 6. d. With success = observation greater than 10, x = # of successes = 4, and

CHAPTER 6. d. With success = observation greater than 10, x = # of successes = 4, and CHAPTR 6 Secto 6.. a. We use the samle mea, to estmate the oulato mea µ. Σ 9.80 µ 8.407 7 ~ 7. b. We use the samle meda, 7 (the mddle observato whe arraged ascedg order. c. We use the samle stadard devato,

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 3423 Simple Linear Regression Page 12-01 ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable

More information

ρ < 1 be five real numbers. The

ρ < 1 be five real numbers. The Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace

More information

Beam Warming Second-Order Upwind Method

Beam Warming Second-Order Upwind Method Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet

More information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

Introduction to local (nonparametric) density estimation. methods

Introduction to local (nonparametric) density estimation. methods Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest

More information

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen.

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen. .5 x 54.5 a. x 7. 786 7 b. The raked observatos are: 7.4, 7.5, 7.7, 7.8, 7.9, 8.0, 8.. Sce the sample sze 7 s odd, the meda s the (+)/ 4 th raked observato, or meda 7.8 c. The cosumer would more lkely

More information

Statistics: Unlocking the Power of Data Lock 5

Statistics: Unlocking the Power of Data Lock 5 STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I- Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re- grades Re-

More information

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

Random Variate Generation ENM 307 SIMULATION. Anadolu Üniversitesi, Endüstri Mühendisliği Bölümü. Yrd. Doç. Dr. Gürkan ÖZTÜRK.

Random Variate Generation ENM 307 SIMULATION. Anadolu Üniversitesi, Endüstri Mühendisliği Bölümü. Yrd. Doç. Dr. Gürkan ÖZTÜRK. adom Varate Geerato ENM 307 SIMULATION Aadolu Üverstes, Edüstr Mühedslğ Bölümü Yrd. Doç. Dr. Gürka ÖZTÜK 0 adom Varate Geerato adom varate geerato s about procedures for samplg from a varety of wdely-used

More information

means the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.

means the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever. 9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 4421 Propagation of Error Page 8-01 ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.

More information

X ε ) = 0, or equivalently, lim

X ε ) = 0, or equivalently, lim Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed

More information

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad

More information

Unsupervised Learning and Other Neural Networks

Unsupervised Learning and Other Neural Networks CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all

More information

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION Samplg Theor MODULE V LECTUE - 4 ATIO AND PODUCT METHODS OF ESTIMATION D. SHALABH DEPATMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPU A mportat objectve a statstcal estmato procedure

More information

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of

More information

ECE 595, Section 10 Numerical Simulations Lecture 19: FEM for Electronic Transport. Prof. Peter Bermel February 22, 2013

ECE 595, Section 10 Numerical Simulations Lecture 19: FEM for Electronic Transport. Prof. Peter Bermel February 22, 2013 ECE 595, Secto 0 Numercal Smulatos Lecture 9: FEM for Electroc Trasport Prof. Peter Bermel February, 03 Outle Recap from Wedesday Physcs-based devce modelg Electroc trasport theory FEM electroc trasport

More information

1 Solution to Problem 6.40

1 Solution to Problem 6.40 1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

STA 105-M BASIC STATISTICS (This is a multiple choice paper.)

STA 105-M BASIC STATISTICS (This is a multiple choice paper.) DCDM BUSINESS SCHOOL September Mock Eamatos STA 0-M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do

More information

CHAPTER 2. = y ˆ β x (.1022) So we can write

CHAPTER 2. = y ˆ β x (.1022) So we can write CHAPTER SOLUTIONS TO PROBLEMS. () Let y = GPA, x = ACT, ad = 8. The x = 5.875, y = 3.5, (x x )(y y ) = 5.85, ad (x x ) = 56.875. From equato (.9), we obta the slope as ˆβ = = 5.85/56.875., rouded to four

More information

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss

More information

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from

More information

Law of Large Numbers

Law of Large Numbers Toss a co tmes. Law of Large Numbers Suppose 0 f f th th toss came up H toss came up T s are Beroull radom varables wth p ½ ad E( ) ½. The proporto of heads s. Itutvely approaches ½ as. week 2 Markov s

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Computational Geometry

Computational Geometry Problem efto omputatoal eometry hapter 6 Pot Locato Preprocess a plaar map S. ve a query pot p, report the face of S cotag p. oal: O()-sze data structure that eables O(log ) query tme. pplcato: Whch state

More information

Queueing Networks II Network Performance

Queueing Networks II Network Performance Queueng Networks II Network Performance Davd Tpper Assocate Professor Graduate Telecommuncatons and Networkng Program Unversty of Pttsburgh Sldes 6 Networks of Queues Many communcaton systems must be modeled

More information

KLT Tracker. Alignment. 1. Detect Harris corners in the first frame. 2. For each Harris corner compute motion between consecutive frames

KLT Tracker. Alignment. 1. Detect Harris corners in the first frame. 2. For each Harris corner compute motion between consecutive frames KLT Tracker Tracker. Detect Harrs corers the frst frame 2. For each Harrs corer compute moto betwee cosecutve frames (Algmet). 3. Lk moto vectors successve frames to get a track 4. Itroduce ew Harrs pots

More information

Third handout: On the Gini Index

Third handout: On the Gini Index Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The

More information

Chapter 13 Student Lecture Notes 13-1

Chapter 13 Student Lecture Notes 13-1 Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato

More information

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d 9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,

More information

Assignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix

Assignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix Assgmet 7/MATH 47/Wter, 00 Due: Frday, March 9 Powers o a square matrx Gve a square matrx A, ts powers A or large, or eve arbtrary, teger expoets ca be calculated by dagoalzg A -- that s possble (!) Namely,

More information

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture) CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.

More information

Statistics MINITAB - Lab 5

Statistics MINITAB - Lab 5 Statstcs 10010 MINITAB - Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of

More information

å 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018

å 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018 Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of

More information

,m = 1,...,n; 2 ; p m (1 p) n m,m = 0,...,n; E[X] = np; n! e λ,n 0; E[X] = λ.

,m = 1,...,n; 2 ; p m (1 p) n m,m = 0,...,n; E[X] = np; n! e λ,n 0; E[X] = λ. CS70: Lecture 21. Revew: Dstrbutos Revew: Idepedece Varace; Iequaltes; WLLN 1. Revew: Dstrbutos 2. Revew: Idepedece 3. Varace 4. Iequaltes Markov Chebyshev 5. Weak Law of Large Numbers U[1,...,] : Pr[X

More information

The Mathematical Appendix

The Mathematical Appendix The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.

More information

L(θ X) s 0 (1 θ 0) m s. (s/m) s (1 s/m) m s

L(θ X) s 0 (1 θ 0) m s. (s/m) s (1 s/m) m s Hw 4 (due March ) 83 The LRT statstcs s λ(x) sup θ θ 0 L(θ X) The lkelhood s L(θ) θ P x ( sup Θ L(θ X) θ) m P x ad ad the log-lkelhood s (θ) x log θ +(m x ) log( θ) Let S X Note that the ucostraed MLE

More information

(b) By independence, the probability that the string 1011 is received correctly is

(b) By independence, the probability that the string 1011 is received correctly is Soluto to Problem 1.31. (a) Let A be the evet that a 0 s trasmtted. Usg the total probablty theorem, the desred probablty s P(A)(1 ɛ ( 0)+ 1 P(A) ) (1 ɛ 1)=p(1 ɛ 0)+(1 p)(1 ɛ 1). (b) By depedece, the probablty

More information

The conformations of linear polymers

The conformations of linear polymers The coformatos of lear polymers Marc R. Roussel Departmet of Chemstry ad Bochemstry Uversty of Lethbrdge February 19, 9 Polymer scece s a rch source of problems appled statstcs ad statstcal mechacs. I

More information

On the Delay-Throughput Tradeoff in Distributed Wireless Networks

On the Delay-Throughput Tradeoff in Distributed Wireless Networks SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 O the Delay-Throughput Tradeoff Dstrbuted Wreless Networks Jamshd Aboue, Alreza Bayesteh, ad Amr K. Khada Codg ad Sgal Trasmsso Laboratory

More information

Kernel-based Methods and Support Vector Machines

Kernel-based Methods and Support Vector Machines Kerel-based Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to Kerel-Based Learg

More information

Convergence of the Desroziers scheme and its relation to the lag innovation diagnostic

Convergence of the Desroziers scheme and its relation to the lag innovation diagnostic Covergece of the Desrozers scheme ad ts relato to the lag ovato dagostc chard Méard Evromet Caada, Ar Qualty esearch Dvso World Weather Ope Scece Coferece Motreal, August 9, 04 o t t O x x x y x y Oservato

More information

Exchangeable Sequences, Laws of Large Numbers, and the Mortgage Crisis.

Exchangeable Sequences, Laws of Large Numbers, and the Mortgage Crisis. Exchageable Sequeces, Laws of Large Numbers, ad the Mortgage Crss. Myug Joo Sog Advsor: Prof. Ja Madel May 2009 Itroducto The law of large umbers for..d. sequece gves covergece of sample meas to a costat,.e.,

More information

Queueing Theory II. Summary. M/M/1 Output process Networks of Queue Method of Stages. General Distributions

Queueing Theory II. Summary. M/M/1 Output process Networks of Queue Method of Stages. General Distributions Queueig Theory II Suary M/M/1 Output process Networks of Queue Method of Stages Erlag Distributio Hyperexpoetial Distributio Geeral Distributios Ebedded Markov Chais 1 M/M/1 Output Process Burke s Theore:

More information

8.1 Hashing Algorithms

8.1 Hashing Algorithms CS787: Advaced Algorthms Scrbe: Mayak Maheshwar, Chrs Hrchs Lecturer: Shuch Chawla Topc: Hashg ad NP-Completeess Date: September 21 2007 Prevously we looked at applcatos of radomzed algorthms, ad bega

More information

Study of Impact of Negative Arrivals in Single. Server Fixed Batch Service Queueing System. with Multiple Vacations

Study of Impact of Negative Arrivals in Single. Server Fixed Batch Service Queueing System. with Multiple Vacations Appled Mathematcal Sceces, Vol. 7, 23, o. 4, 6967-6976 HIKARI Ltd, www.m-hkar.com http://dx.do.org/.2988/ams.23.354 Study of Impact of Negatve Arrvals Sgle Server Fxed Batch Servce Queueg System wth Multple

More information

Ideal multigrades with trigonometric coefficients

Ideal multigrades with trigonometric coefficients Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all

More information

Derivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations

Derivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat

More information

Markov chain. Markov Chain? Markov chain. Discrete-Time Markov Chain (DTMC) Memoryless (Markov) Property: Given the present state, Markov Chain

Markov chain. Markov Chain? Markov chain. Discrete-Time Markov Chain (DTMC) Memoryless (Markov) Property: Given the present state, Markov Chain Outle Marov chas ad some reewal theory Marov cha Revew: Dscrete Evet Radom Processes Hogwe Zhag htt://www.cs.waye.edu/~hzhag Reewal rocesses, reewal reward rocesses, Marov reewal rocesses The excess dstrbuto

More information

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several

More information