EP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN
|
|
- Betty Cole
- 6 years ago
- Views:
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
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 informationIS 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 informationOutline. 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 informationNetwork 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 informationFunctions 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 informationSimulation 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 informationThe 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 informationQueueing 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 informationPTAS 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 informationRandom 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 informationLecture 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 informationChapter 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 informationSpecial 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 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 informationEconometric 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 informationLecture 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 informationMultiple 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 informationChapter 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 informationCHAPTER 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 informationSimple 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 informationDiscrete 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 informationThe 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 informationC-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 informationLecture 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 informationNaïve 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 informationOutline. 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 informationOpen 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 informationRare 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 informationSummary 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 informationUNIVERSITY 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 informationClass 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 informationSteady-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 informationAPPENDIX 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 informationOptimal 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 informationLecture 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 information1. 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 informationUNIVERSITY 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 informationTHE 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 informationSTATISTICAL 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 informationSTA 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 informationCHAPTER 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 informationENGI 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 informationρ < 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 informationBeam 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 informationParameter, 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 informationChapter 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 informationbest 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 informationIntroduction 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 information2.28 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 informationStatistics: 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 informationTHE 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 informationModule 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 informationRandom 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 informationmeans 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 informationENGI 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 informationX 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 informationLecture 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 information12.2 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 informationTHE 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 informationUnsupervised 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 informationESS 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 informationSampling 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 informationObjectives 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 informationECE 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 information1 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 informationMultiple 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 informationLecture 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 informationENGI 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 informationSTA 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 informationCHAPTER 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 informationChapter 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 informationSTA302/1001-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 informationLaw 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 informationChapter 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 informationComputational 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 informationQueueing 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 informationKLT 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 informationThird 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 informationChapter 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 information9 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 informationAssignment 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 informationFeature 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 informationStatistics 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 informationå 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.
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 informationThe 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 informationL(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
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 informationThe 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 informationOn 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 informationKernel-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 informationConvergence 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 informationExchangeable 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 informationQueueing 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 information8.1 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 informationStudy 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 informationIdeal 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 informationDerivation 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 informationMarkov 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 informationhp 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