Mixed Criticality Systems with Weakly-Hard Constraints
|
|
- Jordan Lynch
- 5 years ago
- Views:
Transcription
1 Mixed Criticality Systems with Weakly-Hard Costraits Oliver Gettigs Uiversity of York Sophie Quito INRIA Greoble Rob Davis Uiversity of York
2 Mixed Criticality Systems Mixed Criticality Criticality is the required level of assurace agaist failure Mixed Criticality Systems cotai applicatios of at least two criticality levels Examples: Aerospace Flight Cotrol Systems v. Surveillace Motivatio for MCS Automotive Electric Power Steerig v. Cruise Cotrol Drive by Size, Weight ad Power (SWaP) ad cost requiremets Applicatios with differet criticalities (safety critical, missio critical etc.) o the same HW platform This research: Dual-Criticality - Applicatios of HI ad LO criticality 2
3 Mixed Criticality Systems Key requiremets Separatio must esure that LO-criticality applicatios caot impige o those of HI-criticality Sharig wat to allow LO- ad HI-criticality applicatios to use the same resources for efficiecy Real-Time behaviour Cocept of a criticality mode (LO or HI) LO ad HI-criticality applicatios must meet their time costraits i LO-criticality mode Oly HI-criticality applicatios eed meet their time costraits i HIcriticality mode (?) Iitial Research (Vestal 2007) Idea of differet LO- ad HI-criticality WCET estimates for the same code Certificatio authority requires pessimistic approach to C "# System desigers take a more realistic approach to C $% 3
4 System Model Uiprocessor, fixed priority pre-emptive schedulig Sporadic task sets where a task, τ ( = (T (, D (, C (, L ( ) T ( - Task period or miimum iter-arrival time D ( - Relative deadlie C / ( - WCET of τ ( at criticality level l L ( - Desigated criticality level for τ ( hp(i) - Set of higher priority tasks (tha τ ( ) hphi(i) - Set of higher priority, HI criticality tasks hplo(i) - Set of higher priority, LO criticality tasks 4
5 Recap: Adaptive Mixed Criticality AMC schedulig scheme If a HI-criticality task executes for its C $% without sigallig completio the o further jobs of LO-criticality tasks are started 1 ad the system eters HI-criticality mode This frees up processor badwidth to esure that HI-criticality tasks ca meet their deadlies i HI-criticality mode But, it has the drawback that LO-criticality fuctioality is completely abadoed 1 Ay partially executed job of each LO-criticality task may complete 5
6 Recap: Adaptive Mixed Criticality HI criticality task τ i LO Mode y HI Mode After Criticality chage, τ ( assumed to execute up to C ( "# 0 t C i LO C i HI Job released Deadlie Met τ i Executig LO Mode y HI Mode LO criticality task τ k 0 t LO C k τ k Preempted τ k Executig No more releases of τ 7 after criticality chage 6
7 Recap: AMC-rtb Aalysis LO-criticality mode R ( $% = C ( $% + ; R ( $% < >?(() T < C < $% HI-criticality mode R ( "# = C ( "# + ; R ( "# < hphi(() T < C < "# Iterferece from higher priority LO-criticality tasks oly up to R LO Mode chage trasitio R ( = C "# ( + ; R ( "# C T < < < hphi(() + ; R ( $% 7 hplo(() T 7 C 7 $% 7
8 Recap: AMC-max Aalysis AMC-rtb aalysis assumes (pessimistically) that all jobs of HIcriticality tasks execute with their C "# values AMC-max removes this pessimism LO Mode y HI Mode τ i 0 t C i LO C i HI Job released Deadlie Met τ i Executig Calculates umber of releases after criticality chage up to t M i, y, t = mi t + y + D ( T (, t T ( 8
9 Recap: AMC-max Aalysis AMC-max Criticality Mode Chage (LO HI) at time y R M ( = C "# ( + ; y + 1 C T $% 7 + ; M j, y, R M ( C "# < + R M ( 7 7 hplo(() < hphi (() T < M j, y, R ( M C < $% Values of y that eed to be assessed are bouded by 0 ad R $%. Values of y at which respose time may chage correspod to releases of higher priority, LO-criticality tasks: R ( = max R ( M y where y kt< j hplo i y R ( $% k N 9
10 AMC Abadomet Problem Abadoig all LO-criticality jobs Is ot acceptable i may real systems May lead to loss of importat fuctioality as LO-criticality tasks are still critical (ot o-critical) This work: Aims to address the abadomet problem by combiig AMC with a existig cocept called Weakly-Hard Provides a guarateed miimum quality of service for LO-criticality tasks i HI-criticality mode graceful degradatio 10
11 AMC-Weakly Hard Weakly Hard Model Proposed i 2001 by Guillem Berat et al. Guaratees that (m s ) out of ay m deadlies are met via (somewhat complex) offlie aalysis AMC-Weakly Hard Combies a simple iterpretatio of the weakly-hard cocept with existig AMC policy ad schedulability aalysis Allows s out of m LO-criticality jobs to be skipped i HI-criticality mode to reduce the load o the system Still provides a level of service to LO-criticality applicatios, sice (m s ) out of m deadlies are met Gives system desiger flexibility to provide graceful degradatio for LO-criticality applicatios 11
12 AMC-Weakly Hard Skips a umber of cosecutive jobs i a cycle LO Mode Criticality Mode Chage HI Mode LO criticality task τ k t Job released Deadlie Met τ k Executig τ k Job Skipped After criticality mode chage: Skip s jobs i ext m releases Repeat this cycle idefiitely i HI-criticality mode Number of skipped jobs is strictly bouded (m s ) out of m deadlies met 12
13 AMCrtb-WH Aalysis =3 =2 =1 τ k t m k T k τ k Executig Job released τ k Job Skipped Deadlie Met \ ] t ; t m 7 T 7 T 7 m 7 T 7 ^_` C 7 τ ( = T (, D (, C (, L (, s (, m ( m is legth of a cycle s is umber of skipped jobs i a cycle is idex of a skipped job 13
14 AMCrtb-WH Aalysis LO Criticality Mode R ( $% = C ( $% + < hp(() HI Criticality Mode b c de f g C < $% Worst case assumes skips are at the ed of each cycle R ( "# = C ( $ c + ; R ( "# < hphi(() T < C < "# + ; R ( "# ; R ( "# m 7 T 7 T 7 m 7 T 7 7 hplo ( \ ] ^_` h C 7 $% 14
15 AMCrtb-WH Aalysis Criticality Mode Chage (LO HI) LO Mode R i LO HI Mode Skips starts o first release after mode chage τ k t x k m k T k m k T k τ k Executig Job released τ k Job Skipped Deadlie Met First release of job after Criticality Mode Chage x 7 = R ( $% T 7 T 7 15
16 AMCrtb-WH Aalysis Criticality Mode Chage (LO HI) : HI Criticality Tasks R ( = C ( "# + ; R ( < hphi (() T < C < "# + ; R ( ; R ( m 7 T 7 x 7 T 7 m 7 T 7 7 hplo ( j ] ^_\ ] h C 7 $% Criticality Mode Chage (LO HI) : LO Criticality Tasks Assumes skips are at the start of each cycle R ( = C ( $% + ; R ( < hphi(() T < C < "# + ; R ( 7 hplo(() T 7 C 7 $% No skippig assumed for higher priority LOcriticality task. 16
17 AMCmax-WH Aalysis AMCrtb-WH criticality mode chage aalysis is pessimistic AND Aalysig HI-criticality: Assumes all HI-criticality jobs up to R execute with their C "# values Aalysig LO-criticality: Assumes o skippig of LO-criticality jobs up to R. AMCmax-WH aalysis remove these sources of pessimism by takig ito accout the poits at which a criticality mode chage could occur Aalysis for LO- ad HI-criticality modes is same as AMCrtb-WH 17
18 AMCmax-WH Aalysis Criticality Mode Chage (LO HI) at time y y LO Mode HI Mode τ k t z k m k T k m k T k τ k Executig Job released τ k Job Skipped Deadlie Met First release of job after Criticality Mode Chage z 7 = M f ] T 7 18
19 AMCmax-WH Aalysis Criticality Mode Chage (LO HI) : All Tasks Jobs of LO-criticality task k skipped after the criticality mode chage at time y M R M R ( = C $ ( ( + ; ( ; R M ( m 7 T 7 z 7 T 7 m 7 T 7 7 hplo ( j ] ^_\ ] h C 7 $% + ; M j, y, R M ( C "# < + R M ( < hphi (() T < M j, y, R ( M C < $% Jobs of HI-criticality task k oly take C HI values after the criticality mode chage at time y R M ( = max R ( M where y kt < j hplo i y R $% ( k N For HI-criticality tasks, y checked for values up to R $% For LO-criticality tasks y is icreased util R coverges below the curret value of y 19
20 Evaluatio Compared existig policies: UB-H&L - Composite upper-boud o schedulability AMC-max Baruah et al [3] AMC-rtb - Baruah et al. [3] SMC SMC-NO with budget eforced executio for LO-criticality tasks [3] SMC-NO - Vestal s origial aalysis [29] AMCmax-WH - Weakly-Hard versio of AMC-max AMCrtb-WH - Weakly-Hard versio of AMC-rtb FPPS Fixed priority preemptive schedulig with ru-time moitorig to prevet LO-criticality tasks overruig CrMPO Criticality Mootoic Priority Orderig. Tasks ordered by criticality the by DMPO withi the two partitios 20
21 Evaluatio Taskset geeratio: Uiformly distributed utilisatio values geerated with UUifast T radomly assiged from a Log uiform distributio betwee 10 ad 1000 C $% ( = U ( /T ( Criticality Factor (CF) C "# ( = C $% ( CF Criticality Probability (CP) - probability that a task will be HI-criticality Notes about graphs Plotted agaist LO-criticality utilisatio Solid lies represet policies that guaratee some LO-criticality task deadlies are met i HI-criticality mode. Dashed lies represet polices that de-schedule or permit deadlie misses of LO-criticality tasks i HI criticality mode. 21
22 1: Percetage of Schedulable Tasksets AMC-WH domiates CrMPO ad FPPS s = 1 m = 2 CP = 0.5 CF = 2.0 D = T 20 Tasks AMC-WH domiated by AMC 22
23 Weighted Schedulability Weighted Schedulability Eables overall comparisos whe varyig a specific parameter (ot just utilisatio) Combies results form of a set of equally spaced utilisatio levels W φ p = U τ S ~ τ, P U(τ) Collapses all data o a success ratio plot for a give method, ito a sigle poit o a weighted schedulability graph Weighted schedulability is effectively a weighted versio of the area uder a success ratio curve biased towards schedulig higher utilisatio message sets 23
24 2: Varyig the Criticality Mix Less pessimistic aalysis of LOcriticality tasks i HI-criticality mode with AMCmax-WH v. AMCrtb-WH s = 1 m = 2 CP = 0.05 to 0.95 CF = 2.0 D = T 20 Tasks 24
25 3: Varyig the Number of Skips (fixed cycle) s = 0 => FPPS s = m => AMC s = 0 to 10 m = 10 CP = 0.5 CF = 2.0 D = T 20 Tasks 25
26 Summary ad Coclusios AMC-WH Combies AMC protocol, with a simple iterpretatio of Weakly Hard costraits Provides guarateed miimum Quality of Service (QoS) for LO-criticality tasks HI-criticality mode, meet (m - s) out of m deadlies Performace scales betwee AMC ad FPPS Schedulability tests developed based o AMC-rtb ad AMC-max. Scope for future work: Permit weakly-hard behaviour i ay criticality mode, where each task is assiged a set of weakly hard costraits per criticality level Ivestigate recovery to LO-criticality mode 26
27 Questios? 27
Rank Modulation with Multiplicity
Rak Modulatio with Multiplicity Axiao (Adrew) Jiag Computer Sciece ad Eg. Dept. Texas A&M Uiversity College Statio, TX 778 ajiag@cse.tamu.edu Abstract Rak modulatio is a scheme that uses the relative order
More informationAdvanced Course of Algorithm Design and Analysis
Differet complexity measures Advaced Course of Algorithm Desig ad Aalysis Asymptotic complexity Big-Oh otatio Properties of O otatio Aalysis of simple algorithms A algorithm may may have differet executio
More informationProbabilistic Analysis for Mixed Criticality Scheduling with SMC and AMC
Probabilistic Analysis for Mixed Criticality Scheduling with SMC and AMC Dorin Maxim 1, Robert I. Davis 1,2, Liliana Cucu-Grosjean 1, and Arvind Easwaran 3 1 INRIA, France 2 University of York, UK 3 Nanyang
More informationProbabilistic Analysis for Mixed Criticality Systems using Fixed Priority Preemptive Scheduling
Probabilistic Analysis for Mixed Criticality Systems using Fixed Priority Preemptive Scheduling Dorin Maxim LORIA - University of Lorraine, Nancy, France dorin.maxim@loria.fr Liliana Cucu-Grosjean Inria,
More informationProbabilistic Analysis for Mixed Criticality Systems using Fixed Priority Preemptive Scheduling
Probabilistic Analysis for Mixed Criticality Systems using Fixed Priority Preemptive Scheduling Dorin Maxim LORIA - University of Lorraine, Nancy, France dorin.maxim@loria.fr Liliana Cucu-Grosjean Inria,
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationLecture 14: Graph Entropy
15-859: Iformatio Theory ad Applicatios i TCS Sprig 2013 Lecture 14: Graph Etropy March 19, 2013 Lecturer: Mahdi Cheraghchi Scribe: Euiwoog Lee 1 Recap Bergma s boud o the permaet Shearer s Lemma Number
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationOblivious Gradient Clock Synchronization
Motivatio: Clock Sychroizatio Oblivious Gradiet Clock Sychroizatio Thomas Locher, ETH Zurich Roger Wattehofer, ETH Zurich Clock sychroizatio is a classic, importat problem! May results have bee published
More informationDisjoint set (Union-Find)
CS124 Lecture 7 Fall 2018 Disjoit set (Uio-Fid) For Kruskal s algorithm for the miimum spaig tree problem, we foud that we eeded a data structure for maitaiig a collectio of disjoit sets. That is, we eed
More informationSpare CASH: Reclaiming Holes to Minimize Aperiodic Response Times in a Firm Real-Time Environment
Spare : Reclaimig Holes to Miimize Aperiodic Respose Times i a Firm Real-Time Eviromet Deepu C. Thomas Sathish Gopalakrisha Marco Caccamo Chag-Gu Lee Abstract Schedulig periodic tasks that allow some istaces
More informationCS 270 Algorithms. Oliver Kullmann. Growth of Functions. Divide-and- Conquer Min-Max- Problem. Tutorial. Reading from CLRS for week 2
Geeral remarks Week 2 1 Divide ad First we cosider a importat tool for the aalysis of algorithms: Big-Oh. The we itroduce a importat algorithmic paradigm:. We coclude by presetig ad aalysig two examples.
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe
More informationOptimal Design of Accelerated Life Tests with Multiple Stresses
Optimal Desig of Accelerated Life Tests with Multiple Stresses Y. Zhu ad E. A. Elsayed Departmet of Idustrial ad Systems Egieerig Rutgers Uiversity 2009 Quality & Productivity Research Coferece IBM T.
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationRobust Resource Allocation in Parallel and Distributed Computing Systems (tentative)
Robust Resource Allocatio i Parallel ad Distributed Computig Systems (tetative) Ph.D. cadidate V. Shestak Colorado State Uiversity Electrical ad Computer Egieerig Departmet Fort Collis, Colorado, USA shestak@colostate.edu
More informationOPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES
OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES Peter M. Maurer Why Hashig is θ(). As i biary search, hashig assumes that keys are stored i a array which is idexed by a iteger. However, hashig attempts to bypass
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationUC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 17 Lecturer: David Wagner April 3, Notes 17 for CS 170
UC Berkeley CS 170: Efficiet Algorithms ad Itractable Problems Hadout 17 Lecturer: David Wager April 3, 2003 Notes 17 for CS 170 1 The Lempel-Ziv algorithm There is a sese i which the Huffma codig was
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationΩ ). Then the following inequality takes place:
Lecture 8 Lemma 5. Let f : R R be a cotiuously differetiable covex fuctio. Choose a costat δ > ad cosider the subset Ωδ = { R f δ } R. Let Ωδ ad assume that f < δ, i.e., is ot o the boudary of f = δ, i.e.,
More informationCS284A: Representations and Algorithms in Molecular Biology
CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by
More informationMixtures of Gaussians and the EM Algorithm
Mixtures of Gaussias ad the EM Algorithm CSE 6363 Machie Learig Vassilis Athitsos Computer Sciece ad Egieerig Departmet Uiversity of Texas at Arligto 1 Gaussias A popular way to estimate probability desity
More informationDesign and Analysis of ALGORITHM (Topic 2)
DR. Gatot F. Hertoo, MSc. Desig ad Aalysis of ALGORITHM (Topic 2) Algorithms + Data Structures = Programs Lessos Leared 1 Our Machie Model: Assumptios Geeric Radom Access Machie (RAM) Executes operatios
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationFirst come, first served (FCFS) Batch
Queuig Theory Prelimiaries A flow of customers comig towards the service facility forms a queue o accout of lack of capacity to serve them all at a time. RK Jaa Some Examples: Persos waitig at doctor s
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More information>>> SOLUTIONS <<< a) What is capacity planning? Identify the key inputs, methods, and outputs of capacity planning.
Fial Exam for Capacity Plaig (CIS 493/693) Sprig 1 >>> SOLUTIONS
More informationA proposed discrete distribution for the statistical modeling of
It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationCSE 202 Homework 1 Matthias Springer, A Yes, there does always exist a perfect matching without a strong instability.
CSE 0 Homework 1 Matthias Spriger, A9950078 1 Problem 1 Notatio a b meas that a is matched to b. a < b c meas that b likes c more tha a. Equality idicates a tie. Strog istability Yes, there does always
More informationScheduling under Uncertainty using MILP Sensitivity Analysis
Schedulig uder Ucertaity usig MILP Sesitivity Aalysis M. Ierapetritou ad Zheya Jia Departmet of Chemical & Biochemical Egieerig Rutgers, the State Uiversity of New Jersey Piscataway, NJ Abstract The aim
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationIntro to Learning Theory
Lecture 1, October 18, 2016 Itro to Learig Theory Ruth Urer 1 Machie Learig ad Learig Theory Comig soo 2 Formal Framework 21 Basic otios I our formal model for machie learig, the istaces to be classified
More informationAchieving Stationary Distributions in Markov Chains. Monday, November 17, 2008 Rice University
Istructor: Achievig Statioary Distributios i Markov Chais Moday, November 1, 008 Rice Uiversity Dr. Volka Cevher STAT 1 / ELEC 9: Graphical Models Scribe: Rya E. Guerra, Tahira N. Saleem, Terrace D. Savitsky
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 12
CS 70 Discrete Mathematics ad Probability Theory Fall 2009 Satish Rao,David Tse Note 12 Two Killer Applicatios I this lecture, we will see two killer apps of elemetary probability i Computer Sciece. 1.
More informationCS/ECE 715 Spring 2004 Homework 5 (Due date: March 16)
CS/ECE 75 Sprig 004 Homework 5 (Due date: March 6) Problem 0 (For fu). M/G/ Queue with Radom-Sized Batch Arrivals. Cosider the M/G/ system with the differece that customers are arrivig i batches accordig
More informationClassification of problem & problem solving strategies. classification of time complexities (linear, logarithmic etc)
Classificatio of problem & problem solvig strategies classificatio of time complexities (liear, arithmic etc) Problem subdivisio Divide ad Coquer strategy. Asymptotic otatios, lower boud ad upper boud:
More informationSection 8.5 Alternating Series and Absolute Convergence
Sectio 85 Alteratig Series ad Absolute Covergece AP E XC II Versio 20 Authors Gregory Hartma, PhD Departmet of Applied Mathema cs Virgiia Military Is tute Bria Heiold, PhD Departmet of Mathema cs ad Computer
More informationCS322: Network Analysis. Problem Set 2 - Fall 2009
Due October 9 009 i class CS3: Network Aalysis Problem Set - Fall 009 If you have ay questios regardig the problems set, sed a email to the course assistats: simlac@staford.edu ad peleato@staford.edu.
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationModule 5 EMBEDDED WAVELET CODING. Version 2 ECE IIT, Kharagpur
Module 5 EMBEDDED WAVELET CODING Versio ECE IIT, Kharagpur Lesso 4 SPIHT algorithm Versio ECE IIT, Kharagpur Istructioal Objectives At the ed of this lesso, the studets should be able to:. State the limitatios
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationCS 332: Algorithms. Linear-Time Sorting. Order statistics. Slide credit: David Luebke (Virginia)
1 CS 332: Algorithms Liear-Time Sortig. Order statistics. Slide credit: David Luebke (Virgiia) Quicksort: Partitio I Words Partitio(A, p, r): Select a elemet to act as the pivot (which?) Grow two regios,
More informationExam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234
STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationLecture 10: Universal coding and prediction
0-704: Iformatio Processig ad Learig Sprig 0 Lecture 0: Uiversal codig ad predictio Lecturer: Aarti Sigh Scribes: Georg M. Goerg Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved
More informationInteger Programming (IP)
Iteger Programmig (IP) The geeral liear mathematical programmig problem where Mied IP Problem - MIP ma c T + h Z T y A + G y + y b R p + vector of positive iteger variables y vector of positive real variables
More informationPower supplies for parallel operation Power supplies
Power supplies Power supplies for parallel operatio U 1 2 U Parallel operatio of switchmode power supplies Techical details for passive curret-sharig The aim of operatig switchmode power supplies (SMPS)
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationPower and Type II Error
Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error
More informationDesign and Analysis of Algorithms
Desig ad Aalysis of Algorithms Probabilistic aalysis ad Radomized algorithms Referece: CLRS Chapter 5 Topics: Hirig problem Idicatio radom variables Radomized algorithms Huo Hogwei 1 The hirig problem
More informationCS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
More informationSimulation of Discrete Event Systems
Simulatio of Discrete Evet Systems Uit 9 Queueig Models Fall Witer 2014/2015 Uiv.-Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Christopher M. Schlick Chair ad Istitute of Idustrial Egieerig ad Ergoomics RWTH Aache Uiversity
More informationAlgorithm Analysis. Chapter 3
Data Structures Dr Ahmed Rafat Abas Computer Sciece Dept, Faculty of Computer ad Iformatio, Zagazig Uiversity arabas@zu.edu.eg http://www.arsaliem.faculty.zu.edu.eg/ Algorithm Aalysis Chapter 3 3. Itroductio
More informationAnalysis of Algorithms -Quicksort-
Aalysis of Algorithms -- Adreas Ermedahl MRTC (Mälardales Real-Time Research Ceter) adreas.ermedahl@mdh.se Autum 2004 Proposed by C.A.R. Hoare i 962 Worst- case ruig time: Θ( 2 ) Expected ruig time: Θ(
More information16 Riemann Sums and Integrals
16 Riema Sums ad Itegrals Defiitio: A partitio P of a closed iterval [a, b], (b >a)isasetof 1 distict poits x i (a, b) togetherwitha = x 0 ad b = x, together with the covetio that i>j x i >x j. Defiitio:
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationCSE 5311 Notes 1: Mathematical Preliminaries
Chapter 1 - Algorithms Computig CSE 5311 Notes 1: Mathematical Prelimiaries Last updated 1/20/18 12:56 PM) Relatioship betwee complexity classes, eg log,, log, 2, 2, etc Chapter 2 - Gettig Started Loop
More information11. Hash Tables. m is not too large. Many applications require a dynamic set that supports only the directory operations INSERT, SEARCH and DELETE.
11. Hash Tables May applicatios require a dyamic set that supports oly the directory operatios INSERT, SEARCH ad DELETE. A hash table is a geeralizatio of the simpler otio of a ordiary array. Directly
More informationBER results for a narrowband multiuser receiver based on successive subtraction for M-PSK modulated signals
results for a arrowbad multiuser receiver based o successive subtractio for M-PSK modulated sigals Gerard J.M. Jasse Telecomm. ad Traffic-Cotrol Systems Group Dept. of Iformatio Techology ad Systems Delft
More informationNotes for Lecture 5. 1 Grover Search. 1.1 The Setting. 1.2 Motivation. Lecture 5 (September 26, 2018)
COS 597A: Quatum Cryptography Lecture 5 (September 6, 08) Lecturer: Mark Zhadry Priceto Uiversity Scribe: Fermi Ma Notes for Lecture 5 Today we ll move o from the slightly cotrived applicatios of quatum
More informationDaniel Lee Muhammad Naeem Chingyu Hsu
omplexity Aalysis of Optimal Statioary all Admissio Policy ad Fixed Set Partitioig Policy for OVSF-DMA ellular Systems Daiel Lee Muhammad Naeem higyu Hsu Backgroud Presetatio Outlie System Model all Admissio
More informationWeek 1, Lecture 2. Neural Network Basics. Announcements: HW 1 Due on 10/8 Data sets for HW 1 are online Project selection 10/11. Suggested reading :
ME 537: Learig-Based Cotrol Week 1, Lecture 2 Neural Network Basics Aoucemets: HW 1 Due o 10/8 Data sets for HW 1 are olie Proect selectio 10/11 Suggested readig : NN survey paper (Zhag Chap 1, 2 ad Sectios
More informationIntroduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric.
Itrodctio Qestio: Why do we eed ew forms of parametric crves? Aswer: Those parametric crves discssed are ot very geometric. Itrodctio Give sch a parametric form, it is difficlt to kow the derlyig geometry
More informationSkip Lists. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 S 3 S S 1
Presetatio for use with the textbook, Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Skip Lists S 3 15 15 23 10 15 23 36 Skip Lists 1 What is a Skip List A skip list for
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationPolynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
More information10.6 ALTERNATING SERIES
0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose
More informationCSE 4095/5095 Topics in Big Data Analytics Spring 2017; Homework 1 Solutions
CSE 09/09 Topics i ig Data Aalytics Sprig 2017; Homework 1 Solutios Note: Solutios to problems,, ad 6 are due to Marius Nicolae. 1. Cosider the followig algorithm: for i := 1 to α log e do Pick a radom
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k
More informationStatistical Pattern Recognition
Statistical Patter Recogitio Classificatio: No-Parametric Modelig Hamid R. Rabiee Jafar Muhammadi Sprig 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2/ Ageda Parametric Modelig No-Parametric Modelig
More informationMath 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals
Math 1314 Lesso 16 Area ad Riema Sums ad Lesso 17 Riema Sums Usig GeoGebra; Defiite Itegrals The secod questio studied i calculus is the area questio. If a regio coforms to a kow formula from geometry,
More informationChapter 12 - Quality Cotrol Example: The process of llig 12 ouce cas of Dr. Pepper is beig moitored. The compay does ot wat to uderll the cas. Hece, a target llig rate of 12.1-12.5 ouces was established.
More informationME 539, Fall 2008: Learning-Based Control
ME 539, Fall 2008: Learig-Based Cotrol Neural Network Basics 10/1/2008 & 10/6/2008 Uiversity Orego State Neural Network Basics Questios??? Aoucemet: Homework 1 has bee posted Due Friday 10/10/08 at oo
More informationPixel Recurrent Neural Networks
Pixel Recurret Neural Networks Aa ro va de Oord, Nal Kalchbreer, Koray Kavukcuoglu Google DeepMid August 2016 Preseter - Neha M Example problem (completig a image) Give the first half of the image, create
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationAsymptotic Coupling and Its Applications in Information Theory
Asymptotic Couplig ad Its Applicatios i Iformatio Theory Vicet Y. F. Ta Joit Work with Lei Yu Departmet of Electrical ad Computer Egieerig, Departmet of Mathematics, Natioal Uiversity of Sigapore IMS-APRM
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationAnalysis of the Chow-Robbins Game with Biased Coins
Aalysis of the Chow-Robbis Game with Biased Cois Arju Mithal May 7, 208 Cotets Itroductio to Chow-Robbis 2 2 Recursive Framework for Chow-Robbis 2 3 Geeralizig the Lower Boud 3 4 Geeralizig the Upper Boud
More informationProvläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE
TECHNICAL REPORT CISPR 16-4-3 2004 AMENDMENT 1 2006-10 INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE Amedmet 1 Specificatio for radio disturbace ad immuity measurig apparatus ad methods Part 4-3:
More informationThe axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.
5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =
More information5. INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY
IA Probability Let Term 5 INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY 51 Iequalities Suppose that X 0 is a radom variable takig o-egative values ad that c > 0 is a costat The P X c E X, c is
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationCSI 5163 (95.573) ALGORITHM ANALYSIS AND DESIGN
CSI 5163 (95.573) ALGORITHM ANALYSIS AND DESIGN CSI 5163 (95.5703) ALGORITHM ANALYSIS AND DESIGN (3 cr.) (T) Topics of curret iterest i the desig ad aalysis of computer algorithms for graphtheoretical
More informationHow to Maximize a Function without Really Trying
How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet
More information