Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness
|
|
- Jonathan Greer
- 5 years ago
- Views:
Transcription
1 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 Costrutio of Cotrol Chart for Radom Queue Legth for (M / M / ): ( / FCFS) Queueig Model Usig Skewess Dr.(Mrs.) A.R. Sudamai Ramaswamy, Mrs. B.Veila *Departmet of Mathematis, Aviashiligam Istitute for HomeSiee ad Higher Eduatio for Wome, Coimbatore-4. **Departmet of Mathematis, Sri Ramakrisha Egieerig College, Coimbatore-. Dr. (Mrs.) A.R. Sudamai Ramaswamy, Ph.D, arsudamairamaswamy@hotmail.om Correspodee Author: Mrs. B.Veila, M.Phil, veila.bs@gmail.om ABSTRACT I this paper, first-ome, first served, multiple hael Poisso/epoetial queueig system M / M / with ifiite waitig apaity is osidered. For this model, we itrodue a ew proedure for ostrutio of i) Shewhart s otrol hart C, ii) Cotrol hart C for radom queue legth N usig the method o skewess suggested by Shore() ad the otrol harts C ad C are ompared. Ide Terms: Radom queue legth, Average ru legth, False alarm rate, Average queue legth.. INTRODUCTION Waitig i lie for servie is oe of the most upleasat eperiees of life o this world. Barrer (957) says, I ertai queueig proesses a potetial ustomer is osidered lost if the system is busy at the time servie is demaded. If ot served durig this time, the ustomer leaves the system ad is osidered lost. I queueig system the ustomer satisfatio a be ireased by ostrutig otrol harts for N ad providig otrol limits for N so as to make effetive utilizatio of time. If otrol limits are displayed so that ustomer a have prior idea about otrol limits. With this fator, a attempt is made to fid otrol limits for radom queue legth N for (M/M/) : ( /FCFS) queueig model. The pioeerig work i this diretio was made by Haim Shore i. Two otrol harts C ad C are ostruted for radom queue legth N for (M/M/) : ( /FCFS)
2 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 queueig model by Khaparde ad Dhabe () ad also C ad C 4 are ostruted for radom queue legth N for (M/M/) : ( /FCFS) queueig model by Khaparde ad Dhabe () where Cotrol hart C : is the Shewhart otrol hart, Cotrol hart C : is the otrol hart usig method of weighted variae, Cotrol hart C : is the otrol hart for radom queue legth N based o skewess ad Cotrol hart C 4 : is the otrol hart usig Nelso s power trasformatio. I this paper ostrutio of otrol hart C for radom queue legth N for (M/M/) : ( /FCFS) queueig model is doe usig method based o skewess. Performae of otrol hart C is ompared with otrol hart C.. (M / M / ) : ( / FCFS) QUEUEING MODEL I this model arrival follows Poisso distributio with parameter ad servie time follows idepedetly ad idetially distributed epoetial distributio with parameter. There are idetial servers preset i the system. Figure I represet the state-trasitio diagram for this model. + () FIGURE I : STATE-TRANSITION RATE DIAGRAM FOR M / M / MODEL Let radom variable N deote the umber of ustomers i the system both waitig ad i servie ad P deote the steady state distributio of radom variable N give by P = P,! P,! () where P =,!! is the mea arrival rate ad is the mea servie rate. Let radom variable W S deote the waitig time spet i the system by the ustomer. This iludes both the waitig time ad servie time. The p.d.f. of radom variable W S is give by
3 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 ( ) WS f(w S ) = ( ) e, W S > () Thus W S follows a epoetial distributio with mea E(W S ) = = W S f(ws ) d WS ( )W S ( ) e d WS W S E(W S ) = ( ) () E(W S ) = ( ) (4) Usig (.) ad (.4), the variae is give by V(W S ) = ( ) (5) The distributio futio of W S is F() = P(W S ) = f(w ) d S W S = ( )W S ( ) e d W S = e ( ), >. F() = e ( ), >. The probability distributio of N is a geometri distributio with parameter ( ). The steady state distributio of the radom variable N depeds o two parameters ad oly through their ratio. Usig (), the average umber of ustomers ad variae of queue legth i the system is give by E(N) = = P + P = P +!! P
4 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 4 ISSN 5-5 = P )! ( )! ( Here P =!! E(N) = =!!!! (6) E(N ) = = ) P )P (( = )P ( + )P ( + P + P = ) (! P + ) (! P +! P +! P E(N ) = =!!!! (7) Usig (6) ad (7), the variae of queue legth i the system is give by V(N) =!!!!
5 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 5 ISSN 5-5!! )! ( )! ( (8) Usig (8), the stadard deviatio is alulated as = /!! )! ( )! (!!!! (9) E(N ) = = )P ) ( ) )( (( = P! ) )( ( +! ) )( ( +! ) ( +! ) ( +!! =!!!! ()
6 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 6 ISSN 5-5 Usig (), the third etral momet is give by =!!!! -!!!!! ) ( )! ( +!! =!!!! )! ( )! (!!!!!! () Let S k (N) deote the traditioal measure of skewess for radom queue legth amely, S k (N) =. where is the third etral momet.
7 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 7 ISSN 5-5 Usig (9) ad (), the skewess for radom queue legth is give by S k (N) = / )! ( )! (!!!! )! ( )! (!! )! ( )! (!!!!!! (). SHEWHART S CONTROL CHART C Whe it is possible to speify stadard values for the proess mea ad stadard deviatio, we may use these stadards to establish the otrol hart. The the parameters of the hart are UCL = + A CL = LCL = A where the quatity = A, say, is a ostat that depeds o. Usig epressios (6) ad (9), the otrol limits for (M / M / ) : ( / FCFS) model are alulated as follows.
8 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 8 ISSN 5-5 UCL =!! )! ( )! (!!!! A )! ( )! ( / CL =!! )! ( )! ( LCL =!! )! ( )! (!!!! A )! ( )! ( / TABLE I MEAN, STANDARD DEVIATION, UPPER CONTROL LIMIT FOR THE RANDOM VARIABLE N FOR DIFFERENT VALUES OF USING SHEWHART S CONTROL CHART C WITH A = AND C =
9 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 9 ISSN 5-5 S.No. CL UCL u ARL CONSTRUCTION OF CONTROL CHART C FOR RANDOM VARIABLE N Before ostrutig the hart, it is essetial to otie that the probability futio of a geometri distributio is a mootoously dereasig futio of ad sie otrol limits for queueig system are to be obtaied, there is o meaig i defiig lower otrol limit (LCL) for this distributio. Also upper otrol limit (UCL) obviously makes sese sie it will eable moitorig improbable high value of N.. CONTROL CHART C I the queueig model (M / M / ) : ( / FCFS), the radom variable N deote the umber of ustomers i the system both waitig ad i servie. The steady state distributio of radom variable N is give by epressio (). I order to obtai otrol limits usig Shore s method (), the first three momets of the radom variable N give by epressios (6), (8) ad () respetively, are eeded. The epressio () shows that skewess of the distributio of N depeds o,, ad. Takig ito aout the mea, stadard deviatio ad skewess of the uderlyig geometri distributio of N the epressio for the ew otrol limits for the umber of ustomers i (M / M / ) : ( / FCFS) system usig Shore s method () are as follows : Cotrol Limits for Cotrol Chart C whe S k <.5
10 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 UCL = E(N) + V(N) +.4 V(N) S k (N). CL = E(N) LCL = E(N) V(N) +.4 V(N) S k (N) +. Cotrol Limits for Cotrol Chart C whe S k >.5 UCL = E(N) +.64 V(N) V(N) S k (N). CL = E(N) LCL = E(N).64 V(N) + (.4) (.946) V(N) S k (N) +. Cotrol Limits for Cotrol Chart C for (M / M / ) : ( / FCFS) Model whe S k <.5 UCL = )! ( )! (!! +!!!! )! ( )! (!!
11 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN !!!! )! ( )! (!!!!!!!! )! ( )! (!! + )! ( )! ( CL = )! ( )! (
12 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5!! LCL = )! ( )! (!!!!!! )! ( )! (!! +.4!!!! )! ( )! (!!
13 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5!!!!!! )! ( )! (!! + )! ( )! ( Cotrol Limits for Cotrol Chart C for (M / M / ) : ( / FCFS) Model whe S k >.5 UCL = )! ( )! (!! +.64!!!!
14 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 4 ISSN 5-5 )! ( )! (!! + (.946)!!!! )! ( )! (!!!!!!!! )! ( )! (!!
15 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 5 ISSN )! ( )! ( CL = )! ( )! (!! LCL = )! ( )! (!!.64!!!! )! ( )! (!! + (.4) (.946)!!!!
16 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 6 ISSN 5-5 )! ( )! (!!!!!!!! )! ( )! (!! + )! ( )! ( Table. shows that the miimum value of skewess is early. for high values of. TABLE II SKEWNESS, CONTROL LIMITS FOR THE RANDOM VARIABLE N FOR DIFFERENT VALUES OF USING CONTROL CHART C WITH L = ad C = S.No. Skewess CL UCL u ARL
17 Upper Cotrol Limit (UCL) Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 7 ISSN Cotrol hart C Cotrol hart C Traffi itesity (ρ) FIGURE II :COMPARISON OF UPPER CONTROL LIMIT (UCL) OF CONTROL CHARTS C AND C WITH L = AND C = FOR DIFFERENT VALUES OF It is otied that the upper otrol limit of otrol hart C outperforms the otrol hart C for all values of. Upper otrol limit values i otrol hart C are very very large tha that of hart C. Therefore for skewed populatio the use of otrol hart C is advisable. This a be observed from Figure II
18 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 8 ISSN 5-5 FIGURE III: COMPARISON OF AVERAGE RUN LENGTH OF CONTROL CHARTS C AND C WITH L = AND C = FOR DIFFERENT VALUES OF CONCLUSON The urves of ARL values are displayed i Figure III for harts C ad C. The horizotal ais of spas.5 to.45, Figure III demostrates that the ARL produed by hart C is superior to the hart C. REFERENCES. Arold, A.O. (978). Probability, Statistis ad Queueig Theory with Computer Siee Appliatios, Seod Editio, Aademi Press Limited, New York.. Barrer, D.Y. (957). Queueig with Impatiet Customers ad Idifferet Clerks, Operatios Researh, 5, Khaparde, M.V. ad Dhabe, S.D. (). Cotrol Charts for Radom Queue Legth N for (M / M / ) : ( / FCFS) Model, Iteratioal Joural of Agriultural ad Statistial Siees,, Khaparde, M.V. ad Dhabe, S.D. (). Cotrol Charts for Radom Queue Legth N for (M / M / ) : ( / FCFS) Queueig Model Usig Skewess ad Power Trasformatio, Bulleti of Pure ad Applied Siees, E, Motgomery, D.C. (5). Itrodutio to Statistial Quality Cotrol, Fifth Editio, Joh Wiley ad Sos I. 6. Shore, H. (). Geeral Cotrol Charts for Attributes, IIE Trasatios,, Sudamai Ramaswamy, A.R. (). Statistial Aalysis of M / M / Model, Researh Highlights,, 7-.
Control chart for number of customers in the system of M [X] / M / 1 Queueing system
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.
More informationB. Maddah ENMG 622 ENMG /20/09
B. Maddah ENMG 6 ENMG 5 5//9 Queueig Theory () Distributio of waitig time i M/M/ Let T q be the waitig time i queue of a ustomer. The it a be show that, ( ) t { q > } =. T t e Let T be the total time of
More informationOptimal design of N-Policy batch arrival queueing system with server's single vacation, setup time, second optional service and break down
Ameria. Jr. of Mathematis ad Siees Vol., o.,(jauary Copyright Mid Reader Publiatios www.jouralshub.om Optimal desig of -Poliy bath arrival queueig system with server's sigle vaatio, setup time, seod optioal
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 informationON STEREOGRAPHIC CIRCULAR WEIBULL DISTRIBUTION
htt://www.ewtheory.org ISSN: 9- Reeived:..6 Published:.6.6 Year: 6 Number: Pages: -9 Origial Artile ** ON STEREOGRAPHIC CIRCULAR WEIBULL DISTRIBUTION Phai Yedlaalli * Sagi Vekata Sesha Girija Akkavajhula
More informationModeling and Performance Analysis with Discrete-Event Simulation
Simulatio Modelig ad Performae Aalysis with Disrete-Evet Simulatio Chapter 8 Queueig Models Cotets Charateristis of Queueig Systems Queueig Notatio Kedall Notatio Log-ru Measures of Performae of Queueig
More informationTHE MEASUREMENT OF THE SPEED OF THE LIGHT
THE MEASUREMENT OF THE SPEED OF THE LIGHT Nyamjav, Dorjderem Abstrat The oe of the physis fudametal issues is a ature of the light. I this experimet we measured the speed of the light usig MihelsoÕs lassial
More informationQueuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues
Queuig Theory Basic properties, Markovia models, Networks of queues, Geeral service time distributios, Fiite source models, Multiserver queues Chapter 8 Kedall s Notatio for Queuig Systems A/B/X/Y/Z: A
More informationOptimal Management of the Spare Parts Stock at Their Regular Distribution
Joural of Evirometal Siee ad Egieerig 7 (018) 55-60 doi:10.1765/16-598/018.06.005 D DVID PUBLISHING Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio Svetozar Madzhov Forest Researh
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationChapter 8 Hypothesis Testing
Chapter 8 for BST 695: Speial Topis i Statistial Theory Kui Zhag, Chapter 8 Hypothesis Testig Setio 8 Itrodutio Defiitio 8 A hypothesis is a statemet about a populatio parameter Defiitio 8 The two omplemetary
More informationNonparametric Goodness-of-Fit Tests for Discrete, Grouped or Censored Data 1
Noparametri Goodess-of-Fit Tests for Disrete, Grouped or Cesored Data Boris Yu. Lemeshko, Ekateria V. Chimitova ad Stepa S. Kolesikov Novosibirsk State Tehial Uiversity Departmet of Applied Mathematis
More informationB. Maddah ENMG 622 ENMG /27/07
B. Maddah ENMG 622 ENMG 5 3/27/7 Queueig Theory () What is a queueig system? A queueig system cosists of servers (resources) that provide service to customers (etities). A Customer requestig service will
More informationA queueing system can be described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service,
Queuig System A queueig system ca be described as customers arrivig for service, waitig for service if it is ot immediate, ad if havig waited for service, leavig the service after beig served. The basic
More informationAnnouncements. Queueing Systems: Lecture 1. Lecture Outline. Topics in Queueing Theory
Aoucemets Queueig Systems: Lecture Amedeo R. Odoi October 4, 2006 PS #3 out this afteroo Due: October 9 (graded by 0/23) Office hours Odoi: Mo. 2:30-4:30 - Wed. 2:30-4:30 o Oct. 8 (No office hrs 0/6) _
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 informationSummation Method for Some Special Series Exactly
The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue Summatio Method for Some Speial Series Eatly D.A.Gismalla Deptt. Of Mathematis & omputer Studies Faulty of Siee
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 informationSINGLE-CHANNEL QUEUING PROBLEMS APPROACH
SINGLE-CHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT E-mail: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper
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 informationCertain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator
Iteratioal Joural of Pure ad Applied Mathematial Siees. ISSN 0972-9828 Volume 0, Number (207), pp. 85-97 Researh Idia Publiatios http://www.ripubliatio.om Certai ilusio properties of sublass of starlike
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationBasic Probability/Statistical Theory I
Basi Probability/Statistial Theory I Epetatio The epetatio or epeted values of a disrete radom variable X is the arithmeti mea of the radom variable s distributio. E[ X ] p( X ) all Epetatio by oditioig
More informationQueueing theory and Replacement model
Queueig theory ad Replacemet model. Trucks at a sigle platform weigh-bridge arrive accordig to Poisso probability distributio. The time required to weigh the truck follows a expoetial probability distributio.
More informationWhat is a Hypothesis? Hypothesis is a statement about a population parameter developed for the purpose of testing.
What is a ypothesis? ypothesis is a statemet about a populatio parameter developed for the purpose of testig. What is ypothesis Testig? ypothesis testig is a proedure, based o sample evidee ad probability
More informationInternational Journal of Advanced Research in Computer Science and Software Engineering
Volume 7, Issue 3, Marh 2017 ISSN: 2277 128X Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig Researh Paper Available olie at: wwwijarsseom O Size-Biased Weighted Trasmuted Weibull
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
More informationPass-Fail Testing: Statistical Requirements and Interpretations
Joural of Researh of the Natioal Istitute of Stadards ad Tehology [J. Res. Natl. Ist. Stad. Tehol. 4, 95-99 (2009)] Pass-Fail Testig: Statistial Requiremets ad Iterpretatios Volume 4 Number 3 May-Jue 2009
More informationCONSIDERATION OF SOME PERFORMANCE OF CONTAINERS FLOW AT YARD
CONSIDERATION OF SOME PERFORMANCE OF CONTAINERS FLOW AT YARD Maja Škurić (a), Braislav Dragović (b), Nead Dj. Zrić (), Romeo Meštrović (d) (a), (b), (d) Maritime Faulty Kotor, Uiversity of Moteegro ()
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More information(# x) 2 n. (" x) 2 = 30 2 = 900. = sum. " x 2 = =174. " x. Chapter 12. Quick math overview. #(x " x ) 2 = # x 2 "
Chapter 12 Describig Distributios with Numbers Chapter 12 1 Quick math overview = sum These expressios are algebraically equivalet #(x " x ) 2 = # x 2 " (# x) 2 Examples x :{ 2,3,5,6,6,8 } " x = 2 + 3+
More informationCONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION
CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION Petros Maravelakis, Joh Paaretos ad Stelios Psarakis Departmet of Statistics Athes Uiversity of Ecoomics ad Busiess 76 Patisio St., 4 34, Athes, GREECE. Itroductio
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 informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More informationBirth-Death Processes. Outline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Relationship Among Stochastic Processes.
EEC 686/785 Modelig & Perforace Evaluatio of Couter Systes Lecture Webig Zhao Deartet of Electrical ad Couter Egieerig Clevelad State Uiversity webig@ieee.org based o Dr. Raj jai s lecture otes Relatioshi
More informationSample Size Requirements For Stratified Random Sampling Of Pond Water With Cost Considerations Using A Bayesian Methodology
Sample Size Requiremets For Stratified Radom Samplig Of Pod Water Wit Cost Cosideratios Usig A Bayesia Metodology A.A. Bartolui a, A.D. Bartolui b, ad K.P. Sig a Departmet of Biostatistis, Uiversity of
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationProbability & Statistics Chapter 8
I. Estimatig with Large Samples Probability & Statistis Poit Estimate of a parameter is a estimate of a populatio parameter give by a sigle umber. Use x (the sample mea) as a poit estimate for (the populatio
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationUsing the IML Procedure to Examine the Efficacy of a New Control Charting Technique
Paper 2894-2018 Usig the IML Procedure to Examie the Efficacy of a New Cotrol Chartig Techique Austi Brow, M.S., Uiversity of Norther Colorado; Bryce Whitehead, M.S., Uiversity of Norther Colorado ABSTRACT
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationRecord Values from T-X Family of. Pareto-Exponential Distribution with. Properties and Simulations
Applied Mathematical Scieces, Vol. 3, 209, o., 33-44 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.209.879 Record Values from T-X Family of Pareto-Epoetial Distributio with Properties ad Simulatios
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 information(Dependent or paired samples) Step (1): State the null and alternate hypotheses: Case1: One-tailed test (Right)
(epedet or paired samples) Step (1): State the ull ad alterate hypotheses: Case1: Oe-tailed test (Right) Upper tail ritial (where u1> u or u1 -u> 0) H0: 0 H1: > 0 Case: Oe-tailed test (Left) Lower tail
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 3 January 29, 2003 Prof. Yannis A. Korilis
TCOM 5: Networkig Theory & Fudametals Lecture 3 Jauary 29, 23 Prof. Yais A. Korilis 3-2 Topics Markov Chais Discrete-Time Markov Chais Calculatig Statioary Distributio Global Balace Equatios Detailed Balace
More informationOn Poisson Bulk Arrival Queue: M / M /2/ N with. Balking, Reneging and Heterogeneous servers
Applied Mathematical Scieces, Vol., 008, o. 4, 69-75 O oisso Bulk Arrival Queue: M / M // with Balkig, Reegig ad Heterogeeous servers M. S. El-aoumy Statistics Departmet, Faculty of Commerce, Dkhlia, Egypt
More informationM/M/1 queueing system with server start up, unreliable server and balking
Research Joural of Mathematical ad Statistical Scieces ISS 232-647 Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. Aalysis of a -policy M/M/ queueig system with server start up, ureliable
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
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 informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationThe statistical pattern of the arrival can be indicated through the probability distribution of the number of the arrivals in an interval.
Itroductio Queuig are the most freuetly ecoutered roblems i everyday life. For examle, ueue at a cafeteria, library, bak, etc. Commo to all of these cases are the arrivals of objects reuirig service ad
More informationIncreasing timing capacity using packet coloring
003 Coferece o Iformatio Scieces ad Systems, The Johs Hopkis Uiversity, March 4, 003 Icreasig timig capacity usig packet colorig Xi Liu ad R Srikat[] Coordiated Sciece Laboratory Uiversity of Illiois e-mail:
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided
More informationSTAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)
STAT 515 fa 2016 Lec 15-16 Samplig distributio of the mea, part 2 cetral limit theorem Karl B. Gregory Moday, Sep 26th Cotets 1 The cetral limit theorem 1 1.1 The most importat theorem i statistics.............
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationMixed Acceptance Sampling Plans for Multiple Products Indexed by Cost of Inspection
Mied ace Samplig Plas for Multiple Products Ideed by Cost of Ispectio Paitoo Howyig ), Prapaisri Sudasa Na - Ayudthya ) ) Dhurakijpudit Uiversity, Faculty of Egieerig (howyig@yahoo.com) ) Kasetsart Uiversity,
More informationModeling and Performance Analysis with Discrete-Event Simulation
Simulatio Modelig ad Performace Aalysis with Discrete-Evet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous
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 informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationEconoQuantum ISSN: Universidad de Guadalajara México
EooQuatum ISSN: 1870-6622 equatum@uea.udg.mx Uiversidad de Guadalajara Méxio Plata Pérez, Leobardo; Calderó, Eduardo A modified versio of Solow-Ramsey model usig Rihard's growth futio EooQuatum, vol. 6,
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationMATHEMATICS Paper 2 22 nd September 20. Answer Papers List of Formulae (MF15)
NANYANG JUNIOR COLLEGE JC PRELIMINARY EXAMINATION Higher MATHEMATICS 9740 Paper d September 0 3 Ho Additioal Materials: Cover Sheet Aswer Papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write
More informationThe Poisson Distribution
MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This
More informationE. Teimoury *, I.G. Khondabi & M. Fathi
Iteratioal Joural of Idustrial Egieerig & Produtio Researh eptember 2 Volume 22 umber 3 pp. 5-58 I: 28-4889 http://ijiepr.iust.a.ir/ A Itegrated Queuig Model for ite eletio ad Ivetory torage Plaig of a
More informationCOMP26120: Introducing Complexity Analysis (2018/19) Lucas Cordeiro
COMP60: Itroduig Complexity Aalysis (08/9) Luas Cordeiro luas.ordeiro@mahester.a.uk Itroduig Complexity Aalysis Textbook: Algorithm Desig ad Appliatios, Goodrih, Mihael T. ad Roberto Tamassia (hapter )
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 informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationTHE POTENTIALS METHOD FOR A CLOSED QUEUEING SYSTEM WITH HYSTERETIC STRATEGY OF THE SERVICE TIME CHANGE
Joural of Applied Mathematics ad Computatioal Mechaics 5, 4(), 3-43 www.amcm.pcz.pl p-issn 99-9965 DOI:.75/amcm.5..4 e-issn 353-588 THE POTENTIALS METHOD FOR A CLOSED QUEUEING SYSTEM WITH HYSTERETIC STRATEGY
More informationOptimization design in Wind Turbine Blade Based on Specific Wind Characteristics
Iteratioal Sietifi Joural Joural of Evirometal Siee http://eviromet.sietifi-joural.om/ Optimizatio desig i Wid Turbie Blade Based o Speifi Wid Charateristis Yuqiao Zheg College of Mehao-Eletroi Egieerig
More informationBayesian Control Charts for the Two-parameter Exponential Distribution
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com 2 Uiversity of the Free State Abstract By usig data that are the mileages
More informationAverage Number of Real Zeros of Random Fractional Polynomial-II
Average Number of Real Zeros of Radom Fractioal Polyomial-II K Kadambavaam, PG ad Research Departmet of Mathematics, Sri Vasavi College, Erode, Tamiladu, Idia M Sudharai, Departmet of Mathematics, Velalar
More informationTest of Statistics - Prof. M. Romanazzi
1 Uiversità di Veezia - Corso di Laurea Ecoomics & Maagemet Test of Statistics - Prof. M. Romaazzi 19 Jauary, 2011 Full Name Matricola Total (omial) score: 30/30 (2 scores for each questio). Pass score:
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationMath Third Midterm Exam November 17, 2010
Math 37 1. Treibergs σιι Third Midterm Exam Name: November 17, 1 1. Suppose that the mahie fillig mii-boxes of Fruitlad Raisis fills boxes so that the weight of the boxes has a populatio mea µ x = 14.1
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
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 informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More information4. Basic probability theory
Cotets Basic cocepts Discrete radom variables Discrete distributios (br distributios) Cotiuous radom variables Cotiuous distributios (time distributios) Other radom variables Lect04.ppt S-38.45 - Itroductio
More informationLecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.
STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7:
0 Multivariate Cotrol Chart 3 Multivariate Normal Distributio 5 Estimatio of the Mea ad Covariace Matrix 6 Hotellig s Cotrol Chart 6 Hotellig s Square 8 Average Value of k Subgroups 0 Example 3 3 Value
More informationStatistical Intervals for a Single Sample
3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More informationClosed book and notes. No calculators. 60 minutes, but essentially unlimited time.
IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationModified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More informationLecture 24 Floods and flood frequency
Lecture 4 Floods ad flood frequecy Oe of the thigs we wat to kow most about rivers is what s the probability that a flood of size will happe this year? I 100 years? There are two ways to do this empirically,
More informationSOLUTIONS. 1. a) Let X and Y be random variables with. σ Y = 9. σ X 30 σ XY σ X 30 ρ σ X σ Y σ X + 24 σ XY
STAT 400 UIU Practice Problems #8 Stepaov Dalpiaz SOLUTIONS The followig are a umber of practice problems that may be helpful for completig the homework, ad will likely be very useful for studyig for exams..
More informationFLUID LIMIT FOR CUMULATIVE IDLE TIME IN MULTIPHASE QUEUES. Akademijos 4, LT-08663, Vilnius, LITHUANIA 1,2 Vilnius University
Iteratioal Joural of Pure ad Applied Mathematics Volume 95 No. 2 2014, 123-129 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v95i2.1
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More information