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# Class Note for EMSE 173 with Professor Dorp at GW

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Date Created: 02/07/15

EMSE 173273 EXAMPLES OUTPUT ANALYSIS Termination Simulations Steady State Simulations THE GEORGE WASHINGTON UNIVERSITY WASHINGTON DC Lecture Notes by J Ren van Dorp1 wwwseasgwuedudorpjr 1 Department of Engineering Management and Systems Egineering School of Engineering and Applied Science The George Washington University 1776 G Street NW Suite 110 Washington DC 20052 Email dorpjrgwuedu EXAMPLES OUTPUT ANALYSIS Terminating Consider five components X1 X5 such that 0 Component i is broken Xi 1 Component 2 works for i 1 5 These five components are used in a circuit or system to form what is called a quotBridge Structurequot 3 This bridge structure is in working condition when current can ow from the left to the right EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 2 EXAMPLES OUTPUT ANALYSIS Terminating We are interested in the random failure time of this bridge structure ie the probability that is works as a function of the random failure time of its components When the bridge structure fails it will have to be replaced We shall assume that components fail independently from one another When the failure times of the components follow an eXponential distribution this problem can in fact be solved Via mathematical analysis Let us suppose that the failure times of the components follow a Triang0 1000 5000 distribution with dimension hours In this case it is more convenient to resort to simulation to estimate the failure time distribution of this bridge structure this is true for most distributions other than the eXponential distribution EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 3 EXAMPLES OUTPUT ANALYSIS Terminating This BridgeStructure has a number of quotbridgesquot Via which current can ow Path 1 G Introducing the binary variabe P1 we define 0 Path 1 is broken P1 X1 X X4 1 Path 1 works EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 4 EXAMPLES OUTPUT ANALYSIS Terminating This BridgeStructure has a number of quotbridgesquot Via which current can ow Introducing the binary variabe P2 we define 0 Path 2 is broken P2 X2 X X5 1 Path 2works EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 5 EXAMPLES OUTPUT ANALYSIS Terminating This BridgeStructure has a number of quotbridgesquot Via which current can ow 00 9 Path 3 Introducing the binary variabe P3 we define 0 Path 3 is broken P3X1gtltX3 XXE 1 Path3works EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 6 EXAMPLES OUTPUT ANALYSIS Terminating This BridgeStructure has a number of quotbridgesquot Via which current can ow 0 6130 Path 4 Introducing the binary variabe P4 we define 0 Path 4 is broken P4X2 gtlt4X3 XX5 1 Path4works EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 7 EXAMPLES OUTPUT ANALYSIS Terminating Introducing the binary variabe S that describes system failure we have 4 0 System is broken ltgt 0 S i1 4 1 System functions ltgt Z 1 i1 We shall now develop a terminating simulation such that a single replication captures one failure time of this bridge structure The simulation terminates when all components have failed because no other discrete events can happen and thus this is a terminating simulation We shall use 1000 replications to observe failure times 1 1000 from which we can construct an empirical failure time distribution a confidence interval for the mean failure times and a confidence interval for the probability that this bridgestructure survives 2500 hours EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 8 EXAMPLES OUTPUT ANALYSIS Terminating Review the notes in the le quot9 Explanation Con dence Intervalpdfquot Nnmes The ARENA model above uses subimodels Each submodel simulates the failure of a particular component and its associated paths EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 9 EXAMPLES OUTPUT ANALYSIS Terminating le 0 Beam Process 0 Advanced Process 0 Reports I Navigate 7 amp TapLava Model Cumu Q Comp 1 I I a Comp 2 De zw 5 Comp 3 Q Comp 1 FFHCHMDTEHU 4 Comp 5 Pams I Name Rowsl Columnsl Clear Dptionl mmal Valuesl Report Sla s cs Name 1 1 v 7 System 1 rows Iquot lFaiI Cnmp1 and Pathsl L X2 7 Sysiim 4913 I X3 System 1 mws Assignments39 x4 39 System W F ValiableX1AU Add x5 Sys em wLJ l Variable P1Slategtlt1 gtlt4 p1 State Sysiem 1mwsl r Valltable P3Slate X1 X356 Editquot pzsme Sysxem 1 ruws ltEnd 9 quotSquot Pssme 39 39 System 1 rows Iquot P sme 39 7 System 1 raw NBndges System 1 rnws r LB 5 Sysiem 5 rows r ML 5 System Raw r 13 U3 5 System 5 mws l 14 Immas 39 System 0 rows f EMSE 173273 Spring 2006 JR van Dorp 030606 dorpjrgwuedu Page 10 EXAMPLES OUTPUT ANALYSIS Terminating We save the failure times of the bridgestucture to a text le quotBridgeFailTimebt Name mMmsnn m r shaman EJ 5112 m 912W men Fame ram 9 Type Am Me Name V wmemme v r121 v Egg l glval x 3 19 a w 39 A1 v a 1429 393755 A a 1 C 1 D Aswnmems 1 14m 384 Add A 2994 1352 mum Q i 1358 571 E91 L anus 511 5 12113945 3 E 125m 339 7 27231513 1 1947 m9 Ban22 Help 9 1139 455 19 1519275 Next we draw a empirical cd histogram in the same mariner as in the simple queue simulations that we conducted prior to the midterm exam EMSE 173273 Spring 2006 JR van Dorp 03l06l06 dorpjrgwuedu Page 11 EXAMPLES OUTPUT ANALYSIS Terminating Failure Time Distribution of Bridge Structure Fai39ure Time Dfs ribu i m 2f Bridge Struc ure 1000 Simulated Failure Times 100 S39quot quot Fem T39mes 74426 2 000 100000 200000 300000 400000 500000 Failure Time in Hours 112879 189785 39 Probability 6 g 359 74 1 1 1 1 1 1 1 3 151332 1 1 39 l 1 1 1 1 1 1 1 1 1 I l 1 228238 ilure Time in Hours From this sample we can also construct a confidence interval for the mean time to failure E 1921037 hours 8 736999 n 1000 tn11a2 1962 E tn 11 a2 X 3 2 tn 11 a2 X 8 x5 xE m 1875334 1960744 EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 12 EXANII LES OUTPUT ANALYSIS Terminating 95v cremmiwy imemi of System mime Time Prob FaihireTime aner mm 25m 7229 upper um 975m mus Aver e m FaimreTimes aver 1mm Repiimmns 921 an swam Devialmn oi Fnimre Times over mm Repiimmns m Cnn39idence imemi m Mean mmnme Dver iano Raphcalmns lB 137533 Mum HE is 1 674n Snmmesue innn Haivwmm 5 m Sludenll quinine 1 532 I Suppose the mission time for the bridge structure is 2500 hours Given the failure times of 1000 replications we can create 1000 outcomes ofa binary Variable describing whether the bridge structure passed its mission time or not 1 indicating passing 0 indicating not passing I From this sample of 1 000 binary Values we may estimate Hmission time reliability and a con dence interval for Hmission time rehabihty in the same fashion EMSE173273 7 Spring 2006 J R van Domrososos domirgwu edu Page is EXAMPLES OUTPUT ANALYSIS Terminating 7A 1 727 I a 2 L 2 5 i 5 7 5 i 7 9 a mmr 1421 35 was Imam ms 51 12169 175w 272316 was WssmnTune zsau Naurs u u a nzux u an nuns u snsc mm u 7w mam Average m lmssmn Success aver 1mm Ephulmns mu 759 Cnn ence L5 as Hmwmth n 9 ms n n26 smmam nemmn nl mssmn Success at mo Repnmmns n m mum 53mm Sue 5mm 1 mm Emmncnr cur E 263 an 341 25 3 21 m 49 36 57 4mm 95 99 wens mienm m mssmnnme nemmy over mo Raphulmns or a FawureTm n as nun 1 962 EMSE173273 r Spnng mus J R van DurprEIBEIEEIE durpjrgwu Edu Page 14 EXAMPLES OUTPUT ANALYSIS Homework 1 Consider the following system of components Let us suppose that the failure times of the components follow a Triang0 1000 5000 distribution with dimension hours m 0 6 C3 9 m 7 a Develop an ARENA simulation that simulates the failure times of the system above and generate a le with 1000 failure times Hint Modify the file BridgeStruCtureNoMaintenancedoe EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 15 EXAMPLES OUTPUT ANALYSIS Homework 1 b Estimate the failure time distribution of the system the average failure time and mission time reliability at a mission time of 2500 hours Modify the spreadsheet quotSystemReliabilityXlsquot where necessary c Develop confidence intervals for the average mean time to failure and the mission time reliability at a mission time of 2500 hours EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 16 EXAMPLES OUTPUT ANALYSIS SteadyState Reconsider the five components X1 X 5 such that Xi 0 Component i is broken 1 Component i works for i 1 5 These five components are used in a circuit or system to form what is called a quotBridge Structurequot 3 This bridge structure is in working condition when current can ow from the left to the right EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 17 EXAMPLES OUTPUT ANALYSIS SteadyState We shall change the set up slightly We shall assume that these components are maintained by a single server called quotfixerquot When a component fails it is repaired or replaced with a random repair time in hours that follows a Triang250500750 distribution We are now interested in the long term availability of this quotbridge structurequot given the component failure times and the repair maintenance times of the single fixer Availability can be interpreted as the probability that the bridgestructure is available at any given time At T 1 At 1 System is available at time t 0 System is unavailable at time t EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 18 EXAMPLES OUTPUT ANALYSIS SteadyState PrltSystem is available le39m T H 00 Hence we need a nonrterrninating simulation or steadyistate simulation to estimate the availability PrltSystem is available System Availabilllv tillzation xer El 0 a Q 0 Path mummy HEtogrzm of Path Availablllty c m cm circzrcs Charm 2 an M 7m Four 5 o n m o m a m 3 MU NH NH M m EMSE 173273 Spring 2006 JR van Dorp 03l06l06 dorpjrgwuedu Page 19 EXAMPLES OUTPUT ANALYSIS SteadyState The ARENA simulation above uses again subimodels ErmgeZ CumpZ Resmred nd Update Paths dy 0 Basic Process 0 Advanced Frucess 6 Report I Nawgate j TopLevelModel y Camp 1 5 Comp 2 y Comm 3 5 Comp 4 5 Comm 5 Fa ure 2 A 1 Resture 2 We save the systernistate Via ARENA in a le HSysAV211121bi1ityc121tquot SHinE dva nqed Flute u J Name Type M Report Label nutpm me 1 NPaths Frequency 2 Reliabvmy Frequency v 3 System Ava abxllty TunePersistenk Sys1em Availability SysAvailabmyjat EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 20 EXAMPLES OUTPUT ANALYSIS SteadyState I We convert the HSysAVailahilitydatH to a text le using the output analyzer The end of replications are indicated by negative numbers 1LM1 i 717 77L 3 EN 9 13721 403905 1001900 Unnamed 0me E1 4991305 0 00300 mm 70313010 13741 0001500 000E000 000E000 912E402 100E400 U 124E403 0 quaun 1375 U 00E00 203E403 100E400 137 131E03 100E00 23333 33333 1370 5 00903 0 00500 3 3 7 0 075303 0 005300 1375 57775403 1 005100 12 705E303 1005300 1300 0251303 0001300 I Nexg We Write a Visual Basic Macro to HSeperate these replicationsH 7L A B L 1 D 1 E F 7767 1 H L 1 J 1 1 10200000011 1 Rephcahen 2 Rephcatmn 3 Rep0cam A Repheauun 5 i 0 0 0 u 0 0 0 0 0 0 3 10115002 1 1332 374 1 070 3040 1 1311015 1 2502000 1 7L1 1230 005 0 1031 642 0 1000 071 0 5052 012 0 3120 711 0 5 2032007 1 2301950 1 1090457 1 5705117 1 3034 522 1 70 4033 703 0 2304 552 0 2000 203 0 0251057 0 3002577 0 7 5007 322 1 202100 1 3330 73 1 0933210 1 4130 534 1 EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 21 EXAMPLES OUTPUT ANALYSIS SteadyState Sub SeparateReplications ClearReplicatlons i 5 RowCounter 2 ColCounter 2 Replication el 1 While sheetsquotAvailabilityquotCellsi lValue ltgt 75 If sheetsquotAvailabilityquotCellsi lValue Replication Then Rowcounter 2 Colcounter Colcounter 2 i i l Replication Replication e 1 End If SheetsquotAvailabilityMutlipleReplicationquotCellsRowCounter Colcounter 7 lValue sheetstquotAvailability gtCellsti lgtValue SheetsquotAvailabllityMutlipleReplicatlonquotCellSR0wCounter ColCounterValue sheets1quotAvailebilityquotCells1i 2Value i i l RowCounter Rowcounter l Wend CopyRepllcatlons End Sub EMSE 173273 Spring 2006 JR van Dorp 030606 dorpjrgwuedu Page 22 EXAMPLES OUTPUT ANALYSIS SteadyState Next we calculate the area under the At curve upto time t and estimate the availability upto that time by dividing this area by t A B H C 7 W W Dquot H Hr E 1 1 System State AREA CUMULATIVE AREA Al t 2 0 CI EIDUEHZIU UUUECIU 000 3 9115113020 1A4A383 32TE3U2 2641 4 123850481 El UEIOEEIU 32TEEI2 1242 5 263296 1 221EE3 253EU 5235 Given that we choose the initial states of the components we introduce an quotinitialization biasquot in the assessment of the availability Hence we would like to determine a warm up time Twarmup beyond which one can reasonably say that the quotinitialization biasquot is not present anymore Finally one estimates the availability by selecting a time T quotlarge enoughquot and a warm up time Twarmup and set EM PrSystem is available W warmup EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 23 EXAMPLES OUTPUT ANALYSIS SteadyState 0 STEP 1 Find a warmup time Twarmup beyond which one can reasonably say that initialization bias has been removed 090 1 1 1 1 1 1 1 1 1 1 1 1 7 7777 7777 1 7777 quot1 7777 1 1 1 1 1 1 1 1 1 1 1 1 7 7 7 7 77P77777777777777777777777777777777777777 1 1 1 1 1 1 g 1 1 1 1 1 1 5 7777 7777 1 7777 7777 1 7777 1 77777 1 7777 3 1 1 1 1 1 1 g 7777 7777 1 7777 7777 1 7777 1 77777 1 7777 lt 1 1 1 1 1 1 030 777777777 7777 1 7777 7777 1 7777 1 77777 1 7777 1 1 1 1 1 1 020 777777777 7777 1 7777 7777 1 7777 1 rrrrr 1 7777 1 1 1 1 1 1 110 1 1 1 1 1 1 l l l l l l 000 D D D D D D D D D D D D D D D 8 8 8 8 8 8 8 P E R 3 3 3 9 Time Transient Phase SteadyState Phase The figure above show the behavior of system availability in a particular replication as a function of time It shows a Transient Phase at which initialization bias is still present and a SteadyState phase Without apparent initialization bias Does this gure change for different replications EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 24 EXAMPLES OUTPUT ANALYSIS SteadyState From the previous figure above it appears we can set Twarmup 50000 hours for this replication Would this be true for other replications as well Bridge Structure Availability 50000 4 4 4 150000 4 4 4 200000 4 4 4 250000 4 4 4 4 4 4 300000 4 4 4 Ti me Replication 1 Replica on 2 Repication 3 Repication 4 Repication 5 Transient Phase SteadyState Phase A From the figure above including 4 additional replications it seems more prudent to set Tmmup 100 000 hours EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 25 EXAMPLES OUTPUT ANALYSIS SteadyState The analysis above may not be computationally efficient In fact at Twarmup 100 000 hours the simulation may have reached its steady state for quite some time since the availability already stabalized For a graphical procedure to determine a more computationally efficient warmup threshold Twarmup see Welch PD 1983 The Statistical Analysis of Simulation Results in The Computer Performance Modeling Handbook S Lavenberg Ed Academic Press New York NY pp 268328 or more recently Alexopoulos C and Seila AF 1998 Output Data Analysis in Handbook of Simulations Chapter 7 Jerry Banks Ed John Wiley and Sons New York NY pp 225272 EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 26 EXAMPLES OUTPUT ANALYSIS SteadyState 0 STEP 2 Find a simulation time T such that the accuracy in is suf cient Jim T 7 T WWW P System is available In a steadyestate output analysis the simulation time T is typically chosen such that the halfewidth of an output statistic of interest is suf ciently small after removing initialization bias 3 47 mm W User Speci ed March ai 2005 Unnamed Project Repiicatiuris 3 Replication 1 start Time 900000 00 stop Time 500 000 00 Time uriits Hours Time Persistent Time Persistent Average HailWidm Minimum Maximum system Avaiiabiiity 0 013i 0 020631860 0 i 0000 EMSE 173273 Spring 2006 JR van Dorp03l06l06 dorpjrgwuedu Page 27 EXAMPLES OUTPUT ANALYSIS SteadyState 3 47 4900 User Speci ed March 31 2005 Unnamed Project Rephcatmns 3 Replication 2 Start Trrrre 100 000 00 Stop Trrrre 500 000 00 Trrrre umts Hours Time Persistent Trrrre Persrstem Average Hawwmm Mrmrrrurrr Maxrrrrurrr System Avauanmw 0 7922 0 030751430 0 1 0000 3 5205er User Speci ed Apni 3 2005 Unnamed Project Rephcatmns 5 I Replication 3 Start Trrrre 100000 00 Stop Trrrre 500000 00 Trrrre 0005 Hours Time Persistent Trrrre Persrstem Average Hawwmm Mrmmum Maxrmurrr System Avauammy 0 0043 0 027934093 0 1 0000 EMSE 173273 Spring 2006 JR van Dorp 03l06l06 dorpjrgwuedu Page 28 EXAMPLES OUTPUT ANALYSIS SteadyState 3 52 05PM User Speci ed Apni 3 2005 Unnamed Project Repiications 5 Replication 4 Start Time 100 000 00 Stop Time 500000 00 Time units Hours Time Persistent Time Persistent Average HairWimn Minimum Maximum System Avaiianiiiuy 0 7735 0 039200530 0 1 0000 3 52 05PM User Speci ed Apni 3 2005 Unnamed Project Repiications 5 Replication 5 Start Time 100 000 00 Stop Time 500 000 00 Time units Hours Time Persistent Time Persistent Average HairWimn Minimum Maximum System Avaiiabiiiw 0 0042 0 029452020 0 i 0000 EMSE 173273 Spring 2006 JR van Dorp 03l06l06 dorpjrgwuedu Page 29 EXAMPLES OUTPUT ANALYSIS SteadyState I The mlmmurn halfevndth observed over these ve repllcattohs equals 0020 The mzxn39num halfevndth observed over these ve IepllCathl equals 0039 Ah overvlew ofthese ve rephcauohs 1s provlded m ARENA 1n the CATEGORY OVERVIEW REPORT 15244y Calegol y Oven ie WW am Unnamed Prnjen 5 Mam my User SD ed I Time Persislem HalfWuith by ARENA reported only 002 lt5USP1C10USgtgt EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 30 EXAMPLES OUTPUT ANALYSIS SteadyState If the halfwidth is not small enough one would increase the simulation length Tgtllt EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 31 EXAMPLES OUTPUT ANALYSIS Homework 2 Consider the following system of components Let us suppose that the failure times of the components follow a Triang0 1000 5000 distribution with dimension hours We shall assume that these components are maintained by a single server called quot xerquot When a component fails it is repaired or replaced with a random repair time in hours that follows a Triang250500750 distribution Q 00 V G 09 a Develop an ARENA simulation that simulates the operation of the system above by modifying the simulation quotBridgeStructureMaintenancedoequot Set the initial state of each component to its failed state EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 32 EXAMPLES OUTPUT ANALYSIS Homework 2 b Generate 5 replications to estimate a warm up period to estimate system availability Next use the output analyzer to export the file quotSysAvailabilitydatquot to a text file quotSysAvailabilitytxtquot Finally modify the spreadsheet quotAvailabilityMutlipleReplicationsxlsquot where necessary and observe a reasonable warm up period c Develop a con dence interval for system availability using ARENA from 5 replications by using the warm up period determined under b and running the simulation for 1000000 hours EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 33 EMSE 173273 Spring 2006 JR van Dorp030606 dorpjrgwuedu Page 34

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