Package Reliability. R topics documented: February 19, Version Date

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1 Version Dae Package Reliabiliy February 19, 2015 Tile Funcions for esimaing parameers in sofware reliabiliy models Auhor Andreas Wimann Mainainer Andreas Wimann Depends R (>= 2.4.0) Funcions for esimaing parameers in sofware reliabiliy models. Only infinie failure models are implemened so far. License Unlimied NeedsCompilaion no Reposiory CRAN Dae/Publicaion :39:40 R opics documened: duane duane.plo lilewood.verall lilewood.verall.plo moranda.geomeric moranda.geomeric.plo musa.okumoo musa.okumoo.plo mvf.duane mvf.mor mvf.musa mvf.ver.lin mvf.ver.quad rel.plo oal.plo Index 27 1

2 2 duane duane Maximum Likelihood esimaion of mean value funcion for Duane model duane compues he Maximum Likelihood esimaes for he parameers rho and hea of he mean value funcion for he Duane model. duane(, ini = c(1, 1), mehod = "Nelder-Mead", maxi = 10000,...) ini Deails mehod maxi ime beween failure daa iniial values for Maximum Likelihood fi of he mean value funcion for he Duane model. he mehod o be used for opimizaion, see opim for deails. he maximum number of ieraions, see opim for deails.... conrol parameers and plo parameers opionally passed o he opimizaion and/or plo funcion. Parameers for he opimizaion funcion are passed o componens of he conrol argumen of opim. This funcion esimaes he parameers rho and hea of he mean value funcion for he Duane model. Wih Maximum Likelihood esimaion one ges he following equaions, which have o be minimized. This is equaion 1 := ρ n θ = 0 n and n equaion 2 := θ n 1 i=1 (log( n/ i )) = 0. Where is he ime beween failure daa and n is he lengh or in oher words he size of he ime beween failure daa. So he simulaneous minimizaion of hese equaions happens by minimizaion of he equaion equaion equaion 2 2 = 0. A lis conaining following componens: rho hea Maximum Likelihood esimae for rho Maximum Likelihood esimae for hea

3 duane.plo 3 duane.plo, mvf.duane # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, duane() duane.plo Ploing he mean value funcion for he Duane model duane.plo plos he mean value funcion for he Duane model and he raw daa ino one window. duane.plo(rho, hea,, xlab = "ime", ylab = "Cumulaed failures and esimaed mean value funcion", main = NULL)

4 4 duane.plo rho hea xlab ylab main parameer value for rho parameer value for hea ime beween failure daa a ile for he x axis a ile for he y axis an overall ile for he plo Deails This funcion gives a plo of he mean value funcion for he Duane model. Here he esimaed parameer values for rho and hea, which are obained by using duane, can be pu in. Inernally he funcion mvf.duane is used o ge he mean value funcion for he Duane model. A graph of he mean value funcion for he Duane model and of he raw daa. duane, mvf.duane # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24,

5 lilewood.verall 5 rho <- duane()$rho hea <- duane()$hea duane.plo(rho, hea,, xlab = "ime (in seconds)", main = "Duane model") lilewood.verall Maximum Likelihood esimaion of mean value funcion for Lilewood-Verall model lilewood.verall compues he Maximum Likelihood esimaes for he parameers hea0, hea1 and rho of he mean value funcion for he Lilewood-Verall model. lilewood.verall(, linear = T, ini = c(1, 1, 1), mehod = "Nelder-Mead", maxi = 10000,...) linear ini mehod maxi ime beween failure daa logical. Should he linear or he quadraic form of he mean value funcion for he Lilewood-Verrall model be used of compuaion? If TRUE, which is he defaul, he linear form of he mean value funcion is used. iniial values for Maximum Likelihood fi of he mean value funcion for he Lilewood-Verall model. he mehod o be used for opimizaion, see opim for deails. he maximum number of ieraions, see opim for deails.... conrol parameers and plo parameers opionally passed o he opimizaion and/or plo funcion. Parameers for he opimizaion funcion are passed o componens of he conrol argumen of opim. Deails This funcion esimaes he parameers hea0, hea1 and rho of he mean value funcion in he linear or he quadraic form for he Lilewood-Verall model. Firs, he compuaion wih he mean value funcion in he linear form is explained. Wih Maximum Likelihood esimaion one ges he following equaions, which have o be minimized. This is equaion 1 := n n ρ + log(θ 0 + θ 1 i) i=1 n log(θ 0 + θ 1 i + i ) = 0, i=1

6 6 lilewood.verall and equaion 2 := ρ equaion 3 := ρ n i=1 n i=1 1 n θ 0 + θ 1 i ρ = 0 θ 0 + θ 1 i + i i=1 i n θ 0 + θ 1 i ρ + 1 i = 0. θ 0 + θ 1 i + i i=1 Second, he compuaion wih he mean value funcion in he quadraic form is explained. Wih Maximum Likelihood esimaion one ges he following equaions, which have o be minimized. This is equaion 1 := n n ρ + n log(θ 0 + θ 1 i 2 ) log(θ 0 + θ 1 i 2 + i ) = 0, and equaion 2 := ρ equaion 3 := ρ i=1 n i=1 n i=1 i=1 n 1 θ 0 + θ 1 i 2 ρ θ 0 + θ 1 i 2 = 0 + i i=1 i 2 n θ 0 + θ 1 i 2 ρ + 1 i 2 θ 0 + θ 1 i 2 = 0. + i Where is he ime beween failure daa and n is he lengh or in oher words he size of he ime beween failure daa. So he simulaneous minimizaion of hese equaions happens by minimizaion of he equaion equaion equaion equaion 2 3 = 0. i=1 A lis conaining following componens: hea0 hea1 rho Maximum Likelihood esimae for hea0 Maximum Likelihood esimae for hea1 Maximum Likelihood esimae for rho lilewood.verall.plo, mvf.ver.lin, mvf.ver.quad

7 lilewood.verall.plo 7 # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, lilewood.verall(, linear = TRUE) lilewood.verall(, linear = FALSE) lilewood.verall.plo Ploing he mean value funcion for he Lilewood-Verall model lilewood.verall.plo plos he mean value funcion for he Lilewood-Verall model and he raw daa ino one window. lilewood.verall.plo(hea0, hea1, rho,, linear = T, xlab = "ime", ylab = "Cumulaed failures and esimaed mean value funcion", main = NULL) hea0 hea1 rho linear xlab ylab main parameer value for hea0 parameer value for hea1 parameer value for rho ime beween failure daa logical. Should he linear or he quadraic form of he mean value funcion for he Lilewood-Verrall model be used of compuaion? If TRUE, which is he defaul, he linear form of he mean value funcion is used. a ile for he x axis a ile for he y axis an overall ile for he plo

8 8 lilewood.verall.plo Deails This funcion gives a plo of he mean value funcion for he Lilewood-Verall model. Here he esimaed parameer values for hea0, hea1 and hea, which are obained by using lilewood.verall, can be pu in. Inernally he funcions mvf.ver.lin or mvf.ver.quad are used o ge he mean value funcion for he Lilewood-Verall model. This depends on he calibraion, if he linear or he quadraic form of he mean value funcion for he Lilewood-Verall model should be used. A graph of he mean value funcion for he Lilewood-Verall model and of he raw daa. lilewood.verall, mvf.ver.lin, mvf.ver.quad # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, hea0 <- lilewood.verall(, linear = TRUE)$hea0 hea1 <- lilewood.verall(, linear = TRUE)$hea1 rho <- lilewood.verall(, linear = TRUE)$rho lilewood.verall.plo(hea0, hea1, rho,, linear = TRUE, xlab = "ime (in seconds)", main = "Lilewood-Verall model (linear)") ## No run: ## hea0 <- lilewood.verall(, linear = FALSE)$hea0

9 moranda.geomeric 9 ## hea1 <- lilewood.verall(, linear = FALSE)$hea1 ## rho <- lilewood.verall(, linear = FALSE)$rho ## lilewood.verall.plo(hea0, hea1, rho,, linear = FALSE, ## xlab = "ime (in seconds)", main = "Lilewood-Verall modell (quadraic)") ## End(No run) moranda.geomeric Maximum Likelihood esimaion of mean value funcion for Moranda- Geomeric model moranda.geomeric compues he Maximum Likelihood esimaes for he parameers D and hea of he mean value funcion for he Moranda-Geomeric model. moranda.geomeric(, ini = c(0, 1), ol =.Machine$double.eps^0.25) ini ol ime beween failure daa iniial values for Maximum Likelihood fi of he mean value funcion for he Moranda-Geomeric model. he desired accuracy Deails This funcion esimaes he parameers D and hea of he mean value funcion for he Moranda- Geomeric model. Wih Maximum Likelihood esimaion one ges he following equaion, which have o be minimized, o ge phi. This is n i=1 iφi i n n + 1 = 0. i=1 φi i 2 The soluion of hese is hen pu in in he following equaion in order o ge D D = φn n i=1 φi i. Where is he ime beween failure daa and n is he lengh or in oher words he size of he ime beween failure daa. A lis conaining following componens: rho hea Maximum Likelihood esimae for rho Maximum Likelihood esimae for hea

10 10 moranda.geomeric.plo moranda.geomeric.plo, mvf.mor # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, moranda.geomeric() moranda.geomeric.plo Ploing he mean value funcion for he Moranda-Geomeric model moranda.geomeric.plo plos he mean value funcion for he Moranda-Geomeric model and he raw daa ino one window. moranda.geomeric.plo(d, hea,, xlab = "ime", ylab = "Cumulaed failures and esimaed mean value funcion", main = NULL)

11 moranda.geomeric.plo 11 D hea xlab ylab main parameer value for D parameer value for hea ime beween failure daa a ile for he x axis a ile for he y axis an overall ile for he plo Deails This funcion gives a plo of he mean value funcion for he Moranda-Geomeric model. Here he esimaed values for D and hea, which are obained by using moranda.geomeric, can be pu in. Inernally he funcion mvf.mor is used o ge he mean value funcion for he Moranda-Geomeric model. A graph of he mean value funcion for he Moranda-Geomeric model and of he raw daa. moranda.geomeric, mvf.mor # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24,

12 12 musa.okumoo D <- moranda.geomeric()$d hea <- moranda.geomeric()$hea moranda.geomeric.plo(d, hea,, xlab = "ime (in seconds)", main = "Moranda-Geomeric model") musa.okumoo Maximum Likelihood esimaion of mean value funcion for Musa- Okumoo model musa.okumoo compues he Maximum Likelihood esimaes for he parameers hea0 and hea1 of he mean value funcion for he Musa-Okumoo model. musa.okumoo(, ini = c(0, 1), ol =.Machine$double.eps^0.25) ini ol ime beween failure daa iniial values for Maximum Likelihood fi of he mean value funcion for he Musa-Okumoo model. he desired accuracy Deails This funcion esimaes he parameers hea0 and hea1 of he mean value funcion for he Musa-Okumoo model. Wih Maximum Likelihood esimaion one ges he following equaion, which have o be minimized, o ge hea1. This is 1 θ 1 n i=1 1 n n 1 + θ 1 i (1 + θ 1 n ) log(1 + θ 1 n ) = 0. The soluion of hese is hen pu in in he following equaion in order o ge hea0 θ 0 = n log(1 + θ 1 n ). Where is he ime beween failure daa and n is he lengh or in oher words he size of he ime beween failure daa.

13 musa.okumoo.plo 13 A lis conaining following componens: hea0 hea1 Maximum Likelihood esimae for hea0 Maximum Likelihood esimae for hea1 musa.okumoo.plo, mvf.musa # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, musa.okumoo() musa.okumoo.plo Ploing he mean value funcion for he Musa-Okumoo model musa.okumoo.plo plos he esimaed mean value funcion for he Musa-Okumoo model and he raw daa ino one window.

14 14 musa.okumoo.plo musa.okumoo.plo(hea0, hea1,, xlab = "ime", ylab = "Cumulaed failures and esimaed mean value funcion", main = NULL) hea0 hea1 xlab ylab main parameer value for hea0 parameer value for hea1 ime beween failure daa a ile for he x axis a ile for he y axis an overall ile for he plo Deails This funcion gives a plo of he mean value funcion for he Musa-Okumoo model. Here he esimaed parameer values for hea0 and hea1, which are obained by using musa.okumoo, can be pu in. Inernally he funcion mvf.musa is used o ge he mean value funcion for he Musa-Okumoo model. A graph of he mean value funcion for he Musa-Okumoo model and of he raw daa. musa.okumoo, mvf.musa # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24,

15 mvf.duane 15 hea0 <- musa.okumoo()$hea0 hea1 <- musa.okumoo()$hea1 musa.okumoo.plo(hea0, hea1,, xlab = "ime (in seconds)", main = "Musa-Okumoo model") mvf.duane Mean value funcion for he Duane model mvf.duane reurns he mean value funcion for he Duane model. mvf.duane(rho, hea, ) rho hea parameer value for rho parameer value for hea ime beween failure daa Deails This funcion gives he values of he mean value funcion for he Duane model, his is wrien as µ() = ρ θ. Furher here is a verifying if he parameers rho and hea saisfy he assumpions for he Duane model. So he paramers rho and hea have o be larger han zero, in equaions ρ > 0 and θ > 0. The mean value funcion for he Duane model.

16 16 mvf.mor duane, duane.plo # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, duane.par1 <- duane()$rho duane.par2 <- duane()$hea mvf.duane(duane.par1, duane.par2, ) mvf.mor Mean value funcion for he Moranda-Geomeric model mvf.mor reurns he mean value funcion for he Moranda-Geomeric model. mvf.mor(d, hea, ) D hea parameer value for D parameer value for hea ime beween failure daa

17 mvf.mor 17 Deails This funcion gives he values of he mean value funcion for he Moranda-Geomeric model, his is wrien as µ() = 1 log{[dθ exp(θ)] + 1}. θ Furher here is a verifying if he parameer hea saisfy he assumpions of he Moranda-Geomeric model. So he paramer hea have o be larger han zero, in equaion θ > 0. The mean value funcion for he Moranda-Geomeric model. moranda.geomeric, moranda.geomeric.plo # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, mor.par1 <- moranda.geomeric()$d mor.par2 <- moranda.geomeric()$hea mvf.mor(mor.par1, mor.par2, )

18 18 mvf.musa mvf.musa Mean value funcion for he Musa-Okumoo model mvf.musa reurns he mean value funcion for he Musa-Okumoo model. mvf.musa(hea0, hea1, ) hea0 hea1 parameer value for hea0 parameer value for hea1 ime beween failure daa Deails This funcion gives he values of he mean value funcion for he Musa-Okumoo model, his is wrien as µ() = θ 0 log(θ 1 + 1). The mean value funcion for he Musa-Okumoo model. musa.okumoo, musa.okumoo.plo

19 mvf.ver.lin 19 # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, musa.par1 <- musa.okumoo()$hea0 musa.par2 <- musa.okumoo()$hea1 mvf.musa(musa.par1, musa.par2, ) mvf.ver.lin Mean value funcion in he linear form for he Lilewood-Verall model mvf.ver.lin reurns he mean value funcion in he linear form for he Lilewood-Verall model. mvf.ver.lin(hea0, hea1, rho, ) hea0 hea1 rho parameer value for hea0 parameer value for hea1 parameer value for rho ime beween failure daa Deails This funcion gives he values of he mean value funcion in he linear form for he Lilewood-Verall model, his is wrien as µ() = 1 θ 1 θ θ 1ρ. Furher here is a verifying if he parameer hea1 saisfy he assumpions for he Lilewood-Verall model. So he paramer hea1 should no be equal zero, in equaion θ 1 0.

20 20 mvf.ver.quad The mean value funcion in he linear form for he Lilewood-Verall model. lilewood.verall, lilewood.verall.plo, mvf.ver.quad # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, li.par1 <- lilewood.verall(, linear = TRUE)$hea0 li.par2 <- lilewood.verall(, linear = TRUE)$hea1 li.par3 <- lilewood.verall(, linear = TRUE)$rho mvf.ver.lin(li.par1, li.par2, li.par3, ) mvf.ver.quad Mean value funcion in he quadraic form for he Lilewood-Verall model mvf.ver.quad reurns mean value funcion in he quadraic form for he Lilewood-Verall model.

21 mvf.ver.quad 21 mvf.ver.quad(hea0, hea1, rho, ) hea0 hea1 rho parameer value for hea0 parameer value for hea1 parameer value for rho ime beween failure daa Deails This funcion gives he values of he mean value funcion in he quadraic form for he Lilewood- Verall model, his is wrien as µ() = 3v 1 (Q 1 + Q 2 ), where v 1 = (ρ 1)1/3 (18θ 1 ) 1/3, and v 2 = 4θ 3 0 9(ρ 1) 2 θ 1, Q 1 = [ + ( 2 + v 2 ) 1/2 ] 1/3 Q 2 = [ ( 2 + v 2 ) 1/2 ] 1/3. Furher here is a verifying if he parameer hea1 saisfy he assumpions for he Lilewood-Verall model. So he paramer hea1 should ne be equal zero, in equaion θ 1 0. The mean value funcion in he quadraic form for he Lilewood-Verall model. lilewood.verall, lilewood.verall.plo, mvf.ver.lin

22 22 rel.plo # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, li.par1 <- lilewood.verall(, linear = TRUE)$hea0 li.par2 <- lilewood.verall(, linear = TRUE)$hea1 li.par3 <- lilewood.verall(, linear = TRUE)$rho mvf.ver.quad(li.par1, li.par2, li.par3, ) rel.plo Ploing he relaive error for he mean value funcions for all models oal.plo plos he relaive error for he he mean value funcion for all models ino one window. rel.plo(duane.par1, duane.par2, li.par1, li.par2, li.par3, mor.par1, mor.par2, musa.par1, musa.par2,, linear = T, ymin, ymax, xlab = "ime", ylab = "relaive error", main = NULL) duane.par1 duane.par2 li.par1 li.par2 li.par3 mor.par1 mor.par2 musa.par1 musa.par2 parameer value for rho for Duane model parameer value for hea for Duane model parameer value for hea0 for Lilewood-Verall model parameer value for hea1 for Lilewood-Verall model parameer value for rho for Lilewood-Verall model parameer value for D for Moranda-Geomeric model parameer value for hea for Moranda-Geomeric model parameer value for hea0 for Musa-Okumoo model parameer value for hea1 for Musa-Okumoo model

23 rel.plo 23 linear ymin ymax xlab ylab main ime beween failure daa logical. Should he linear or he quadraic form of he mean value funcion for he Lilewood-Verrall model be used of compuaion? If TRUE, which is he defaul, he linear form of he mean value funcion is used. he minimal y limi of he plo he maximal y limi of he plo a ile for he x axis a ile for he y axis an overall ile for he plo Deails This funcion gives a plo of he relaive error for he mean value funcions for all models, his is relaive error = µ( i) i, i = 1, 2,..., i where µ() is a mean value funcion and i is he number of failures. Here he esimaed parameer values, which are obained by using duane, lilewood.verall, moranda.geomeric und musa.okumoo can be pu in. Inernally he funcions mvf.duane, mvf.ver.lin, mvf.ver.quad, mvf.mor and mvf.musa are used o ge he mean value funcions for all models. A graph of he relaive error for he mean value funcions for all models. duane.plo, lilewood.verall.plo, moranda.geomeric.plo, musa.okumoo.plo, oal.plo # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24,

24 24 oal.plo duane.par1 <- duane()$rho duane.par2 <- duane()$hea li.par1 <- lilewood.verall(, linear = TRUE)$hea0 li.par2 <- lilewood.verall(, linear = TRUE)$hea1 li.par3 <- lilewood.verall(, linear = TRUE)$rho mor.par1 <- moranda.geomeric()$d mor.par2 <- moranda.geomeric()$hea musa.par1 <- musa.okumoo()$hea0 musa.par2 <- musa.okumoo()$hea1 rel.plo(duane.par1, duane.par2, li.par1, li.par2, li.par3, mor.par1, mor.par2, musa.par1, musa.par2,, linear = TRUE, ymin = -1, ymax = 2.5, xlab = "ime (in seconds)", main = "relaive error") ## No run: ## rel.plo(duane.par1, duane.par2, li.par1, li.par2, li.par3, mor.par1, ## mor.par2, musa.par1, musa.par2,, linear = TRUE, ## xlab = "ime (in seconds)", main = "relaive error") ## End(No run) oal.plo Ploing he mean value funcions for all models oal.plo plos he mean value funcion for all models and he raw daa ino one window. oal.plo(duane.par1, duane.par2, li.par1, li.par2, li.par3, mor.par1, mor.par2, musa.par1, musa.par2,, linear = T, xlab = "ime", ylab = "Cumulaed failures and esimaed mean value funcions", main = NULL) duane.par1 parameer value for rho for Duane model

25 oal.plo 25 duane.par2 li.par1 li.par2 li.par3 mor.par1 mor.par2 musa.par1 musa.par2 linear xlab ylab main parameer value for hea for Duane model parameer value for hea0 for Lilewood-Verall model parameer value for hea1 for Lilewood-Verall model parameer value for rho for Lilewood-Verall model parameer value for D for Moranda-Geomeric model parameer value for hea for Moranda-Geomeric model parameer value for hea0 for Musa-Okumoo model parameer value for hea1 for Musa-Okumoo model ime beween failure daa logical. Should he linear or he quadraic form of he mean value funcion for he Lilewood-Verrall model be used of compuaion? If TRUE, which is he defaul, he linear form of he mean value funcion is used. a ile for he x axis a ile for he y axis an overall ile for he plo Deails This funcion gives a plo of he mean value funcions for all models. Here he esimaed parameer values, which are obained by using duane, lilewood.verall, moranda.geomeric und musa.okumoo can be pu in. Inernally he funcions mvf.duane, mvf.ver.lin, mvf.ver.quad, mvf.mor and mvf.musa are used o ge he mean value funcions for all models. A graph of he mean value funcions for all models and of he raw daa. duane.plo, lilewood.verall.plo, moranda.geomeric.plo, musa.okumoo.plo

26 26 oal.plo # ime beween-failure-daa from DACS Sofware Reliabiliy Daase # homepage, see sysem code 1. Number of failures is 136. <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, duane.par1 <- duane()$rho duane.par2 <- duane()$hea li.par1 <- lilewood.verall(, linear = TRUE)$hea0 li.par2 <- lilewood.verall(, linear = TRUE)$hea1 li.par3 <- lilewood.verall(, linear = TRUE)$rho mor.par1 <- moranda.geomeric()$d mor.par2 <- moranda.geomeric()$hea musa.par1 <- musa.okumoo()$hea0 musa.par2 <- musa.okumoo()$hea1 oal.plo(duane.par1, duane.par2, li.par1, li.par2, li.par3, mor.par1, mor.par2, musa.par1, musa.par2,, linear = TRUE, xlab = "ime (in seconds)", main = "all models")

27 Index Topic models duane, 2 duane.plo, 3 lilewood.verall, 5 lilewood.verall.plo, 7 moranda.geomeric, 9 moranda.geomeric.plo, 10 musa.okumoo, 12 musa.okumoo.plo, 13 mvf.duane, 15 mvf.mor, 16 mvf.musa, 18 mvf.ver.lin, 19 mvf.ver.quad, 20 rel.plo, 22 oal.plo, 24 duane, 2, 4, 16, 23, 25 duane.plo, 3, 3, 16, 23, 25 lilewood.verall, 5, 8, 20, 21, 23, 25 lilewood.verall.plo, 6, 7, 20, 21, 23, 25 moranda.geomeric, 9, 11, 17, 23, 25 moranda.geomeric.plo, 10, 10, 17, 23, 25 musa.okumoo, 12, 14, 18, 23, 25 musa.okumoo.plo, 13, 13, 18, 23, 25 mvf.duane, 3, 4, 15, 23, 25 mvf.mor, 10, 11, 16, 23, 25 mvf.musa, 13, 14, 18, 23, 25 mvf.ver.lin, 6, 8, 19, 21, 23, 25 mvf.ver.quad, 6, 8, 20, 20, 23, 25 opim, 2, 5 rel.plo, 22 oal.plo, 23, 24 27