Reliability growth via testing Lawrence M. Leemis a a

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1 Ths arcle was downloaded by: [College of Wllam & Mary] n: 9 March 200 Access deals: Access Deals: [subscrpon number 978] Publsher Taylor & Francs Informa Ld Regsered n England and Wales Regsered Number: Regsered offce: Mormer House, 7-4 Mormer Sree, London WT JH, UK IIE Transacons Publcaon deals, ncludng nsrucons for auhors and subscrpon nformaon: hp:// Relably growh va esng Lawrence M. Leems a a The College of Wllam & Mary, Wllamsburg, VA, USA nlne publcaon dae: 02 February 200 To ce hs Arcle Leems, Lawrence M.(200) 'Relably growh va esng', IIE Transacons, 42: 4, 7 24 To lnk o hs Arcle: DI: 0.080/ URL: hp://dx.do.org/0.080/ PLEASE SCRLL DWN FR ARTICLE Full erms and condons of use: hp:// Ths arcle may be used for research, eachng and prvae sudy purposes. Any subsanal or sysemac reproducon, re-dsrbuon, re-sellng, loan or sub-lcensng, sysemac supply or dsrbuon n any form o anyone s expressly forbdden. The publsher does no gve any warrany express or mpled or make any represenaon ha he conens wll be complee or accurae or up o dae. The accuracy of any nsrucons, formulae and drug doses should be ndependenly verfed wh prmary sources. The publsher shall no be lable for any loss, acons, clams, proceedngs, demand or coss or damages whasoever or howsoever caused arsng drecly or ndrecly n connecon wh or arsng ou of he use of hs maeral.

2 IIE Transacons (200) 42, 7 24 Copyrgh C IIE ISSN: prn / onlne DI: 0.080/ Relably growh va esng LAWRENCE M. LEEMIS The College of Wllam & Mary, Wllamsburg, VA 287, USA E-mal: leems@mah.wm.edu Receved Aprl 2008 and acceped January 2009 Downloaded By: [College of Wllam & Mary] A: 2:0 9 March 200 bserved daa values are ypcally assumed o come from an nfne populaon of ems n relably and survval analyss applcaons. The case of a fne populaon of ems wh exponenally dsrbued lfemes s consdered here. The daa se consss of he lfemes of a large number of ems ha are known o have exponenally dsrbued falure mes wh a falure rae ha s known wh hgh precson. Falure of he ems s no self-announcng, as s he case wh a smoke deecor. A sgnfcan fracon of he ems are sampled perodcally, and he ems ha have faled are repared o a lke-new condon wh respec o her survval dsrbuon. The goal s o assess he mpac of hs perodc samplng and repar on he overall fne populaon relably over me. Keywords: Exponenal dsrbuon, fne populaon, relably esmaon, survvor funcon. Inroducon Consder a fne populaon of n dencal reparable ems ha operae connuously unl falure. We assume ha each em has an exponenal (λ) me o falure, where he falure rae λ>0 s assumed o be a known fxed consan. Ths assumpon s based on a long hsory of esng and he fac ha he ems conss manly of elecrcal componens arranged n seres. If here s no nervenon (e.g., perodc esng and repar) n he fne populaon of ems, he populaon survvor funcon S() s (Ross, 2007, p. 282): S() = e λ, > 0. Assume ha falure can only be deeced by esng. A ordered mes, 2,..., k, selec n, n 2,...,n k ems a random and whou replacemen a each samplng me from he fne populaon, and observe Y, Y 2,...,Y k of hese ems ha pass he es. Tesng s nsananeous. Any em ha fals a es undergoes a perfec repar wh no me delay, and s mmedaely reurned o he populaon wh hose ha pass he es. We consder he esmaon of he populaon survvor funcon a me under hs perodc samplng scheme by presenng a seres of models of ncreasng complexy by progressvely relaxng condons n order o arrve a a general model for survvor funcon esmaon for fne populaons. 2. Models The goal of each of he four models presened n hs secon s o gve he survvor funcon for a fne populaon of ems, each wh exponenal (λ) mes o falure. The frs model consders a sngle (k = ) samplng me. The second model consders wo (k = 2) samplng mes. The hrd model consders he general case of k samplng mes. Fnally, he fourh model consders he case of a varyng populaon sze. 2.. Sngle samplng me Assume ha here s only a sngle (k = ) samplng me. The number of passng ems Y (ou of he n ems sampled a me ) has he hypergeomerc dsrbuon wh condonal probably mass funcon: ( g )( n g ) y n f Y G =g (y G = g ) = y ( n ), y = 0,, 2,...,n. n Ths probably mass funcon s condoned on G = g good ems and n g bad ems n he populaon a me. Noe ha ( ) r = 0, s by convenon, whenever s < 0ors > r, whch allows he suppor of he random varable Y o be wren n he smple fashon presened above. The random varable G has he bnomal dsrbuon wh parameers n and e λ, and probably mass funcon: ( ) n f G (g) = (e λ ) g ( e λ ) n g, g = 0,, 2,...,n. g Hence, he number of ems Y ha pass he es a me has a bnomal hypergeomerc dsrbuon wh C 200 IIE

3 8 Leems uncondonal probably mass funcon: n f Y (y) = f Y G =g(y G = g) f G (g) g=0 ( n g n g )( ) y)( n = y n ( n ) (e λ ) g ( e λ ) n g, g g=0 n y = 0,, 2,...,n S () Downloaded By: [College of Wllam & Mary] A: 2:0 9 March 200 Example. Assume ha: () he sngle samplng me occurs a me = ; () he populaon falure rae s λ = ln(2/); () here are n = 5 ems n he populaon; and (v) here are n = ems sampled a me =. In hs case, he condonal probably mass funcon of Y s f Y G =g (y G = g ) = ( g y )( 5 g y ) ( 5 ), y = 0,, 2,, for g = 0,, 2,, 4, 5. The probably mass funcon of G s ( )( ) 5 2 g ( ) 5 g f G (g) =, g = 0,, 2,, 4, 5. g The uncondonal probably mass funcon of he number of passng ems Y s ( 5 g 5 g )( )( ) y)( y 5 2 g ( ) 5 g f Y (y) = ( 5, y = 0,, 2,, ) g g=0 whch smplfes o /27 for y = 0, 2/9 for y =, f Y (y) = 4/9 for y = 2, 8/27 for y =. Now consder he effec of he samplng, esng, and possble repars ha occur a me on he populaon survvor funcon a me, whch s denoed by S(). Pror o me, he survvor funcon s S() = e λ, 0 <. Afer he es ha occurs a me, he populaon s a fne mxure of he esed group of n ems and he unesed group of n n ems, so he populaon survvor funcon s S() = n n n e λ + n n e λ( ), >. The ( ) erm accouns for he renewal of he n ems ha were esed a me. Those ems ha were found o be faled are assumed o be repared o a good-as-new condon; hose ems ha were found o be operang are Fg.. Survvor funcon for k = esng me =. as good as new because of he memoryless propery assocaed wh he exponenal dsrbuon. The esng resuls n a dsconnuous ncrease n S() a he esng me. Example 2. As before, le =, λ = ln(2/), n = 5, n =. The survvor funcon s e ln(2/) for 0 <, S() = 2 5 eln(2/) + 5 eln(2/)( ) for >, or ( ) 2 for 0 <, S() = ( 2 5 )( ) 2 + ( 5 )( ) 2 for >, whch s ploed n Fg.. The upward jump a me accouns for he benef (.e., relably growh) assocaed wh he esng ha occurs a me = Two samplng mes When here are n and n 2 ems sampled a mes and 2, he survvor funcon s dencal o he prevous case up o me 2, and s a mxure of he mxure for values exceedng 2 : e λ for 0 <, n n e λ + n n n e λ( ) for < 2, ( S() = n n 2 n n e λ + n ) n n n e λ( ) for > 2. + n 2 n e λ( 2)

4 Relably growh va esng S () used o defne S() for compacness. The survvor funcon S() assumes a pecewse form S (), for =, 2,..., k +, due o he esng ha occurs a mes, 2,..., k. For me values less han or equal o, he frs esng me, he survvor funcon s S () = e λ, 0 <. The survvor funcon on each subsequen me nerval s defned recursvely as Downloaded By: [College of Wllam & Mary] A: 2:0 9 March Fg. 2. Survvor funcon for k = 2 esng mes = and 2 = 2. Example. Le =, 2 = 2, n = 5, n =, n 2 =, λ = ln(2/). The survvor funcon e ln(2/) for 0 <, 2 5 eln(2/) + 5 eln(2/)( ) for < 2, ( S() = eln(2/) + ) 5 eln(2/)( ) for > 2, + 5 eln(2/)( 2) or ( ) 2 for 0 <, ( ) ( ) 2 for < 2, 5 5 S() = ( (2 )( ) 4 2 ( )( ) ) 2 + for > 2, ( ) s ploed n Fg. 2. Ths resul, and he resuls n Examples and 2, have been verfed by Mone Carlo smulaon. The upward jump a = accouns for he ncrease assocaed wh he esng of n = ems ha occurs a me = ; he smaller upward jump a 2 = 2 accouns for he ncrease assocaed wh he esng of jus n 2 = em ha occurs a me 2 = 2. The jump n S() a = brngs he survvor funcon n /n = /5 of he vercal dsance o one; he jump n S() a 2 = 2 brngs he survvor funcon n 2 /n = /5 of he vercal dsance o one. 2.. The general case: k samplng mes The general case (any posve neger k) s a smple exenson of he k = 2 case, alhough a recursve formula wll be S ()= n n n S () + n n e λ( ), <, for = 2,,..., k +, where k+. Even for moderae values of k, calculaon of he survvor funcon becomes unweldy snce here are k erms n he kh pecewse segmen. The followng resul, alluded o n he example wh k = 2 esng mes, wll be used o decrease he compuaonal me requred o calculae S(). Resul. For he pecewse survvor funcon S() defned above: S( + ) S( ) = n S( ) n, =, 2,...,k, where + s a me value ha s an nfnesmal amoun larger han. The proof of a generalzaon of hs resul (Resul 2) s gven n he Appendx. The nuon assocaed wh hs resul s ha he hegh of he dsconnuy n he survvor funcon a me s a recovery fracon n /n oward one, for =, 2,..., k. In he exreme case of n = n, he survvor funcon reses o one, as should, because he enre populaon has been sampled a me Varyng populaon sze Insances mgh arse when he populaon sze does no reman consan because ems mgh be expended, dscarded or rgh censored. We agan assume ha each em has an exponenal(λ) me o falure, where he falure rae λ>0 s assumed o be a fxed consan deermned from a long hsory of esng. Jus pror o he esng mes < 2 < < k,herearer, r 2,...,r k ems a rsk and n, n 2,...,n k of hese ems are seleced a random and whou replacemen a each samplng me from he fne populaon and esed, where n r, =, 2,..., k. The esng and he reurn o he populaon are agan nsananeous. The ne effec of he esng s o perform a perfec repar on any faled ems. As before, for me values less han or equal o, he survvor funcon s S () = e λ, 0 <.

5 20 Leems Downloaded By: [College of Wllam & Mary] A: 2:0 9 March Fg.. Tesng and rgh-censorng mes The survvor funcon on each subsequen me nerval s defned recursvely as S () = r n S () + n e λ( ), r r <, for = 2,,..., k +, where k+. An analogous resul (proven n he Appendx) o he fxed populaon sze case s gven below. Resul 2. For he pecewse survvor funcon S() defned above: S( + ) S( ) = n, =, 2,...,k, S( ) r where + s a me value ha s an nfnesmal amoun larger han. Example 4. Fgure shows he esng and rgh-censorng hsory for a fne populaon of ems, where me s measured n monhs. The symbols correspond o esng mes and he o symbols correspond o rgh-censorng mes. The symbols n Fg. correspond o he k = 6 dsnc esng mes. There are q = dsnc esng and rgh-censorng mes. The numbers gven o he rgh of each lne segmen are he mulplces of each observaon. If hese mulplces are added, one can conclude ha here are 25 unque ems n he populaon. The goal here s o esmae he survvor funcon assocaed wh hs sequence of esng mes and rgh-censorng mes for a fxed falure rae λ. The rgh-censored values mgh correspond o expended ems, dscarded ems or ems currenly n use. Thus, rgh censorng effecvely corresponds o alerng he populaon sze. For each em, = 0 corresponds o he purchase dae. Snce each em s renewed upon esng (eher by passng he es or by falng he es and beng repared), he daa se can be reduced o hese 25 ordered values: 4., 26.6, 26.6, 26.6, 26.6, 27.2, 27.2, 27.2,,,,, 9.0, 4.5, 4.2, 4.2, 4.2, 45.8, 45.8, 56.8, 6.7, 6.7, 62.6, 62.6, 62.6, where an unmarked daa value denoes a me o esng and he * superscrp denoes a me o rgh censorng. Le x be he esng or rgh-censorng me, δ = 0fhe h daa value s rgh censored, δ = fheh daa value s a esng me, and m be he mulplcy of he number of dsnc (x,δ ) pars. By convenon, he prmary sorng creron s n ascendng values of x (chronologcal) wh δ used as a secondary sorng creron (censored observaons placed frs for ed me values). Ths convenon s necessary for he algorhm (whch follows) o process he daa properly. In he case of a ed censorng me and esng me, he censored ems are ncluded n he number a rsk jus pror o he esng me. Usng hs noaon, he number a rsk a me 0 s q = m because each esng me corresponds o a renewal. The (x,δ, m )rples for he daa se n hs example are gven n Table, for =, 2,..., q. The algorhm gven below s used o plo he survvor funcon S() assocaed wh a daa se smlar o he 2

6 Relably growh va esng 2 Table. Tes daa (k = 6, q = ) x δ m.00 S() Downloaded By: [College of Wllam & Mary] A: 2:0 9 March one gven n he prevous example. Indenaon s used o denoe nesng. The algorhm assumes he exsence of wo generc hgh-level plong funcons: Plo, for plong a curve, and PloPon, for plong a pon. A pon s ploed raher han he convenonal hash mark assocaed wh he Kaplan Meer produc lm esmae (Kaplan and Meer, 958) so as no o obscure he dsconnues n S(). By convenon, he survvor funcon s no ploed afer he las observed falure me. The assumpon ha m 0 = 0 s made n order o make he frs pass hrough he loop correcly. The algorhm reles on Resul 2 for he adjusmen of he survvor funcon assocaed wh a esng me. Inpu: The falure rae λ, henumberof daa rplesq,and he (x,δ, m )rples,for =, 2,..., q Algorhm: m 0 0 nalze m 0 for frs pass hrough loop 0 0 nalze me S nalze S(0) r q = m nalze he number a rsk a me 0 for from o q loop hrough daa rples Plo(Se λ( 0), 0 < x ) plo survvor funcon segmen S Se λ(x 0 ) updae S() f (δ = 0) f rgh censored PloPon(x, S) plo a pon a censorng me else else es me (δ = ) f (x = x ) smulaneous esng and rgh censorng m S S + ( S) updae S() r + m else dsnc esng me S S + m ( S) updae S() r 0 x updae me r r m decremen he number a rsk Fg. 4. Survvor funcon for k = 6 dsnc esng mes. The survvorfunconconsss ofk = 6 pecewse segmens. Alhough he survvor funcon s cu off afer me 62.6, he survvor funcon would rese o one due o he fac ha all hree of he survvng ems were esed a me Survvor funcons of hs ype wll characerscally have smaller jumps n S() for smaller values of (e.g., = 4.n Fg. 4) snce here are more ems a rsk. The daa se used n he prevous example can be approached from a second perspecve. Ths new perspecve allows us o consder he esmaon of he fne populaon relably over me. Example 5. When hs algorhm s appled o he sample daa from Table wh λ = 0.006, he resul s he plo n Fg. 4 showng he survvor funcon S() on0< Example 6. Fgure 5 shows he esng and rgh-censorng hsory for a fne populaon of 2 ems. There are 25 lne

7 22 Leems Downloaded By: [College of Wllam & Mary] A: 2:0 9 March Fg. 5. Tesng hsory for 2 reparable ems segmens shown n Fg. 5, 2 of whch correspond o a rghcensorng me and of whch correspond o a esng me. Ths daa se, alhough se n a reparable conex, produces he same daa as ha n Table and he same survvor funcon as n Fg. 4.. Case sudy We conclude wh an analyss of he daa se of esng and rgh-censorng mes (n monhs) ha frs promped he developmen of he model presened here. A long hsory of daa collecon ndcaes an exponenal me o falure wh λ = for a populaon of ems. The daa se consss of q = 094 records wh q = m = 274 dsnc evens (rgh censorngs and esngs). In addon: q = δ m q = m = 0.226, whch ndcaes ha 22.6% of he evens were assocaed wh esngs; he remander were rgh censorngs. Fgure 6 s a plo of wo survvor funcons he lower survvor funcon S() = e λ assocaed wh no esng of he ems n he populaon and he upper survvor funcon compued by he algorhm assocaed wh he relably growh accrued wh he esng. The upper survvor funcon cus off a he larges observed daa value, 95.8, where an ncrease n relably of abou 0.06 has accrued due o he esng and subsequen repar of faled ems n he populaon. The perodc esng has resuled n a sgnfcan ncrease n he sockple (.e., he fne populaon) relably. The survvor funcon assocaed wh he perodc esng exhbs ncreasng varably for larger values S() Fg. 6. Survvor funcon wh and whou esng. of due o he fac ha he sample sze s smaller n he rgh-hand al of he dsrbuon. A second queson ha arose concernng hs sockple of ems was o deermne he frequency and fracon of ems esed n order o acheve a sockple relably ha exceeds a prescrbed hreshold p, where0 < p <. Assume ha some fracon q of he sockple wll be sampled perodcally, where 0 < q <. Agan assumng ha λ s a known fxed consan, a sockple of new ems does no need o be esed for he frs 0 = (/λ)lnp me uns (found by solvng p = e λ 0 for 0 ). A me 0, some fracon q of he populaon s sampled (and faled ems are mmedaely repared as before), whch nsananeously ncreases he sockple relably o p + q( p). The me of he nex requred es (o keep he sockple relably a or above p) can be found by solvng: [p + q( p)]e λ( 0) = p, for. Ths yelds a me beween ess c = 0 : c = ( ) λ ln p. p + q( p) The paern of esng a fracon q of he populaon every c me uns connues ndefnely and resuls n a sockple relably ha exceeds p. In some suaons, he frequency of samplng may be dcaed by he problem seng. Ths equaon can be solved for he fracon of ems sampled q yeldng: q = p(eλc ). p Boh cases are llusraed numercally n he nex paragraph. Reurnng o he sockple of ems wh exponenal mes o falure wh a known falure rae λ = falures per monh, we consder he calculaon of c and q when he oher parameer s specfed for a sockple relably hreshold p = Frs, assume ha here s he

8 Relably growh va esng 2 Downloaded By: [College of Wllam & Mary] A: 2:0 9 March S() Fg. 7. Sockple relably a me for λ = 0.05, p = 0.85 and q = ably o sample only 5% of he ems a a me (q = 0.05); he frs samplng of a group of new ems should occur a monhs, hen every c = ( λ ln 0 = λ ln p = 09 ) p = 5.9 p + q( p) monhs hereafer. The survvor funcon assocaed wh hese parameers s graphed on 0 < < 200 n Fg. 7, where he sockple relably always remans above p = Second, consder he occason when he samplng me s consraned. Assume agan a falure rae λ = falures per monh and a relably hreshold p = If samplng can only occur annually (c = 2), hen a larger proporon q of he populaon mus be esed n order o manan he q p = 0.98 p = p = 0.90 p = 0.85 Fg. 8. Fracon esed q versus me beween ess c for λ = c relably hreshold: q = p ( e λc ) = 0.0. p Hence, 0% of he populaon mus be esed annually o manan a long-erm sockple relably ha exceeds Fgure 8 gves a graph showng level surfaces of q and c for several values of p assocaed wh a falure rae of λ = falures per monh. The pons consdered earler n hs paragraph are ploed on he p = 0.85 curve. The curves mach he nuon assocaed wh he process; n order o acheve a hgh relably hreshold, one mus sample a large fracon of he populaon frequenly. 4. Conclusons The model and assocaed algorhm presened here yeld a survvor funcon ha reflecs he relably growh over me assocaed wh perodc esng and repar for a fne populaon of sascally dencal reparable ems havng exponenal falure mes wh known falure rae λ. Ifa sgnfcan fracon of he populaon s esed perodcally for falure, hs procedure can be an effecve means for keepng he sockple relably a a prescrbed hreshold. Acknowledegmen The auhor graefully acknowledges he helpful commens provded by he hree referees. References Kaplan, E.L. and Meer, P. (958) Nonparamerc esmaon from ncomplee observaons. Journal of he Amercan Sascal Assocaon, 5(282), Ross, S.M. (2007) Inroducon o Probably Models, nnh edon, Academc Press, London. Appendx Ths Appendx conans a proof of he resul concernng he heghs of he dsconnues n he survvor funcon for he case of a fne populaon of varyng sze wh assocaed perodc esng. Resul. Le he pecewse survvor funcon S() be defned by and S () = e λ, 0 <, S () = r n S () + n e λ( ), r r <,

9 24 Leems Downloaded By: [College of Wllam & Mary] A: 2:0 9 March 200 for = 2,,..., k +, where k+. Jus pror o he esng mes < 2 < < k when n, n 2,...,n k ems are esed, here are r, r 2,...,r k ems a rsk (n r for =, 2,...,k). Then S( + ) S( ) = n, =, 2,..., k, S( ) r where + s a me value an nfnesmal amoun larger han. Proof. Le + = +. The quany of neres s S( + ) S( ) S + ( + ) S ( ) = lm S( ) 0 S ( ) ((r n )/r ) S ( + )+(n /r )e λ( + ) S ( ) = lm 0 S ( ) [ (n /r ) e λ S ( + ) ] + S ( + ) S ( ) = lm 0 S ( ) = (n /r ) [ S ( )] + S ( ) S ( ) S ( ) = n, r for =, 2,..., k, whch proves he resul. Bography Lawrence M. Leems s a Professor n he Mahemacs Deparmen a he College of Wllam & Mary. He receved hs B.S. and M.S. degrees n Mahemacs and hs Ph.D. n Indusral Engneerng from Purdue Unversy. He has also augh courses a Purdue Unversy, The Unversy of klahoma and Baylor Unversy. He has served as an Assocae Edor for IEEE Transacons on Relably and Naval Research Logscs and Book Revew Edor for he Journal of Qualy Technology. He s a member of IIE, INFRMS and ASA.