SAR Selected opcs Assurace Related echologes able of Cotets Itroducto Relablty of Seres Systems of Idetcal ad Idepedet Compoets Numercal Examples he Case of Dfferet Compoet Relabltes Relablty of Parallel Systems Numercal Examples Relablty of K out of N Redudat Systems wth Idetcal Compoets Numercal Example Combatos of Cofguratos Summary For Further Study About the Author Other SAR Sheets Avalable Itroducto Relablty egeers ofte eed to work wth systems havg elemets coected parallel ad seres, ad to calculate ther relablty. o ths ed, whe a system cossts of a combato of seres ad parallel segmets, egeers ofte apply very covoluted block relablty formulas ad use software calculato packages. As the uderlyg statstcal theory behd the formulas s ot always well uderstood, errors or msapplcatos may occur. he obectve of ths SAR sheet s to help the reader better uderstad the statstcal reasog behd relablty block formulas for seres ad parallel systems ad provde examples of the practcal ways of usg them. hs kowledge wll allow egeers to more correctly use the software packages ad terpret the results. We start ths SAR sheet by provdg some otato ad deftos that we wll use dscussg oreparable systems tegrated by seres or parallel cofguratos:. All the system compoet lves (X) are Expoetally dstrbuted: Volume, Number Uderstadg Seres ad Parallel Systems Relablty F δ dt () P{ X } ; f() F() e. herefore, every th compoet < < Falure Rate (FR) s costat ( (t) ). 3. All system compoets are detcal; hece, FR are equal ( ; < < ). 4. All compoets (ad ther falure tmes) are statstcally depedet: P P { X ad X ad...x } > { X > } P{ X > }...P{ X } >. Deote system msso tme. Hece, ay th compoet ( < < ) relablty R () : l () ( ) R P X > e ( R () ) Summarzg, ths SAR sheet we cosder the case where lfe s expoetally dstrbuted (.e., compoet FR s tme depedet). Frst, examples wll be gve usg detcal compoets, ad the examples wll be cosdered usg compoets wth dfferet FR. Idepedet compoets are those whose falure does ot affect the performace of ay other system compoet. Relablty s the probablty of a compoet (or system) of survvg ts msso tme. hs allows us to obta both, compoet ad system FR, from ther relablty specfcato. We wll frst dscuss seres systems, the parallel ad redudat systems, ad fally a combato of all these cofguratos, for oreparable systems ad the case of expoetally dstrbuted lves. Examples of aalyses ad uses of relablty, FR, ad survval fuctos, to llustrate the theory, are provded. Relablty of Seres Systems of Idetcal ad Idepedet Compoets A seres system s a cofgurato such that, f ay oe of the system compoets fals, the etre system fals. Coceptually, a seres system s oe that s as weak as ts weakest lk. A graphcal descrpto of a seres system s show Fgure. SAR 004, S&P SYSREL A publcato of the Relablty Aalyss Ceter
Fgure. Represetato of a Seres System of Compoets Egeers are traed to work wth system relablty [R S ] cocepts usg blocks for each system elemet, each block havg ts ow relablty for a gve msso tme : R S R R R (f the compoet relabltes dffer, or) R S [R ] (f all,, compoets are detcal) However, behd the relablty block symbols les a whole body of statstcal kowledge. For, a seres system of compoets, the followg are two equvalet evets : System Success Success of every dvdual compoet herefore, the probablty of the two equvalet evets, that defe total system relablty for msso tme (deoted R()), must be the same: P R() P { System Succeeds} P{ Compad Comp...ad Comp Succeed} { Comp Suc}...P{ Comp Suc} R e...e (e ()...R () ) System FR s s the, the sum ( tmes) of all compoet falure rates (): R() Exp{(+++ + ) } Exp{ }) Exp{ s } 3. Compoet FR () ca be obtaed from system relablty R(): [ l (R())] / (vertg the relablty results gve ) Compoet FR ca also be obtaed from compoet relablty R (): l [R ()] / l [R ()] / Prevous expresso s used for allocatg system FR s, amog the system compoets 4. otal system FR s ca also be obtaed from 3: s [ l (R())] / l [R ()] / s remas tmedepedet seres cofgurato. Allocato of compoet relablty R () from systems requremets s obtaed by solvg for R () the prevous R() equatos. 6. System urelablty U() R() relablty. Oe ca calculate the varous relablty ad FR values for the specal case of ut msso tme ( ) by lettg vash from all the formulas (e.g., substtutg by ). Oe ca obta relablty R() for ay msso tme, from R(), relablty for ut msso tme: [ R ()] [ Comp Relablty() ] (e ) [ ()] [ Comp Relablty() ] ( ) s s [ ] R R() P X..., X > e (e ) R() he precedg asserto holds because R (), the probablty of ay compoet succeedg msso tme, s ts relablty. All system compoets are assumed detcal wth the same FR ad depedet. Hece, the product of all compoet relabltes R () yelds the etre system relablty R(). hs allows us to calculate R() usg system FR ( s ), or the power of ut tme compoet relablty [R ()], or the th power of compoet relablty [R ()], for ay msso tme. We wll dscuss, later ths SAR sheet, the case where dfferet compoets have dfferet relabltes or FR. From all of the precedg cosderatos, we ca summarze the followg results whe all elemets, whch are detcal, of a system are coected seres:. he relablty of the etre system ca be obtaed oe of two ways: R() [R ()] ;.e., the relablty () of ay compoet to the power R() [R ()] ; ut relablty of ay compoet to the power. System relablty ca also be obtaed by usg system FR s : R() exp{ σ }: Sce s +++ + (all compoet FR are detcal) Numercal Examples he cocepts dscussed are best explaed ad uderstood by workg out smple umercal examples. Let a computer system be composed of fve detcal termals seres. Let the requred system relablty, for ut msso tme ( ) be R() 0.999. We wll ow calculate each compoet s relablty, urelablty, ad falure rate values. From the data ad formulas ust gve, each termal relablty R () ca be obtaed by vertg the system relablty R() equato for ut msso tme ( ): R() e s (e ) (e ) [ R ()] 0.999 / / [ R() ] (0.999) 0.9998 R () Compoet urelablty s: U () R () 0.9998 0.000.
Compoet FR s obtaed by solvg for the equato for compoet relablty: l Now, assume, that compoet relablty for msso tme s gve: R () 0.999. Now, we are asked to obta total system relablty, urelablty, ad FR, for the (computer) system ad msso tme 0 hours. Frst, for ut tme: R() e s (e ) (e ) Hece, system FR s: ( R ()) l( 0.9998) 0.000 If we requre system relablty for msso tme 0 hours, R(0), ad the ut tme relablty s R() 0.99, we ca use ether the 0 th power or the FR s : If msso tme s arbtrary, the R() s called Survval Fucto (of ). R() ca the be used to fd msso tme that accomplshes a prespecfed relablty. Assume that R() 0.98 s requred ad we eed to fd out maxmum tme : Hece, a Msso me of 4.03 hours (or less) meets the requremet of relablty 0.98 (or more). Let s ow assume that a ew system, a shp, wll be propelled by fve detcal eges. he system must meet a relablty requremet R() 0.9048 for a msso tme 0. We eed to allocate relablty by ege (compoet relablty), for the requred msso tme. We vert the formula for system relablty R(0), expressed as a fucto of compoet relablty. he, we solve for compoet relablty R (0): [ R ()] (0.999) 0.99 ( R() ) l( 0.99) 0.0003 l s [ R() ] 0 (0.99) 0 0s R(0) e (e s ) 0 0 e s (e 0x0.000 ) e 0.0 0.9 lr() R() e s e 0.98; s 0 R(0) e s (e 0x ) (e 0 ) l(0.98) 0.0003 4.03 [ R (0)] 0.9048 ( 0) [ R( 0) ] / ( 0.9048) 0. 0.980 R We ow calculate system FR ( s ) ad MF (µ) for the fveege system. hese are obtaed for msso tme 0 hours ad requred system relablty R(0) 0.9048: ( R() ) l( 0.9048) l s 0 0.00 0 0.0000 MF µ 99.96 s FR ad MF values, equvaletly, ca be obtaed usg FR per compoet, yeldg the same results: ( R ()) l( 0.980) l 0.09999 0.009999 0 0 s x x 0.009999 0.009999 0.0 MF R() d e s d µ 99.96 0 0 s Fally, assume that the requred shp FR s 0.0000 s gve. We ow eed compoet relablty, Urelablty ad FR, by ut msso tme ( ): R() Exp{ s } Exp {0.0000} 0.99 Exp{ } [Exp()] [R ()] Compoet relablty: R () [R()] / [0.99] 0. 0.998 Compoet urelablty: U () R () 0.998 0.00 Compoet FR: [ l (R())]/ [l(0.99)]/ 0.00 he Case of Dfferet Compoet Relabltes Now, assume that dfferet system compoets have dfferet relabltes ad FR. he: Σ R() R...e e e s s ()...R () e he system Mea me o Falure, MF, µ / s /Σ For example, assume that the fve eges (compoets), the above system (shp) have dfferet relabltes (maybe they come from dfferet maufacturers, or exhbt dfferet ages). Let ther relabltes, for msso tme ( 0) be 0.99, 0.97, 3
0.9, 0.93, ad 0.9, respectvely. he, total system relablty R() for 0 ad FR are: R () R ()...R () 0.99 x 0.97 x 0.9 x 0.93 x 0.9 0.7636 ( R() ) l{ R( 0) } l s 0 0.697 0 0.0697 Sce the system FR s s 0.0697, the the system MF s µ / σ /Σ /0.0697 37.077. Relablty of Parallel Systems A parallel system s a cofgurato such that, as log as ot all of the system compoets fal, the etre system works. Coceptually, a parallel cofgurato the total system relablty s hgher tha the relablty of ay sgle system compoet. A graphcal descrpto of a parallel system of compoets s show Fgure. { X }{ P X } P [ ][ ] ; If R () R () [ P{ X > } ] [ ] ; If R () R () X > X > X ad X > Fgure 3. Ve Dagram Represetg the Evet of Ether Devce or Both Survvg Msso me hs approach easly ca be exteded to a arbtrary umber of parallel compoets, detcal or dfferet. By expadg the formula R S ( R ) ( R ) ( R ) to products, the wellkow relablty block formulas are obtaed. For example, for 3 blocks, whe oly oe s eeded: Fgure. Represetato of a Parallel System of Compoets Relablty egeers are traed to work wth parallel systems usg block cocepts: R S Π ( R ) ( R ) ( R ) ( R ); the compoet relabltes dffer, or R S Π ( R ) [ R] ; f all compoets are detcal: [R R;,, ] However, behd the relablty block symbols les a whole body of statstcal kowledge. o llustrate, we aalyze a smple parallel system composed of detcal compoets. he system ca survve msso tme oly f the frst compoet, or the secod compoet, or both compoets, survve msso tme (Fgure 3). I the laguage of statstcal evets : () P{ System Survves } P{ > > or BOH } R X or X > { > } + P{ > } P{ > } P X X X ad X > R () + R () R () x R () [ R ()] + R () + () R ()[ R ()][ R ()] R () + [ R ()][ R ()] [ P{ > } ][ P{ } ] X X > f R S ( R ) ( R ) ( R 3 ) R + R +R 3 R R R R 3 R R 3 + R R R 3 or R S ( R) ( R) ( R) 3R 3R + R 3 (f all compoets are detcal: R R;,, Usg stead, the statstcal formulato of the Survval Fucto R(), we ca obta system MF (µ) for a arbtrary msso tme. For, say arbtrary compoets: R() [ ][ ] R() R () ( + ) e + e MF R() d e µ + e 0 0 + + ( + ) d Fally, oe ca calculate system FR s from the theoretcal defto of FR. For : FR s Desty Fucto Survval Fucto δ R() dt R() ( ) ( e + ) + ( + ) e e + s e + e () 4
Notce from ths dervato that, eve whe every compoet FR() s costat, the resultg parallel system Hazard Rate s () s tmedepedet. hs result s very mportat! Numercal Examples Let a parallel system be composed of detcal compoets, each wth FR 0.0 ad msso tme 0 hours, oly oe of whch s eeded for system success. he, total system relablty, by both calculatos, s: { X > 0} e 0 e 0. 0.9048;, R (0) P [ R (0)][ R ()] [ R (0 ] R(0) ) Relablty of K out of N Redudat Systems wth Idetcal Compoets A k out of redudat system s a parallel cofgurato where k of the system compoets, as a mmum, are requred to be fully operatoal at the completo tme of the msso, for the system to succeed (for k t reduces to a parallel system; for k, to a seres oe). We llustrate ths usg the example of a system operato depcted Fgure. he Probablty p for ay system ut or compoet, < <, to survve msso tme s: () P( X > ) e P R R() e + e Mea me to Falure ( hours): ( 0.9048) 0.9909 ( + ) he falure (hazard) rate for the twocompoet parallel system s ow a fucto of : hs system hazard rate s () ca be calculated as a fucto of ay msso tme, as show Fgure 4. e R(0) e 0 0 e 0. 0. 0.9909; for 0; MF µ + + e e () + s e + e 0.0 0.0 0.0 0.0e 0.0e e 0.0 0.0 0.0 0 ( ) ( e + ) + ( + ) Compoet Operato mes (Lves) X Msso Fgure. Uts Ether Fal/Survve Msso me me All uts are detcal ad k or more uts, out of the total, are requred to be operatoal at msso tme, for the etre system to fulfll the msso. herefore, the Probablty of Msso Success (.e., system relablty) s equvalet to the probablty of obtag k or more successes out of the possble trals, wth success probablty p. Hazard() 0.003 0.00 hs probablty s descrbed by the Bomal (, p) Dstrbuto. I our case, the probablty of success p s ust the relablty R () of ay depedet ut or compoet, for the requred msso tme. herefore, total system relablty R(), for a arbtrary msso tme, s calculated by: 0.00 0.000 0 0 0 Msso Fgure 4. Plot of the Hazard s () as a Fucto of Msso me. Hazard Rate s () creases as tme creases. hs plot ca be used to fd the s () requred to meet a Msso me of. Say 0, the s () about 0.008 ( ; ot. ; Ut Rel. p) R() k P Succ. k p ( p) B( ;, p) k ( ) Sometmes the formula: k 0 p p hs holds true because: ( ) p C p ( ) k + 0 p k p s used stead.
he summato values are obtaed usg the Bomal Dstrbuto tables or the correspodg Excel algorthm (formula). Followg the same approach of the seres system case, we obta the MF (µ). R() k e ( ) MF µ R()dt k e ( ) dt 0 0 k We ca obta all parameters for a arbtrary, by recalculatg probablty p e of a compoet survvg ths ew msso tme. I the specal case of msso tme, the vashes from all these formulas (e.g., substtuted by ). Applyg the mmedately precedg assumptos ad formulas, we obta the followg results: he relablty R() of the etre system, for specfed, s obtaed by: Provdg the total umber of system compoets () ad requred oes (k) Provdg the relablty (for msso tme ) of oe compoet: R () p Alteratvely, provdg the Falure Rate (FR) of oe ut or compoet System MF ca be obtaed from R() usg the precedg puts ad: MF k he Urelablty U() Relablty R() Numercal Example Let there be detcal compoets (computers) a system (shuttle). Defe system success f k or more compoets (computers) are rug durg reetry. Let every compoet (computer) have a relablty R () 0.9. Let msso reetry tme be. If each compoet has a relablty R () p 0.9, the total system (shuttle) relablty R(), the compoet FR () ad the MF (µ) are obtaed as: R() P(Succ. ; ot. ; Ut Rel. 0.9) 0.00046 0.9994 e s R () 0.9 e l l(0.9) 0.036 { R ()} MF µ x + + + k 0.036 3 4 9.49 x.83.77 Now, assume that a less expesve desg s beg cosdered, cosstg of 8 detcal compoets parallel. he ew desg requres that at least k uts are workg for a successful completo of the msso. Assume that msso tme s ad the ew compoet FR 0.344. Compare the two system relabltes ad MFs. Frst, we eed to obta the ew compoet relablty R () p for : R () P(X ) e e 0.344 > 0.79999 0.8 p Proceedg as before, we obta the ew total system relablty for ut msso tme: 0.9 ( 0.9) 0 0.9 ( 0.9) 8 8 8 R() 0.8 ( 0.8) 0 0.8 ( 0.8) k 0.068 0.9437 e s MF µ x + + + k 0.344 3 4 4.48 x.83.7497 he cheaper (secod) desg s, therefore, less relable (ad has a lower MF) tha the frst desg. Combatos of Cofguratos Some systems are made up of combatos of several seres ad parallel cofguratos. he way to obta system relablty such cases s to break the total system cofgurato dow to homogeeous subsystems. he, cosder each of these subsystems separately as a ut, ad calculate ther relabltes. Fally, put these smple uts back (va seres or parallel recombato) to a sgle system ad obta ts relablty. For example, assume that we have a system composed of the combato, seres, of the examples developed the prevous two sectos. he frst subsystem, therefore, cossts of two detcal compoets parallel. he secod subsystem cossts of a out of (parallel) redudat cofgurato, composed of also fve detcal compoets (Fgure 6). Assume also that Msso me s 0 hours. 6
R R 3 R 4 hs result mmedately shows whch subsystem s drvg dow the total system relablty ad sheds lght about possble measures that ca be take to correct ths stuato. R Subsystem A Subsystem B Fgure 6. A Combed Cofgurato of wo Parallel Subsystems Seres Usg the same values as before, for subsystem, A (two detcal compoets parallel, wth FR 0.0 ad msso tme 0 hours), we ca calculate relablty as: Smlarly, subsystem B ( out of redudat) has fve detcal compoets, of whch at least two are requred for the subsystem msso success. R 3 () R 4 () R () R 6 () R 7 () 0.9, for. We frst recalculate the compoet relablty for the ew msso tme 0 ad the calculate subsystem B relablty as follows: Recombg both subsystems, we get a seres system, cosstg of subsystems A ad B. herefore, the combed system relablty, for msso tme 0, s: R R 6 R 7 [ R (0)][ R (0)] R A (0) { X > } e 0.9 R () P { R ()} l(0.9) 0.036 l { X > 0} e p R (0) P e 0.036x0 e.036 0.3487 p R (0) B P(Succ. ; ot. ; p 0.3487) [ R (0)] ( 0.9048) 0.9909 0.3487) 0 C 0.3487 ( 0 0.4309 0.69 e s R(0) R A (0) x R B(0) 0.9909 x 0.69 0.639 Summary he relablty aalyss for the case of oreparable systems, for cofguratos seres, parallel, k out of redudat ad ther combatos, has bee revewed for the case of expoetallydstrbuted lves. Whe compoet lves follow other dstrbutos, we substtute the desty fucto the correspodg relablty formulas R() ad redevelop the algebra. Of partcular terest s the case whe compoet lves have a uderlyg Webull dstrbuto: f F () P{ X } δ dt () F () β α β α β β β e α Here, we substtute these values to equatos through of the frst secto ad through 6 of the secod secto ad redevelop the algebra. Due to ts complexty, ths case wll be the topc of a separate SAR sheet. Fally, for those readers terested pursug these studes at a more advaced level, we provde a useful bblography For Further Study. For Further Study. Kececoglu, D., Relablty ad Lfe estg Hadbook, Pretce Hall, 993.. Hoylad, A. ad M. Rausad, System Relablty heory: Models ad Statstcal Methods, Wley, NY, 994. 3. Nelso, W., Appled Lfe Data Aalyss, Wley, NY, 98. 4. Ma, N., R. Schafer, ad N. Sgpurwalla, Methods for Statstcal Aalyss of Relablty ad Lfe Data, Joh Wley, NY, 974.. O Coor, P., P ractcal Relablty Egeerg, Wley, NY, 003. 6. Romeu, J.L. Relablty Estmatos for Expoetal Lfe, RAC SAR, Volume 0, Number 7. <http://rac. aloscece.com/pdf/r_exp.pdf>. About the Author Dr. Jorge Lus Romeu has over thrty years of statstcal ad operatos research experece cosultg, research, ad teachg. He was a cosultat for the petrochemcal, costructo, ad agrcultural dustres. Dr. Romeu has also worked statstcal ad smulato modelg ad data aalyss of software ad hardware relablty, software egeerg, ad ecologcal problems. 7
Dr. Romeu has taught udergraduate ad graduate statstcs, operatos research, ad computer scece several Amerca ad foreg uverstes. He teaches short, tesve professoal trag courses. He s curretly a Aduct Professor of Statstcs ad Operatos Research for Syracuse Uversty ad a Practcg Faculty of that school s Isttute for Maufacturg Eterprses. For hs work educato ad research ad for hs publcatos ad presetatos, Dr. Romeu has bee elected Chartered Statstca Fellow of the Royal Statstcal Socety, Full Member of the Operatos Research Socety of Amerca, ad Fellow of the Isttute of Statstcas. Romeu has receved several teratoal grats ad awards, cludg a Fulbrght Seor Lectureshp ad a Speaker Specalst Grat from the Departmet of State, Mexco. He has extesve experece teratoal assgmets Spa ad Lat Amerca ad s fluet Spash, Eglsh, ad Frech. Romeu s a seor techcal advsor for relablty ad advaced formato techology research wth Alo Scece ad echology prevously II Research Isttute (IIRI). Sce reog Alo 998, Romeu has provded cosultg for several statstcal ad operatos research proects. He has wrtte a State of the Art Report o Statstcal Aalyss of Materals Data, desged ad taught a threeday tesve statstcs course for practcg egeers, ad wrtte a seres of artcles o statstcs ad data aalyss for the AMPIAC Newsletter ad RAC Joural. Other SAR Sheets Avalable May Selected opcs Assurace Related echologes (SAR) sheets have bee publshed o subects of terest relablty, mataablty, qualty, ad supportablty. SAR sheets are avalable ole ther etrety at <http://rac. aloscece.com/rac/sp/start/startsheet.sp>. For further formato o RAC SAR Sheets cotact the: Relablty Aalyss Ceter 0 Mll Street Rome, NY 3440696 oll Free: (888) RACUSER Fax: (3) 337993 or vst our web ste at: <http://rac.aloscece.com> About the Relablty Aalyss Ceter he Relablty Aalyss Ceter s a worldwde focal pot for efforts to mprove the relablty, mataablty, supportablty ad qualty of maufactured compoets ad systems. o ths ed, RAC collects, aalyzes, archves computerzed databases, ad publshes data cocerg the qualty ad relablty of equpmets ad systems, as well as the mcrocrcut, dscrete semcoductor, electrocs, ad electromechacal ad mechacal compoets that comprse them. RAC also evaluates ad publshes formato o egeerg techques ad methods. Iformato s dstrbuted through data complatos, applcato gudes, data products ad programs o computer meda, publc ad prvate trag courses, ad cosultg servces. Alo, ad ts predecessor compay II Research Isttute, have operated the RAC cotuously sce ts creato 968. 8