Inrnaional Journal of Compuaional Sin and Mahmais. ISSN 97-3189 Volum 3, Numr 3 11, pp. 351-3 Inrnaional Rsarh Puliaion Hous hp://www.irphous.om Rliailiy Analysis of a Bridg and Paralll Sris Nworks wih Criial and Non- Criial Human Errors: A Blok Diagram Approah. M.V. Phanikumari*, A. Mallikarjuna Rddy 1 and R. Bhuvanavijaya *Dparmn of Mahmais, Malla Rddy Collg of Engg. and Thnology, Maisammaguda, Dhulapally, Sundraad, India E-mail: phanin_mouli@yahoo.o.in 1 Dparmn of Mahmais, S.K. Univrsiy, Ananapur, India Dparmn of Mahmais, J.N.T.U. Collg of Enginring, Ananapur, India Asra This papr prsns an approah for prforming rliailiy analysis of ridg and paralll- sris nworks wih riial and non-riial human rrors. Rliailiy and man im o failur formulas ar dvlopd for xponnial and Rayligh disriud failur ims. Sliv plos ar shown for dmonsraing h ff of human rrors on sysm rliailiy and man im o failur. Kywords: Rliailiy, Availailiy, Human Errors, Man im o failur, Bridg Sysm Inroduion Humans play a pivoal rol in h dsign, dvlopmn and opraional phass of nginring sysms. Rliailiy valuaion of sysms wihou aking ino onsidraion h human lmn dos no provid a ralisi piur. Hn, hr is a dfini nd for inorporaing h ourrn of human rrors in sysm rliailiy valuaion. A Human rror is dfind as a failur o prform a prsrid ask or h prforman of a prohiid aion whih ould lad o disrupion of shduld opraions or rsuls in damag o propry and quipmn. Furhrmor, dpnding upon h svriy of human rror onsquns, human rrors an lassifid ino wo agoris, namly, riial and non-riial. For our purpos h ourrn of a
35 M.V. Phanikumari al riial human rror auss h nir sysm o fail whr as h ourrn of a nonriial human rror rsuls in a singl uni failur only. This papr prsns rliailiy analysis of ridg and paralll- sris nworks wih riial and non-riial human rrors [1-].A nwly dvlopd approah [1,-5,]. is usd o prform sysm rliailiy analysis. This approah is a modifid vrsion of h lok diagram approah and is dmonsrad in his whih onsiss of a svn uni ridg nwork. Assumpions Th following assumpions ar assoiad wih analysis givn low: 1. A uni an fail ihr du o a hardwar failur or du o a non-riial human rror.. Th ourrn of a riial human rror an rsul in oal sysm failur u h ourrn of a non-riial human rror an aus h failur of a singl uni only. 3. Eah uni failur is indpndn of ohrs. Figur 1.1: Blok diagram for ridg nwork. Analysis Bridg Nwork This papr rprsns a svn uni ridg nwork wih riial and non-riial human rrors as shown in Figur 1.1. In his figur, ah ral uni is rprsnd y a
Rliailiy Analysis of a Bridg and Paralll Sris Nworks 353 rangl. Th failur proailiy of ah uni is dividd ino wo omponns, namly, hardwar failur proailiy and non-riial human rror proailiy. Ths failur proailiis ar rprsnd y lok onnd in sris as shown in ah rangl in figur 1.1. A hypohial uni rprsning riial human rrors is onnd in sris wih h ridg nwork. Th oal sysm an fail du o h failur of his hypohial uni. Noaion Th following symols ar assoiad wih his: F j - Hardwar failur proailiy of jh uni, for j=1,,3,,5,7. f j - jh uni failur proailiy wih rsp o non-riial human rrors, for j=1,,3,,5,,7. f - riial human rror ourrn proailiy assoiad wih h sysm R Hj hardwar rliailiy of h jh uni. R NCj - rliailiy of h jh uni wih rsp o non-riial human rrors. R j - rliailiy of h jh uni wih rsp o hardwar failurs and non-riial human rrors. R - sysm rliailiy wih rsp o riial human rrors. R H,NC - sysm rliailiy wih rsp o hardwar failurs and non-riial human rrors. R - ridg sysm rliailiy wih rsp o hardwar failur, riial and non-riial human rrors. s- Lapla ransform varial - im Th im-indpndn rliailiy analysis ar dvlopd for h following wo ass: Cas1 : Non-idnial unis Th hardwar rliailiy of jh uni is givn y R Hj =1-F j, for j=1,,3,,5,,7. 1 Th rliailiy of jh uni wih rsp o non-riial human rrors is R NCj =1-f j, for j=1,,3,,5,,7. Th rliailiy of h jh uni wih rsp o hardwar failurs and non-riial human rrors is R j =R Hj R NCj, for j=1,,3,,5,,7. 3 Th ridg nwork s rliailiy wih rsp o hardwar failurs and non-riial human rrors is:
35 M.V. Phanikumari al R H,NC =R 1 R R 3 R R 5 R R 7 - R R 3 R R 5 R R 7 - R 1 R 3 R R 5 R R 7 - R 1 R R R 5 R R 7 -R 1 R R 3 R 5 R R 7 - R 1 R R 3 R R R 3 R 5 R R 7 R 1 R R 5 R R 7 R R R 1 R 3 Th rliailiy of h ridg nwork wih rsp o riial human rrors only is R =1-f 5 Finally, using quaions and 5, w g R =R * R H,NC Cas : Idnial unis By sing R j =Ri.., F j =F and f j =f, for j=1,,3,,5,,7 in quaion yilds R =R R 7 _ R R 5 R R, 7 Whr R=R H R NC, R H =1-F and R NC =1-f. Th plos of quaion 7 ar shown in figur 1. for h spifid valus of F, f and f. Ths plos larly show h impa of varying riial human rror proailiy f and non-riial human rror proailiy f on ridg sysm rliailiy. I is vidn from hs plos ha h sysm rliailiy drass wih inrasing valus of f and f. Rliailiy Plos Typ - 1 Bridg Sysm Rliailiy*.1 1 1 8....8 Criial Human Error Proailiy Sris1 Sris Sris3 Sris Sris5 Figur 1.: F=.1. Sris 1 3 5 f= f=.5 f=.1 f=.15 f=. Tim dpndn analysis for h following wo ass ar dvlopd:
Rliailiy Analysis of a Bridg and Paralll Sris Nworks 355 Cas A: Exponnially disriud failur ims For xponnially disriud hardwar failur, riial and non-riial human rror ims h im dpndn quaions for R H, R NC,R and R ar as follows: R H = λ H 8 Whr λ H is h onsan hardwar failur ra of a uni. R NC = λ NC 9 Whr λ NC is h onsan non-riial human rror ra assoiad wih a uni. R= -X,. 1 Whr X=λ H λ NC R = λ 11 Whr λ is h onsan riial human rror ra assoiad wih h sysm. Using quaions 7 o 11, w g h rliailiy of h svn idnial uni nworks as follows: 7 X λ X λ 5X λ X λ X λ R = 1 Th plos of quaion 1 ar shown in figur 1.3. for h assumd valus of h modl paramrs. Rliailiy Plos Typ - II Bridg Sysm Rliailiy 1. 1.8... 5 1 15 Tim - Sris1 Sris Sris3 Figur 1.3: λ H =.1, λ NC =.1. Sris 1 3 λ C = λ C =.3 λ C =.7
35 M.V. Phanikumari al Ths plos larly dmonsra h ff of varying im and ridg sysm rliailiy. I is vidn from hs plos ha h sysm rliailiy drass wih inrasing valus of im. Th man im o failur of h ridg sysm is givn y MTTF = R d = 1 7X λ X λ 5X λ X λ X λ Th plos of h aov quaion ar shown in figur 1. 13 MEAN TIME TO FAILURE PLOTS BRIDGE SSTEM MEAN TIME TO FAILURE 9 8 7 5 3 1....8 CRITICAL HUMAN ERROR Sris1 Sris Sris3 Sris Sris5 Figur 1.: λ H =.1. Sris 1 3 5 λ NC = λ NC =.1 λ NC =. λ NC =.3 λ NC =. Ths plos larly dmonsra h ff of varying riial and non-riial human rror ras λ and λ NC on h ridg sysm man im o failur. Th varian of im o failur of h ridg sysm wih human rrors is σ = R s MTTF lim s = 8 -- -- 7X λ X λ 5X λ X λ X λ 1 [ 7X λ X λ 5X λ X λ X λ Whr ] 1 R is h driva of h lapla ransform of R wih rsp o s. -
Rliailiy Analysis of a Bridg and Paralll Sris Nworks 357 Th ridg sysm failur dnsiy funion is givn y f =- R =7Xλ 7X λ -X λ -X λ -5X λ 5X λ -X λ -X λ X λ -X λ 15 Whr d R = R d Th hazard ra funion of h ridg sysm is f h = R 7 X X 5 X X 7 X λ X λ 5 X λ X λ X λ = 7X X 5 X X X X 1 Th plos of quaion 1 ar shown in figur 1.5. Hazard Ra Plos Typ-I Bridg sysm Hazard Ra.5.5..35.3.5..15.1.5 5 1 15 Tim- Sris1 Sris Sris3 Sris Sris5 Figur 1.5: λ H =.1 λ NC =.8. Sris 1 3 5 λ = λ =.1 λ =.3 λ =.5 λ =.7 Ths plos larly show h impa of varying im and onsan riial human rror ra λ. I is vidn from hs plos ha h sysm hazard ra inrass wih h inrasing valus of and λ. Cas B: Rayligh disriud failur ims For Rayligh disriud failur ims h im dpndn quaions for R H,R NC, R and
358 M.V. Phanikumari al R ar: R H = H 17 Whr H = 1 ; αh is h sal paramr assoiad wih α H h Rayligh disriuion rprsning hardwar failur ims of a uni R NC = NC 18 Whr 1 NC= ; α NC is h sal paramr assoiad wih α NC h Rayligh disriuion rprsning failur ims of a uni du o non-riial human rrors. R= - 19 Whr = H NC R = 1 Whr = ; α is h sal paramr assoiad wih h Rayligh disriuion α rprsning ridg sysm failur ims wih rsp o riial human rrors. Using quaions 7, 17 o, h ridg sysm rliailiy wih human rror is R = 7 5 1 Th numrial rsuls praining o quaion 1 ar aulad in h al1 Th sysm man im o failur is givn y MTTF = R d = π 1/ 1/ π π 1 π π 7 5 1/ 1/ 1/ Tal 1: Rliailiy valus for Bridg sysm H =.13, NC =.9. Bridg sysm rliailiy Tim =. =. =. =. =.8. 1. 1. 1. 1. 1..3.999.997.995.993.99..98.979.97.95.958.9.93.917.9.888.873 1..819.787.75.73.7 1.5.31.3.57.551.57 1.8.38.1.38.3.338.1.73.9.9.9.19..15.317.1.19.97.7.79.9.59.51. 3..38.3...18
Rliailiy Analysis of a Bridg and Paralll Sris Nworks 359 Th failur dnsiy funion of h ridg sysm is f =- R d d = 5 7 5 8 7 3 Th sysm hazard ra funion is xprssd as h = B A R f = Whr A= 5 7 5 7 B= 7 - - -5 - - - Th plos of quaion ar shown in figur1.. Figur 1.: H =.1 NC =.8. Sris 1 3 5 = =. =. =. =.8 Hazard Ra Plos Typ-II 8 1 1 5 1 15 Tim- Bridg Sysm Hazard ra Sris1 Sris Sris3 Sris Sris5
3 M.V. Phanikumari al Ths plos larly show h impa of varying im and. I is vidn from hs plos ha h sysm Hazard ra inrass wih inrasing valus of and. Rfrns [1] B.S. Dhilon, Human Rliailiy : Wih Human Faors, Prgamon Prss In., Nw ork,198. [] B.S. Dhilon, On human rliailiy- Bilipgraphy, Mirolronis and Rliailiy, Vol..pp.371-373. 198. [3] B.S. Dhilon and S.N. Rayapai, Analysis of rdundan wih human rrors, Prodings of h Annual Rliailiy and Mainainailiy Symposium, IEEE. Nw ork, U.S.A., pp315-31, 1985. [] B.S. Dhilon and S.N. Rayapai, Human rliailiy analysis mhods, Prodings Of h Third Inrnaional Confrn on Human Faor Manufauring HUMAN - 3, Sraford-Upon_Avon, U.K., 198. [5] B.S. Dhilon and S.N. Rayapai, Rliailiy modling of sysms wih human rrors, Prodings of h Twnih Hawaii Inrnaional Confrn on Sysm Sins HICSS-, Honolulu, Hawaii, U.S.A., Vol. III, PP.93-51,1987. [] B.S. Dhilon and S.N. Rayapai, Rliailiy Analysis of Nworks wih Human Errors: A Blok diagram approah, Mirolronis and Rliailiy, Vol 7, No, PP.981-999,1987.