Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog Real Tme Study of a out of System: Idetcal Reparable Elemets wth Costat Falure ad Repar Rates Ma Sharf, Azzollah Memara, Behoush Kazem Ahar Abstract- Relablty models based o Marov cha (Except queug systems) have extesve applcatos electrcal ad electrocally devces. I ths paper we cosder a system wth parallel ad detcal reparable elemets wth costat falure ad repar rates (falure ad repar rates are expoetally dstrbuted. The falure rates crease whe some elemets are faled. The system wors utl at least elemets wor. The system of equatos are establshed ad the exact equatos are sought for the parameters le MTTF ad the probablty that system worg at the tme t. A umercal example has bee solved to demostrate the procedure whch clarfes the theoretcal developmet. It seems that ths model ca tacle more realstc stuatos. Keywords: Marov cha, out of models, Expoetal dstrbuto, Reparable elemets. Nomeclature The otatos used ths paper are as follows: : Number of elemets, λ : Falure rate of the elemets st category, μ : Repar rate of each elemets each category, P : Probablty that the system s state at the tme t, Mauscrpt receved December, 008. Ma Sharf s a member of Idustral Egeerg departmet, Islamc Azad Uversty, Qazv brach (correspodg author to provde phoe: +98-9-85776; fax: +98-8-3675787; e-mal: m.sharf@ Qazvau.ac.r). Azzollah Memara s wth the Idustral Egeerg departmet, Bu Al Sa Uversty (e-mal: memar@rphe.r). Behoush Kazem Ahar s the IT Maager, Brtsh Coucl, Ira (e-mal: Behoush.Kazem@Ir.Brtshcoucl.Org). R : Probablty that system wors at tme t, MTTF : Mea tme to falure of the system,. Itroducto out of models, are oe of the most useful models to calculate the relablty of electrcal ad electrocally devces ad systems. I the lterature there are may studes ths area. We try to categorcally classfy them. At the frst glace, they may be classfed to two ma groups amely out of : F ad out of : G systems. If the falure of a system wth compoets s abadoed to the falure of at least compoets ( ), the the system s called out of : F. O the other had, f the worg of ths system s abadoed to the worg of at least compoets ( ), the the system s called out of : G. Both these systems ca be cosdered steady state ad real tme stuatos. The elemets both stuatos may be reparable or o reparable. The falure rates of the elemets may be costat, creasg or decreasg whereas the repar rate s costat. Each compoet ca be expressed bary or multple states Bary models: Bolad ad Papastavrds [], study a stuato where there are dstct compoets wth falure probabltes q for,,..., where the falure probablty of the j th compoet ( j m + ( ) ) s q. They obtaed exact expressos for the falure probablty of a r cosecutve out of : F system. ISBN: 978-988-70-7-5 IMECS 009
Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog Gera [3], study the relablty of a cosecutve out of : G system s ad the problem has bee solved va a matrx formulato usg state space method. Lam ad Toy N.G [5] troduce a geeral model for cosecutve out of : F reparable system wth expoetal dstrbuto ad ( ) -step Marov depedece. Sarha ad Abouammoh [6], vestgate the relablty of o reparable out of system wth o detcal elemets subject to depedet ad commo shoc. Dutut ad Rauzy [7], study the performace of bary decso dagram for all out of system ad proposed a ew approxmato scheme. Krshamoorthy ad Ushaumart [8], obta the system state dstrbuto, system relablty ad several other measures of performace for a out of : G system wth repar uder D-polcy. Gupta [9], calculates relablty fucto ad the falure rate of the out of system, wth ad wthout corporatg the evrometal effect. Cu [0], presets a boud for for whch the system does ot preserve IFR whe >. Arulmozh [], propose a expresso for relablty of out of : G system ad developed a algorthm for computg relablty of out of system. Yam ad Zuo [], derve the state trasto probabltes of the reparable crcular cosecutve out of : F system wth oe reparma. Mlcze [3], presets a class of lmt relablty fuctos of homogeeous seres out of systems. Che [4], develop a method for aalyzg compoet relablty ad system relablty of out of systems wth depedet ad detcally dstrbuted compoets based o system lfetme data. [5], Koucy, deals wth relablty of geeral out of systems whose compoet falures eed ot be depedet ad detcally dstrbuted ad author gves the approxmatos for the system relablty. Arulmozh [6], gves smple ad effcet computatoal method for determg the system relablty of out of systems havg uequal ad equal relabltes for compoets. Smth-Destombes [7], presets both a exact ad a approxmate approach to aalyze a out of system wth detcal, reparable compoets. Fly ad Chug [8], preset a heurstc algorthm for determg replacemet polces a dscrete-tme, fte horzo, dyamc programmg model of a bary coheret system wth statstcally depedet compoets, ad the specalzed the algorthm to cosecutve out of systems. Hseh ad Che [9], preset a smple formula for the relablty lower boud of the two-dmesoal cosecutve out of : F system. Jalal ad Hawes [0], prove two theorems for the optmal cosecutve out of : G le for. Da Costa Bueo [], defe mmal stadby redudacy ad use the reverse rule of order ( RR ) property betwee compesator processes to vestgate the problem of where to allocate a spare a out of : F system of depedet compoets through mmal stadby redudacy. L ad Zuo [3], derved formulas for varous relablty dces of the out of system wth depedet expoetal compoets cludg mea tme betwee falures ad some other parameters. Gua ad Wu [4], study a reparable cosecutve out of : F system wth fuzzy states. Mult state models: Moustafa [], usg the marov method, develope a closed form avalablty soluto for two out of systems wth M falure modes. Huag ad Zuo [4] vestgate two types of mult-state out of : G systems (.e. creasg ad decreasg systems). The authors developed a aalytcal model o the propertes of the bary-state out of system. Jeab ad Dhllo [] preset a flow-graph-based approach to aalyze a mult-state out of : G / F / load-sharg systems. The multstate out of : G / F / load-sharg systems comprse detcal uts, that are uder state motorg ad recovery fucto. I ths paper we wor o a system wth parallel ad detcal reparable bary elemets wth costat falure ad repar rates for real tme codtos ad the umber of ISBN: 978-988-70-7-5 IMECS 009
Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog reparma s ulmted. The system wors utl all elemets fal (at least elemet wors). The paper s dvded to four parts. The secod part explores the models. Numercal examples are preseted the thrd part ad the fal secto deals wth the cocluso.. Modelg Assume a system wth parallel ad detcal reparable elemets. The system wors utl all elemets fal. Therefore each elemet has two states ad cosequetly the system wll have where the elemets A, A, ( ) states. Let A... A be the state A are worg ad other elemets are faled. Also A... A η j s the state that the elemet η j wors addto to other elemets. The state O dcates that all the elemets are faled. There s C state that,,,...,, elemets are worg ad the probablty of all states category are equal. The state structure of the system s show fgure. I ths fgure, the category, 0,,,...,, dcates that the states ths category are wth elemets worg ad ( ) elemets are faled. Each state s closely related to the states the atecedet ad precedet category..e., f a elemet s faled ay state, the the state s trasferred to the ext category ad f a elemet repars ay state, the the state s trasferred to the past category. I other words, f the system s ay states category, wth the falure of oe elemet, the state wll be category ( ) ad wth the repar of oe elemet, the state wll be category ( +). Whe some elemets are faled, other elemets wor wth more loads ad therefore the falure rate s creased for the remag elemets. The we have λ < λ( ) <... < λ. The system wors f at least elemets wor. Therefore we have: A... A A < A <... < A P + P0 ( 0) The frst part of equato ( 0 ), s related to states wth at least elemets are worg ad the secod part deals wth the state wth 0 elemet s worg. I order to fd R from equato ( ) P t for each states. 0, we must calculate () From the state A... A through O fgure we have: P ( t + Δt) P P A... A λ Δt P μ Δt P j j A,,..., + j η j A,,..., μ Δt P A... A A... A λ( + ) Δt PA A... A η + A A A... A A... A η ( 03) A... A j s the probablty that the system be the state A... A at tme t. Also η j A j λ ( + ) Δt PA A A... η,,..., s the trasfer rate from states of category ( +) to ths state ad λ Δt P s the trasfer rate from ths A... A state to the states of category ( ), μ Δt P s the trasfer rate from states of A A A A... category ( ) to ths state ad μ Δt P η j A j,,..., A... A s the trasfer rate from ths state to the states of category ( +) at tme Δ t. j () PA A A... ( 0) R t We ow that: A < A <... < A ISBN: 978-988-70-7-5 IMECS 009
Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog Fgure : The structure of the model The dfferetal equato of equato ( 03 ) s as follow: Ad: P + { λ + ( ) μ} P ( ) λp P + { λ + ( ) μ} P ( ) λ ( + ) P ( + ) + μp( )() t () C P R t,3,..., ( 04 ) ( 05 ) Whe a compoet fals each category, the other compoet must wor harder ad the falure rate of these compoets crease. We ca calculate the falure rate of each category as follows: λ λ γ ( ) ( ) 07 where 0 γ. If γ 0 ad costat ad f γ the λ λ., the the falure rates are equal The MTTF of the system s also calculated as follows: + t 0 MTTF R() t dt ( 06 ) 3. Numercal example I ths Example, we cosder a system wth detcal reparable elemets. Let γ 0. 5 ad based o a depedet sample, the falure rate of compoet ISBN: 978-988-70-7-5 IMECS 009
Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog category s λ 0. 00ad the 4 λ λ λ 0.5( ) 3 ad also repar rate s μ 0. 0005. The fgure, Show the states of ths system. falure of the remag elemets. Necessary relatos have bee developed for the falure rates. The ambguty of the data for falure rates s demostrated by fuzzy tragular umbers. It seems that ths model provdes more relable solutos. Ths model ca be appled to a wde varety of complex dustral problems le the eges of a arplae. As a exteso to ths wor, the falure rates may be cosdered dfferet for each elemet. I that case ether a system of out of model should be tae to cosderato or a certa polcy should be developed whe whole system fals. Hece, the cost for procuremet, repar of the elemets ad the falure of the whole system may be tae to accout such a way that the optmum umber of elemets wth mmum cost ad maxmum relablty wll be determed. These varatos could be studed for the systems wth creasg falure rate of the elemets too. Fgure : the states of the system example 3 The system of dfferetal equato s as follow: P + ( λ + μ) P λp P + λp μp ( 08 ) Usg Laplace trasformato to solvg the system of dfferetal equato ad usg vlaplace commad Maple 0, we have: () t 0.0096t 45Cosh( 0.00003466t ) e + 08.37435Sh( 0.00003466t) 0.0096t 0.996546e Sh( 0.00003466t) P 0.00689655 P Ad: () C P P + P R t Also the MTTF calculated as follow: + ( 0 ) MTTF R dt { P + P } dt ( ) t 0 t 0.66406499 0 A real tme 6 + 4. Cocluso ( 09 ) out of system has bee studed whch the falure rate of the elemets s costat ad the elemets are reparable. That s by falure of oe elemet, the load o the remag elemets, creases the chace of 5. Refereces [] M. S. Moustafa. Avalablty of -out-of-:g system wth M falure models. Mcroelectroc relablty, 996, 36, 385-389. [] P. J. Bolad, S. Papastavrds. Cosecutve -out-of-:f systems wth cycle. Statstcs & Probablty Letters, 999, 44, 55-60. [3] A. E. Gera. A Cosecutve -out-of- system wth depedet elemets A matrx formulato ad soluto. Relablty Egeerg ad System safety, 000,68, 6-67. [4] J. Huag, M. J. Zuo. Mult-state -out-of- system models ad ts applcato. Proceedg of the aual relablty ad mataablty symposum, 000, 64-68. [5] Y. Lam, K. H. Toy N.G. A geeral model for cosecutve -out-of-: F reparable system wth expoetal dstrbuto ad (-) - step Marov depedece. Europea Joural of Operato Research, 00, 9, 663-68. [6] A. M. Sarha, A. M. Abouammoh. Relablty of -out-of- oreparable system wth o depedet compoets subject to commo shocs. Mcroelectroc Relablty, 00, 4, 67-6. [7] Y. Dutut, A. Rauzy. New sghts to the assessmet o -out-of- ad related systems. Relablty egeerg ad system safety, 00, 7, 303-34. [8] A. Krshamoorthy, P. V. Ushaumart. K-out-of-:G system wth repar: The D-polcy. Computer & Operatos Research, 00, 8, 973-98. [9] R. C. Gupta. Relablty of a -out-of- system of compoets sharg a commo evromet. Appled mathematcs letters, 00,5, 837-844. [0] L. Cu. The IFR property for cosecutve -out-of-:f systems. Statstcs & Probablty letters, 00, 59, 405-44. ISBN: 978-988-70-7-5 IMECS 009
Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog [] G. Arulmozh. Exact equato ad a algorthm for relablty evaluato of -out-of-: G system. Relablty Egeerg ad System safety, 00, 78, 87-9. [] R. C. M. Yam, M. J. Zuo, Y. L. Z. A method for evaluatg of relablty dces for reparable crcular cosecutve -out-of-:f systems. Relablty egeerg & System safety, 003, 79, -9. [3] B. Mlcze. O the class of lmt relablty fuctos of homogeeous seres-"-out-of-" systems. Appled mathematcs ad computatos, 003, 37, 6-76. [4] Z. Che. Compoet relablty aalyss of -out-of- systems wth cesored data. Joural of statstcal plag ad ferece, 003, 6, 305-35. [5] M. Koucy. Exact relablty formula ad bouds for geeral -out-of systems. Relablty Egeerg ad System safety, 003, 8, 9-3. [6] G. Arulmozh. Drect method for relablty computato of -out-of-: G systems. Appled mathematcs ad computato, 003, 43, 4-49. [7] K. D. de Smth-Destombes, M. C. va der Hejde, A. va Harte. O the relablty of -out-of- system gve lmted spares ad repar capacty uder a codto based mateace strategy. Relablty egeerg & System safety, 004, 83, 87-300. [8] J. Fly, C. S. Chug. A heurstc algorthm for determg replacemet polces cosecutve -out-of- systems. Computer & Operatos Research, 004, 3, 335-348. [9] Y. C. Hseh, T. C. Che. Relablty lower bouds for two-dmesoal cosecutve -out-of-: F systems. Computers & Operatos research, 004, 3, 59-7. [0] A. Jalal, A. G. Hawes, L. R. Cu, F. K. Hwag. The optmal cosecutve -out-of-:g le for. Joural of statstcal plag ad ferece, 005, 8, 8-87. [] V. da Costa Bueo. Mmal stadby redudacy allocato a -outof-:f system of depedet compoets. Europea joural of operato research, 005, 65, 786-793. [] K. Jeab, B. S. Dhllo. Assessmet of reversble mult-state -out-of:g/f/load sharg systems wth flow-graph models. Relablty egeerg & System safety, 006, 9, 765-77. [3] X. L, M. J. Zuo, R. C. M. Yam. Relablty aalyss of a reparable - out-of- system wth some compoet beg suspeded whe the system s dow. Relablty egeerg & System safety, 006, 9,305-30. [4] J. Gua, Y. Wu. Reparable cosecutve -out-of-: F system wth fuzzy states. Fuzzy Sets ad Systems, 006, 57, -4. ISBN: 978-988-70-7-5 IMECS 009