Stochastic Repair and Replacement with a single repair channel

Transcription

1 Sochasc Repar and Replacemen wh a sngle repar channel MOHAMMED A. HAJEEH Techno-Economcs Dvson Kuwa Insue for Scenfc Research P.O. Box 4885; Safa-309, KUWAIT mhajeeh@s.edu.w hp:// Absrac: Sysems conss of one or more componen arranged n dfferen confguraons. In a seres confguraon, a sysem fals upon he falure of any componen, n a parallel confguraon, he sysem fals when all componens fal, whle n a sandby confguraon, a faled componen s replaced by an avalable sandby. Upon falure of a sysem/componen s perfecly, mnmally, or mperfecly repared. Perfec repar s he mos commonly used n leraure, bu n real lfe, sysems undergo several mperfec repars before complee replacemen. Ths research wor examnes a one componen sysem where upon falure s eher replaced or mperfecly repared wh dfferen probables. In hs regard, we provde an analycal expresson for he seady-sae avalably. An example s provded for demonsraon. Key Words: reparable models, mperfec, avalably, exponenal dsrbuon Inroducon Operang sysems such as mechancal sysems conss of componens; hese componens may fal n dfferen modes such as fague, lea, rupure, creep, wear, corroson, deformaon. Faled componens may eher be replaced or repared; reparable sysems when faled usually receve repar or manenance acons ha resore her funcons. These acons affec he sysem behavor and aler he maeup of he sysem. Repar eher brngs he sysem o as good as new perfec repar or a saus before falure mnmal repar or o an nferor sae called mperfec repar. The am of he dfferen manenance acons s o enhance sysems performance; relably s one of he performance measures used o nsure sysem effecveness and produce qualy characerscs producs requred by consumers. Anoher mporan performance creron s avalably ha encompasses boh relably and mananably; s he probably ha a sysem s operaonal and execues s requred funcons a a gven pon of me Sysems performance has been addressed exensvely n he leraure; for example, Abdel- Hameed [] examned an age dependen mnmal repar model where he faled sysem s eher s perfecly repared wh probably p() or mnmally repared wh probably p (). Brown and Proschan [] developed opmal replacemen polces for a sysem ha upon falure s eher undergoes perfec repar wh probably p or s mnmally repared wh probably (-p). Meanwhle, Bechel [3] consdered he oal repar cos lm replacemen polcy, where he sysem s replaced as soon as s oal repar cos reaches a specfc level. Mousafa [4] suded Marov models for he ransen analyss of he relably of -ou-of-n: G sysems wh and whou repar subjec o M falure modes. Zhao [5] suded he falure paern of reparable componens when a faled componen s eher perfecly or mperfecly repared. In hs wor, he lfeme of a componen s assumed o behave accordng o a general dsrbuon. Several asympoc quanes were derved, such as he mean number of falures n a specfc componen poson, he fracon of me he sysem s down due o falure n a parcular componen poson and he avalably of he sysem ISSN: X 33 Volume, 07

2 Dmrov e al. [6] examned an agedependen repar model wh mperfec repar, obanng he warrany coss for he producs under a nonhomogeneous Posson process scenaro. The behavor of mulple reparable sysems was nspeced by Pan and Rgdon [7] usng Bayesan mehods for models ha are beween as bad-as-old and he good-as-new. Mone Carlo mehods were used o approxmae properes of he poseror dsrbuons. Pandey e al. [8] developed a mahemacal model o asss decson maers n selecng proper manenance scenaros under mperfec repar. Examples were used o valdae he applcably of he proposed mehod. Resuls ndcaed ha nroducng of mperfec repar beer faclaes he allocaon of manenance resources. Meanwhle, El-Damcese and Shama [9] suded he performance of a -sae reparable sysem wh wo ypes of falure. Laplace ransform echnques were ulzed o develop expressons for several performance measures ncludng avalably, relably and mean me o falure under exponenally dsrbued mes beween falures and repar mes. Neguyen e al. [0] suded reparable sysems under mperfec repar where he me beween falures of a new sysem follows Webull dsrbuon. Furhermore, an analycal approach for he dsrbuon of he ner-falure mes was obaned n addon o producng under seady condons an opmal prevenve manenance polcy under a sac, a dynamc or a falure lm polcy. repared p. Afer perfec repar he sysem becomes as new, whle followng mperfec repar he sysem s performance decreases as me progresses. Thus he falure rae ncreases afer each repar ( + for =,, n); smlarly, he rae of repar declnes afer each repar; + for =,,.., n. The sysem s replaced afer undergong a specfed number n of repars. A pcoral presenaon of he sae ransons s shown n Fgure. The recangular shapes n he fgure represen he operaonal sae, whle he oval shapes are he faled saes. The process sars a sae, where he sysem s new, afer falure ( ) a sae, he sysems moves o sae for repar. A sae he sysem s eher perfecly repared wh probably n+ o sae, or mperfecly repared wh probably q (q+p =) o sae 3. Smlarly, afer falure wh falure rae, he sysem ransons o sae 4 A sae 4, s sysem s eher perfecly repared o sae, or mperfecly o sae wh probably q. Fnally, afer he n+ falure, he sysem s replaced and moves o sae and he process s regeneraed. Sysem Descrpons The purpose of hs paper s o sudy he avalably of a sysem ha upon falure eher s replaced wh ceran probably or s or mperfecly. The falure and repar rae mes are assumed o follow exponenal dsrbuon several. The sysem s analyzed usng Kolmogorov's forward equaons mehod. Oher assumpons are: he ravel me o repar saon s neglgble, avalably of only one repar person... A each falure, he sysem s mperfecly repared wh probably q or s perfecly Fg.. A pcoral presenaon of he ranson process of he sysem. Avalably Analyss The ranson probably P () for he sysems s as follows: ISSN: X 333 Volume, 07

3 p () p () n p () n p 4()... n pn() () p() ( n q) p() : n() np n() ( n qn) p n() n() np n() n pn() () p () p () n n n n n The seady sae sysem wh exponenal Marov wh connuous me s expressed as: Qπ = 0, where π s he sae ranson probably marx, and Q s he sae ranson rae marx, Q s: (6) q ( n q) ( n q) Where 0 ( q ) n for n Ulzng equaons 4, 5, 6, he expresson for he probably of beng n he nal sae (π ) s obaned. Defne a q b ( ) n q n n 0 n 0... n 0 n ( n q ) q ( n q) Q0 0 0 q : : : : : : : : : ( n qn) qn n n n n n aq b( ) q b( ) n q b( ) A n n aq b( ) q b( ) (7) (8) Solvng he sysem of equaons Qπ = 0, he seady sae ranson probably s: ;for n j ( qj n ) j J n () q for n () (3) n n n n (4) Equaons can be wren n erms of π as follows: n q b( ) A n n aq b( ) q b( ) 3. Examples (9) To llusrae he process an example s presened for a sysem ha undergoes wo mperfec repars and s replaced afer he hrd falure. The ranson saes for hs sysem are gven n Fgure. (5) q ( n q) ( n q) for n ISSN: X 334 Volume, 07

4 q q ( q) 3( q) A q q ( q) 3( q) q q ( q) ( q) Fg.. A pcoral presenaon of a sysem wh mperfec repars before replacemen Fgures 3 and 4 exhb he sysem avalably for varous values of repar and falure raes; n fgure 4, he falure raes are ncrease for he same repar raes ncreased The sae ranson marx for hs problem s as presened below: ( q) q Q ( q) q From he above marx, he sysems ranson probables are presened and he expresson for he seady sae probably of beng a sae s derved as follows: Fg. 3. Avalably vs. replacemen probably for he sysem ( = 0.05, = 0.30, 3= 0.70; =.0, =.75, 3=.5; = 3.0). 3( q )( q ) 3( q)( q) 3( q) 3q( q) 3q q q 3q q Summng he probables of he operaonal saes, we oban he seady sae avalably of he sysems: Dvdng he numeraor and he denomnaor by 3 ( q)( q) and rearrangng and smplfyng, he followng expresson s derved: Fg. 4. Avalably vs. replacemen probably for he sysem (= 0.6 = 0.90, 3 =.5; =.0, =.75, 3 =.5; = 3.0) 4 Conclusons and Fuure Research Majory of manenance problems n leraure address he opc of perfec. Ths ype of repar s useful for nexpensve ems and s a mus for crcal componens. However, n generally s cosly and unpraccal. Ths sudy examnes he behavor of reparable sysems ha upon each falure could be eher perfecly or ISSN: X 335 Volume, 07

5 mperfecly repared. An analycal expresson for seady sae avalably s derved. Ths expresson s applcable o boh perfec and mperfec repar. The expresson become more complcaed as he number of saes ncreases. Furhermore, for fxed values of repar raes, avalably declnes as he falure raes ncreases. Conversely, for fxed falure raes, avalably rases as he repar raes ncreases. Fuure research should address cases were he falure raes or he repar raes or boh are no exponenally dsrbued and wh more han one repar saons complemens Smulaon may be ulzed o address large scale dynamc manenance models and problems wh mulple compounds... References [] Abdel-Hameed, M. (978). Inspecon and manenance polces of devces subjeced o deeroraon. Advances n Appled Probably 0, [] Brown M. & Proschan F. (983). Imperfec repar, Journal of Appled Probably, 0: 85l 859. [3] Bechel F. (997). Manenance polces under sochasc repar cos developmen. Economc Qualy Conrol, [4] Mousafa, M.S. (998). Transen analyss of relably wh and whou repar for K-ouof-N: G sysems wh M falure modes. Relably Engneerng and Sysem Safey 59, [5] Zhao, M. (994). Avalably for reparable componens and seres sysems. IEEE Transacons on Relably 43(), [6] Dmrov B, Chuova S, Khall Z. (004). Warranee coss: An age-dependen falure/repar model, Naval Research Logscs, 5, [7] Pan, R., Rgdon, S.E. (009). Bayes nference for general reparable sysems, Journal of Qualy Technology, 4(), 8-94 [8] Pandey, M., Zuo, M. J., Moghaddass R. & Twar, M.K.(03). Selecve manenance for bnary sysems under mperfec repar, Relably Engneerng and Sysem Safey,. 3, 4 5. [9] El-Damcese, M. A. & Shama M. S. (05) Relably and avalably analyss of a - sae reparable sysem wh wo-ypes of falures, Eng. Mah. Leers,, -9. [0] Nguyen, D. T., Djoux, Y. & Mra F. (07). Analycal properes of an mperfec repar model and applcaon n prevenve manenance schedulng, European Journal of Operaonal Research, Vol. 56, No., 07, pp ISSN: X 336 Volume, 07