Applied Mathematics and Materials

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1 Appled Mathematcs ad Materals O a Two-No-Idetcal Stadby Reparable System wth Imperfect Repar MOHAMMED A. HAJEEH, ABDUL-WAHAB ALOTHMAN 2 Techo-Ecoomcs Dvso Kuwat Isttute for Scetfc Research P.O. Box 24885; Safat-309 KUWAIT mhaeeh@ksr.edu.kw;wothma@ ksr.edu.kw; Abstract: Systems after cotuous operato fal ad hece requre repar or replacemet. I ths research work, faled systems/compoets are repared mperfectly, ths type of repar; the faled compoet s repared several tmes before complete replacemet. Ths study exames the behavour of a stadby system cosstg of two o-detcal compoets oe operato ad the other cold stadby; the compoets are of dfferet ages ad hece exhbt dfferet falure ad repar rates. Asymptotc avalablty s used to measure the performace of the system. The behavour patter s demostrated through a real lfe example. Moreover, comparsos are made betwee the o-detcal compoet system ad aother wth detcal compoets. Keywords- Steady state avalablty, Falure, Repar, Performace, Trasto probablty Itroducto Redudacy s a fudametal cocept used relablty egeerg ad s wdely used for supplemetg the operatg system. Although t brgs about a extra cost, however, t mproves the performace ad reduces dow tme. I stadby redudacy, the compoets are dvded to two types, Actve ad Stadby. The redudat compoets start operatg wheever the actve compoets fal. The stadby compoets have two falure rates; oe whe stadby ad oe for whe operato. Whe the falure rate of the stadby compoet s less tha the operatg compoet, t s warm stadby. Whe t the same as the operatg compoet, t hot stadby. I ths research work, the redudat compoet s assumed to be cold status, ad hece t does ot fal whle stadby. Addtoally, we address the behavour of two compoets system wth dfferet age. May artcles related to stadby systems are publshed the lterature; for example, Srdhara ad. Kalya used the Graphcal Evaluato Revew Techque (GERT) to aalyze a parallel system wth two odetcal uts subected to commo cause falure uder costat falure ad repar rates, the steady state avalablty ad mea tme to falure for the system were derved []. Arulmozh [2] examed the performace of a k-out-of-:g system wth odetcal compoets. A algorthm was developed whch proved to be more effcet calculatg relablty compared to other methods. Wu ad Zhag [3] appled a Bayesa approach to estmate the survval fuctos of a odepedet ad o-detcal seres system subected to + sources of fatal shocks. Sad ad El-Sherbey vestgated the performace of two parallel cofguratos wth dssmlar compoets; prevetve mateace was added the secod assumg the falure of ay compoet to cause a crease the falure rate of the secod compoet. I ths regard, the secod system s performace was proved to be superor to the frst [4]. Meawhle, others researched a k-outof-n:g system wth N categores, o-detcal compoets, ad of R repar facltes; assumg compoets each category to have a dfferet falure ad repar dstrbutos. The model was preseted as a stochastc etwork wth state depedet trastos [5]. O the other had, Zag ad Wag examed the behavour of a cold stadby system wth two dssmlar compoets ad oe reparma. The ma obectve was to mmze the log-ru average cost per ut tme for the optmal replacemet polcy [6]. Lee ad. Wag cosdered a two o-detcal compoet parallel system wth the obectve of developg a optmal polcy that mmzes the total cost ad maxmzes avalablty [7]. Fodella ad Xg modelled the relablty of systems wth correlated detcal compoets, where compoets possess the same relablty, addto to exhbtg a commo falure correlato parameter [8]. ISBN:

2 Appled Mathematcs ad Materals I ths artcle, we study the performace of a twostadby compoet wth dssmlar compoets. Real lfe examples of systems wth o-detcal compoets or parts are may; power geerators ad trasmsso systems, pumps ad heat exchagers power plats, vehcle tres, ad chllers ad so o. I all these examples ew compoets are usually added for replacg some of the old oes order to mprove the overall system performace. 2 Problem Descrptos 2. Assumptos ad Notatos 2.. Assumptos The assumptos lsted ths secto are appled to all models developed the paper. Other modelspecfc assumptos are preseted whe dscussg the relevat model. The geeral assumptos are as follows:. The tme betwee falures ad the repar rates are expoetally dstrbuted. 2. Falure rates ad repar rates are costat for all compoets. 3. All falures are statstcally depedet. 4. The travel tmes to ad from the repar faclty are eglgble. 5. The system becomes as good as ew after each replacemet Notatos Several otatos are gve for the two-compoet wth o-detcal compoets ad for the specal case of two detcal compoet systems. I the ma problem, each compoet has a dfferet age; oe compoet s desgated as the ew whle the other by the old. The old has spet more tme the system ad operato, whle the ew oe has bee stalled afterwards. Therefore, the ew has a lower falure rate ad a hgher repar rate whe compared to the old. The termologes used are defed as follows: The falure rate of the ew compoet after the th falure,,2,,. α The falure rate of the older compoet after the th falure,,2,,. μ Repar rate of the ewer compoet after the th repar,,2,,. ν Repar rate of the older compoet after the th repar,,2,..,. The steady state probablty of state,,2,, 6, where s the umber of the falures allowed before complete replacemet. a Compoet after the th falure, 0,,,; where 0 meas the compoet s ew. b Compoet 2 after the th falure, 0,,,; where 0 meas the compoet s ew. A s The Steady state avalablty of the system. 2.2 System Aalyss Mechacal systems deterorate over tme ad evetually fal to perform ther teded fucto ad requre repar or replacemet. The type of repar coducted defes the performace of the system. Repar s ether perfect where the system s restored to a as good as ew state, mmal whch brg the system to ts status mmedately before falure, or mperfect where the system becomes feror after each repar. The topc of perfect repar has bee thoroughly vestgated the lterature sce t s smple ad less complcated whe compared to mperfect repar. The latter s represetatve of the maorty of real lfe systems. Smlarly, systems real lfe cosst of compoets whch are o-detcal, these systems oly faled compoets are repared, hece over tme there wll be compoets of dfferet ages ad thus havg dfferet falure rates. Therefore, dervg aalytcal expressos for systems wth o-detcal compoets udergog mperfect repar s a dffcult oe. I ths secto, the performace of a two-compoet system s aalyzed usg steady state avalablty (the proporto of tme a system s a fuctog codto). Studyg larger scale systems wth multple o-detcal compoets s more tedous ad hece s addressed usg smulato. I ths artcle, a attempt s made to derve a aalytcal expresso for the steady state avalablty (log ru avalablty) of a system cosstg of two o-detcal compoets udergog mperfect repar. A pctoral presetato of the system state trasto s gve Fgure, where the oval shapes represet operatoal states ad rectagular shapes dcate faled states. I ths fgure, states to 4 are the operatoal states, whle states 4+ to 6 costtute the faled states. The system kcks off at state where both compoets are ew, compoet (a) s operato ad compoet (b) s cold stadby. The system trastos to state 2 upo falure of the operatg compoet (a) at a falure rate. At state 2, compoet (b) s put to operato ad compoet ISBN:

3 Appled Mathematcs ad Materals (a) s repar. From state 2, the trasto s ether to the operato state 3 by the repar of compoet (a) wth repar rate or to the falure state 4+ by the falure of compoet (b) wth a falure rate of α. Fg.. Pctoral presetato of a two dssmlar compoet cold stadby system. The trasto to state 4 could take place oe of two ways, ether from state 3 (operatoal state) by the falure of compoet (b) wth a falure rate α, or from the faled state 4+ by reparg compoet (a) wth repar ate of. All other trastos are carred out accordgly. After falures of both compoets, they are replaced by ew oes ad the system moves from state 4 to state, ad the process regeerates. 3. Problem Soluto 3.. A two-compoet stadby system wth o-detcal compoets The Chapma Kolomogorov trastoal relatoshp for the model Fgure s as follows: () t p() t + vp4() t 2() t p() t ( α ) p2() t + vp6() t 2(2 + ) () t ( α + + ) p2(2 + ) () t + + p4 + () t + vp4+ 2 (); t,2,..,. 4 () t ( + + v) p4 () t + α p4 ) t) p4+ 2 () t,2,..,. 4 ( t) α p4 ( t) p4 2( t) 4 + () t + p4 + () t + vp4 () t,2,..,.,2,..,. () 4+ () t 4+ p4+ () t + α p4 2() t,2,..,. () t v p () t + p () t,2,.., ; I the above expressos p () t s the probablty of beg at state at tme t ad () t s the dervatve of p () t wth respect to tme t ( p ()/ t () t ). Solvg the above set of equatos uder steady state codtos, the geeral relatoshps amog the varous states are expressed as: ( ),,..., + 2(2 + ) 4 ( α ) ( α ) 4 4 2,,,...,, + ( + ),,,..., (2) α ν,,..., 4 + 4, + α,.., ,2,..,, + ν From the above relatos, the probabltes of the system beg the dfferet states terms of the tal state ( ) are derved, they are as follows: ( ) ; 0,..., 2(2 + ) ν( α ) ( ) ;,,...,, 4 + ν( + ) ( ), 0,..., 2(2 + ) ν( α ) ( ) 4,,...,, + ν( + ) ( ),,..., 3 4 αν ( α ) ν ( ),,..., 4+ ( + + ν ) α + ( + ν 3) 4 + (2 + ), 0,,.., 3( α ) + ( + ν) 4+ 2,2,..,, + νν ( ) + I order to derve the expresso for, the above set of expressos s used alog wth the followg relatoshp: ( ) ISBN:

4 Appled Mathematcs ad Materals A closed formula for becomes as follows: ν α ν ( + ) ( + + ν) ν ( + )( α ) α α + ( + )( α ) ν( + ) + α ν α + ( + )( α ) ( α ) + (4) I order to drve the expresso for the avalablty, the state probabltes are derved terms of as formula (4), the avalablty s derved by addg the probabltes of the operatg states, ad t has the followg structure: ν ( + )( α ) α α As ν ( + )( α ) α α + α + α ν ( + )( α ) + ν( + ) ( α ) + (5) Dvdg both the umerator ad the deomator the above expresso by: α ν ( )( α ) + The expresso for steady state avalablty smplfes to: A + α (6) α α ν( + ) ( α ) s A two-compoet stadby system wth detcal compoets A specal case s preseted to show that systems wth o-detcal compoets are more complex tha those wth detcal compoets. Moreover, the avalablty dervato process s more volved the o-detcal case. Ths system s smlar to the stadby system of the two o-detcal compoets prevously dscussed, except ths case both compoets are detcal,.e., have the same falure rate ( ) ad repar rate ( ). Solvg the system of equatos results the followg expressos for the log ru probablty of the system beg state : ( + ) + 2 ( ) k l + + k, k l, l ( ) ( 7) All the other probabltes of the systems are derved term of from (4). The steady state avalablty s the sum of the operatg probabltes, t has the followg expresso: A s ( ) + k l ( ) k, k l, l Smplfyg ad rearragg, the above expresso reduces to the followg: + A s ( + ) + ( + +!) + 2, Numercal Examples Moder reverse osmoss (RO) desalato plats operate wth a hgh base load capacty ad therefore the mechacal relablty ad effcecy of hghpressure membrae feed pumps deserve thorough cosderato. Producto losses owg to mechacal falure or hgher eergy cosumpto due to lower hydraulc effcecy ca be sgfcat factors the plat operatg costs. RO hgh pressure feed pump feeds water through the prmary reverse osmoss membraes. Ths fltrato method s the most commoly used method of desalato the world ad cossts removg molecules ad os from solutos such as bracksh or sea water by applyg pressure; makg t passes through a permeable membrae. Fresh water oe sde ad a bre stream the other sde are the results of ths (8) ISBN:

5 Appled Mathematcs ad Materals process. The ma drver of a Reverse Osmoss plat s the hgh pressure pump whch has to be a relable heavy-duty ad corrosve resstat. Hghpressure pumps supply the pressure eeded to eable the water to pass through the membrae ad have the salt reected. The reverse pressure rages from 7 to 27 bars for bracksh water, ad from 52 to 69 bars for seawater. The pump rases the pressure of the pretreated feed water to a operatg pressure approprate for the membrae ad the salty of the feed water. As specfc RO desalato plats use two 60 bar hgh pressure pumps that help pushg the water through the membrae, oe ew ad oe old, the ew pump s put tally operato whle the old s cold stadby. The falures rates of the ew () ad the old (α) ad ther respectve repar rates ( for ew, ν for old) are show Table. The operato maager has to decde upo falure of ay pump whether to replace both pumps wth ew oes (smlar), or oly replace the faled compoet wth a ew old (dssmlar). The log ru avalabltes of both optos are calculated for varous falure ad repar rates as show Table ad graphcally preseted Fgure 2. Number of Repars Table. Falure Rates of the ew () ad the old (α) ad ther respectve repar rates ( for ew, ν for old) Falure rate () Falure rate (α) Repar Rate () Repar Rate (ν) Avalablty Smlar Avalablty Dssmlar Fg. 2. Avalablty for systems wth smlar ad dssmlar compoets uder dfferet umber of repars. Fg.3. Total cost for a system wth dssmlar compoets uder varous avalablty. I addto to the avalablty, the cost of the dfferet optos s show Fgures 3, 4 for dssmlar compoets ad smlar compoets, respectvely. The cost of purchasg two ew pumps s $200,000, whle the cost of purchasg oe ew pump s $25,000. The replacemet polcy s depedet o the avalablty ad the total cost. The obectve s to fd a polcy that provdes good avalablty at least cost. Fg. 4. Total cost for a system wth smlar compoets uder dfferet avalablty levels. It s a fact that a system wth ew compoets s superor to other combatos wth regard to performace. Moreover, the decle the avalablty s sharper the case wth two dssmlar compoets (oe old ad oe ew) whe compared to a system wth two ew compoets especally whe the umber of repars creases beyod sx repars. Numercally, oe could observe from Fgure that f the operato maager RO of the plat desres the system to operate 99% of the tme or more, he should choose the opto of two ISBN:

6 Appled Mathematcs ad Materals ew compoets ad replace both upo falure of ayoe compoet. However, f he would lke the system to be operable 95% of the tme he has two choces, ether he uses oe ew compoet ad oe old opto, ad replace the faled wth a ew at each falure, or use a system wth two ew compoets ad replace both after every thrd falure ad so o as show Table ad Fgure 2. Now, f we ot oly cosdered performace (avalablty), but also cost the a combato of both has to be take to accout wheever selectg ay opto. The cost usually creases as the umber of mperfect repars creases for both optos. I the begg, a system wth two smlar ew compoets provdes a better performace but a at hgher cost whe compared to the other opto (oe ew, oe old); however, as tme passes t becomes the best opto both performace ad cost. Numercally, f we compare the two optos through the example, Fgures 3, ad 4 provde the cost of the two optos versus the system s avalablty. It s observed that tally the cost s hgher for the frst system (2 ew) whe compared to the other system ( ew ad old), but over tme as the umber of allowable repars creases, the frst opto becomes more attractve. 5. Cocluso Ths artcle exames the performace of a twocompoet stadby system wth two o-detcal compoets where upo falure of ay oe compoet, t s repared several tmes (mperfect repar) before complete replacemet. I order to assess the behavor of a two-compoet stadby system, geeral aalytcal expressos were derved for the state probabltes ad the steady state avalablty for o-detcal compoets. I addto, the derved expresso for the avalablty for a two-compoet system s compared to the oe wth two detcal compoets. It was foud that the performace of the system wth detcal (smlar) compoets s superor f at least oe compoet s ew. Whe comparg both cost ad performace, t s observed that although the system wth detcal compoets has hgher cost ad hgher performace, hece t s ot preferred. However, the log ru as both systems udergo may mperfect repars t becomes a better opto both cost ad performace. Future research ths area should vestgate the behavor of other ad more complex cofguratos. It should also address systems where the tmes betwee falure ad repar tme are ot expoetally dstrbuted. Refereces [] V.Srdhara ad T.V. Kalya. Stochastc aalyss of a o-detcal two-ut parallel system wth commo-cause falure usg GERT techque. Iformato ad Maagemet sceces, 3(), (2002), pp [2] G. Arulmozh. Drect method relablty computato of k-out-of-: G systems. Appled Mathematcs ad Computato. 42(2-3), (2003), pp [3] H. Wu ad G. Zhag. Bayesa estmato of the parameters o-depedet ad odetcal seres system. Appled Mathematcs ad Computato, 74(), (2006), pp [4] K. Sad ad M.S. El-Sherbey. O the evaluato of relablty ad avalablty characterstcs of two dfferet systems. Iteratoal Joural of Iformato ad Maagemet sceces, 8(), (2007), pp [5] A. Khatab, N. Nahas ad M. Nourelfath. Avalablty of a K-out-of-N G systems wth o-detcal compoets subect to repar prortes. Relablty Egeerg & System Safety, 94(2), (2009), pp [6] Y.L. Zag ad G.J. Wag. A optmal reparreplacemet polcy for a cold stadby system wth use prorty. Appled Mathematcal Modellg, 35(3), (20), pp [7] B.L. Lee ad M. Wag. Approxmately optmal testg polcy for two-ut parallel system. Iteratoal Joural of Appled Scece ad Egeerg, 0(3), (202), [8] L. Fodella ad L. Xg. Dscrete ad cotuous relablty models for systems wth detcally dstrbuted correlated compoets. Relablty Egeerg & System Safety 33, (205), pp.-0. ISBN: