A Monotone Process Replacement Model for a Two Unit Cold Standby Repairable System

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1 Itrtol Jorl of Egrg Rsrch d Dlopmt -ISS: p-iss: Volm 7 Iss 8 J 3 PP A Mooto Procss Rplcmt Modl for Two Ut Cold Std Rprl Sstm Dr.B.Vt Rmd Prof.A. Mllrj Rdd M. Bhg Lshm 3 Assstt Profssor Dpt. of Sttstcs S.S.B.. Dgr collg Atoomos Atpr-55. A.P Id Profssor of Mthmtcs Dpt. of MthmtcsS..Urst Atpr A.P Id. 3 Rsrch Scholr Dpt. of Mthmtcs S.. Urst Atpr A.P Id. Astrct:- I ths ppr two t cold std rprl sstm wth two mooto procsss xposg to xpotl flr lw s stdd dr th ssmpto tht ch compot ftr rpr s ot s good s w. Udr ths ssmpto w std optml rplcmt polc whch w rplc th sstm wh th mr of flrs of compot rchs. W dtrm optml rpr rplcmt polc * sch tht th log r rg los s mmzd. W dr xplct xprsso of th log-r rg los d th corrspodg optml rplcmt polc c dtrmd ltcll. mrcl rslts r lso stlshd to hghlght th thortcl rslts. words:- Rwl procss Gomtrc procss α-srs procss Rpr rplcmt polc Rwl ccl Rwl rwrd thorm. I. ITRODUCTIO I th rlr ds most rpr rplcmt modls ssm tht flr sstm ftr rpr wll ld fcto sstm whch s s good s w d th rpr tms r glctd so tht th sccss oprtg tms forms rwl procss. Ths tps of modls m clld s prfct rpr modls. Brlow d Htr trodcd [] mml rpr modl whch mml rpr dos ot chg th g of th sstm. Thrftr mprfct rpr modl ws dlopd Brow d Prosch [3] dr whch rpr wth prolt p s prfct rpr d wth prolt -p s mml rpr. M othrs word ths drcto d dlopd corrspodg optml rplcmt polcs Blc t.l [] Pr [] jm [6] McCll [9] gw [] Stdj d rm [3] d so o. I grl for dtrortg sstm t s rsol to ssm tht th sccss worg tms r stochstcll dcrsg whl th cosct rpr tms ftr flrs r stochstcll crsg d to th gg d ccmltd wrg m sstms. To modl sch smpl rprl dtrortg sstm Lm [78] frst trodcd gomtrc procss rpr modl dr th ssmptos tht th sstm ftr rpr s ot s good s w. Udr ths ssmptos h cosdrd two ds of rplcmt polcs -o sd o th worg g T of th sstm d othr sd o th mr of flrs of th sstm. Ltr hg [5] dlopd rt rplcmt polc T to grlz Lm s wor. Othr rplcmt polcs dr gomtrc procss rpr modl wr rportd Stdj d crm [3] Stl [4] hg [6] hg d Wg [78] d so o. All ths rsrch wors dscssd o r rltd to o compot rprl sstm. Howr m prctcl pplctos th std tchqs r sll sd for mprog th rllt or rsg th llt of th sstm. hg [6] ppld th gomtrc procss rpr modl to sgl cold std rprl sstm wth o rprm d stdd rplcmt polc d corrspodg optml rplcmt polc * s dtrmd sch tht th log-r-rg cost pr t tm s mmm. Ltr hg [7] ppld th gomtrc procss rpr modl to two t cold std rprl sstm wth o rprm d stdd rplcmt polc. Br t.l [4] stdd som mportt proprts of mooto procsss d prod tht lph srs procsss s mor pproprt to modl th p tms. O ths drstdg ths chptr w h proposd to dlop two mooto procsss mtc modl d otd optml rplcmt polc. Th ojct of ths chptr s to dtrm optml rplcmt polc for two t cold std sstms wth o rprm sg two mooto procsss xposg to Wll flr lw. It ssmd tht th sccss worg tms { = } of sstm form dcrsg α-srs procss whl th cosct rpr tms {Y = } form crsg gomtrc procss. Udr ths ssmptos w stdd rplcmt polc d corrspodg optml rplcmt polc * s dtrmd sch tht th log-rrg cost pr t tm s mmzd. 4

2 A Mooto Procss Rplcmt Modl for Two Ut Cold Std I modlg of ths dtrortg sstms w tlz dftos g Lm [7]. Dfto : G two rdom rls d Y f P>t > PY>t for ll rl t th s clld stochstcll lrgr th Y or Y s stochstcll lss th. Ths s dotd > st Y or Y < st rspctl. Dfto : Assm tht {Y =.} s sqc of dpdt o-gt rdom rls. If th dstrto fcto of s F t = F - t for som > d ll =3. th {Y = } s clld gomtrc procss s th rto of th gomtrc procss. Oosl: f > th {Y =.} s stochstcll dcrsg. Y > st Y + = ; f << th {Y =.} s stochstcll crsg. Y < st Y + =; f = th th gomtrc procss coms rwl procss. Dfto 3: Assm tht { =.} s sqc of dpdt o-gt rdom rls. If th dstrto fcto of s t F t for som > d ll = 3 th { = } s F clld srs procss s clld xpot of th procss. Br t. l [4]. Oosl: f > th { =.} s stochstcll dcrsg. > st + = ; f < th { =.} s stochstcll crsg.. < st + =; f = th th srs procss coms rwl procss. II. THE MODEL I ths scto w dlopd modl for two compot cold std rprl sstm wth o rprm sg two mooto procsss d xposg to Wll flr lw sch w tht th log-r rg cost pr t tm s mmzd wth th followg ssmptos. ASSUMPTIOS At th gg oth th compots r w d compot s worg stt whl th othr compot s cold std stt. Th two compots ppr ltrtl th sstm... wh th worg compot fls mmdtl th std compot gs to wor d th fld o s rprd th rprm. Whr th rpr of th fld o s compltd t coms cold std. If o fls d th othr s stll dr rpr t mst wt for rpr d th sstm rs dow. 3 A compot th sstm s rplcd som tm dtcl o d th rplcmt tm s glgl. 4 Th compots ftr rpr r ot s good s w. Th tm trl tw th complto of th - th rpr d th complto of th th rpr o compot s clld th ccl of compot for = d =. 5 A compot th sstm c t prodc th worg rwrd drg cold std d o cost s crrd drg th wtg prod for rpr. 6 Lt d Y for = d = r ll S-dpdt. 7 Lt worg tm follow dcrsg α-srs procsss xposg to Wdll flr lw d Y th rpr tm follow crsg gomtrc procsss xposg to Wll flr lw of compot th th ccl for = d =.. Lt E d E Y for =. 8 9 Lt F α x =F x d G - =G th dstrto fctos of 43 d Y rspctl for = d =3 whr α> d <<. Lt th rpr cost rt of ch compot s C r th worg rwrd pr t tm of ch compot s C w d th rplcmt cost of th sstm s C. I th xt scto w fd optml solto for polc sd o th ssmptos of th modl d dtrm optml solto for sch tht th log-r rg cost s mmm. III. OPTIMAL SOLUTIO W cosdr optml rplcmt polc dr whch th mr of flrs of compot rchs. Accordg to th ssmptos of th modl two compots ppr ltrtl th sstm. Wh th mr of flrs of compot rchs compot m th rpr stt of th - th

3 A Mooto Procss Rplcmt Modl for Two Ut Cold Std ccl or th cold std stt th th ccl. trll rsol rplcmt polc shold tht compot c t rprd mor wh th mr of ts flrs rchs d compot wors tl flr th th ccl. Accordg to th rwl rwrd thorm s Ross [] th log-r rg cost pr t tm of th sstm dr polc s g : Th xpctd cost crrd rwl ccl C 3. Th xpctd lgth of rwl ccl Whr th lgth of sstm rwl ccl dr polc s: - - L Y Y I Y Y I Y = = = = 3. whr th frst scod thrd forth d th ffth trms rfrs to worg g rpr tm wtg for rpr tm std tm of compot d worg g of compot th th ccl rspctl. Th totl tm drto wh th compot s cold std s clld std tm. Th xpctd lgth of rwl ccl s - E L E E Y E Y I Y = = = E Y I E Y = Whr I s th dctor fcto sch tht f t A occrs I A f t A dos't occrs. Accordg to th ssmptos of th modl coolto d Jco trsformtos th prolt dst fcto of Y d Y r rspctl. g f d Whr Y sch tht Y 3.4 d g f d 3.5 ; Y. whr sch tht Y 3.6 Thrfor dfto of mthmtcl xpctto w h: E Y. I g d { Y }. 3.7 E Y I g d { Y }. 3.8 ow th xpctd lgth of worg tm c otd s follows: Lt W x : for 3... d. ~ Th th dstrto fcto of F x F x for =3.d = s : x ; x 3.9 B dfto th xpctd lgth of worg tm s g : 44

4 A Mooto Procss Rplcmt Modl for Two Ut Cold Std 45. x xdf E x 3.. whr 3. Th xpctd lgth of rpr tm of compot c otd s follows: Lt : ~ W Y th th dstrto fcto of Y for = d = 3. s. ; F F 3. B dfto th xpctd lgth of rpr tm s s g :. df E Y x. whr 3.3 Th xpctd lgth of wtg tm for rpr c comptd s follows: Lt g th prolt dst fcto of Y th dfto of prolt dst fcto d sg Jco trsformto w h: g f d whr Y sch tht. Y 3.4 Sc d Y r ll dpdt for = d = g f f d 3.5 From qtos 3.4 d 3.5w h: d g. 3.6 O smplfcto qto 3.6 coms:. g 3.7 Lt { } Y E Y I g d 3.8 d. O smplfcto qto 3.8 coms:

5 A Mooto Procss Rplcmt Modl for Two Ut Cold Std 46 { } Y E Y I. 3.9 Whr. Smlrl th xpctd lgth of cold std tm c comptd s follows: { } Y E Y I g d. 3. Whr g th prolt dst fcto p.d.f of Y. B dfto of p.d.f d sg Jco Trsformto w h: g f d. 3. Y sch tht. whr Y 3. Sc d Y for = r ll dpdt d form gomtrc procss.. g f f d 3.3 Usg qtos 3. d 3.3 w h: d g 3.4 O smplfcto qto 3.4 coms:. g 3.5 From qtos 3.8 d 3.5 w h: { } Y E Y I g d d. whr. 3.6 From qtos d 3.6 qto 3.3 coms: L E 3.7 Usg th qtos 3. d 3.7 w h:

6 A Mooto Procss Rplcmt Modl for Two Ut Cold Std CrE Y Y C CwE C EL Cr C wr C C Crl C C wl C 5.8 l l Ths s th log-r rg cost fcto dr polc. Whr l l. Usg C w dtrmd optml rplcmt polc * sch tht th log-r rg cost pr t tm s mmm. I th xt scto w prod mrcl wor to hghlght th thortcl rslts IV. UMERICAL RESULTS AD COCLUSIOS V. For g hpothtcl ls of th prmtrs α Cw C Cr th optml rplcmt polc * s clcltd from xplct xprsso qto sch tht th log-r rg cost s mmm : Tl: 5.4.: Vls of log-r rg cost pr t of tm For g hpothtcl ls of α =.5=.8 5 5Cr=5 Cw=5C=5 For g hpothtcl ls of α =.35 =.8 5 5Cr=5 Cw=5C=5 C C

7 A Mooto Procss Rplcmt Modl for Two Ut Cold Std VI. COCLUSIOS From th tl 4. d grph 4. w osr tht C 6= s th mmm of th log r rg cost ldg to th optml polc * = 6 whch dcts w shold rplc th sstm t th d of 6 th flr. W osrd tht th log r rg cost pr t tm s dcrss s th l of s crss d c rs cs of th l of α. It s osrd tht for smll crs thr s crs * d dcrs rg log-r cost pr t tm. Smlrl coclso c drw cs of th prmtr α. At dffrt ls of th prmtrs of th modl cosdrd th l of log-r rg cost pr t tm s corgs to costt l. Ths rslt s cocdg wth thortcl rslt. REFERECES []. Brlow R.E d Htr L.C Optmm Prt Mtc Polcs Oprtos Rsrch Vol. 8 pp []. Bloc H.W. Borgs W.S. d Sts T.H. Ag-dpdt mml Rpr Jorl of Appld prolt Vol. pp [3]. Borw M. d Prosch F. Imprfct Rpr Jorl of Appld Prolt Vol. PP [4]. Br W.J L W d ho Y.Q Proprts of th gomtrc d rltd procss l Rsrch Logstcs Vol.5 pp [5]. o W. d o M.J. Optml Rllt Modlg : Prcpls d Applctos Joh Wl & Sos Ic. 3. [6]. jmm. Som Rslts for Rprl sstms wth Grl Rpr Jorl of Appld prolt Vol. 6 pp [7]. Lm Yh. Gomtrc Procsss d Rplcmt Prolms Act Mthmtc Applct Sc Vol.4 pp [8]. Lm Yh. A ot o th Optml Rplcmt Prolm Adcd Appld Prolt Vol. pp

8 A Mooto Procss Rplcmt Modl for Two Ut Cold Std [9]. McCll J.J Mtc polcs for stochstcll flg qpmt : A sr Mgmt Scc Vol. pp []. Pr.S. Optml mr of Mml Rprs for Rplcmt IEEE Trsctos o Rllt ol. R-8 o. pp []. gwt. Grlzd modls for dtrmg optml mr of mml rpr d ts pplcto to rplcmt polc Erop Jorl of oprtol Rsrch Vol.o. 983 pp []. Ross S.M. Appld Prolt Modls wth optmzto Applctos S Frcsco Hold-D 97. [3]. Stdj W. d crm D. Optml Strtgs for som Rpr Rplcmt Modls Adcd Appld Prolt Vol. pp [4]. Stl A.D.J. O Gomtrc Procsss d Rpr Rplcmt Prolms Mcrolctrocs Rllt Vol.33 pp [5]. hgy.l. A Brt Optml Rplcmt Polc for Rprl Sstm Jorl of Appld Prolt Vol.3 pp [6]. hgy.l. A Optml Gomtrc Procss Modl for Cold Std Rprl Sstm Rllt Egrg d Sstm Sft Vol.63 pp [7]. hgy.l. Wg G.J. d J C. Rplcmt Prolms for Cold Std Rprl Sstm Itrtol Jorl of sstms Scc Vol.37 pp [8]. hgy.l. d Wg G.J. A rt optml rpr rplcmt modl sg gomtrc procsss for cold std rprl sstm Egrg optmzto Vol.385 pp

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Section 5.1/5.2: Areas and Distances the Definite Integral Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w