Laplace Transformation Techniques vs. Extended Semi-Markov Processes Method

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1 Innionl Jounl of Alid Sin nd hnology Vol. No.4; July Ll nsfomion hniqus vs. Endd Smi-Mkov Posss Mhod Abs Ling Hong ASA Ph.D. Assisn Pofsso of Mhmis nd Auil sin Dmn of Mhmis dly Univsiy 5 Ws dly Avnu Poi IL 665 USA. E-mil: lhong@bdly.du Phon: (9) Jyoimoy Sk Ph.D. Pofsso of Sisis Dmn of Mhmil Sins Indin Univsiy Pudu Univsiy Indinolis 4 N. lkfod S. Indinolis Indin 46 USA E-mil: jsk@mh.iuui.du Phon: (7)74-8 W onsid oninuously moniod on-uni sysm suod by n idnil s uni nd fly id by n in-hous gul i wihin dmind fid in im o by visiing i who ivs whn h sysm fils o whn h in im is ov nd who is ll fild unis. W dmons h diffiulis nd h shoomings of h Ll nsfomion hniqu nd how hs ovom by h ndd smi-mkov oss mhod. MSC: imy 95; sondy 6K Kywods: Ribl sysm; Ll nsfomion; Endd smi-mkov oss; Limiing vilbiliy. Inoduion In h liu mos ibl modls sudid by using h Ll nsfomion hniqu. (S fo ml Kumu nd nj (995) Oski nd Asku (97) Sn nd hhj (986) Sk nd Chudhui (999) Sidhn nd Mohnvdivu (998) Sidhn () Wng K nd L (7) Zhng nd Wng (7).) h mjoiy of hs modls ssum onnil lif-ims o/nd onnil i-ims. yilly on sblishs sysm of nwl-y quions whih ly on h mmoylss oy fom onnil disibuion. hn h Ll nsfomion hniqu is usd o solv h sysm. Nonhlss h sm sk n b omlishd mo ffiinly by using h mhod of smi-mkov oss (SMP). u nih mhod n ommod h mo lisi biy lifim disibuion. Rnly ih Hong nd Sk (S ih Hong nd Sk (9) ih Hong nd Sk ()) inodud n ndd smi-mkov oss (ESMP) mhod o solv sohsi ibl modls h llow biy lif- nd i- ims. hy nd h limiing obbiliy hom of n SMP o h of n ESMP. h oh is libl o wid sum of siuions. In his w m o sudy h ibl modl in ih Hong nd Sk () und biy lif nd ibl ims by mloying h diionl Ll nsfomion hniqu. W dmons only il suss: h Ll nsfomion hniqu divs foml soluion o h limiing vilbiliy hough h mhmil mniulion is qui omlid. Howv i sms o b fomidbl hllng o div h limiing ooion of ims h sysm snds in vious ss whih is ssnil fo ying ou os nlysis. On h oh hnd his modl n b solvd omlly using h ESMP mhod inluding dminion of h limiing ooion ims in h s. hus h EMSP mhod no only yilds h limiing vilbiliy mo onvninly bu lso i offs h oion o ondu os nlysis. h mining of his ognizd s follows. Sion inodus h modl. Sion sblishs h nwl-y quions. Sion 4 ovids foml soluion o h limiing vilbiliy using h Ll nsfomion hniqu. Sion 5 silizs h foml soluion o h s of h onnil lif- nd i- ims s n ml. Sion 6 summizs h soluion using h ESMP mhod. Sion 7 onluds h wih bif disussion.

2 Cn fo Pomoing Ids USA Modl Sing W onsid oninuously moniod wo-idnil-uni old sndby sysm. As soon s h oing uni fils i undgos i by n in-hous gul imn whil h s is ld on oion immdily. h gul i is llowd dmind fid in im o oml i. If h oing uni fils whil h oh uni is sill und i h sysm fils. A visiing i son is lld in s soon s ih h in im is ov o h sysm fils. h i filiy n ommod only on imn im nd h bnfi of il i don by h gul i is fofid whn h ks ov. Fo h i h is ll fild uni bfo h lvs. A fshly fild uni wis i whil h viously fild uni is bing id. h wo imn k sohsilly diffn mouns of im o oml i f whih h uni boms s good s nw. Viions of his modl hv bn sudid by svl uhos. (S fo ml Kum u nd nj (995) Sidhn nd Mohnvdivu (998) nd uj Ao nd nj (99).) hs s ssum h lifim is onnilly disibud. hy ly on h mmoylss oy of onnil disibuion nd us h Ll nsfomion hniqu o obin im-dndn vilbiliy libiliy busy iods fo h wo is nd ol ofi. On h oh hnd ih Hong nd Sk (9) shows h fo onnil lif- nd onnil i im disibuions h sdy-s obbiliis n b obind by ognizing h h oninuous im sohsi oss (CSP) is in f smi- Mkov oss (SMP). Nih mhod n ommod biy lif- nd i- ims. h mo lisi oblm is solvd in ih Hong nd Sk () whih sysmilly dvlos h ESMP mhod. Consisn wih h noion of ih Hong nd Sk (9) nd ih Hong nd Sk () w l dno h in im nd l h lifim X of n oing uni hv umuliv disibuion funion (CDF) F h i im Y by h gul i son hv CDF nd h i im Y by h i son hv CDF. L h osonding suvivl funions (sf) b F ndom vibls b bsoluly oninuous wih obbiliy dnsiy funions (df). L h of hs f g g sivly. W ssum ll lifims nd i ims sohsilly indndn. A ny insn h sus of uni is s (on sndby) (in oion) (und i by h gul i son) (und i by h o w (wiing i). Consqunly dnding on h sus of h wo unis h sysm is in on of h following fou ss: ( s ) ( ) ( ) nd ( w). h sysm is u in ss nd nd down in s. Sin h wo unis idnil in hi sohsi bhvio i is ilvn whih uni is on oion nd whih is und i. Also whn boh unis down hy id in h od in whih hy fild. = (s ) X Y Y = ( ) = ( ) X X Y = ( w) Figu : h shmi digm of nsiions

3 Innionl Jounl of Alid Sin nd hnology Vol. No.4; July Figu givs h shmi digm of nsiions fom on s o noh. A im h sysm ss in s. I sys h fo ndom duion X X nd hn i ns s. h sojoun im in s is h smlls mong Y X.h sysm movs o s if Y is h smlls o s if X is h smlls o o s if is h smlls. h sojoun im in s is h smll of Y nd X ~ wh X ~ quls X if h sysm hd ivd s fom s nd i quls X if fom s. Fom s h sysm movs o s ~ if Y X o o s ohwis. Fom s h sysm n only mov o s s soon s h finishs i on h uni h hs fild li. Hn h sojoun im in s is Y ~ whih quls Y if h sysm ~ hd ivd s fom s nd i quls Y X if fom s. Finlly h sojoun im in s ( fo h iniil s im ) is X X Y if h sysm hd ivd s fom s nd i is X X ~ Y if fom s~. Fom s h sysm lwys movs o s. (In his gh whv wo ndom vibls qud w mn h hy hv h sm CDF.). Rnwl-y Equions A full dsiion of h oninuous im sohsi oss quis doumning ly how long h uni is in oion/und i/wiing i. Howv w shll onn on n mbddd dis-im sohsi oss (DSP) by fousing nion o h ohs whn on uni is jus u on i (by h gul o h i son) nd h oh uni jus ss o o o o wi fo i. W shll no k k of oh ohs whn nsiion fom on s o noh ks l. o b is ou DSP ks k of ll ohs whn h sysm ns s ( ) (lbi fom s ( s ) ). u i omlly bysss ll ohs whn h sysm ns s ih fom s o fom s ( ) s h nssis king od of h ongoing oing im. h DSP ks ks of ohs whn h sysm gos down by ning s ( w) fom s bu i ignos ll ohs whn h sysm gos down by ning s fom s sin h quis king od of h ongoing i im. Finlly h DSP ks k of ohs whn h down sysm is vivd nd i ns s fom s bu i ignos ll ohs whn h sysm ns s fom s. y fousing nion on h bov-mniond DSP w bl o onsu sysm of nwl-y quions. Fo ) ( s )( )( )( ) l ( ) dno h obbiliy h h sysm is down unis of ( w ( im f h oh whn i jus ns s ). hn lly A( ) ( ).In od o find ssion fo ( ) w s jusify nd solv h following sysm of fou ingl quions s ( ( involving s ) ) ( ) nd w ( ). s ( ) ( (.) ( ). ( ( ( ) ( ( ) ( ) ( ) F( ) F( ) w w ( ) ( y) F( y) F( ) d ( ( ( ) ( y ) if if ( ) ( ( y) F( y) d ( y) F( ) ( ) (.) w( ) ( y) d ( y) ( ) (.4) s (.)

4 Cn fo Pomoing Ids USA h jusifiion fo ingl quions (.)-(.4) follows long simil lins. Fo ml w jusify (.) fo s follows: If h oing uni fils im ( ] dnding on whh o no h gul i son finishs i by im h sysm jus ns s o. his ouns fo h fis wo ingls on h igh hnd sid (hs). On h oh hnd if h filu hns im ( ] nd h i is oml duing ( ) whih hns wih obbiliy ( ) ( ) ( ) sin im h i son ls h gul on hn h sysm jus ns s im. his onibus h hid ingl on h hs. h fouh ingl on h hs oms fom h vn h h filu hns im ( ] nd h i is omld fwds im y ( ). Suly his vn imlis h h gul i son did no oml i by im whih hs obbiliy ( ) nd h i son finishd i in y ddiionl im. h vn lso imlis h h filu oud duing ( y] whih hns wih obbiliy F( y) F( ) ; nd h sysm jus ns s im y. Finlly h ls m on h hs ouns fo h vn h filu did hn somim in ( ) wih obbiliy F( ) F( ) bu h i ws no omld by h gul i son by im no by h in ddiionl im. No w my wi h ingl quion (.) in h following fom: ( ) ( ( ) wh F is shif of F( downwd by F ( ) nd sivly; h is 4 w ( y) F( y) F( ) d F( ) F( ) ( ( y ) ( ) ( ) wh if ( ) ( ) ( ) if if if his n b jusifid s follows: Whn y w hv d ( y ) nd [ F ( ) F( )]. So quion (.5) dus o (.). Whn h fis ingl on h hs of (.5) n b win s ( X ) ( ) whih by dfiniion of quls ( X ) ( ( ) ( ) ( ) ) (.5). Also no h fo so h h sond ingl on h hs of (.5) quls (. hus h w ) fis wo ingls on h hs of (.5) ogh qul h fis h ingls on h hs on (.). h mining ms on h hs of (.5) obviously qul h mining ms on h hs of in (.). 4. Soluion o A W us h f h h Ll nsfomion of onvoluion of wo funions quls h odu of hi Ll nsfomions. king Ll nsfomions in quions (.) (.) (.) (.4) w obin (by sussing h gumn s) s f (4.6) [ f] w[ f ] ( ) [ g F ] ( )[ ( F ) ] (4.7) [ f ] ( )[ gf] [ F ] (4.8) w ( ) g ( ) (4.9) g shifs of g ( ( of h igh by

5 Innionl Jounl of Alid Sin nd hnology Vol. No.4; July F g F( F( ) g ( ) ( ). W fis solv h sysm onsising of quions (4.7)-(4.9). Puing hos h quions in mi fom w hv w w wh [ f] [ f ] Solving his sysm fo ( )[ ( F ) ] ( )[ g F ] [ gf] [ F ] nd subsiuing i in (4.6) w obin [ f ] [ g ] (4.) s f (4.) s s No h s sa s) s A( ) d A( ) d A() ( A( ) d. Hn king limi s s ohs nd inhnging limi nd ingl w hv lim sa ( s) A( ) lim A( ) s Sin A( ) s ( ) w hv A ( s) s ( s). hfo s A( ) lim sa ( s) s lim s s s lim s s s ( s) ( s). 5. Eml Eml (Eonnil lifim onnil i ims). Suos h X Y Y hv onnil disibuions wih sl ms sivly. Wihou loss of gnliy w my ssum. In his s quion (4.) yilds s s s s s s s s s s 5

6 Cn fo Pomoing Ids USA s s s s s s s s s s. s Subsiuing hs in (4.) nd hn using (4.) w g A wh. his gs wih h suls in ih Hong nd Sk (9). 6. h ESMP Mhod. (5.) W dsib h ESMP mhod in gnl ms fis. hn w ly h ESMP mhod o h ibl modl sudid in his. Finlly w siliz i o Eml. Und h ssumion of biy lif- nd iims ofnims h CSP is no long n SMP bus h mbddd DSP h ks vy nsiion is no Mkovin. h is h sojoun im in s nd h nsiion obbiliis ou of his s my dnd on h vious s(s). Howv whn on n idnify noh DSP whih is Mkovin (h is boh h sojoun im in s nd h nsiion obbiliis ou of his s dnd only on h un s) hn h CSP is lld n ESMP. Fo n ESMP hom blow givs mhod of omuing h obbiliy h CSP snds in vious ss. W nd som noion. Suos h w hv n ESMP wih s s S suh h h DSP sid o only odd nsiions is osiiv un Mkov hin wih s s S S P. h nsiion obbiliis obind by lising ll ossibl hs i i... i m ss i i... i m P i j nd nsiion mi onsising of s of (his s my b my) h visis o whih no odd h h CSP n follow sing fom odd visi o s i o h h n odd visi o s j (whih my b h sm s s i). L Pi j dno h ondiionl obbiliy h sing fom odd visi o s i h CSP movs long h o mk h n odd visi o s j. Adding ov ll hs bwn odd visi o s i nd h n odd visi o s j w hv No h P fo ll js i j P i j Pi j (6.) j S sin P is sohsi mi. h siony disibuion fo his DSP sid o only odd nsiions is obind (s fo ml Ross (996) ) by solving h sysm of quions 6

7 Innionl Jounl of Alid Sin nd hnology Vol. No.4; July P (6.4) j is i i j js j No h i is h long un obbiliy h h nsiions n (o i) s i. L k i j dno h d im h CSP snds in s k i i i... i m s i movs fom odd visi o s i o h n odd visi o s j vi. hn w hv h following hom. (S ih Hong nd Sk () fo dild oof.) P P i j i js hom Fo n ESMP if h nsiion mi of visis h odd is osiiv un wih siony disibuion i i hn h limiing ooion of im h ESMP snds in ny s S k S (visi o whih my b odd o unodd) is givn by k P i jk i j (6.5) is js h ooionliy onsn is obind fom h onsin. hf h limiing vilbiliy whih is h long-un ooion of ims h sysm is u n b obind s A whu S onsiss of ll h u ss. ks k k ku Und h modl sudid in his h sohsi bhvio of h sysm is CSP wih s s { } of whih s is h only down s. Indd his CSP is n ESMP s lind in h n gh. Hn lying hom w obin ssions fo k h ooion of im h CSP snds in s k (k= ). hf w obin h limiing vilbiliy s A. (6.6) Likwis h limiing ooion of busy im fo h gul i is nd h fo h i is. Oh usful quniis n b omud using h s. How is ou CSP n ESMP? W mus hibi DSP h is Mkovin. Consid s odd nsiions hos ohs whn on uni is jus u on i (by h gul o h i son) nd h oh uni jus ss o o o o wi fo i. o b is ll ohs whn h sysm ns s (lbi fom s ) odd bu ll ohs whn h sysm ns s ih fom s o fom s unodd sin h sojoun im in s dnds on h g of h oing uni (whih quls h im sn in h vious s). Also ohs whn h sysm ns s fom s odd bu ohs whn h sysm ns s fom s no sin h sojoun im in s is h ddiionl im h nds o finish h i sh hs sd in s. Finlly ohs whn h down sysm is vivd nd i ns s fom s odd bu ohs whn h sysm ns s fom s no odd. hn h DSP sid o odd nsions only fom Mkov oss on S. h is boh h sojoun im nd h nsiion obbiliis bwn wo sussiv odd nsiions dnd only on h un s. Indd his is ly h sm DSP h w uilizd o dvlo h nwl-y quions of Sion. N w ly hom o ou ESMP. h nsiion hs h ondiionl obbiliis nd h ondiionl d sojoun ims h CSP snds in h s long hs hs givn in bl wh Z min X Y nd Z Y k Z Y X 7

8 Cn fo Pomoing Ids USA Z Y X Z X Y X Y X. bl : Abiy oninuous lif- nd i ims Ph Condiionl d im in s i j Pi j () P EX Y E Y () P EX Y E Y () P EX EY X P E X () P EX Y E Y () P E X EY X E N fom bl w g h nsiion mi fo h Mkovin DSP s P P P P P P P Solving (6.4) h siony disibuion fo h Mkovin DSP is Y wh D P P ( ). P D P D ( P D PP hf using hom nd h f h P EW EWI vn whih quls on nd on ) w hv E X Y I E X Y I E X Y I EY I EXI PZ EZ E Y I E X I Emin X Y Y X I EY X I EY E ) (wh I() is h indio funion of wh h ooionliy onsn is h sum of h ssions on h igh hnd sid. Eml Rvisid. Suos h X Y Y hv onnil disibuions wih sl ms sivly. hn bl silizs o h following bl. 8

9 Innionl Jounl of Alid Sin nd hnology Vol. No.4; July 9 bl : Eonnil lif- nd onnil i ims Ph Condiionl d im in s j i j i P () () () () () N using (6.7) i is sy o vify h D D D D wh. D Lsly using quion (6.6) w hv A onsisn wih quion (5.) obind using h Ll nsfomion hniqu. 7. Disussion In his w sudy sohsi ibl modl using h diionl Ll nsfomion hniqu. Using Mkovin DSP w bl o s u sysm of nwl-y quions (.)-(.4) nd obin foml soluion of h limiing vilbiliy. y silizing o h onnil lif- nd i ims w odu h limiing vilbiliy divd in ih Hong nd Sk (9). Howv o div h limiing ooion of ims sn in h s sms o b fomidbl hllng using his diionl oh. In gnl sing u suh nwl-y quions fo oh mo omlid sohsi modls is likly o b h hllnging nd solving h sysm omiss o b qui fomidbl.

10 Cn fo Pomoing Ids USA On h oh hnd hving obind h Mkovin DSP i is sy o ly h ESMP mhod inodud in ih Hong nd Sk (9) nd ih Hong nd Sk (). Asid fom omuionl onvnin h ESMP mhod yilds h limiing ooion of ims h CSP snds in h s. hfo w n no only obin h limiing vilbiliy bu lso obin h nssy ingdins fo ying ou os-bnfi nlysis. hus in gnl h ESMP mhod hs svl dvngs ov h Ll nsfomion hniqu. REFERENCES ih. Hong L. nd Sk J. (9). A sndby sysm wih wo ys of is. Alid Sohsi Modls in usinss nd Indusy; 6 (): ih. Hong L. nd Sk J. (). A sndby sysm wih wo is und biy lif- nd i ims. Mhmil nd Comu Modling; 5 (5-6): isws A. nd Sk J. (). Avilbiliy of sysm minind hough svl imf is bfo lmn o f i. Sisis & Pobbiliy Ls; 5 (): 5-4. Kum A. u S. K. nd nj. (995). Comiv sudy of h ofi of wo sv sysm inluding in im nd insuion im. Miolonis Rlibiliy; 6: Nkgw. nd Oski S. (974). Oimum vniv minnn oliis mimizing h mn im o h fis sysm filu fo wo-uni sndby dundn sysm. Jounl of Oimizion hoy nd Aliion; 4 (): 5-9. Oski S. nd Asku. (97). A wo-uni sndby dundn sysm wih i nd vniv minnn. J. Al. Pob.; 7: Ross S (996). Sohsi Posss. Wily: Nw Yok. Sk J. nd Chudhui. (999). Avilbiliy of sysm wih gmm lif nd onnil i im und f i oliy. Sisis & Pobbiliy Ls; 4: Sn P.K. nd hhj M.C. (986). Non-mi simos of vilbiliy und ovisions of s nd i. In: su A.P. (Ed.) Rlibiliy nd Quliy Conol}; Elsvi Amsdm Sidhn V. nd Mohnvdivu P. (998). Sohsi bhvio of wo-uni sndby wih wo ys of imn nd in im. Mhmil nd Comu Modling}; 8: 6-7. Sidhn V. (). Pobbilisi msus of dundn sysm wih wo ys of imn snsing dvi nd nlyil oh o find h oimum inhnging im. Innl Jounl of Quliy & Rlibiliy Mngmn; 7 (9): uj R. K. Ao R.. nd nj. (99). Sohsi bhvio of wo-uni sysm wih wo ys of imn nd subj o ommon insion. Miolonis Rlibiliy; : Wng K.H. K J.. nd L W.C. (7). Rlibiliy nd snsiiviy nlysis of ibl sysm wih wm sndbys nd R unlibl svi sions. In. J. Adv. Mnuf. hnol.; : -. Zhng Y.L. nd Wng.J. (7). A dioing old sndby ibl sysm wih ioiy in us. Euon Jounl of Oionl Rsh; 8: