JOINT IMPORTANCE MEASURES IN NETWORK SYSTEM

Transcription

1 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December JOINT IPORTANCE EASRES IN NETWORK SYSTE V Chaco and anoharan Deartment of Stattc StThoma Coege Thrur-68000Keraa Inda Deartment of Stattc nverty of Cacut Keraa Inda e-ma: chacovm@gmacom Abtract: any rea word ytem (eectrc ower tranortaton teecommuncaton etc) are muttate ytem comoed of muttate comonent n whch ytem reabty can be comuted n term reabte of t comonent Such ytem may be regarded a fow networ whoe arc (comonent) have ndeendent dcrete and mut-vaued random caacte An arc can at dfferent condton be characterzed by dfferent erformance eve caung networ ytem to wor wth dfferent eve of outut erformance The crtcaty of uch arc mut be meaured wth reference to ther erformance eve and reabty and t contrbuton to the overa ytem outut erformance meaure(op) In th aer we ntroduce a generazed concet of mortance meaure and ont mortance meaure for the fow networ made u of muttate arc wth reect to outut erformance meaure (exected erformance reabty and avaabty) An aroach baed on the unvera generatng functon (GF) for the evauaton of the rooed ont mortance meaure ntroduced An utratve exame gven Keyword: Networ reabty avaabty dcrete tate arc ont mortance meaure GF SC B5 90B5 Introducton Snce the very eary tme of reabty engneerng the networ reabty one of the man ubect of reearch The networ reabty theory ha been aed extenvey n many rea-word ytem uch a comuter and communcaton ytem ower tranmon and dtrbuton ytem tranortaton ytem o/ga roducton ytem etc [8] Networ reabty evauaton aroache exot a varety of too for ytem modeng and reabty ndex cacuaton Networ reabty robem are generay cafed baed on the method ued to tranfer the fow (or gna) and how the fow conervaton aw atfed Tycay there are two categore; the muttate arc networ (AN) and the muttate node networ (NN) In AN each arc ha a non-negatve nteger vaued dcrete random varabe caacty (muttate arc) and a fow n the networ obey the conervaton aw Aarenty n NN each node a muttate node wth dcrete tate determned by a et of node recevng the gna drecty from t wthout atfyng conervaton aw Both have ther own acaton; for exame eectrca ower dtrbuton ytem can be modeed by AN and comuter networ or ceuar hone networ can be modeed a NN The tandard mathematca and tattca theory of ytem reabty aume both ytem and comonent behavor are of bnary nature functonng (tate ) and faed (tate 0) [] However n ome ytem when comonent may be oeratng n a degraded tate the ytem may be oeratng n degraded tate and the ytem may t rovde an accetabe eve of ervce [] The networ reabty evauaton for comex degn ree on enumeratve technque [] The fow reabty robem for the drected caacted-fow networ n whch the caacty of each arc ha + vaue from ource to n generazed a a muttate ytem mode [0] A grah theoretc method ued for the reabty evauaton of muttage nterconnecton networ wth muttate eement [4] 9

2 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December Imortance meaure (I) quantfe the crtcaty of a artcuar comonent wthn a ytem degn They have been wdey ued a too for dentfyng ytem weane and to rortze reabty mrovement actvte [6] They can ao rovde vauabe nformaton for the afety and effcent oeraton of the ytem In muttate ytem (SS) I characterze for a gven comonent the mot mortant comonent tate wth regard to t mact on ytem reabty The nowedge about the I can be ued a a gude to rovde redundancy o that ytem reabty ncreaed Thu meaure that can dfferentate uch an mact are hghy derabe In genera there are two way to mrove the reabty of a bnary ytem ) ncreae the reabty of ndvdua comonent and/or ) add redundant comonent to the ytem Comote mortance meaure are deveoed wth the am of dentfyng and ranng artcuar arc (comonent) n a networ ytem deendng on ther mact on the muttate networ reabty behavor [3] Jont reabty mortance (JRI) of two or more comonent a quanttatve meaure of the nteracton of two or more comonent or tate of two or more comonent [5] It nvetgated to rovde nformaton on the tye and degree of nteracton between two or more comonent by dentfyng the gn and ze of t [55] The vaue of JRI rereent the degree of nteracton between two or more comonent wth reect to ytem reabty Jont tructura mortance (JSI) ued when the comonent reabte are not avaabe [5] Jont tructura and ont reabty mortance meaure for any number of muttate comonent n the SS are uefu for the degn engneer [5] [7] For the SS wth muttate comonent the robem reated to SS reabty mrovement t evovng The robem of fndng the ont mortance of more than two arc n a networ ytem wth varou outut erformance meaure (eg reabty avaabty etc) t reman unoved To ove th robem method deendent on the nformaton obtaned from muttate I and ont mortance meaure(ji) can be deveoed for effcent reource aocaton any of the engneerng ytem are modeed by networ (eectrc ower generaton ytem tranortaton ytem teecommuncaton networ ytem etc) (ee [7] [0] and [4]) hence the deveoment of ont mortance meaure of two or more arc wth dfferent outut erformance (eg roductvty caacty etc) n a drected networ wth muttate erformance eve qute derabe In order to anwer th robem we ntroduce the JI of two or more arc n muttate drected networ ytem wth varou outut erformance meaure (exected erformance reabty and avaabty) We rovde an agorthm baed on unvera generatng functon (GF) for the evauaton of ont mortance meaure when networ ytem ha dfferent outut erformance Th aer organzed a foow In ecton we defne the JI n networ ytem wth varou outut erformance meaure (exected erformance reabty and avaabty) Secton 3 conder the acaton of GF for the JI evauaton Iutraton gven n ecton 4 foowed by concuon n the at ecton Jont mortance meaure of arc n muttate networ ytem Conder a drected muttate networ made u of n arc Each arc may be n one of tate { 0 } { n } Let W (t) outut erformance of the muttate networ at tme t whch tae the vaue w =0 where max{ } deendng on the ytem tate at tme t The two vector of the ytem erformance reazaton w={w 0 } and of the ytem tate robabte ={ 0 } defne the ytem outut erformance dtrbuton Let (t) the tate of the SS at tme t We ue ome meaure of the erformance of a SS for obtanng ont mortance meaure The teady-tate the robabty dtrbuton of the ytem tate : m Pr{ (t) } m Pr{ W ( t) w } 0 t t 30

3 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December An aocated me meaure of ytem outut erformance t exected vaue of ytem tate defned n the teady-tate a: E [ ( t )] A mar meaure of ytem outut erformance t exected vaue of ytem outut erformance n the teady-tate a: E w When aed to SS the concet of avaabty reated to the abty of the ytem to meet a requred demand w correondng to tate The genera defnton of ntantaneou muttate ytem avaabty then: A( t) Pr{ ( t) } Pr{ W ( t) w} If the ytem under oeraton wthout brea u to tme t then A ( t ) the ytem reabty: R( t) Pr{ ( t) } Pr{ W ( t) w} The SS tatonary avaabty defned a A 0 (w w ) Let G ( N A) rereent a tochatc caacted networ wth nown demand d from a ecfed ource node to a ecfed n node t N rereent the et of a node and A { a n} rereent the et of a arc The current tate (caacty) of arc a rereented by x { 0 } the range of tate of arc a The vector ( 0 ) rereent the robabty aocated to each of the vaue taen by x The ytem tate vector x ( x x xn) denote the tate of a the arc of the networ ytem Functon ( x) : n where { 0 } max{ } ma the ytem tate vector nto ytem tate That (x) the networ caacty from ource to n under ytem tate vector x whch rereent a muttate tructure functon [] Networ reabty may be defned a the robabty that a demand of d unt can be ued from ource to n through the muttate arc We ha mae the foowng aumton for the networ reabty ytem Arc tate are tochatcay ndeendent The tructure functon (x) tattcay coherent That mrovng an arc erformance cannot caue to degrade the erformance of the networ ytem and a arc are reevant Jont reabty mortance (JRI) of the two edge n an undrected networ n bnary nature an extenon of the margna reabty mortance (RI) of edge [6] In an undrected networ reabty the robabty that ource and termna are connected by worng edge [6] For an undrected tochatc networ G ( N E) where E { e n} et of a edge N the et of node et R (G) rereent the robabty that the ource and termna are connected by worng edge and q ( q q qn ) where q Pr{ e E worng } RI of edge e n an R( G ) undrected networ defned a I G( ) [6] Agan JRI of two edge defned a foow q 3

4 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December Defnton The JRI of two edge e and e the econd order arta dervatve of reabty of an undrected networ wth reect to reabte of both edge: R( G) IG ( ) qq An exct exreon for th JRI of two edge * * * * ( ) R( G ) R( G ) R( G ) R( G ) I G * where G rereent G wth edge e contracted and edge e deeted The JRI exreed n term of RI of edge n ame ub-grah a IG ( ) I * ( ) IG ( ) G and IG ( ) I * ( ) IG ( ) G Aternatvey the foowng reatonh are obtaned IG( ) IG ( ) IG( ) IG ( ) IG ( ) and IG ( ) We now roceed wth the robem of meaurng ont mortance n the drected networ ytem wth reect to exected erformance Frt we fnd the JRI of any number of arc n the networ where the arc caacty rereented a fnte dcrete tate n nature That each arc can tae the vaue n a dcrete tate ace { 0 } where rereent the maxmum fow (bet tate) through the arc For fndng the JRI of more than two arc we foow the method for fndng JRI n SS [5] Suoe for ntance the robabty dtrbuton of each arc unnown then we ue the ont tructura mortance of muttate ytem (JSI) of more than two comonent [5] The JSI ( ) for two comonent and gven by JSI( ) m q ( ( m { SI(;m ) SI(;m )} ) X where SI ( ;m ) = n ( ) X ( m X ) q ) m m and X ( x x m xn ) the tate ace vector of ytem comonent Here χ(true)= and χ(fae)=0 ( m X ) ( m X ) q determne the crtca ath vector to the eve wth tate m of comonent The JSI () for three comonent JSI( r ) r nm { JSI( r;mn ) JSI( r;mn )} where JSI( r;mn ) SI( r;mn ) SI(r;m n ) So n order to fnd the JSI of three arc we have to fnd JSI of two arc for each tate of thrd arc and tae ucceve dfference and tota um Agan the change n JSI of three comonent wth fourth comonent rovde JSI of four comonent Thu roceedng e th we can fnd JSI of any number of arc Suoe that the arc robabte are nown Then to fnd the ont reabty mortance of more than two muttate arc for the networ one may roceed a foow The ont reabty mortance (JRI) of SS for comonent defned a foow [5] The ont reabty mortance of tate b of comonent a tate b of comonent a tate b of the comonent a ( n) of the SS 3

5 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December JRI( a a ; b b ) R a E b R b R a a n b where E the exected erformance of ytem and { } P( ( x ) ) 0 Ra b P( xa b ) are the comonent reabte Thu n fndng the JRI of muttate arc we have to fnd the th arta dervatve of the overa exected erformance of networ wth reect to the reabte of each arc under conderaton For ntance we conder the JRI of tate of three comonent Let Pmn P( ( m nr X r ) ) and m P( X m ) For =3 e dfferentatng E artay wth reect to and we get m 3 E [ P P mn m n rn P mn Pmn Pmn Pm n P m m n P m n rn ] [ P But oberve that JRI of tate of two comonent when thrd comonent n tate n Therefore JRI of tate three comonent are exreed n term of JRI of tate of two comonent a foow: 3 E E E [ ] n [ ] r n R m R R n R m R R m R r r R( G) JRI worng tate for edge can be wrtten a where R It how that the R R above reut hod wth bnary nature of edge e = Hence the reut of JRI of two edge n a bnary networ [5] can be condered a a generazaton of the reut JRI [6] to any number of bnary and muttate edge when conderng undrected networ ytem Thu we have the foowng theorem for three arc of a drected networ ytem Theorem The ont reabty mortance of three arc n a muttate networ wth muttate arc m n P m n P 3 r r E E ([ ] n r m n R m R R r n m n R m R E the exected outut erformance of networ R m R and Rrn where m n P m n E [ R m R ] ] n r are the reabte of arc and r wth reect to erformance eve m and n reectvey In the above dcued ont reabty mortance meaure and ont tructura mortance meaure we ued the exected erformance of the networ a outut erformance meaure But n order to fnd the JI wth reect to other outut erformance meaure reabty and avaabty of the muttate networ ytem we roceed a foow When the generc -th muttate arc condered one can ntroduce a erformance threhod and dvde the comete ordered et of t tate nto two ordered ubet correondng to the arc erformance above and beow the eve reectvey By o dong we re-ntroduce a coectvey bnary ogc for the arc tate Let arc be contraned to erformance beow whe the ret of arc of the networ ytem are not contraned: we denote by the networ ytem OP (reabty or avaabty) obtaned n th tuaton Smary we denote by the networ ytem OP reutng from the dua tuaton n whch arc contraned to erformance ) 33

6 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December above The networ ytem erformance meaure o ntroduced rey on a retrcton of the achevabe erformance of the networ arc Dfferent modeng aumton n the enforcement of th retrcton w ead to dfferent erformance vaue ng the meaure and we can defne Brnubaum mortance meaure for muttate eement Suoe that rereent the Brnubaum mortance of the comonent when comonent retrcted to the erformance above eve Smary defne the Brnubaum mortance of the comonent when comonent retrcted to beow eve Thu we can defne the ont mortance of two comonent and to the networ ytem erformance a Smary we can obtan the hgher order ont mortance meaure for more than two arc e for exame to meaure the mrovement of ont mortance of two arc wth reect to the nteractve effect of more than two arc at frt we ha cacuate change n the ont mortance of two arc wth reect to the change of thrd arc If there any change n the ont mortance of two arc due to change n erformance of thrd arc from uer tate to ower tate we can ay that there an nteractve effect for three arc for the networ OP mrovement We ha fnd the ont mortance meaure at teady tate ytem erformance n the foowng ecton 3 Acaton of GF In SS modeed by networ wth reect to varou outut erformance we modfy the above ont mortance meaure The GF found to be a uefu too n aement of outut erformance meaure of the networ ytem [] The method of GF generaze the technque that baed on ung a we nown ordnary generatng functon The bac dea are ntroduced by haov n 987 [] The aroach roved to be very convenent for numerca reazaton It requre reatvey ma comutatona reource for evauatng SS reabty ndce and therefore can be ued n comex reabty oeraton Imortance meaure evauaton n SS ung GF can be ee n Ref [9] The SS mode ncude the erformance dtrbuton of a arc and the ytem tructure functon: x n ( x x xn ) where any ytem eement can have fnte number of dcrete tate and t erformance dtrbuton rereented by ordered et x : ( x x x ) and : ( ) that reate the robabty of each tate wth erformance correondng to th tate The GF of a dcrete varabe X correondng to the tate of an arc defned a the oynoma ( ) 0 x where the dcrete random varabe X ha obe vaue and the robabty that X equa to x In order to rereent a the obe combnaton of tate of the two arc a and a one ha to reate the correondng robabte of tate of two muttate arc ubytem wth vaue of the vector ( xa xa ) n thee tate For thee uroe we conder a comoton oerator Ω over GF of ndvdua muttate arc whch tae the foowng form for a ar ( ) of muttate arc 34

7 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December ( ) ( ( ) ( )) ( x n n x n ) n n ( x xn ) The reutng oynoma ( ) rereent the robabty dtrbuton of the ubytem contanng arc and + Ayng the oerator Ω to a other arc one by one we get the reutng oynoma that tae the form n ( ) 0 The oynoma rereent dtrbuton of tate of connecton between ource and n of the entre networ Th oynoma reate the robabte of a the obe tate of whoe networ q wth the outut erformance correondng to thee tate y Thu we can obtan the reabty of networ a R 0 q I( demand of d unt u ed from to t ) where I () the ndcator functon To evauate the ont mortance meaure we need the teady tate dtrbuton of the oberved erformance of the networ ytem under ome contrant In order to ue the GF n ont mortance meaure evauaton we ue the foowng aroach Let O be the outut erformance of muttate networ ytem when arc n fxed tate whe the ret of the arc evove tochatcay among ther correondng tate wth teady-tate erformance dtrbuton { x } n 0 Aume that the arc n one of t tate wth erformance not greater than We denote by the tate n the ordered et of tate of arc whoe erformance x equa or mmedatey beow e x x The condtona robabty of the arc beng n a generc tate characterzed by a erformance X not greater than a re-ecfed eve threhod ) : x q y ( * x ] r 0 r Smary the condtona robabty of arc beng n a tate when t nown that x ] * Pr[ ] r r Now conder the ont robabty dtrbuton of two arc and for X x X x h gven four addtona retrcton () h h () h h (3) h h and (4) h h Thu now under the conderaton that the arc are ndeendent we coud arrve at robabty dtrbuton gven beow That can be comuted a earer reut for ndeendent arc a foow Let h x X x h h h ] h h r r 0 m 0 m 35

8 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December x X x h h h ] h h r m r 0 m h x X x h h h ] 3 h h h r r m 0 m and x X x h h h ] 4 h h r m r m h Smary we can fnd ont dtrbuton of any number of arc wth the ecfed retrcton We defne a the networ outut erformance meaure ( eg reabty or avaabty) obtaned when arc forced to vt ony tate wth erformance not greater than : O 0 Smary we defne a the networ outut erformance meaure obtaned under the condton that the arc tay ony n tate wth erformance greater than : O Thu the Brnubaum mortance tae the form In order to comute the ont mortance of two arc and e et denote the ont mortance of two arc wth reect to the networ outut erformance meaure then e h h ho h ho h 0h 0 0h h h 0 3 h O h 4 h h h O h 36

9 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December Smary by fndng change n ont mortance of two arc wth reect to thrd arc we get the ont mortance of three arc Contnung e th we get the ont mortance of any number of arc wth reect to networ outut erformance meaure and tate ace retrcted robabte of a arc In order to obtan the tate ace retrcted meaure one ha to modfy the GF of arc a foow ( ) ( ) 0 hh x x ( ) 0 ( h x x ) h0 h h x x h ( x x h h h ( ) h0 0 h0 ) h h x xh ( x x h) h h ( ) h0 h0 x h ( x x h x h h ( ) hh h h ) and x h x h h ( x x h ) ( ) 0 hh 0hh when evauatng GF of foowng agorthm for evauaton of and and We ue the Obtan the u-functon of a of the ytem eement If the ytem contan a ar of eement connected n arae or n ere reace th ar wth an equvaent macro-eement wth u-functon obtaned by um or mn oerator for ( ) If the ytem contan more than one eement do t agan and agan Then determne the u-functon of the entre ere-arae ytem a the u-functon of the remanng nge equvaent macro-eement The ytem robabty and erformance dtrbuton are rereented by the coeffcent and exonent of th u-functon correondng to the tate robabte and erformance eve reectvey Comute the ytem OP for the gven eve wth the gven vector of the tate robabte and erformance eve 4 Iutratve exame For the networ n fgure t dered to obtan the robabty that a demand of 0 unt can be ued from ource to n [3] Here the ytem can be condered a the SS n Ref [5] Tabe reent arc robabte In tabe we comuted JRI of ar of arc wth α= β= whch nfuence ytem mot wth reect to ytem outut met the demand or not 37

10 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December It how the ar (89) ha arget JRI wth reect to ytem reabty 4 Concuon In th aer ont mortance meaure of two or more arc n muttate arc networ wth varou outut erformance meaure are deveoed The rocedure of evauatng ont mortance meaure ung GF rooed The rooed meaure can be ued n any ytem modeed a muttate networ havng varou outut erformance meaure wth muttate arc Tabe Arc State State robabte Tabe Par () (89) (4) (6) JRI

11 V CHACKO ANOHARAN JOINT IPORTANCE EASRES IN NETWORK SYSTE RT&A # 04 (3) (Vo) 0 December Reference [] R E Barow and F Prochan Stattca theory of reabty and fe tetng Hot Rnheart NewYor975 [] R E Barow and A Wu Coherent ytem wth muttate comonent ath Oer Re 3 (978) 75-8 [3] W Brnbaum On the mortance of dfferent comonent n a mut-comonent ytem P R Krhnaah (Ed) utvarate anay II Acadamc Pre NewYor 969 [4] V C Bueno On the mortance of comonent for muttate monotone ytem Stat Prob Lett 7 (989) 5-59 [5] V Chaco and anoharan Jont mortance meaure for the muttate ytem Advance n Performance and Safety of Comex ytem A KVerma P K Kaur and S G Ghadge (Ed) acman Inda 008 [6] J S Hong and C H Le Jont reabty mortance of two edge n an undrected networ IEEE Tran Reab 47 (993) 97-0 [7] G Levtn Otma reabty enhancement for muttate tranmon networ wth fxed tranmon tme Reab Eng Syt Saft76 (00) [8] G Levtn Reabty of acycc muttate networ wth contant tranmon charactertc of ne Reab Eng Syt Saft 78 (00) [9] G Levtn and A Lnan Imortance and entvty anay of muttate ytem ung the unvera generatng functon Reab Eng Syt Saf 65 (999) 7-8 [0] Y K Ln ng mnma cut to tudy the ytem caacty for a tochatc fow networ n two-commodty cae Comuter and Oer Reer 30 (003) [] A Lnan and G Levtn uttate Sytem Reabty-aement otmzaton and evauaton Word Scentfc Sngaore 003 [] S Patra and R B ra Evauaton of robabty ma functon of fow n a communcaton networ conderng a muttate mode of networ n croereab36 (996) 45-6 [3] J E Ramrez-arquez and D W Cot uttate comonent crtcaty anay n muttate ytem Adv Saf Reab (Ed K Koowroc) Tayor and Franc grou London 005 [4] C R Thrathy S Patra R B ra and N ahaatra Reabty evauaton of muttate nterconnecton networ wth muttate eement croereab 36 (996)43-48 [5] S Wu Jont mortance of muttate ytem Com Indut Eng 49 (005) [6] W C Yeh The -out-of-n acycc muttate node networ reabty evauaton ung unvera generatng functon method Reab Eng Syt Saf 9 (006) [7] V Chaco and anoharan Jont mortance meaure for muttate reabty ytem OPSEARCH Voume 48 Number DOI: 0007/ z 39