Dependence In Network Reliability

Transcription

1 Depedece I Network Relablty Nozer D. Sgpurwalla George Washgto Uversty Washgto, DC, U.S.A. ozer@reasearch.crc.gwu.edu Abstract It s farly easy to calculate the relablty of a etwork wth depedet odes va the use of a techque called pvotg. However whe there s depedece, ths calculato wll prove accurate ad a model for the depedece s requred. Ths paper cosders the ssues volved developg a sutable model. Partcular emphass s placed o esurg the calculatos ad dstrbutos volved do ot become tractable for large etworks. Varous potetal dstrbutos are dscussed. Strateges are suggested for smplfyg the depedece. A model for cascadg falures s proposed. Keywords: Network relablty, relablty fucto, depedece, codtoal depedece, multvarate Beroull dstrbutos, cascadg falure. Relablty Uder Idepedece. Structure Fucto Cosder the brdge etwork of fgure ; t has fve odes labeled,,,. Let the dcator varable ( ) = f ode s fuctog at tme ad ( ) = otherwse, for some fxed tme. Let ( ) = f the etwork s fuctog at tme ad ( ) = otherwse. The etwork s sad to be fuctog f t s able to provde a flow from the source to the sk. Arrows dcate the drecto of the flow. Source Fgure. Brdge Structure Network Sk Clearly (,,, ), say, for VRPHQRQGHFUHDVLQJIQFWLRQ RIWKH s. Ths fucto s called the structure fucto. Let, = C = K, ad = = ( ) = The for the brdge etwork structure t ca be easly verfed that,, or ) = ( )( )( ) or () ( )( ) + ( )( ). Gve that there are several dfferet ways of expressg the structure fucto, a mportat ssue s whch oe of the dfferet forms s to be preferred.. Relablty Polyomal Let P( = p ) = p = E ( p ), the p s the relablty of ode. The relablty of the etwork ca be smlarly expressed as P ) = ). Suppose that for x =,, = x p,, p ) = P( = x p ) = = = P( = x,, p ;.e. the s are codtoally (gve the p s) mutually depedet. The the relablty of the etwork, gve p = (p,,p ) s, P( Φ ( ) = p) = h( p) = E( ) p) ; h(p) s kow as the relablty polyomal. The key message here s that t s oly uder the codtoal depedece of the ode dcator varables, that the relablty of the etwork s a fucto of the ode relabltes aloe. Whe the s are depedet we eed to kow more tha the p s so that the etwork s relablty s h(p,t), where t ecapsulates the depedece. 98

2 Let us retur to our brdge structure, ad let us suppose that the s are mutually depedet, wth p, =,, kow. We shall cosder usg the forms of the structure fucto () to derve the relablty polyomal. The frst form does ot gve us a aswer sce clearly, h( p) p p C p p C p p p C p p p, as the evets ( = ) ad ( = ) both volve ad so are ot depedet. Smlarly we caot use the secod form to calculate the relablty polyomal, sce t cludes the terms ( C ) ad ( C C ), whch are depedet. However t s possble to use the thrd form, h( p) = p( pc p )( p C p ) + ( p)( p p C p p ). The message here s that the assumpto of ode depedece does ot automatcally smplfy a evaluato of the relablty polyomal. The thrd form, whch does lead drectly to the relablty polyomal, s tself foud by pvotg [] o ode. Ths process ca be performed for ay ode ad repeated for dfferet odes utl the result leads drectly to the relablty polyomal. For stace pvotg frst o ode stead of ode results a structure fucto that cludes a term volvg ad, so that the process would eed to be repeated o a further ode. We must detfy the rght odes to pvot upo. A mportat problem s, for large etworks how ca we effcetly detfy the odes to pvot o? Relablty Uder Depedece. Beroull Dstrbutos Suppose that the s of the brdge structure are depedet. I order to corporate ths depedece to our evaluato of the relablty polyomal, we eed to have a -varate Beroull dstrbuto. Ths leads to a umber of questos cocerg ths dstrbuto.. How shall we choose a verso?. What happes f the etwork has may, say odes?. Does depedece matter ad how does t mafest tself?. Ca we smplfy or approxmate? I order to choose a verso of mult-varate Beroull dstrbuto, we eed to develop models for depedet lfe-legths. The followg are examples:. Freud s Bvarate Expoetal []. Marshall ad Olk s Multvarate Expoetal []. The Multvarate Pareto of Ldley ad Sgpurwalla [] v. The Multvarate Webull/Gamma v. A Multvarate Expoetal of Sgpurwalla ad Yougree [] There are a umber of resultg questos ad problems lked to these dstrbutos.. Whch of them are realstc?. Whch are tractable (some have sgulartes). Whch ca be geeralzed?. The umber of parameters requred explode for some dstrbutos as the umber of odes creases.. Noe of the models are approprate for descrbg cascadg falures The last problem requres a aswer to the questos, what s meat by cascadg falure? Ca ths oto be made precse? For etworks wth may odes, the assumpto of depedece betwee all the odes leads to specfcatos that are cumbersome, ad calculatos that are umaageable, f ot mpossble. However t s mportat for depedece to be take to accout. It s well kow that depedece uder-estmates system relablty for seres systems ad over-estmates t for parallel systems, whe ode dcators are postvely depedet. I order or cosder depedece a way that s maageable we cosder three strateges.. Strategy : Codtoal Idepedece.. M-Path Depedece Assume that t s oly the odes a m path set that are depedet. A set of odes s a m-path set f the etwork wll be workg f all the odes the set are workg ad may ot be workg f all but oe of the odes the set 98

3 are workg. Thus the example of the brdge etwork, the m path sets are: (,) (,), (,,) ad (,,) 8VLQJ WR GHQRWH GHSHQGHQFH DQG Ø to deote depedece we have,,, etc. but Ø ad Ø. Whle ths wll smplfy calculato of the relablty of the etwork, t ca stll become horredous. For example, wth the brdge etwork pvotg o ode, ad suppressg all probabltes, P(,..., E( E( E( ) + E( ) = ) = E ) E( [ )] ) + E( ) + E( = ) + E( ) E( Sce ad E( ) = P( = ad = ) we wll eed a bvarate Beroull model for evaluatg E( ). It wll have parameters oe each for P( = ), P( = ) ad P( =, = ). Smlarly we wll eed a bvarate Beroull model for E( ). However for evaluatg E( ) we wll eed to have a trvarate Beroull model wth 8 parameters. Ths s maageable, but cumbersome. If a m-path set cossts of odes, we eed to have a -varate Beroull model, whch would etal parameters... Further Smplfcato Oe way to smplfy matters further s to order the etres each m-path set the drecto of the etwork, ad assume depedece gve the terveg adjacet eghbors. The drected m-path sets are: (,) (,), (,,) ad (,,). Thus the assumpto gves us Ø ad Ø. I geeral for a drected m-path set (,,, k-, k ) we assume that ( Ø k, k- ) Wth such a assumpto of codtoal depedece, we ca show that E( ) = P( = = ) P( = = ) P( = so that all that s eeded s a par of bvarate bomals. I order to evaluate E( ) ad E( ) we eed several trvarate ad bvarate Beroull dstrbutos. Thus for the brdge structure etwork wth codtoal depedece all the drected m-path sets, the ). ) ) ). dmeso of the largest multvarate Beroull dstrbuto s, as opposed to wthout the assumpto. However, for the multple brdge etwork gve Fgure below, a evaluato of the relablty polyomal volves computg 9 expectatos, the largest oe beg E( 6 7 ). To evaluate ths expectato requres a -varate Beroull dstrbuto. 6 Fgure. Multple Brdge Network 7.. Costructo of Multvarate Beroull Dstrbutos. A smple way to proceed s to costruct bary versos of well-kow multvarate falure dstrbutos. For example: Marshall ad Olk s Multvarate Expoetal Model: For =, the umber of parameters = =, " " " " = 7 =, " " " " = = N, " " " " = N - Thus the model s ot scalable. Multvarate Pareto Model: For =, the umber of parameters = =, " " " " = =, " " " " = 6 = N, " " " " = N+ Thus, ths model s more scalable tha the oe by Marshall ad Olk, but s t realstc, ad ca we develop falure models that are more scalable? 98

4 . Strategy : Exchageablty Exchageablty mples a mld form of depedece. We may therefore costruct a exchageable sequece of Beroull trals va De Fett s represetato theorem as: P x x ( π = x,..., = x ) = p ( p) ( p) dp where (p) s a beta desty o (,). Ths stll leaves us wth the questo as to whether ths s realstc. Also t does ot corporate the oto of cascadg. S ( u) = θue F θu Its falure rate s: h F + e θu ( u) = θ u /(θu + ); ; u. Its mea tme to falure s: -.. Cascadg Falures Suppose that Freud s model, we troduce a threshold, say, ad suppose that the falure rate of the survvg ode s for t [t, t + ], ad for t > t.. Strategy : Neural Net If the exact relablty s ot eeded the t may be possble to use a eural et approxmato. Ths wll avod tractable mathematcs, but the cocept lacks a mathematcal foudato. FAILURE RATE PROFILE d. Summary Evaluatg the relablty polyomal of a etwork wth ode depedece opes up several problems, all of whch are more computatoal tha coceptual. The oly coceptual problem s the oto of cascadg. A Model For Cascadg Falures. Causal Falures Cosder a -ode parallel redudat system. Suppose that the falure rate of each ode whe both odes are operatve s >. Whe the frst ode fals, say at t, the falure rate of the ode permaetly creases to. Thus falure of oe compoet causes the falure rate of the other compoet to crease, other words, the system expereces causal falure. The model descrbed here s due to Freud, who dd ot put t the cotext of causal falures. t t + Fgure. Falure Rate For Cascadg Falure Model TIME It ca be show that for ths cascadg falure system, that the relablty s: S θu θu θue + e, θu ( u+ δ ) θ θue δ + e e ( = C u) θu ad, that the falure rate fucto s: h c θ u /(θu + ), ( u) = somethg ugly, u < δ u δ ; The mea tme to system falure s:, u < δ u δ θδ + e. θ θ FAILURE RATE PROFILE Note that the mea s whe δ ad s whe θ θ δ. Thus the mea s sestve to threshold. Uder depedece of the odes we have: t TIME I I θu S ( u) = (e h ( u) = θ ( e ) e θu θu ; ) /( e θu ); ad Fgure. Falure Rate For Causal Falure Model The relablty of the causal falure system s: 98 the mea s. θ

5 Comparso of the relablty fuctos shows that causalty ad depedece boud cascadg, ad that depedece overestmates relablty. Ths s llustrated fgures ad I geeral ca we develop a theory for the stochastc behavor of cascadg falures? INDEPENDENCE CASCADING CAUSAL Refereces [] Barlow, R. E. ad Proscha, F. Statstcal Teory of Relablty ad Lfe Testg, Holt, Rehart ad Wsto, Ic., New York, 97. [] Freud, J. E., A Bvarate Exteso of the Expoetal Dstrbuto, Joural of the Amerca Statstcal Assocato, Vol 6, pp , Fgure. Survval (Relablty) Uder Causal, Cascadg ad Idepedece CAUSAL CASCADING INDEPENDENCE [] Marsall, A. W. ad Olk, I. A Multvarate ExpoetalDstrbuto, Joural of the Amerca Statstcal Assocato, Vol 6, pp. -, 967. [] Ldley, D.V. ad Sgpurwalla, N.D., Multvarate Dstrbutos for the Lfe Legths of Compoets, Joural of Appled Probablty, Vol, Part, pp. 8-. [] Sgpurwalla, N.D. ad Yougre, M. A., Multvarate Dstrbutos Iduced by Dyamc Evromets, Joural of Statstcs, Vol, pp. -6. Fgure 6. Falure Rates Uder Causal, Cascadg ad Idepedece. Ope Problems There are a umber of ope problems that our cocept of cascadg falures leads to.. Geeralze results to cover mult-ode etworks.. Cosder falure rates that are mootoc creasg.. Cosder radom thresholds.. Istead of, let the falure rate crease to C, where C s radom ad has a dstrbuto dexed by the umber of faled odes at the tme of the jump.. Are the relablty bouds vald geeral? 6. Ca the falure tme pots a etwork experecg cascadg falures be descrbed by a clustered pot process? 98