Computing ll-terminl reliility of stohsti networks with Binry Deision Digrms Gry Hry 1, Corinne Luet 1, n Nikolos Limnios 2 1 LRIA, FRE 2733, 5 rue u Moulin Neuf 80000 AMIENS emil:(orinne.luet, gry.hry)@u-pirie.fr 2 LMAC, UTC, BP 20529 60205 COMPIEGNE Ceex emil:nikolos.limnios@ut.fr Astrt. In this pper, we propose n lgorithm se on Binry Deision Digrm (BDD) for omputing ll-terminl reliility. It is efine s the proility tht the noes in the network n ommunite to eh other, tking into ount the possile filures of network links. The effetiveness of this pproh is emonstrte y performing experiments on severl lrge networks represente y stohsti grphs. 1 Keywors: Network reliility, Binry Deision Digrm (BDD), Stohsti grph. 1 Introution A stohsti network is moele y n unirete grph G = (V, E) where V is the vertex set n E is the ege set. Sites orrespon to verties n links to eges. The ll-terminl reliility R(G) is the proility tht G remins onnete ssuming ll eges n fil inepenently with known proility n noes re perfet. Provn [Provn, 1986] showe tht even for plnr grphs this prolem is still NP-hr. In literture, two lsses of lgorithms for omputing the network reliility n e istinguishe. The first lss els with the enumertion of ll the minimum pths. The inlusionexlusion or sum of isjoint prouts methos hve to e pplie sine this enumertion provies non-isjoint events. The lgorithms in the seon lss re ftoring lgorithms improve y reutions. It onsists in reuing the size of the network while preserving its reliility. When no reution is llowe, the ftoring metho is use. The ie is to hoose omponent n eompose the prolem into two su-prolems: the first ssumes the omponent hs file, the seon ssumes it is funtioning. Stynryn n Chng [Stynryn n Chng, 1983] n Woo [Woo, 1985] hve shown tht the ftoring lgorithms with reutions re more effiient thn the lssil pth or ut enumertion metho for solving this prolem. This ws onfirme y the experimentl works of Theologou n Crlier [Theologou n Crlier, 1991]. 1 Aknowlegment: This reserh ws supporte y the Conseil Regionl e Pirie.
Computing ll-terminl Reliility using Binry Deision Digrms 1469 This pper is orgnize s follows. First, we give rief introution to BDD in Setion 2. Then, in Setion 3 we propose esription of our metho for omputing network reliility. In Setion 4, we introue n other importnt reliility mesure (Birnum importne mesure) n its fst omputtion vi BDD. Next, we present experimentl results in Setion 5. Finlly, we rw some onlusions n outline the iretion of futur works in Setion 6. 2 Binry Deision Digrm (BDD) Akers [Akers, 1978] first introue BDD for representing oolen funtion. Brynt populrize the use of BDD y introuing set of lgorithms for effiient onstrution n mnipultion of the BDD struture [Brynt, 1992]. Nowys, BDD re use in wie rnge of re, inluing hrwre synthesis n verifition, moel heking n protool vlition. Their use in the reliility nlysis frmework hs een introue y Mre n Couert [Couert n Mre, 1992] [Couert n Mre, 1992] n eveloppe y Ruzy [Ruzy, 1993]. Sekine n Imi were the first to use the BDD struture in network reliility [Sekine n Imi, 1998]. A BDD is irete yli grph (DAG) se on Shnnon s eomposition. The Shnnon s eomposition is efine s follows: f = xf x=1 + xf x=0 where x is one of eision vriles n f x=i is the oolen funtion f evlute t x = i. The grph hs two sink noes lele with 0 n 1 representing the two orresponing onstnt expressions. Eh internl noe is lele with oolen vrile x n hs two out-eges lle 0-ege n 1-ege. The noe linke y 1-ege represents the oolen expression when x = 1, i.e f x=1 while the noe linke y 0-ege represents the oolen expression when x = 0, i.e f x=0. An orere inry eision igrm (OBDD) is BDD where vriles re orere oring to known totl orering n every pth visits vriles in n sening orer. Afterwrs, BDDs will e onsiere s orere. Leves of the BDD give the vlue of f for the ssignment orresponing to pth from the root to the lef. The size of BDD struture epens ritilly on vrile orering. Fining n orering tht minimizes the size of BDD is lso NP-omplete prolem [Friemn n Supowit, 1990]. 3 Computing ll-terminl reliility Definitions n nottions A grph G is onnete if there exists t lest one pth etween ny two verties. Our network moel is n unirete stohsti grph G = (V, E).
1470 Hry et l. x 1 x 2 x 3 f 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 x 1 x 3 () x 2 x 3 x 2 x 1 () Fig.1. Funtion f(x 1, x 2, x 3) = (x 1 x 3) (x 2 x 3) represente y its truth tle n BDDs with orer: () x 1 < x 2 < x 3 n (): x 3 < x 2 < x 1. A she (soli) line represents the vlue 0 (1). Eh ege e i of E (i {1, 2,..., m} where m = E ) n fil inepenently with known proility q i (p i = 1 q i is the funtioning proility of e i ) n we onsier tht verties of G re perfetly relile. A stte G of the stohsti grph G is enote y (x 1, x 2,..., x m ) where x i stns for the stte of ege e i, i.e, x i = 0 when ege e i fils n x i = 1 when it funtions. The ssoite proility of G is efine s: Pr(G) = m (x i.p i + (1 x i ).q i ) i=1 At eh stte G is ssoite prtil grph G(G) = (V, E ) suh tht e i E if n only if e i E n x i = 1. The ll-terminl reliility n e efine s follows: R(G) = Pr(G) G(G) is onnete We enote y G e the grph G with ontrte ege e n y G e the grph G with elete ege e. Constrution of the ll-terminl reliility funtion Our lgorithm follows three steps: 1 The eges re orere y using heuristi. 2 The BDD is generte to enoe the network reliility. 3 From this BDD, we otin the ll-terminl reliility. We pply reursively the ftoring lgorithm in the orer of e 1, e 2,..., e m in top-own wy. The omputtion proess n e represente s inry tree suh tht the root orrespons to the originl grph G n hilren orrespon to grphs otine y eletion /ontrtion of eges. Noes in
Computing ll-terminl Reliility using Binry Deision Digrms 1471 the inry tree orrespon to sugrphs of G. We use the metho introue y Crlier [Crlier n Luet, 1996] for represententing grph y prtition. It is n effiient wy for representing grph n fining isomorphi grphs uring the omputtion proess. By shring the isomorphi sugrphs n expnsion tree is moifie s roote yli grph (therefore BDD). Shring isomorphi grphs Consier tht E k = {e 1, e 1,..., e k } n E k = {e k+1,..., e m }. The grphs in the k-th level of the BDD re su-grphs of G with the ege set E k. For eh level k, we efine the ounry set F k s vertex set suh tht eh vertex of F k is inient to t lest one ege in E k n one ege in Ēk. Then we gther verties in loks oring the following rule: two verties s n t of F k re in the sme lok if n only if there exists pth me of funtioning eges linking s to t. For instne in figure 3(), in the first level, the ounry set is equl to {, }. G e1 n e represente y prtition [] n G e1 y prtition [][]. Now, we orer prtitions in the sme level k in orer to ientify n stok them in n effiient wy. We numer the prtition from 1 to Bell( F k ) where Bell( F k ) (known s the Bell numer) is the theoretil mximum numer of prtitions in level k. This numer grows exponentilly with i, onsequently the numer of lsses grows exponentilly with the size of the ounry set. From now on, we only mnipulte prtitions inste of grphs uring the ll-terminl reliility omputtion. e2 e2 e1 e3 e4 e1 e3 e4 e3 e4 e5 e5 e5 G = (V, E) G 1 =< 1, 0, 1, 1, 1 > () G 2 =< 0, 1, 1, 1, 1 > Fig.2. G(G 1) n G(G 2) represent su-grphs in level 2 in the omputtion proess illustre in figure 3(). G(G 1) n G(G 2) hs the sme prtition: [][] uring the omputtion. e i = 1 mens the stte of e i is not yet fixe. () All-terminl reliility omputtion In the previous setion, BDD of the ll-terminl reliility funtion ws onstrute. The BDD n e reognize s grph-se set of isjoint prouts. Bse on the isjoint property of this struture, we n now esily ompute the ll-terminl reliility of G. Given the non-filure proilty p k
1472 Hry et l. F 0 = [ ] x 1 F 1 = {, } [] [][] x 2 x 2 e 2 F 2 = {, } [][] x 3 e 1 e 3 e 4 F 3 = {, } [] [][] x 4 x 4 e 5 F 4 = {, } [][] x 5 G=(V,E) () Fig.3. Grph G n its BDD (). A she (soli) line represents the vlue 0 (1). () illustrtes the omputtion proess of the BDD. () (k {1, 2,..., m}) of ege e k, the ll-terminl reliility of BDD-se funtion f n e reursively otin y: R(G) = Pr(f = 1) = Pr(x k.f xk =1 = 1) + Pr( x k.f xk =0 = 1) (isjoint property) R(G) = Pr(f = 1) = p k.pr(f xk =1 = 1) + q k.pr(f xk =0 = 1) (inepenent property) The reliility is evlute y trversing the BDD from the root to the leves. 4 Importne mesure Fining the ritil omponents is lso n importnt issue for reliility nlysis n the optimiztion esign of network topology. The im is to otin informtion onerning omponent s ontriution to the system reliility. The three most use importne mesures re: Birnum, Critilly n Fussell-Vesely. We riefly explin here the Birnum importne mesure. The Birnum importne mesure of omponent e k is the proility tht system is in ritil stte with respet to e k n tht the filure of omponent e k will then use the system to fil. Here, the Birnum importne mesure of ege e k, note Ik B, is efine s: I B k = Pr(f x k =1 = 1) Pr(f xk =0 = 1) The figure 4 shows the importne mesures for the reliility grph G. 5 Experimentl results Computtions re one y using Pentium 4 with 512 MB memory. Our progrm is written in C lnguge. The experimentl results re shown in Tles 1 n 2. The unit of time is in seon. The running time inlues the BDD genertion n the ll-terminl reliility omputtion. The heuristi
Computing ll-terminl Reliility using Binry Deision Digrms 1473 e 2 e 1 Grph G = (V, E) e 5 e 6 e 9 e 8 e 7 e 10 e 3 e 4 orering: e 5,e 1,e 2,e 9,e 10,e 4,e 6,e 3,e 8,e 7 e k qk e.3 e 2 0.3 e 3 0.3 e 4 0.4 e 5 0.05 e 6 0.4 e 7 0.4 e 8 0.2 e 9 0.3 e 10 0.3 Ik B 0.2849 0.2849 0.1611 0.2285 0.3534 0.1872 0.1497 0.1543 0.2625 0.2625 Fig. 4. Sensiility nlysis of grph G. Aoring to the Birnum importne mesure, e 5 hs the highest egree of ontriution to the grph reliility. use for orering eges (n so vriles in BDD) in the experiments is known s reth-first-serh (BFS) orering. We give two hrteristis of the generte BDD: its size (numer of noes) n its with (if W i is the numer of noes in the ith level then the with is: mx i W i ). F mx orrespons to the mximl size of the ounry set uring the omputtion proess. The omputtion spee hevily epens on F mx n so the ege orering. 6 Conlusion A metho for evluting the ll-terminl reliility vi BDD hs een propose in this pper. Bse on this pproh, our futur works will fous on omputing other kins of reliility n reusing the BDD struture in orer to optimize esign of network topology. Referenes [Akers, 1978]B. Akers. Binry eision igrms. IEEE Trns. On Computers, vol. C-27:509 516, 1978. [Brynt, 1992]R. E. Brynt. Symoli Boolen mnipultion with orere inryeision igrms. ACM Computing Surveys, 24(3):293 318, 1992. [Crlier n Luet, 1996]J. Crlier n C. Luet. A eomposition lgorithm for network reliility evlution. In Disrete Applie Mthemtis, volume 65, pges 141 156, 1996. [Couert n Mre, 1992]O. Couert n J. C. Mre. Impliit n inrementl omputtion of primes n essentil primes of oolen funtions. In Proeeings of the 29th ACM/IEEE Design Automtion Conferene (DAC 92), pges 36 39. IEEE Computer Soiety Press, June 1992. [Couert n Mre, 1992]O. Couert n J. C. Mre. A new metho to ompute prime n essentil prime implints of oolen funtions. In Avne Reserh in VLSI n Prllel Systems, pges 113 128, Mrh 1992.
1474 Hry et l. type n m time size with F mx K08 8 28 0.03 2745 405 7 K09 9 36 0.06 10265 1265 8 K10 10 45 0.13 39856 3925 9 K11 11 55 0.52 160793 15105 10 K12 12 66 2.14 673934 652 11 K13 13 78 9.97 2932248 279981 12 K14 14 91 50.00 13227624 1191235 13 K15 15 105 490 61780095 5021561 14 Tle 1. Benhmrk on omplete grphs type n m time size with F mx 6x6 49 84 0.15 39523 858 8 7x7 64 112 0.7 179410 2860 9 8x8 81 144 3.16 797916 9724 10 9x9 100 180 14.75 3495491 33592 11 10x10 121 220 67.94 15137188 117572 12 15x10 176 325 101.2 33360848 117572 12 20x10 231 430 101.2 24249018 117572 12 11x11 144 264 321.3 64959137 416024 13 Tle 2. Benhmrk on lttie grphs [Friemn n Supowit, 1990]S. J. Friemn n K. J. Supowit. Fining n optiml vrile orering for inry eision igrms. IEEE Trns. On Computers, vol. C-39:710 713, 1990. [Provn, 1986]J.S. Provn. The omplexity of reliility omputtions on plnr n yli grphs. SIAM J. Computing, 15(3):694 702, 1986. [Ruzy, 1993]A. Ruzy. New lgorithms for fult tolernt trees nlysis. Reliility Engineering n System Sfety, pges 203 211, 1993. [Stynryn n Chng, 1983]A. Stynryn n M.K. Chng. Network reliility n the ftoring theorem. Networks, 13:107 120, 1983. [Sekine n Imi, 1998]K. Sekine n H. Imi. Computtion of the network reliility (extene strt). Tehnil report, Deprtment of Informtion Siene, University of Tokyo, 1998. [Theologou n Crlier, 1991]O. Theologou n J. Crlier. Ftoring n reutions for networks with imperfet verties. In IEEE Trns. on Reliility, volume 40, pges 210 217, 1991. [Woo, 1985]R.K. Woo. A ftoring lgorithm using polygon-to-hin reutions for omputing k-terminl network reliility. In Networks, volume 15, pges 173 190, 1985.