Reliability Analysis of Sparsely Connected Consecutive-k Systems: GERT Approach

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1 Relably Aalyss of Sparsely Coece Cosecuve- Sysems: GERT Approach Pooa Moha RMSI Pv. L Noa-2131 poalovely@yahoo.com Mau Agarwal Deparme of Operaoal Research Uversy of Delh Delh-117, Ia Agarwal_maulaa@yahoo.com Kawar Se c/o Deparme of Sascs Uversy of Delh Delh-117, Ia Kawarse25@yahoo.com Absrac Durg he las quarer ceury, a mass of research has bee vesgae o cosecuve--ou-of-:f sysems, a s varous geeralzaos, aressg problems of relably compuao. Beses, several ew relably moels by combg he -ou-of- moel wh oher cosecuve--ou-of moels have also bee propose. Whe he umber of worg compoes bewee he wo falures s a maxmum, he he wo fale compoes are calle cosecuve falures wh sparse, Zhao e al. [8]. I hs paper, Graphcal Evaluao a Revew Techque (GERT has bee apple o suy cosecuve--ouof- -ou-of- :F sysems wh sparse, m-cosecuve--ou-of:f sysems wh sparse, a (, f, :F sysems wh sparse. Keywors- Cosecuve--ou-of-:F sysems wh sparse, M cosecuve--ou-of-:f sysems wh sparse, (, f, :F sysems wh sparse, ree srucure, graphcal evaluao a revew echque, Maso s rule. A. Noaos umber of compoes he sysem mmum umber of cosecuve fale compoes wh sparse causg sysem falure m umber of o-overlappg rus of cosecuve fale fale compoes wh sparse causg sysem falure f umber of fale compoes p, q probably ha a compoe s worg, fale T ree srucure x smalles eger greaer ha or equal o x v umber of chl oes possble from each pare oe of T L maxmum level he ree ha s he sysem has L + 1 levels cae by =, 1, 2,..., L. x a oe of he ree srucure T Π( x pare oe of x H( x se of chl oes of x =, value of he l from a pare oe a level 1 h o s chl oe a level, x = 1, 2,..., v z W, (x geerag fuco of he wag me for he occurrece of sysem falure a roo oe x a level W, z x geerag fuco of he wag me for he h occurrece of sysem falure a chl oe, eoe by x a level, of a pare oe Π( x a level 1 F (, falure probably of cosecuve- -ou-of- :F sysems wh sparse F m (,,m falure probably of m-cosecuve- -ou-of :F sysems wh sparse F (, falure probably of (, :F sysems wh sparse. I. INTRODUCTION Cosecuve- sysems frs sue by Kooleo 198 [4], have arouse grea eres relably leraure. I has we applcaos mcrowave saos of a elecom ewor, ol ppele sysems, mlary sysems, vacuum sysems elecro acceleraors, ec. A survey of cosecuve- sysems a s varous geeralzaos, alog wh mul-sae moels, ca also be fou Chag e al. [2], a Kuo a Zuo [5]. Recely, Zhao e al. [8] propose several ew relably moels by exeg exsg cosecuve- sysems o sparsely coece cosecuve- sysems: cosecuve- - ou-of- :F sysems wh sparse, m - cosecuve- -ou-of :F sysems wh sparse, a (, :F sysems wh sparse usg fe Marov Cha mbeg echque. I hs paper relably evaluao of he above hree geeralze cosecuve- sysems wh sparse, have bee sue usg GERT. The compoes are assume o be epee a ecal. For he relably evaluao of frs wo sysems a rec formula s prove whereas for he hr sysem a algorhm s geerae /9/$ IEEE 213

2 II. BRIEF DESCRIPTION OF GERT, AND MASON S RULE GERT s a proceure, whch combes he scples of flow graph heory, MGF a PERT o oba a soluo o sochasc ewors havg logcal oes, a rece braches. The resuls ca be obae base o MGF usg Maso s rule, whch aes care of all he possble proucs of rasmaces of o-ersecg loops. The rasmace of a arc a GERT ewor s he correspog W-fuco. I s use o oba he formao of a relaoshp, whch exss bewee he oes. If we efe W ( s r as he cooal W fuco assocae wh a ewor whe he braches agge wh a z are ae r mes, he he equvale W geerag fuco ca be wre as whch mples ha, r= r W ( s, z = W ( s r z (1 r r W (, z = W ( r z = ξ ( r z ( W ( r ξ ( r r= r= W (, z = (2 The fuco s he geerag fuco of he wag me for he ewor realzao. Maso s Rule (Whehouse [7], pp : I a ope flow graph, wre ow he prouc of rasmaces alog each pah from he epee varable o he epee varable. Mulply s rasmace by he sum of he o-ouchg loops o ha pah. Sum hese mofe pah rasmaces, a ve by he sum of all he loops he ope flow graph yelg rasmace T as [ (pah * oouchg loops ] T= (3 loops where loop ( frs orerloops + ( seco orerloops... oouchg loops= 1 ( frs orer oouchg loops + ( ( secoorer oouchg loops hr orer oouchg loops +... For more ecessary eals abou GERT, oe ca see Whehouse [7], Cheg [3], Agarwal e al. [1], a Moha [6]. III. BRIEF DESCRIPTION RELIABILITY EVALUATION OF CONSECUTIVE- SYSTEMS WITH SPARSE Moel 1: Cosecuve- -ou-of- :F sysems wh sparse ( C (, : F A cosecuve- -ou-of- :F sysems wh sparse cosss of compoes orere a le a he sysem fals, ff, here exs a leas cosecuve falures wh sparse,.e., here mus be or more falures, a sparse or less f he sysem fals, Zhao e al. [8]. The GERT ewor s gve Fgure 1 where each oe represes a specfc sae as escrbe below: : o compoe falure 1 : oe compoe falure 2 : wo cosecuve compoes falure wh sparse -1 : -1 cosecuve compoes falure wh sparse : cosecuve compoes falure wh sparse 1 : frs worg compoe precee by a fale compoe 2 : seco cosecuve worg compoe precee by a fale compoe : h cosecuve worg compoe precee by a fale compoe. For C (, : F, he GERT ewor ca be summarze as follows. If oe compoe s ormal, he here s cooal probably p from sae o sae oherwse q from sae o 1. Oce whe frs compoe falure occurs he sysem eers sae 2 eher recly wh cooal probably q or precee by a mos coguous worg compoes wh cooal probables = 1 p q. However, f sysem s sae 1 a ex (+1 coguous compoes are worg sae he sysem eers sae wh cooal +1 probably p. I hs way same proceure s followe ul sae. As soo as sysem eers hs sae, fals. Theorem 1: For posve egers, a, he W, (, z fuco,.e., geerag fuco of he wag me for he occurrece of sysem falure, s gve by: ( (1 ( (1 ( qz pz z + qz pz W, (, z = (1 pz((1 z(1 pz + ( qz ( pz (1 ( pz (4 The relably R (,, of he sysem s gve by: R (, = 1 ξ ( u (5 u= where ξ (u s he coeffce of of (,. W, z u z he power seres expaso Proof: I C (, : F sysem here are ( + +1 oes: 1 2,1,,,...,, 2,..., 1, o he pahs leag o sysem falure. There s oly oe pah from oe o oe 1,.e. occurrece of he frs compoe falure. However, from each of he oes 1, 2,, -1 here are +1 possble pahs o reach The subseque oe,.e., from oe o +1 ( =1, 2,, 1 2 eher recly or va oes,,..., epeg upo 214

3 C (, : F Fgure 1. GERT ewor represeg a. wheher he ex coguous compoe s fale or worg. 1 Thus, here are ( + 1 pahs leag o sysem falure. Furher, frs orer loops = pz + qz( pz ( qz ( pz *(1+ pz ( pz + ( qz ( pz 2 *(1 + pz ( pz ( qz *(1 + pz ( pz ( pz + 1 whle seco a hgher orers o o exs. By he use of Maso s rule (, fuco s obae as gve (4. Now he W, z (, fuco ca be expresse as: W, z W, (, z = ξ ( z + ξ( + 1 z ξ( z + ξ( + 1 z +... a so, he sysem falure probably s obae as ξ (u. Hece, he R (, s gve by (5. u = Moel 2: m -Cosecuve- -ou-of- :F sysems wh sparse, ( C (, : F m A m-cosecuve--ou-of-:f sysems wh sparse cosss of compoes orere a le a he sysem fals, ff, here exs m o-overlappg cosecuve falures wh sparse, Zhao e al. [8]. The GERT ewor s gve Fgure 2, where he followg oes ao o he oes Moel 1, represe specfc saes as: A : occurrece of a leas oe a a mos cosecuve worg compoes : h occurrece of cosecuve falures wh sparse, = 1, 2,, m. Theorem 2: For posve egers, m, a, he W (, z fuco,.e., geerag fuco of he wag me m, for he occurrece of sysem falure, s gve by: W m (, z = W (, z. (6 m, (, Proof: The oes ( =1, 2,,m represes occurrece of h o-overlappg cosecuve falures wh sparse. Thus, geerag fuco of he wag me for he occurrece of sysem falure s gve by (6; see also Agarwal e al. [1]. Moel 3: (, :F sysems wh sparse, ( C (, The falure probably of C (, s vesgae usg GERT, a explae as a ree srucure. The opmos oe a ree s calle he roo oe. I s a he level,.e. a =. Each oe has zero or more chl oes. A oe ha has a chl s calle he chl s pare oe. However, beg he opmos oe, he roo oe oes have ay pare oe. The oes ha o have ay chl oe are calle leaf oes. I oher wors, o furher brachg s possble from he leaf oe. The ree srucure represeg a C (, (Fgure 3 has oe source oe (also calle he roo oe x here, oe or more leaf oes, a some ermeae oes. The reco of calculag falure probably s from he roo oe o a leaf oe va some ermeae oes. The W fuco s obae for each of he oes, whch s he use o oba he 215

4 C (, : F Fgure 2. GERT ewor represeg a m. falure probably of he sysem correspog o ha oe. The sum of he falure probables correspog o all he oes wll eerme he falure probably of he sysem. As a example GERT ewor for C(,5,3 2 s gve Fgure 4. Fgure 3. Tree Srucure represeg a C (,. *Here x 3, x 5, x 6, x 7, x 8, x 9, x 1, x 11, a x 12 are he leaf oes; a x 1, x 2, a x 4 are he ermeae oes. ALGORITHM Coser a compoe sysem whch sysem falure occurs ff here exs eher a leas f fale compoes or a leas (< f cosecuve falures wh sparse. Sep Calculae v = 1, a L = f, so ha =,1,, L. Sep 1 =. For roo oe x ( Π x = φ, he level s =. ( M ( = because < f. Deerme he W fuco correspog o he roo oe W Sep 2 W, z ( x ( qz = x (1 + pz + ( pz The falure probably a oe, z u = F( x = 2 (1 pz x as ( pz s ξ ( u, 1. (7 (8 For = 1 o L by 1, o he followg. F H x, he se of v chl oes of each pare ( oe x locae a level 1. Assg he value of he l from a pare oe x o s chl oe as =, wh =1,2,,v; a for each of he chl oes a level, If < f ( ( x = ( qz s + ( s s= 1 h, he calculae he W fuco as ( pz *( + 1 * Π (1 + pz ( pz (1 pz * (1 + pz ( pz + 1 s

5 Fgure 4. Tree Srucure represeg a C(,5,3 2. The falure probably a oe W fuco s W F ( x = ξ ( u (x u = + ( s + *( + 1 correspog o hs a hs chl oe wll ow become a pare oe a level ; else,, z ( x ( qz f ( pz (1 pz *( + 1 Π (1 + * (1 pz pz ( pz = M ( f s 1, The falure probably correspog o hs W fuco s F ( x = ξ ( u. u = f + *( + 1 (9 s 1 (1 No chl oes woul exs for such oe, a becomes a leaf oe. Sep 3 Calculae he oal falure probably F (, of he sysem as F, = F( x + F ( x. (11 ( geerae all he oes Hece, he relably 1- F (,. R (, of he sysem s gve by. IV. CONCLUSION I hs paper relably evaluao of hree geeralze cosecuve- sysems: cosecuve--ou-of-:f sysems wh sparse, m-cosecuve--ou-of-:f sysems wh sparse, a (, f, :F sysems wh sparse have bee sue usg GERT for... compoes. These sysems have we applcaos mcrowave saos of a elecom ewor, ol ppele sysems, mlary sysems, vacuum sysems elecro acceleraors, ec. GERT has bee a well-esablshe echque apple several areas. However, applcao of GERT relably aalyss of cosecuve- sysems has o bee repore much. Oly, recely, relably evaluao of some mpora sysem srucures clug m-cosecuve- sysems, cosecuve- sysems wh epeece, mul-sae cosecuve- sysems a srbuos of he wag me of paers have bee carre ou by he auhors usg GERT, emosrag ha beg sraghforwar a smple, s geerally more effce ha oher approaches. REFERENCES [1] M. Agarwal, K. Se, a P. Moha, GERT aalyss of m-cosecuve--ou-of- sysems, IEEE Tras. Relably, vol. 56, pp , 27. [2] G.J. Chag, L. Cu a F.K. Hwag, Relables of Cosecuve- Sysems, Dorrech, Neherlas, Kluwer Acaemc Publshers,

6 [3] C.H. Cheg, Fuzzy cosecuve--ou-of-:f sysem relably, Mcroelecrocs a Relably, vol. 34, pp , [4] J.M. Kooleo, Relably eermao of r- successve-ou-of-:f sysem, IEEE Tras. Relably, vol. 29, pp. 437, 198 [5] W. Kuo a M. Zuo, Opmal Relably Moelg Prcples a Applcaos, New Jersey, Joh Wley a Sos, Ic, 23. [6] P. Moha, Relably Aalyss of Cosecuve- Sysems: GERT Approach, PhD. Thess, Uversy of Delh, Delh, 27. [7] G.E. Whehouse, Sysems Aalyss A Desg Usg Newor Techques, Prece-Hall Eglewoo Clffs, New Jersey, [8] X. Zhao, L. Cu W. Kuo, Relably for sparsely coece cosecuve- Sysems, IEEE Tras. Relably, vol. 56, pp ,

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