TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 erformance nalyss for a Nework havng Sby edundan Un wh ang n epar Jendra Sngh 2 abns orwal 2 Deparmen of Compuer Scences NIMS Unversy ajashan Inda jendra.j@gmal.com 2 rabnsp@redffmal.com bsrac In hs paper he auhors deal wh a complex nework havng n-dencal uns n seres wo dencal sby uns for evaluaon of s ably measures. The auhors have been used splemenary varables echnque o formulae he mahemacal equaons for varous flow-saes of fg-(b. These mahemacal equaons have been solved by he applcaon of Laplace ransform. The Laplace ransforms of varous flow-saes probables have obaned. e have calculaed he expressons for relably funcon avalably funcon mean me o falure (M.T.T.F.. nalyss of me-ndependen sae probables a parcular case when all repars follow exponenal me dsrbuon have also gven o mprove he praccal uly of he model. Keywords Compuer Nework elably nalyss Splemenary Varables Laplace Transform Tme Dsrbuon. The followng assumpons have been assocaed wh hs sudy:. Inally he whole nework s good operable wh full effcency. 2. There are n uns conneced n seres n subsysem. 3. There s one sby un n subsysem on falure of man un we can onlne sby un hrough mperfec swchng devce. 4. ll falures follow exponenal me dsrbuon are S-ndependen. 5. ll repars follow general me dsrbuon are perfec. 6. On falure of boh he uns of subsysem he sysem has o wa for repar. 7. epar of any one faled un of subsysem can be done mmedaely. 8. Nohng can fal from a faled sae. I. INTODUCTION The whole sysem consss of wo subsysems conneced n seres. The subsysem has wo dencal uns 2 n sby redundancy. e can onlne he uns of subsysem by he applcaon of mperfec swchng devce. The sysem confguraon has shown n fg-(a. The pcure drawn n fg-(b shows he flow of saes. The whole sysem may fal due o falure of s eher subsysems. The sysem can be repared mmedaely f he subsysem or he un of subsysem or boh are faled. The sysem has o wa for repar n case of repar of whole subsysem. e may use he resuls obaned n hs sudy o every nework sysem of real lfe wh smlar confguraons. numercal example ogeher wh s graphcal llusraon has also appended a he end o hghlgh mporan resuls of he sudy. Fg-(b: Sae-ranson dagram Fg.(a: Sysem confguraon 789
TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 Ls of noaons used n hs sudy s as follows: / Falure rae of subsysem /. ( Falure rae of swchng devce. ang rae for repar of whole subsysem. h ( j Frs order probably ha w falure can be repared n he me nerval ( j j condoned ha was no repared o he me j. r {a me subsysem ( swchng devce s operable.e. he whole sysem s operable}. r {a me sysem s ( F ( F ( z. operable whle one onlne un of subsysem has faled. r {a me sysem s faled due o falure of subsysem s wang for repar}. r (a me sysem s faled due o falure of subsysem s ready o repar}. Elapsed repar me les n he nerval ( z z. ( F x ec. r (a me sysem s faled due o falure of subsysem }. Elapsed repar me les n he nerval ( x x. S ( j ( jexp. ( j dj ( j. Laplace ransform (L.T. of funcon (. D ( j S ( j / j j. II. FOMULTION OF MTHEMTICL MODEL In hs secon auhor has done analyss for mahemacal formulaon of he model s soluon some parcular cases varous resuls relaed o relably esmaon. Usng probably argumens lmng procedure we oban he followng se of dfferencedfferenal equaons governng he bahavour of consdered sysem dscree n space connuous n me: d d x F x xdx F z z ( x F dz x ( (2 d ( ( F ( x ( x dx d F ( y S ( y dy (3 ( x F x x (4 S ( y F y y (5 d w F d (6 ( z F z z (7 oundary condons are: (8 ( w F (9 F ( F F F ( Inal condons are: all oher sae probables are zero a. (2. Soluon of he Model e shall solve he above sysem of dfferencedfferenal equaons wh he ad of Laplace ransform o oban probables of dfferen ranson saes depced n fg-. Thus akng Laplace ransform of equaons ( hrough ( subjeced o nal condons (2 hen on solvng hem one by one we oban: ( (3 ( D( F ( ( (4 ( ( ( s (5 ( ( ( D s s F ( (6 ( DS ( ( F ( ( s (7 ( F ( ( ( s w (8 79
TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 F ( w D ( ( ( ( s w (9 where ( (2 s D( ( DS ( w ( s S ( ( S ( (2 s w w (22 ( s S ( ( S ( s w I s neresng o noe here ha Sum of equaons (3 hrough (9 = s. Long-run ehavour of he Sysem Usng fnal value heorem n L.T. vz ; lm ss say lm s (23 provded lm on LHS exss n equaons (3 hrough (9 we oban he followng long-run behavor of he consdered sysem: F ( M ( (24 (25 (26 ( M (27 F ( ( M (28 S F ( F (29 w( M F (3 ( where M S ( Mean me o repar subsysem ( d ds ( s C. Some arcular Case Case : hen all repars follow exponenal me dsrbuon In hs case seng S j j n ( j equaons (3 hrough (9 we oban he followng L.T. of ranson-sae probables: ( (3 (32 F ( s ( E ( s (33 ( ( C s F ( s (34 ( F ( ( s (35 S F ( F ( ( s w (36 w ( s w( s (37 where ( s (38 s s S s w s (39 s ( s w( s Case : hen swchng devce used s perfec In hs case pung n equaons (3 o (9 we oban he requred resuls. Noe ha n hs case F (. 2.5 elably MTTF of he sysem elably of he consdered nework can be obaned as ( s (4 lso or ( L ( ( exp M. T. T. F. ( d (4 D. valably of he Sysem For he consdered nework s s s ( ( ( (42 ( e e ( ( 79
TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 lso down E. Numercal llusraon For a numercal llusraon le us consder he followng value se for parameers:.45.37. 7 = 2-----. III. ESULTS ND DISCUSSION Table- gves he values of relably of consdered sysem for varous values of me. Is graph has been shown n fg-2. nalyss of able- fg-2 reveal ha he relably of consdered sysem decreases approxmaely n consan manner here are no sudden jumps n he values of relably. TLE (.49236 2.24298 3.994 4.5866 5.28869 6.427 7.6992 8.344 9.693.833 Fg. 3 Table-3 gves he values of M.T.T.F. of consdered sysem for dfferen values of falure rae of subsysem. Is graph has been shown n fg-4. nalyss of able-3 fg-4 yeld ha value of M.T.T.F. decreases caasrophcally. TLE 3 MTTF 3.864. 3.77472.2 3.584229.3 3.4628.4 3.344482.5 3.236246.6 3.34796.7 3.3954.8 2.949853.9 2.86533. 2.78555 Fg. 2 Table-2 gves he values of avalably of consdered sysem for dfferen values of me. Is graph has been shown n fg-3. Crcal examnaon of able-2 fg-3 yeld ha value of avalably decreases rapdly n he begnnng bu hereafer decreases consanly. TLE 2 (.596659 2.32774 3.7242 4.88485 5.4473 6.224 7.5 8.5529 9.2734.35 Fg. 4 Table-4 gves he values of M.T.T.F. of consdered sysem for dfferen values of falure rae of subsysem. Is graph has been shown n fg-5. nalyss of able-4 fg-5 yeld ha value of M.T.T.F. decreases caasrophcally. 792
TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 TLE-4 MTTF 2.222222. 2.2877.2 2.2533.3 2.29.4 2.2848.5 2.2572.6 2.2673.7 2.98285.8 2.9498.9 2.954. 2.8884 Fg. 5 EFEENCES [] arlow.e; roschan F. (965: Mahemacal Theory of elably New York; John ley. [2] Chung K (988: K-ou-of-n:G redundan sysem wh dependan falure raes common cause falures Mcroelecron. elably U.K. Vol 28: 2-23. [3] Gnedenko.V; elayer Y.K; Soloyar (969: Mahemacal Mehods of elably Theory cademc press New York. [4] Ga..; Ga.K. (986: Cos analyss of an elecronc reparable redundan sysem wh crcal human errors Mcroelecron. elab. U.K Vol 26: 47-42. [5] Nagraja H.N.; Kannan N.; Krshnan N.. (24: elably Sprnger ublcaon. [6] ey D.; Jacob Mendus 995: Cos analyss avalably MTTF of a hree sae sby complex sysem under common-cause human falures Mcroelecronc. elab. U.K. Vol. 35: 9-95. [7] Sharma S.K. ;Sharma Deepankar ; Masood Mons (25: valably esmaon of urea manufacurng ferlzer plan wh mperfec swchng envronmenal falure Journal of combnaorcs nformaon & sysem scences Vol.29 (-4 : 35-4. [8] Sharma Deepankar Sharma Jyo (25; Esmaon of relably parameers for elecommuncaon sysem Journal of combnaorcs nformaon & sysem scences Vol.29 (-4: 5-6. [9] Sharma Deepankar (2; Cos Esmaon for TM wh re-empve esume epar ublshed Inernaonal Journal of Compuaonal Inellgence Informaon Secury usrala Vol- No-9 9-27. [] Sharma Deepankar (2; elably Forecas for Telecommuncaon Sysem ublshed Inernaonal Journal of Compuaonal Inellgence Informaon Secury usrala Vol- No-9-8. 793