AN ACCURACY OF ASYMPTOTIC FORMULAS IN CALCULATIONS OF A RANDOM NETWORK RELIABILATY. Tsitsiashvili G.Sh., Losev A.S.

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1 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY Tsitsiashvili GSh, Losev AS Institute for Alie Matheatics, Far Eastern Branch of AS 69004, Vlaivostok, aio str 7, gura@iavoru, alexax@bkru INTODUCTION In this aer a roble of asytotic an nuerical estiates of relative errors for ifferent asytotic forulas in the reliability theory are consiere These asytotic forulas for rano netorks are siilar to calculations of Feynan integrals A secial interest has analytic an nuerical coarison of asytotic forulas for the ost srea Weibull an Goertz istributions in life tie oels In the last case it is shon that an accuracy of asytotic forulas is uch higher AN ASYMPTOTIC ESTIMATE OF A ELATIVE EO IN A DEFINITION OF A ELIABILITY LOGAITHM Consier the nonoriente grah ith fixe initial an final noes an ith the arcs set W Define = {,, n} as the set of all acyclic ays beteen the initial an final noes of the grah Designate P the robability of the ay ork Then in the conition ~ex ( ch ), h 0, W, e have: ~ex P ( C( ) h C ( ) h ( + o( ) )), here C( ) = c an D < D is a next by a quantity after D = ax eleent in the set : = D( ) { }, C ( ) = c If in the ay this eleent is absent e ut then D =,, : = D ( ) C ( ) = 0 Denote D = in D an esignate = { : D = D }, = \, then the robability P of the grah ork satisfies the forulas i P ~ P + P, ~ ex P P = C( ) h C ( ) h ( + o( ) ), i =, By the efinition i i ( ) D P ~ ex( C( ) h ( ) C h ( + o( ))) ~ here C = in C an D < D,, so here : C( ) = C ( Ch D ) ( ( ) D C h ( + o( ))) ~ex ex ( D) ( D ( ())) P ~ex Ch ex C h + o, D = in D < D, C = in C ( ) : C( ) = C, D = D,

2 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber here An consequently D < D, :, P = o( P) ~ ex P ( ( ) ( ) ( ( ))) ~ ex C h C h + o ( C( ) h ( + o( ) )) = So e have: D D D D D ( Ch ) ( Ch Ch ( ) ( o( ))) = ex ex + ~ ( Ch ) ( Ch ( ) ( + o( ))) ~ex ex ~ ( D D Ch ) ( Ch ( o() )) ~ex ex +, D = in D > D, C = in C( ) ( )( ( ( + ())) + ( ( + ()))) ~ex( D D Ch ) ex( Ch ( + o() )) P ~ex C h ex C h o ex C h o ~ As a result obtain that ln P ~ C h Δ ( Ah + ( + o() )), Δ = D D > 0, A = C / C An consequently ln P C( ) h ~ Ah Δ () AN ASYMPTOTIC ESTIMATE OF A ELATIVE EO IN A DEFINITION OF A ELIABILITY Assue that PU ( ) is the robability of the event ork Then e have n = = U that all arcs,, of the ay P P U U () Suose that the robability of the arc W ex ch, h > 0, here c, are soe ositive nubers an for arcs the constants So e have ( ) ork equals ( ) = PU = ex c h Assue that the enueration of the arcs in the ay satisfies the inequalities > > >

3 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber Denote D =,, an introuce on the vectors set { D, n} f, if for soe ( ) the folloing q orer relation Say that D D k in, q the first k coonents of these q vectors coincie an the k coonent in the vector D is larger than in the vector D If there is q q not such k an in the vectors D, D all first in (, q) coonents coincie then D f D for < q eark that for soe q the arcs sets { },{ q } can not satisfy the inclusion { } { q} In the oosite case there is the noe u * in hich the ays q iverge by the arcs ( u*, u ) ( u*, u q ) But as the arc ( u*, u) { q} then the ay q has a cycle This conclusion contraicts ith the assution that the ay q is acyclic q So as the quantities are ifferent then D D, q As a result e obtain the orer relation on the vectors set { n q D,, D }, an if D f D, h 0, so PU ( q) = opu ( ( ) ) It is not ifficult to check that this relation is transitive Consequently the orer relation on the set n D,, D is linear Assue that the enueration of the vectors D satisfies the forula { } n D f f D Fro the forula () e have As the inclusion { } { q} * ( q), P P( U ) * * < q P P U U P P = (3) is not true for q so in the ay there is an arc hich oes not belong to the ay q Consequently e have ( q) opu ( ( ) ) PU =, q = ( ) (4) <, PUU ( ) = opu ( ) < i The forulas (3), (4) give us the folloing asytotic exansion for P ith the first an the secon ebers of the sallness: * P ~ P ~ P( U), P P( U) ~ P( U), PU ( ) = opu ( ( )), h 0 (5) q 3 AN APPLICATION TO LIFE TIME MODELS Suose thaτ are ineenent rano variables an characterize life ties of the arcs W Denote Denote ( h) = P( τ > t) an esignate the life tie of the grah by τ = in axτ If h= / t then e have ith t the Weibull istributions of the arcs life ties an the forula ln P( τ > t) Δ = g () t ~ At = G D () t (6) C( ) t If h= ex( t), t, then e have the Goertz istributions of the arcs life ties an the forula () transfors into

4 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber so G ( t) = o( G( t) ) ln P( τ > t) = g() t ~ G() t = G( ex() t ), (7) C( ) ex( D t) Consequently for the Goertz istributions the convergence rate in the asytotic (7) is uch faster than for the Weibull istributions in (6) If h= / t, t, then for the Weibull istributions of the arcs life ties the forula (5) transfors into P ( t ) ~ex c t τ >, (8) = P( τ > t) = f () t ~ F() t = o(), ex F () t ex c t c t = = + = = c t If h= ex( t), t, then for the Goertz istributions of the arcs life ties the forula (5) transfors into ( τ > ) ~ex ex( ), P( τ > t = ex c ex )( t ) P t c t so F () t = o( F() t ) = = f() t ~ F() t = F( ex() t ), (9) Consequently for the Goertz istributions the convergence rate in the asytotic (9) is uch faster than for the Weibull istributions in (8) * * For h= / t enote P P / P = A() t, an for h= ex( t) esignate P P / P = A () t It is clear that A ( t) = A( ex( t) ) tens to zero for t uch faster than At ( ) Fro this section e see that the Goertz istributions of the arcs life ties (these istributions are referable in life tie oels of alive [] an of colex inforation [] systes), give uch ore accuracy asytotic forulas than the Weibull istributions These both istributions are liit for a schee of a iniu of ineenent an ientically istribute rano variables 4 ESULTS OF NUMEICAL EXPEIMENTS FO BIDGE SCHEMES Fig The brige schee - 6 -

5 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber Consier the brige schee reresente on the Fig ith the araeters = 00, = 009, 3 = 05, 4 = 07, 5 = 0 Calculate the functions f ( t), f( t), A( t), A( t), g( t), g( t ) Fig The relative errors f ( t) an f ( t ) in the reliability P calculations Fig3 The relative errors At ( ) an A ( t ) in the reliability P calculations Fig4 The relative errors g( tan ) g ( t ) in ln P calculations The results of the nuerical exerients reresente above sho that a transition fro the Weibull to the Goertz istribution ecreases significantly relative errors in calculations of the - 6 -

6 Tsitsiashvili G, Losev A AN ACCUACY OF ASYMPTOTIC FOMULAS IN CALCULATIONS OF A ANDOM NETWOK ELIABILATY &ATA # 3 (Vol) 009, Seteber * reliability an its logarith The asytotic estiate P of the reliability P is better than PU ( ) The relative error of the ln P calculation is larger than the relative error of the P calculation But a colexity of the ln P calculation is saller EFEENCES Gavrilov NA, Gavrilova NS Biology of life uration M: Science (In ussian) Dvoeglazov DV, Matchin VT, Morvinov VA, Svechnikov SV,Trifonov NI, Filinov AM, Shlenov AYu Inforation systes in inforation eia control Part III Mosco state institute of raiotechnique, electronic an autoatic (technical university) 00 8 (In ussian)