Tsitsiashvili G. Sh , Vladivostok, Radio st. 7, IAM FEB RAS

Transcription

1 Tstsashvl G. SH. - BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS Tstsashvl G. Sh. guram@am.dvo.ru , Vladvostok, Rado st. 7, IAM FEB RAS In ths paper a model of general type logcal system wth unrelable elements [1], [2] s consdered. An asymptotc analyss of ts work (falure probablty s made n approprate condtons on work (falure probabltes of the system elements. A concept of bottlenecks of ths system s constructed on a suggeston that an ncrease (a decrease of elements relabltes lead to an ncrease (a decrease of the system relablty. A constructon of general type logcal system s founded on concepts of dsjunctve and conjunctve normal forms (DNF and CNF of a logcal functon. Ths approach allows obtanng man results n maxmal general and convenent for engneerng calculatons form comparatvely wth recursve defntons of logcal functons used n [3]. Denote the set whch conssts of ndependent random logcal varables, I { 1,2,...2 }. Consder the logcal functon A represented n DNF {,, I} Here the famly ( (, ( j, j. (1 A= I conssts of the sets pars,. Suppose that p = P( = 1, q P( 0, I = Ø, and for j = =, p + q = 1, and random varables are ndependent. The logcal functon A wth random arguments s denoted by A and called the logcal system. Low relable elements Suppose that for Denote C mn max c( =, I I = I :maxc( = C = mn ( : N S S c(, c( > 0:, S { : c( C} = =, I S S, = { S, I } c p = p h ~exp( h, h 0. (2, = { S, I } S, T, N( T = mn ( T : T T and let S,T are famles of mnmal by an ncluson sets from the famles S, T,

2 Tstsashvl G. SH. - BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS S = { S S : S = N( S }, = T : T = N Theorem 1. If the formulas (1, (2 are true then Proof. Rewrte the logcal functon A as follows T T T. ln P A = 1 ~ N S h, h 0. (3, A= A I A, J { : } k k = k J = =. The formula (2 leads to If the obvous that As for j p = P = 1 0, h 0, so P A = 0 = p 0, h 0. : = k k pp( A = 1 P A = 1 I A = 1 j P = = j I, j I, j ( A 1 p P( A 1 I P( AA j = 1 = P = 1 q 1, h 0 б, k J, n J j kun kun and P A = 1 I A = 1 = P( = 1 p, j I, j j A Aj, j U j (4 So from the formula (4 obtan c P( A = 1 ~ p ~ exp h, h 0. (5 I I Denote C = max c(, K { : c( C } and (5 gve Consequently, = =. The formulas c ~ h h K, h 0, As C ( = 1 ~ exp ( 1+ ( 1 I ( P A h o K, h 0. ( ( ( ( ( A ( P = 1 ~ exp h 1+ o 1 S ~ exp h 1+ o 1 S = I I : S = N( S ( h ( o( N( S I S N( S = exp 1+ 1 : =.

3 Tstsashvl G. SH. - BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS ( ( ( ( ( ( S ( S ( ( ln exp h 1+ o 1 N S I : S = N S ~ ln exp h 1+ o 1 N S = = h 1+ o 1 N ~ h N, h 0. So formula (3 s true. Remark 1. Suppose that τ ( are ndependent random varables equal to lfe tmes of logcal elements, and h= h( t - s monotoncally decreasng and contnuous functon, h 0, Then the asymptotc c P( τ ( > t = p ( h~exp( h, t. t. Is character for the Webull dstrbuton whch s wdely used n lfe tme models of complex systems wth old and so low relable elements [4], [5]. Hghly relable elements Suppose that for c(, c( > 0: Consder the logcal functon A represented n CNF c q = q h ~exp( h, h 0 (6. (7 A= I Theorem 2. If the formulas (6, (7 are true then ln P A = 0 ~ N S h, h 0. (8 Remark 2. Suppose that τ ( are ndependent random varables equal to lfe tmes of logcal elements, и h= h( t - s monotoncally ncreasng and contnuous functon, h 0, t 0. Then the asymptotc ( ~exp ( c P τ t = q h h( t, t 0 Is character for the Webull dstrbuton whch s wdely used n lfe tme models of complex systems wth young and so hgh relable elements. Mxng case

4 Tstsashvl G. SH. - BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS Suppose that the sets X, V, X, V are nonntersectng. For X U X the formula (2 s true and for V U V the formula (6 takng place, I. So low relable and hgh relable elements n the system A are present smultaneously Theorem 3. Suppose that A= I XUV X UV (9 Then for = X UV Ø, = V U X, I, the formula (3 s true. Suppose that A=. I XUV X UV Then for = V U X Ø, = X U V, I, the formula (8 s true. Concept of bottlenecks Defne bottlenecks n logcal system A Theorem 4. Suppose that ( C c ε 0 = mn > 0 :. 1. For any S S and each ε, 0 < < 0 c( ε for all 2. If a set ε ε, the property (B s true: the replacement S leads to the replacement C C ε. S and satsfes the condton (В, then S* S : S* S. 3. For any T T and each ε, 0 < < 0 c by ε ε, the property (С s true: the replacement c + ε for all T leads to the replacement C C+ε. 4. If a set T and satsfes the condton (С, then T* T : T* T. Proof. Proof the statements 1, 3, as the statements 2, 4 are trval. 1. If c( s replace by c( ε, S, then max c( = C ε, max c( C ε, j mn max c( = C ε. I 2. If c( s replace by c( + ε, T, then max c( = C+ε, I, max c( C+ε, mn max j I c = C+ ε. I c by Corollary 1. The statements 2, 4 of the theorem 4 establsh that the famles S, S, T, T and the C, N S, N T do not depend on a vew of DNF (of KNF of the logcal functon A. numbers

5 Tstsashvl G. SH. - BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS Proof. Suppose that the theorem 1 condton s true, all other case s consdered analogcally. Denote by A 1, A2 - DNF, whch defne the logcal functon A, S 1, S 2 are famles of subsets, created by A 1, A 2, and S 1, S 2 are famles of mnmal sets from the famles S 1, S 2, correspondngly. If the set S 1 S 1 then t satsfes the property (В and so S2 S 2 : S2 S1. * * * Analogously f S 2 S 2 then there s S1 S 1 : S1 S2. Consequently S1 S2 S1 and the famles * S1, S 2 defnton leads to the equalty S1 = S2 = S1 and so S1 = S 2. Thus, the famly S does not depend on a vew of logcal functon A DNF. Smlar statements may be proved for the famles T, T, S. For the numbers N ( T, C, N ( S the statements of the corollary 1 may be obtan from the formula (3. Remark 3. The statements 1 (the statements 3 of the theorem 4 establshes that an ncrease of elements S relabltes for any set S S (a decrease of elements T relabltes for any set T T leads to an ncrease (to a decrease of system A relablty. The corollary1 allows to call sets from the famles S, S, T, T by bottlenecks n logcal system A. Remark 4. Suppose that the condton (2 or the condton (6 are replaced by c c(, d(, c( > 0, d( > 0: p = p( h ~exp ( d( h, h 0, or by c c(, d(, c( > 0, d( > 0: q = q( h ~exp ( d( h, h 0, correspondngly. Then to obtan the formula (3 or the formula (8 correspondngly t s enough to redefne S, S, and put (besdes of number of elements n a set S: S = d. S References [1] Rabnn I.A. Logc-probablty calculus as method of relablty and safety nvestgaton n complex systems wth complcated structure// Automatcs and remote control No 7. P (In Russan. [2] Solojentsev E.D. Specfcs of logcal-probablty rsk theory wth groups of antthetcal events// Automatcs and remote control, No 7. P (In Russan. [3] Tstsashvl G.Sh. Asymptotc Analyss of Logcal Systems wth Unrelable Elements// Relablty: Theory and Applcatons, Vol P [4] Rocch P. Boltmann-lke Entropy n Relablty Theory//Entropy Vol. 4. P [5] Rocch P. The Noton of Reversblty and Irreversblty at the Base of the Relablty Theory// Proceedngs of the Internatonal Symposum on Stochastc Models n Relablty, Safety, Securty and Logstcs. Sam Shamoon College of Engneerng. Beer Sheva, February 15-17, P