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BOTTLENECKS IN GENERAL TYPE LOGICAL SISTEMS WITH UNRELIABLE ELEMENTS
Tsitsiashvili G. Sh.
690041, Vladivostok, Radio st. 7, IAM FEB RAS
In this paper a model of general type logical system with unreliable elements [1], [2] is considered. An asymptotic analysis of its work (failure) probability is made in appropriate conditions on work (failure) probabilities of the system elements. A concept of bottlenecks of this system is constructed on a suggestion that an increase (a decrease) of elements reliabilities lead to an increase (a decrease) of the system reliability.
A construction of general type logical system is founded on concepts of disjunctive and conjunctive normal forms (DNF and CNF) of a logical function. This approach allows obtaining main results in maximal general and convenient for engineering calculations form comparatively with recursive definitions of logical functions used in [3].
Denote Z the set which consists of |Z| independent random logical variables z, I c{1,2,..iZ|} . Consider the logical function A represented in DNF
rf ^ f w
(1)
Here the family {(, Zi), i e I} consists of the sets pairs Zi, Zi e Z, Zi I Zi = 0, and for i ^ j
(, Zi , Zj ). Suppose that pz = P (z = 1), qz = P (z = 0), pz + qz = 1, and random variables z e Z are independent. The logical function A with random arguments z e Z is denoted by A and called the logical system.
Low reliable elements
Suppose that for Vz e Z
rf ^ f >
A = V A z A A z
ieI V zeZi V zeZi J
3 c (z), c (z)> 0: pz = pz (h) ~ exp(-h~c(z)), h ^ 0. (2)
Denote C = min max c (z),
ieI zeZ:
I' = ji e I :maxc(z ) = cj, St = {z e Zf : c (z ) = C}, i e I', S = {si, i e I'}, N (s ) = min (( : St e s), T = {{ zi e St, i e I'}}, N(t) = min (( : T e T)
and let S, T' are families of minimal by an inclusion sets from the families S, T,
S" = {Si e S' : \S,\ = N (s)} , T" = {T e T' : \T\ = N (t)}.
Theorem 1. If the formulas (1), (2) are true then
- ln P (A = 1) ~ N(S) h~C, h ^ 0 . Proof. Rewrite the logical function A as follows
A = V
A *
v zeZi
A A
Ai =vîA* ], Ji ={k : Zk = zi} ■
keJi VzeZi J
(3)
The formula (2) leads to pz = P (z = l) — 0, h — 0, so
P(A. = 0)= _n Pz — 0, h — 0.
zeZk : Zk =Zt
If the obvious that
As for i ^ j
and
En PzP (Ai = 1)- Z P
ieI zeZt i, jel, i^ j
(f
Ai n z = 1
zeZ,
\ ( n
J
Aj n z = 1
V zeZj JJ
<
< P (A = 1)< Z n PZP (Ai = 1)
ieI zeZ,
P (AiAj = 1) = P
( ( > >
P Z n_ z = 1 ^ n_ qz — 1, h — 0 6,
keJi neJj V 1 j V zeZk UZn J J zeZkUZn
((
Z P
i,jeI, i*j
Ai n z = 1
zeZ,
\ ( n
Aj n z = 1
V zeZj JJ
P (AiAj = 1) n Pz,
V ' zeZ,. uZj
(4)
So from the formula (4) obtain
P (A = 1)~ Z n Pz ~ Z exp
ieI zeZ, ieI
f
-Z h
iz )
h — 0.
v zeZi J
Denote Ci = max c (z), Ki = {z e Zi : c (z) = Ci}. The formulas
Z h"c(z) ~ h~Ci\K\, h — 0,
zeZ,
(5)
and (5) give Consequently,
P (A = 1) ~ Z exp (-hCi (1 + o(1))|K |), h
— 0.
P (A = 1)~ E exp (-h~C (1 + o
tel ' V
E exp (-h~C (1
tel ': n=n (s ) v v
+ o
= exp
(( (1 + o(1))N(S)))e I' : M = N(S)}}
As
ln
exp(c (1 + o(l))N(S))|{ i g I': = N(S)}|] ~ lnexp(-h"c (l + o(l))N(S)) = = -h~C (l + o(1))N(S)~ -h~cN(S), h ^0.
So formula (3) is true.
Remark 1. Suppose that t(z ) are independent random variables equal to life times of logical elements z, and h = h (t) - is monotonically decreasing and continuous function, h ^ 0, t ^ ro. Then the asymptotic
P(t(z)> t) = pz(h) ~ exp(-h~c(z)), t
^ œ.
Is character for the Weibull distribution which is widely used in life time models of complex systems with old and so low reliable elements [4], [5].
Highly reliable elements
Suppose that for Vz e Z
3c (z), c (z ) > 0: qz = qz (h) ~exp(-h"c(z)), h ^ 0
(6)
Consider the logical function A represented in CNF
A = A
17 > r \
V Z V V z
V zeZi V zeZt y
(7)
Theorem 2. If the formulas (6), (7) are true then
- ln P (A = 0) ~ N (S) h~c, h ^ 0 .
(8)
Remark 2. Suppose that t(z) are independent random variables equal to life times of logical elements z, u h = h (t) - is monotonically increasing and continuous function, h ^ 0, t ^ 0. Then the asymptotic
P (t(z ) < t ) = qz (h) ~ exp (-h (t )-c( z }), t ^ 0
Is character for the Weibull distribution which is widely used in life time models of complex systems with young and so high reliable elements.
Mixing case
Suppose that the sets Xi , V, Xi ,Vi e Z are nonintersecting. For Vz e Xi U Xi the formula (2) is true and for Vz e V U Vi the formula (6) taking place, i e I. So low reliable and high reliable elements in the system A are present simultaneously
Theorem 3. Suppose that
A = V
-gi
\ f
A z A A z
V^X uV y V zgx. uV-
(9)
Then for Zi = Xi U V ^ 0, Zi = Vt U Xi, i e I, the formula (3) is true . Suppose that
A = A
iel
V z V V z
v zGxt uVj y v zgx. u vi y
Then forZi = Vi U Xi ^ 0, Zi = Xi U Vi, i g I, the formula (8) is true.
Concept of bottlenecks
Define bottlenecks in logical system A Theorem 4. Suppose thats0 = min (C - c (z)| > 0 : z g Z) .
1. For any Sj g S and each s, 0 <s<s0, the property (B) is true: the replacement c(z) by c(z)-s for all z g Sj leads to the replacement C ^ C - s.
2. If a set S Œ Z and satisfies the condition (B), then S* g S : S* œ S .
3. For any T g T and each s, 0 < s < s0, the property (C) is true: the replacement c (z) by c (z) + s for all z g T leads to the replacement C ^ C + s .
4. If a set T œ Z and satisfies the condition (C), then 3T* g T : T* œ T. Proof. Proof the statements 1, 3, as the statements 2, 4 are trivial.
1. If c (z) is replace by c (z)- s , z g St, then
max c (z) = C - s, max c (z) > C - s, j ^ i ^ min max c (z) = C - s .
zgz. zgz,. iGl zgz.
2. If c (z) is replace by c (z) + s, z g T, then
max c (z) = C + s , i g I', max c (z)> C + s, j ^ Imin max c (z) = C + s.
r7 \ / r-r \ / • T r7 \ /
zgz. zgz. iGI zGZi
Corollary 1. The statements 2, 4 of the theorem 4 establish that the families SS" , T, T" and the numbers C, N (S ), N (T) do not depend on a view of DNF (of KNF) of the logical function A.
Proof. Suppose that the theorem 1 condition is true, all other case is considered analogically. Denote by A1, A2-DNF, which define the logical function A, s1, S2 are families of subsets Z, created by A1, A2, and Si, S'2 are families of minimal sets from the families St, S2, correspondingly. If the set S1 g Si then it satisfies the property (b) and so 3 S2 g S'2 : S2 e S1. Analogously if S2 g S'2 then there is S* g Si : S* e S2 . Consequently S* e S2 e S1
and the families Si, S2 definition leads to the equality S* = S2 = S1 and so Si = S2. Thus, the family S' does not depend on a view of logical function A DNF. Similar statements may be proved for the families T', TSFor the numbers N (T), C, N (S) the statements of the corollary 1 may be obtain from the formula (3).
Remark 3. The statements 1 (the statements 3) of the theorem 4 establishes that an increase of elements z g S reliabilities for any setS g S (a decrease of elements z g T reliabilities for any set T g T) leads to an increase (to a decrease) of system A reliability. The corollary 1 allows to call sets from the families S, S", T', T" by bottlenecks in logical system A .
Remark 4. Suppose that Vz g Z the condition (2) or the condition (6) are replaced by
3 c(z), d(z), c(z)> 0, d(z)> 0: pz = pz (h) ~ exp(-d(z)h"c(z, h ^ 0,
or by
3 c(z), d(z), c(z)> 0, d(z)> 0: qz = qz (h) ~ exp(-d(z)h"c(z, h ^ 0,
correspondingly. Then to obtain the formula (3) or the formula (8) correspondingly it is enough to redefine | S |, S e Z, and put (besides of number of elements in a set S): | S |= z d(z).
zgS
References
[1] Riabinin I.A. Logic-probability calculus as method of reliability and safety investigation in complex systems with complicated structure// Automatics and remote control. 2003. No 7. P. 178-186. (In Russian).
[2] Solojentsev E.D. Specifics of logical-probability risk theory with groups of antithetical events// Automatics and remote control, 2003. No 7. P. 187-203. (In Russian).
[3] Tsitsiashvili G.Sh. Asymptotic Analysis of Logical Systems with Unreliable Elements// Reliability: Theory and Applications, 2007. Vol. 2. № 1. P. 34-37.
[4] Rocchi P. Boltzmann-like Entropy in Reliability Theory//Entropy. 2002. Vol. 4. P. 142-150.
[5] Rocchi P. The Notion of Reversibility and Irreversibility at the Base of the Reliability Theory// Proceedings of the International Symposium on Stochastic Models in Reliability, Safety, Security and Logistics. Sami Shamoon College of Engineering. Beer Sheva, February 15-17, 2005. P. 287-291.
