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ANALYSIS OF PORTS RELIABILITIES
Tsitsiashvili G.Sh.
Institute for Applied Mathematics, Far Eastern Branch of RAS 690041, Vladivostok, Radio str. 7, e-mail: guram@iam.dvo.ru
Introduction
This paper is devoted to algorithms of a calculation of ports reliabilities. A port is a no oriented graph with fixed initial and final nodes. As accuracy so asymptotic formulas are considered. Suggested algorithms have minimal numbers of arithmetical operations.
Using the article [1] results, in this paper algorithms of a calculation of asymptotic constants for ports reliabilities are constructed. These algorithms allow to estimate an influence of some arc reliability on a port reliability and to obtain an invariance condition when this influence is absent.
In solid state physics, surface physics and in nanotechnologies recursively defined ports are of large interest. An example of such a structure is in the monograph [2, fig. 7.7]: where each arc of a bridge scheme r is replaced by r. In this paper linear upper bounds of arithmetical operations numbers necessary to calculate as a reliability so its asymptotic constants are obtained. For a comparison it is worthy to say that a number of arithmetical operations necessary to calculate a reliability of a port increases as a geometrical progression of port arcs number.
1. Preliminaries
Consider a port r with a final number U of nodes, a set W = {w = (u,v), u,v e U} of arcs and fixed initial u0 and final u0 nodes. Denote R a set of all ways R in the port r, which connect the nodes u0 and u0. Suppose that R . Consider the sets
A = {a c: U, u0 e A, u0 g a}, L = L(A) = {(u,u'): u e A, u'g A}
and L = {l(A), A e a} - the set of all sections in r. Correspond for each arc w e W a logic variable a(w) = I (the arc w works), where I(B) is an indicator function of an event B. Denote a quantity which characterizes a connectivity between the nodes u0, u0 in r by
/ = V A«(w) . (1)
ReR weR
Suppose that a(w), w eW, are independent random variables, P(a(w) = 1) = pw (h), qw (h) = 1 - Pw (h) ,
where h is some small parameter : h ^ 0. In [1] the following statements are proved.
Theorem 1. Suppose that pw (h) ~exp (-h " d (w)), h ^ 0, where d(w)> 0, w e W. Then
-lnP(/ = 1)~ hTD and D = D(r) = minmaxd(w).
v ' v ' ReR weR '
Theorem 2. Suppose that qw (h) ~ exp (-h-d1(w)), h ^ 0, where d1 (w) > 0, w e W. Then - ln P (/3 = 0) ~ h~D1 and D1 = D1 (r) = maxmin d1 (w).
LeL weL
Theorem 3. Suppose that pw (h) ~ hg (w), h ^ 0, where g (w)> 0, w e W. Then ln P (fi = 1)~ G ln h and G = G (r) = min 2 g (w).
ReR weR
Theorem 4. Suppose that qw (h) ~ hgl(w), h ^ 0, where g1 (w) > 0, w e W, then ln P(fi = 0) ~ G1 ln h and G1 = G1 (r) = min 2 g1 (w).
LeL weL
The constants G, G1 [3] may be interpreted as a length of a shortest way or a minimal ability to handle of cross-sections in the port r correspondingly. In a definition of the constants D, D1 a summation is replaced by a maximization. So the constants D, D1may be interpreted as a pseudo-length of the shortest way or a minimal pseudo-ability to handle in the port r.
Remark 1. Suppose that r(w) are independent random variables which characterize life times of the arcs w e W. Denote P(t(w)> t) = pw (h) and put the graph r life time equal to r(r) = min max t(w). If h = h (t) is monotonically decreasing and continuous function and
ReR weR
h ^ 0, t then the theorems 1, 3 remain true if P (fi = 1) is replaced by P (r(r)> t) . If h = h(t) is monotonically increasing and continuous function and h ^ 0, t ^ 0, then the theorems 2, 4 remain true if P(fi = 0) is replaced by P(r(r) < t) . So it is possible to consider widely used in the reliability theory the exponential and the Weibull distributions of arcs life times.
Denote r a port with the nodes set U = {u0, u1, u2, u3} and with the arcs set (fig.1)
W = {w1 =( ^ u1), w2 =( ^ u2 ), w3 =(u^ u3 ), w4 = (u2, u3 ), w5 =(U1, u2 )} v
The node u0 is initial and the node u3 is final. The scheme r [4] is called the bridge scheme and the arc w5 - the bridge element in this scheme
Fig. 1. Bridge scheme r.
The scheme r reliability P = P (p1,..., p5) in a suggestion that the arcs w1,..., w5 work independently with the probabilities p1,..., p5 is calculated by the formula
P = p5 [1 -(1 - p )(1 - p2 )][1 -(1 - p3 )(1 - p4 )] + (1 - p5 )[1 -(1 - p1 p3 )(1 - p2 P^ )] (2)
To make these calculations it is necessary n (r) = 14 arithmetical operations.
2. Element wise analysis
Remark that in an accordance with the formula (1) the logical function fi = fi(a(w),w e W)
has all properties of the monotone structure [2, ra.7]:
a)fi(a(w) = 1,w e W) = 1 ,b)fi(a(w) = 0,w e W) = 0 ,
c)fi(a1 (w),w e W) < fi(a2 (w),w e W) , if a1 (w) <a2 (w),w e W .
u
u
0
3
Fix an arc v e W and using the complete probability formula [2, §7.4] obtain the following formulas:
P(/ = 1) = P(a(v) = 1)F + P(a(v) = 0)Fv° , (3)
Ff = P
(ß(a(w), w eW, w £ v;a(v ) = f) = l) f = 0,1,
and
F° < Z1.
(4)
Define the graph by an exclusion of the arc v = (u,u') from the graph r and the graph rv by a gluing of the nodes u, u' in the graph r°. Using the previous section results and the formulas (3), (4) obtain the following statements.
Theorem 5. Suppose that pw (h) ~ exp h ^ 0, where d (w)> 0, w e W. Then
- ln P (/ = 1)~ h~D where D = min [max (d (v), D (rj,)), D (r0 )], D (rj, )< D (r0).
Theorem 6. Suppose that qw (h) ~ exp
(-h-d1(w)), h ^ 0, where d1 (w) > 0, w e W. Then
- ln P (/ = 0)~ h~D1 where D = min
max I
(d(v),D1 (rv)),D1 (r0)\, D(rv)<D1 (r0).
Theorem 7. If pw (h) ~ hg(w), h ^ 0, where g (w)> 0, w e W .Then ln P(/ = 1)~ G ln h and G = min [g (v) + G (rv ), G (r°° )], G (rv )< G (r0 ).
Theorem 8. If qw (h) ~ hg1(w), h ^ 0 ,where g1 (w)> 0, w e W. Then lnP(/ = 0)~ G1lnh and G1 = min [ g1 (v) + G1 (rv ), G1 (r0 , G1 (rv) < G1 (r 0).
Remark 2. The constants D, D1, G, G1 do not depend on d (v), d1 (v), g (v), g1 (v) correspondingly if and only if D (r^, ) = D (r°v), D, (r^, ) = D, (r^), G (r^, ) = G (r0), G1 (r^, ) = G1 (r0). The fig. 2, 3 show how the parameters d (v), g (v) influence on the constants D (r), G (r).
D (r)
D (rV ) D (r0 )
Fig. 2.
d (v)
G (r0)
G (rV )
G (r)
^ ( v )
Fig. 3.
Example. Consider the port r (fig. 1) with independently working arcs w1,...,w5 and show how the element w5 reliability influences on the port reliability on an example of the constants
D(r), G(r) from the theorems 5, 7. Define the port by an exclusion of the arc w5 from the graph r and the port rl5 by a gluing of the nodes u1, u2 in the graph .
0
0
w.
w-.
wA
Fig. 4. Port rW5
Fig.5. Port rW5
If pw (h) ~ exp(-h di), h ^ 0, with dt = d(wt) > 0, then it is easy to obtain the formulas D(rW5) = min(max(dj,d3),max(d2,d4)), D(rW5) = max(min(dj,d2),min(d3,d4)) , D(r) = min [max(d5,D(rW5)),D(rW5 )] .
Here the equality D (rW5) = D (rW5) is true in one of the following eight conditions:
1) d3 > d1 > d2 , 2) d3 > d1 = d2 , 3) d4 > d1 = d2, 4) d4 > d2 > d1, 5) d1 > d3 > d4 , 6) d1 > d3 = d4 , 7) d2 > d3 = d4, 8) d2 > d4 > d3. If pw (h) ~ hgi, h ^ 0 , with gt = g(wt) > 0,i = 1,...,5 , then it is easy to obtain the formulas
G (rW5) = min (( gi + g3), ( g2 + g4 )) , G (rW5) = min ( g^ g2 ) + min ( g3 , g4 ),
G (?)
= min
" g5 + G (rjw5), G (rW5 )" Here the equality G (rW5) = G (rW5) is true in one of the following two conditions:
j) g4 > gз,g2 > gl, 2) g3 > g4, gi > g2.
Remark 3. If the graph r' is constructed by an addition of the arc w6 = (u0,u3) to the port r then D(r') = min(d6, D (r)), G (r') = min(g6, G (r)) where d6, g6 are appropriate parameters of the arc w6. As the graph r' is complete (each two its nodes is connected by some arc) so these formulas
may be spread to a case when we take interest to a connectivity of each two nodes of the graph r' (this scheme is an analog of a transformer electrical scheme). For this purpose it is necessary to renumber the graph r' nodes.
u
u
3
0
u
u
0
3
3. Ports superposition
Define recursively a class of bridge schemes B :
1) the arcs w1; w2,..., working independently with the probabilities px, p2,..., belong to B,
2) if the ports rl5...,r5 e B consist of nonintersecting sets of arcs then their superposition n = r(rl5...,r5) belongs to B .
A number of arcs in the superposition r' is m(r') = m(r1 ) +... + m(r5) where m(r) is a number of arcs in the port r. The reliability of the superposition r' equals to P(P1;...,P5 ) and is calculated by the formula (2) and needs
np (r') = np (f) + np (r1 ) +... + np (r5)
arithmetical operations where n (ri) is a number of arithmetical operations necessary to
calculate the
reliability p.
If np (ri) < np (r)(m(ri) -1),1 < i < 5, then
np (r')<np (r)(m(r')-1) . (5)
So a number of arithmetical operations necessary to calculate the reliability of the port r' e B has a bound which is linear increasing by a number of the port r' arcs.
For the superposition r' = r(r1,...r5) of the ports r1,...r5 e B it is easy to obtain the recurrent formulas
D (r') = min max D (P), D1 (P) = max min D1 (P) (6)
ReR i:wieR LeL i:wteL
G (r') = min 2 G (ri), G1 (r') = max 2 G1 (ri) (7)
ReR i:wteR LeL i:w,eL
Here R, L are the sets of ways and cross sections in the graph r. The constants
D(ri), D1 (ri), G(ri), G1 (ri), i = 1,...,5, are calculated by the theorems 1-4 formulas. The formulas (6), (7) allow analogously to (5) to construct linear by m (r') upper bounds for numbers of arithmetical operations
nD (r'), nDi (r'), nG (r'), nGi (r') which are necessary to calculate the constants D (r'), D (r'), G (r'), G1 (r'):
nD (r') < nD (r)(m(r') -1), nA (r') < nD1 (r)(m(P) -1) , nG (r') < nG (r)(m(P) -1) , nGi (r') < nG1 (r)(m(P) -1) .
For a comparison remark that a number of arithmetical operations necessary to define the shortest way length or the minimal cross sections ability to handle in general type graphs [5] is significantly larger.
Remark 4. The constructed algorithm of a recursive definition of a port reliability for the class B with the generating scheme r and the upper bound (5) may be spread to a case of a finite set g = {r} of generating schemes with a replacement n(r) in the formula (5) by maxn(r). For
v ' reg
example it is possible to construct g by the graphs with two arcs which are connected parallel and sequentially.
References
[1] Tsitsiashvili G.Sh. Asymptotic Analysis of Logical Systems with Anreliable Elements// Reliability: Theory and Applications, 2007. Vol. 2, № 1. Pp. 34-37.
[2] Reliability of technical systems: Handbook: Editor I.A. Ushakov. Moscow: Radio and Communication, 1985, 608 p. (In Russian).
[3] Belov V.V., Vorobiev E.M., Shatalov V.E. Graph theory. Education guidance for technical universities. Moscow: High School, 1976. (In Russian).
[4] Riabinin I.A. . Reliability and safety of structural complicated systems. Sankt-Petersberg: Edition of Sankt-Petersberg university, 2007. 276 p. (In Russian).
[5] Kormen T., Leizerson Ch., Rivest R. Algorithmes: construction and analysis. Moscow: Laboratory of basic knowledges, 2004, 955 p. (In Russian).
