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AN ACCURACY OF ASYMPTOTIC FORMULAS IN CALCULATIONS OF A RANDOM NETWORK RELIABILATY
Tsitsiashvili G.Sh., Losev A.S. •
Institute for Applied Mathematics, Far Eastern Branch of RAS 690041, Vladivostok, Radio str. 7, guram@iam.dvo.ru, alexax@bk.ru
INTRODUCTION
In this paper a problem of asymptotic and numerical estimates of relative errors for different asymptotic formulas in the reliability theory are considered. These asymptotic formulas for random networks are similar to calculations of Feynman integrals.
A special interest has analytic and numerical comparison of asymptotic formulas for the most spread Weibull and Gompertz distributions in life time models. In the last case it is shown that an accuracy of asymptotic formulas is much higher.
1. AN ASYMPTOTIC ESTIMATE OF A RELATIVE ERROR IN A DEFINITION OF A RELIABILITY LOGARITHM
Consider the nonoriented graph r with fixed initial and final nodes and with the arcs set W . Define R = {R1,..., Rn} as the set of all acyclic ways between the initial and final nodes of the graph r. Designate PR the probability of the way R work. Then in the condition
Pw ~ exp(-cwh~dw), h ^ 0, w eW,
we have:
PR ~exp(-C(R)h"dr -C(R)h~DR (1 + o(1))), where C (R) = 2 cw and DR < DR is a next by a quantity after DR = maxdw element in the set
w:dw =D(R) weR
{dw,w e R}, C (R) = 2 cw . If in the way R this element is absent we put then DR = -oo,
w:dw = D(R)
C ' ( R ) = 0.
Denote Dr = min DR and designate R1 = {R: DR = Dr}, R2 = R \ R1, then the probability Pr of
ReR
the graph r work satisfies the formulas
Pr ~ P1 + Pr2, P{ ~ 2 Pr = 2 exp (-C (R)h~DR - C (R)h"DR (1 + o (1))) , i = 1,2.
ReR,. ReRi
By the definition
Pr1- 2 exp (-C (R)h-Dr - C'(R)h-DR (1 + o (1))) ~
ReR1
~ exp(-Crh-Dr) 2 exp(-C(R)h~DR (1 + o(1))),
ReR1
where Cr = min CR and DR < Dr, R e R1, so
ReRi
P/ ~ exp(-Crh-Dr )exp(-Cf h~D (1 + o (1))), where Df = min DR < Dr, Cf = min C' (R).
ReRi :C(R)=Cr ReRi :C(R)=Cr, DR =Df
And consequently DR < Dr, R e R2, Pr2 = o (p):
Pr2~ Z exp (-C (R)h~DR - C'(R )h ~DR (1 + o (1))) ~ 2 exp (-C (R)h"DR (1 + o (1))} =
ReR 2 ReR 2
= exp(-Crh"Dr) Z exp(-Crh"Dr -C(R)h"DR (1 + o(1))) ~
ReR 2
~ exp(-Crh"Dr) Z exp(-C(R)h~DR (1 + o(1))) ~
ReR2
~ exp(-Crh~Dr )exp(-CjTh(1 + o (1))),
where
Df = min Dr > Dr, Cf = min C (R) .
ReR2 ReR 2
So we have:
Pr ~exp(-Crh-Dr )(exp(-Cf h(1 + o(1))) + exp(-Cf h(1 + o(1)))) ~ exp(-Crh~Dr)exp(-Cfh(1 + o(1))).
As a result obtain that
ln Pr ~ -Cr h"Dr (1 + AhAr (1 + o (1))), Ar= Dr - Df > 0, A = Cf / Cr . And consequently
lnPr__Aba
-C (r) h"D'
^ -1~ AhAr . (1)
2. AN ASYMPTOTIC ESTIMATE OF A RELATIVE ERROR IN A DEFINITION OF A RELIABILITY
Assume that P(Up) is the probability of the event Up that all arcs wf,...,w^ of the way Rp work. Then we have
Pr= P| U Up ,p=1
(2)
Suppose that the probability of the arc w eW work equals exp (-cwh dw), h > 0 , where cw, dw are some positive numbers and for arcs w' ^ w" the constants dw ^ dw . So we have
P (Up ) = exp
( mp -Z c p,
1 wj V J= J
-d „ A
Assume that the enumeration of the arcs in the way Rp satisfies the inequalities
i p >d p >...>< w, w2
Denote Dp = \dwP,..., dwP I and introduce on the vectors set {Dp, 1 < p < n} the following
V wi wmp )
order relation. Say that Dp f Dq, if for some k < min (mp, mq) the first k -1 components of these
vectors coincide and the k component in the vector Dp is larger than in the vector Dq . If there is not such k and in the vectors Dp , Dq all first min (mp, mq) components coincide then Dp f Dq for
mp < mq .
Remark that for some p ^ q the arcs sets {w e Rp}, {w e Rq} can not satisfy the inclusion {w e Rp} c {w e Rq}. In the opposite case there is the node u* in which the ways Rp Rq diverge by the arcs (u*,up)(u*,uq). But as the arc (u*,up)e{w e Rq} then the way Rq has a cycle. This conclusion contradicts with the assumption that the way Rq is acyclic.
So as the quantities dw are different then Dp ^ Dq, p ^ q . As a result we obtain the order relation on the vectors set {D1,...,Dn}, and if Dp f Dq, h ^ 0, so P(Uq) = o(P(Up)). It is not difficult to check that this relation is transitive. Consequently the order relation on the set {D1,...,Dn} is linear. Assume that the enumeration of the vectors Dp satisfies the formula
D1 f... f Dn . From the formula (2) we have
m
Pr* - S P (UpUq )< Pr< Pr* , Pr* = S P (Up). (3)
1< p<q<m p=1
As the inclusion {w e Rp }c{w e Rq} is not true for p ^ q so in the way Rp there is an arc which does not belong to the way Rq . Consequently we have
P(Uq) = o(P(Up)), 1 < p < q < m, S P(UpUq) = o(P(U2)) (4)
1</< j<m
The formulas (3), (4) give us the following asymptotic expansion for Pr with the first and the second members of the smallness:
Pr ~ Pr*~ P (U1), Pr-P (U1) ~ P (U2), P (U2 ) = o (P (U1)), h ^ 0. (5)
3. AN APPLICATION TO LIFE TIME MODELS
Suppose tha tw are independent random variables and characterize life times of the arcs w e W. Denote Denote pw (h) = P (tw > t) and designate the life time of the graph r by
Tr = min max Tw.
ReR weR
If h = 1/1 then we have with t the Weibull distributions of the arcs life times and the formula
'"C^tD1 -1 = * (t) ~ G (t) (6)
If h = exp(-t) , t , then we have the Gompertz distributions of the arcs life times and the formula (1) transforms into
lnP > t) _! = gi (t) ~ G (t) = g(exp(t)), (7)
-C (r)exp (Drt) so G1 (t) = o (G (t)).
Consequently for the Gompertz distributions the convergence rate in the asymptotic (7) is much faster than for the Weibull distributions in (6).
If h = 1/1, t then for the Weibull distributions of the arcs life times the formula (5)
transforms into
( mi d i ^
P(rr> t)~exp^-Z cw]t J, (8)
(P(TY> t) d 1 ^ _ 1 = f (t) ~ F(t) = o(1) , F(t) = exp(-Z;=iCw2tdw22 ;=iCwi tdwj ] . exp[_Z;=1Cwit w '
If h = exp(_t), t then for the Gompertz distributions of the arcs life times the formula (5) transforms into
P(Tr > t) ~ expf- m± Cw1 exp(dw11)), ( pTr >t\-^ _ 1 = fx (t) ~ F1 (t) = F(exp(t)), (9)
V '' ' ^ expexp(t
so F1 (t ) = o (F (t)).
Consequently for the Gompertz distributions the convergence rate in the asymptotic (9) is much faster than for the Weibull distributions in (8).
For h = 1/1 denote |Pr* -Pr|/Pr = A(t), and for h = exp(_t) designate |Pr* -Pr|/Pr = A1 (t). It is clear that A1 (t) = A (exp (t)) tends to zero for t ^<x> much faster than A (t).
From this section we see that the Gompertz distributions of the arcs life times (these distributions are preferable in life time models of alive [1] and of complex information [2] systems), give much more accuracy asymptotic formulas than the Weibull distributions. These both distributions are limit for a scheme of a minimum of independent and identically distributed random variables.
4. RESULTS OF NUMERICAL EXPERIMENTS FOR BRIDGE SCHEMES
U1
Fig.1 The bridge scheme.
Consider the bridge scheme r represented on the Fig. 1 with the parameters d1 = 0.02, d2 = 0.09, d3 = 0.5, d4 = 0.72, d5 = 0.2 . Calculate the functions f (t), f1 (t), A (t), A1 (t), * (t), g1 (t).
Fig.2 The relative errors f (t) and f (t) in the reliability Pr calculations
A(t)
05
06
Aft) 04 •
03 -
02
Fig.3 The relative errors A (t) and A1 (t) in the reliability Pr calculations
Fig.4 The relative errors g (t) and g1 (t) in ln Pr calculations.
The results of the numerical experiments represented above show that a transition from the Weibull to the Gompertz distribution decreases significantly relative errors in calculations of the
reliability and its logarithm. The asymptotic estimate Pr* of the reliability Pr is better than P (U1 ). The relative error of the ln Pr calculation is larger than the relative error of the Pr calculation. But a complexity of the ln Pr calculation is smaller.
REFERENCES
1. Gavrilov N.A., Gavrilova N.S. Biology of life duration. M.: Science. 1991. 280 p. (In Russian).
2. Dvoeglazov D.V., Matchin V.T., Mordvinov V.A., Svechnikov S.V.,Trifonov N.I., Filinov A.M., Shlenov A.Yu. Information systmes in information media control. Part III. Moscow state institute of radiotechnique, electronic and automatic (technical university). 2002. 181 p. (In Russian).
