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12
By maximizing the logarithm of likelihood function for grouped data, we calculate p = 0.643316, a = 0.001284, X = 0.0288. For these values of parameters, we prove Pearson's test of fit and compute X = 0.68. By Proposition 4, we conclude that r (t) is unimodal.
3. Bounds and Approximation
In this section we cover some of the well known bounds and approximations to the MTBF. By the inequality
i
J R(s)ds £ min{t, ET} .
where ET is the mean value of T, we obtain the upper bound for MTBF:
Iff (t) is unimodal then exists tm and ti such that f'(tm) =0, tm<ti and g(t)>0 for t e (0,ti), g(t)<0 for te (ti, <x>).
Proposition 7. If T eMTFR, then 1
MTBF >
r(t)
for t > 0.
Proof. By Definition 1, if M (t) is non-decreasing, then we have
f(t)J R(s)ds - F(t)R(t) [M(t)]' =-5-> 0
(JR(s)ds)7
and
MTBFU =
1
F(t)
min{t, ET} .
MTBF > MTBFL =
r(t)
for te {t: r (t)>0}.
In [1] for MTBF proposed is the following average approximation
MTBFa = -
t_ 1 + R(t) 2 F(t)
Proposition 8. If the lifetime T has unimodal failure rate function r (t), then T e MTFR if and only if
r(rn) ET - 1 > 0.
Proof. Let
Proposition 6. Iff (t) is unimodal, then these exist tj such that MTBFa is a lower bound of MTBF for t e <0, t{) and it is an upper bound of MTBF for t e (ti, <»)
Proof. We consider the difference
t
J R(s)ds
g(t) = -
__t R(t) +1
~Fft) 2 F(t)
i
h (t) = r (t) J R(s)ds - F (t).
It is easy to show that h (0) = 0 and
h (<x>) = r (<x>) ET - 1.
The first derivative of h (t) is
i
h' (t) = r' (t) J R(s)ds.
and
t 1
gi(t) = J R(t)dt - - t(R(t) +1).
It is easy to find that g1(0) = 0, g1(+œ) = - œ. The first derivative of g'1(t) is
g'i(t) =-[tf(t) - F(t)].
If f (t) is decreasing then g'1(t) < 0 and MTBFA is a lower bound for MTBF.
If r (t) is increasing, then h (t) is increasing and if r (t) is decreasing, then h (t) is decreasing. This completes the proof.
Example 2. Consider the system with failure rate function proposed in Example 1. The exact and approximate results for MTBF are shown in Table 1 for varying R (t) with the corresponding t. The results show that the average approximation MTBFA is greater than MTBF. For this data, we compute ET = 34.81 and AET - 1 > 0 and by Proposition 8 we obtain that T e MTFR.
Table 1. The values of the exact and approximate MTBF of lifetime data
1
R(t) t MTBF MTBFu MTBFl MTBFa
0.99999 0.00054 53.97 53.97 53.97 53.97
0.9999 0.0054 53.97 53.97 53.97 53.97
0.999 0.05398 53.95 53.98 53.93 53.95
0,99 0.54031 53.76 54.03 53.55 53.76
0.9 5.43971 51.67 54.40 49.22 51.68
0.8 10.9261 49.15 54.63 44.05 49.17
0.7 16.4696 46.62 54.90 39.17 46.66
0.6 22.1619 44.21 55.40 34.90 44.32
0.5 29.1751 41.98 56.35 34.81 42.26
0.4 34.8007 39.94 58.00 34.81 40.60
0.3 42.5854 38.11 49.73 34.81 39.54
0.2 52.8174 36.50 43.51 34.81 39.61
0.1 70.2655 35.23 38.68 34.81 42.94
4. Conclusion
In this paper we show that, from a practical point view, the unimodal failure rate model can be obtained from a mixture of two common IFR models. This model is flexibility. Practical relevance and applicability have been demonstrated using well known data. In this paper a simple approximation of the MTBF of systems subjected to periodic maintenance has been proposed as well.
[8] Marschall, A. W. & Proschan, F. (1972). Classes of distributions applicable in replacement with renewal theory implications, In: L. LeCam et al eds., Proc. of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. I, University of California Press, Berkeley, 395 -415.
[9] Mondro, M. J. (2002). Approximation of mean time between failures when system has periodic maintenance. IEEE Transaction on Reliability 2, 51, 166 - 167.
[10] Mudholkar, G. S., Srivastava, D. K. & Freiner, M. (1995). The Exponentiated Weibull Family: A Reanalysis of the Bus - Motor - Failure Data. Technometrics 4, 37, 436 - 445.
[11] Wondmagegnehu, E. T., Navarro I. & Hernandez P. J. (2005). Bathtub Shaped Failure Rates from a Mixtures: A Practical Point of View. IEEE Transaction on Reliability 2, 54, 270 - 275.
References
[1] Amari, S. V. (2006). Bounds on MTBF of Systems Subjected to Periodic Maintenance. IEEE Transaction on Reliability 3, 55, 469 - 475.
[2] Barlow, R. E. & Campo, R. (1975). Total time on test processes and applications to failure data analysis. In: R. E. Barlow, J. Fussel & N. P. Singpurwalla, eds. Reliability and Fault Tree Analysis. SIAM, Philadelphia, 451 - 481.
[3] Block, H. & Joe, H. (1997). Tail Behavior of the Failure Rate Functions. Lifetime Data Analysis 3, 209 - 288.
[4] Block, H.W., Savits, T.H. & Wondmagegnehu, E.T. (2003). E. T. Mixtures of distributions with linear failure rates J. Appl. Probab. 40, 485 - 504.
[5] Klefsjo, B. (1982). On ageing properties and total time on test transforms. Scandinavian Journal of Statistics. Theory and Applications 9 (1), 37 - 41.
[6] Klutke, G. A., Kiessler, R. C. & Wortman, M. A. (2003). A critical look at the bathtub curve. IEEE Transactions on Reliability 1, 52, 125 - 129.
[7] Knopik, L. (2006). Characterization of a class of lifetime distributions. Control and Cybernetics 35(2), 407 - 411.
Kolowrocki Krzysztof
Maritime University, Gdynia, Poland
Reliability modelling of complex systems - Part 1
Keywords
reliability, large system, asymptotic approach, limit reliability function Abstract
The paper is concerned with the application of limit reliability functions to the reliability evaluation of large systems. Two-state large non-repaired systems composed of independent components are considered. The asymptotic approach to the system reliability investigation and the system limit reliability function are defined. Two-state homogeneous series, parallel and series-parallel systems are defined and their exact reliability functions are determined. The classes of limit reliability functions of these systems are presented. The results of the investigation concerned with domains of attraction for the limit reliability functions of the considered systems and the investigation concerned with the reliability of large hierarchical systems as well are discussed in the paper. The paper contains exemplary applications of the presented facts to the reliability evaluation of large technical systems.
types are also used in telecommunication, in rope transportation and in transport using belt conveyers and elevators. Rope transportation systems like port elevators and ship-rope elevators used in shipyards during ship docking are model examples of seriesparallel and parallel-series systems. In the case of large systems, the determination of the exact reliability functions of the systems leads us to complicated formulae that are often useless for reliability practitioners. One of the important techniques in this situation is the asymptotic approach to system reliability evaluation. In this approach, instead of the preliminary complex formula for the system reliability function, after assuming that the number of system components tends to infinity and finding the limit reliability of the system, we obtain its simplified form.
The mathematical methods used in the asymptotic approach to the system reliability analysis of large systems are based on limit theorems on order statistics distributions, considered in very wide literature, for instance in [4]-[5], [7], [12]. These theorems have generated the investigation concerned with limit reliability functions of the systems composed of two-state components. The main and fundamental results on this subject that determine the three-element classes
1. Introduction
Many technical systems belong to the class of complex systems as a result of the large number of components they are built of and their complicated operating processes. As a rule these are series systems composed of large number of components. Sometimes the series systems have either components or subsystems reserved and then they become parallel-series or seriesparallel reliability structures. We meet large series systems, for instance, in piping transportation of water, gas, oil and various chemical substances. Large systems of these kinds are also used in electrical energy distribution. A city bus transportation system composed of a number of communication lines each serviced by one bus may be a model series system, if we treat it as not failed, when all its lines are able to transport passengers. If the communication lines have at their disposal several buses we may consider it as either a parallel-series system or an "m out of n" system. The simplest example of a parallel system or an ' 'm out of n" system may be an electrical cable composed of a number of wires, which are its basic components, whereas the transmitting electrical network may be either a parallel-series system or an "m out of n"-series system. Large systems of these
of limit reliability functions for homogeneous series systems and for homogeneous parallel systems have been established by Gniedenko in [6]. These results are also presented, sometimes with different proofs, for instance in subsequent works [1], [8]. The generalizations of these results for homogeneous "m out of n" systems have been formulated and proved by Smirnow in [13], where the seven-element class of possible limit reliability functions for these systems has been fixed. As it has been done for homogeneous series and parallel systems classes of limit reliability functions have been fixed by Chernoff and Teicher in [2] for homogeneous series-parallel and parallel-series systems. Their results were concerned with so-called "quadratic" systems only. They have fixed limit reliability functions for the homogeneous seriesparallel systems with the number of series subsystems equal to the number of components in these subsystems, and for the homogeneous parallel-series systems with the number of parallel subsystems equal to the number of components in these subsystems. Kolowrocki has generalized their results for non-"quadratic" and non-homogeneous series-parallel and parallel-series systems in [8]. These all results may also be found for instance in [9]. The results concerned with the asymptotic approach to system reliability analysis have become the basis for the investigation concerned with domains of attraction ([9], [11]) for the limit reliability functions of the considered systems and the investigation concerned with the reliability of large hierarchical systems as well ([3], [9]). Domains of attraction for limit reliability functions of two-state systems are introduced. They are understood as the conditions that the reliability functions of the particular components of the system have to satisfy in order that the system limit reliability function is one of the limit reliability functions from the previously fixed class for this system. Exemplary theorems concerned with domains of attraction for limit reliability functions of homogeneous series systems are presented here and the application of one of them is illustrated. Hierarchical series-parallel and parallel-series systems of any order are defined, their reliability functions are determined and limit theorems on their reliability functions are applied to reliability evaluation of exemplary hierarchical systems of order two.
All the results so far described have been obtained under the linear normalization of the system lifetimes. The paper contains the results described above and comments on their newest generalizations recently presented in [9].
2. Reliability of two-state systems
We assume that
Ei, i = 1,2,...,«, n eN,
are two-state components of the system having reliability functions
Ri(t) = P(Ti > t), t e (-¥, ¥),
where
Ti, i = 1,2,...,n,
are independent random variables representing the lifetimes of components Ei with distribution functions
Fi(t) = P(Ti £ t), t e (-¥, ¥).
The simplest two-state reliability structures are series and parallel systems. We define these systems first.
Definition 1. We call a two-state system series if its lifetime T is given by
T = min{T }.
1£i£n
The scheme of a series system is given in Figure 1.
Figure 1. The scheme of a series system
and only if all its components are not failed, and therefore its reliability function is given by
Rn (t) = nR (t), t e (-¥, ¥). (1)
i =1
Definition 2. We call a two-state system parallel if its lifetime T is given by
T = max{Ti}.
1<i<n
The scheme of a parallel system is given in Figure 2.
Figure 2. The scheme of a parallel system
Definition 4. We call a two-state series-parallel system regular if
Definition 2 means that the parallel system is failed if and only if all its components are failed and therefore its reliability function is given by
Rn(t) = 1 -nF(t), tÎ (-¥,¥).
(2)
Another basic, a bit more complex, two-state reliability structure is a series-parallel system. To define it, we assume that
Eij, i = 1,2,...,kn, j = 1,2,...,/i, kn, li, l2,...,lkn ÎN,
are two-state components of the system having reliability functions
Rj(t) = P(Tj > t), t Î (-¥, ¥), where
Tj i = 1,2,...,kn, j = ^^...^i,
are independent random variables representing the lifetimes of components E. with distribution functions
Fj(t) = P(Tij < t), t Î (-¥, ¥).
Definition 3. We call a two-state system series-parallel if its lifetime T is given by
T = maximiniT,.,.}.
1<i<kn 1< j<h 1
By joining the formulae (1) and (2) for the reliability functions of two-state series and parallel systems it is easy to conclude that the reliability function of the two-state series-parallel system is given by
/1 = /2 = . . . = lkn = ln, ln îN,
i.e. if the numbers of components in its series subsystems are equal.
The scheme of a regular series-parallel system is given in Figure 3.
Figure 3. The scheme of a regular series-parallel system
Definition 5. We call a two-state system homogeneous if its component lifetimes have an identical distribution function F(t), i.e. if its components have the same reliability function
R(t) = 1 - F(t), t e (-¥, ¥).
The above definition and equations (1)-(3) result in the simplified formulae for the reliability functions of the homogeneous systems stated in the following corollary.
Corollary 1. The reliability function of the homogeneous two-state system is given by
- for a series system
Rn (t)= [R(t)]n, t e (-¥, ¥), (4)
- for a parallel system
Rn(t) =1 - [F(t)]n, t e (-¥, ¥), (5)
R knhh h (t) = 1 -n [1 -n R, (t)], t e (-¥, ¥), (3)
where kn is the number of series subsystems linked in parallel and li are the numbers of components in the series subsystems.
- for a regular series-parallel system
R », (t) = 1 - [1 - [R(t)]/n ]kn, t e (-¥, ¥).
(6)
i=1
3. Asymptotic approach to system reliability
The asymptotic approach to the reliability of two-state systems depends on the investigation of limit distributions of a standardized random variable
(T - bn )/An ,
where T is the lifetime of a system and an > 0 and bn e (-¥, <) are suitably chosen numbers called normalizing constants. Since
P((T - bn)/ an > t) = P(T > ant + bn) = Rn(ant + bn),
where Rn(t) is a reliability function of a system composed of n components, then the following definition becomes natural.
Definition 6. We call a reliability function ft(t) the limit reliability function of a system having a reliability function Rn(t) if there exist normalizing constants an > 0, bn e(-¥, <) such that
lim Rn(ant + bn) = ft(t) for t eCft,
lim nF(ant + bn) = V(t) for t eCV
Proof The proof may be found in [1], [6], [8].
Lemma 1 is an essential tool in finding limit reliability functions of two-state series systems. It also is the basis for fixing the class of all possible limit reliability functions of these systems. This class is determined by the following theorem.
Theorem 1. The only non-degenerate limit reliability functions of the homogeneous two-state series system are:
ft j(t) = exp[-(-t)-a ] for t < 0,
ft j(t) = 0 for t > 0, a > 0;
ft 2(t) = 1 for t < 0,
ft 2(t) = exp[-ta ] for t > 0, a > 0;
ft 3(t)= exp[-exp[t]] for t e (-<, <).
where Cft is the set of continuity points of ft(t).
Thus, if the asymptotic reliability function ft(t) of a system is known, then for sufficiently large n, the approximate formula
Rn(t) @ ft((t - bn) /an), t e (-<, <). (7)
may be used instead of the system exact reliability function Rn(t).
3.1. Reliability of large two-state series systems
The investigations of limit reliability functions of homogeneous two-state series systems are based on the following auxiliary theorem.
Lemma 1. If
(i) ft (t) = exp[-V (t)] is a non-degenerate reliability function,
(ii) Rn (t) is the reliability function of a homogeneous two-state series system defined by (4),
(iii) an > 0, bn e (-¥, <), then
lim Rn (ant + bn) = ft(t) for t eCft
if and only if
Proof The proof may be found in [1], [6], [8].
3.2. Reliability of large two-state parallel
systems
The class of limit reliability functions for homogeneous two-state parallel systems may be determined on the basis of the following auxiliary theorem.
Lemma 2. If
(i) ft(t) = 1 - exp[-V(t)] is a non-degenerate reliability function,
(ii) Rn(t) is the reliability function of a homogeneous two-state parallel system defined by (5),
(iii) an > 0, bn e (-¥, <), then
lim Rn(ant + bn) = ft(t) for t eCft ,
if and only if
lim nR(ant + bn) = V(t) for t eCV . Proof The proof may be found in [1], [6], [8].
By applying Lemma 2 it is possible to fix the class of limit reliability functions for homogeneous two-state
parallel systems. However, it is easier to obtain this result using the duality property of parallel and series systems expressed in the relationship
Rn(t) =1 - Rn (-t) for t e (-¥, ¥),
that results in the following lemma, [1], [6], [8]-[9].
Lemma 3. If ^ (t) is the limit reliability function of a homogeneous two-state series system with reliability functions of particular components R (t), then
^(t) = 1 (-t) for t eC^
is the limit reliability function of a homogeneous two-state parallel system with reliability functions of particular components
R(t) = 1 - R (-t) for t e CR.
At the same time, if (an,bn) is a pair of normalizing constants in the first case, then (an, - bn) is such a pair in the second case.
The application of Lemma 3 and Theorem 1 yields the following result.
Theorem 2. The only non-degenerate limit reliability functions of the homogeneous parallel system are:
^(t) = 1 for t < 0,
^(t) = 1 - exp[-ra] for t > 0, a > 0;
^(t) = 1 - exp[-(-t)a] for t < 0,
^2(t) = 0 for t > 0, a > 0;
^3(t) = 1 - exp[-exp[-t]] for t e(-<x>,<x>).
Proof. The proof may be found in [1], [6], [8].
3.3. Reliability evaluation of large two-state series-parallel systems
The proofs of the theorems on limit reliability functions for homogeneous regular series-parallel systems and methods of finding such functions for individual systems are based on the following essential lemmas. Lemma 4. If
(i) kn ® <,
(ii) ^(t) =1 - exp[-V(t)] is a non-degenerate reliability function,
(iii) R k , (t) is the reliability function of a
kn ,ln
homogeneous regular two-state series-parallel system defined by (6),
(iv) an > 0, bn e(-< ,< ),
then
lim R (ant + bn) = ^(t) for t eCm
n®< n , n
if and only if
lim kn[R(ant + bn) ]ln = V(t) for t eCV .
Proof. The proof may be found in [8].
Lemma 5. If (i) kn ® k, k > 0, ln ® ¥,
(ii) ^(t) is a non-degenerate reliability function,
(iii) R. , (t) is the reliability function of a
kn ,ln
homogeneous regular two-state series-parallel system defined by (6),
(iv) an > 0, bn e(-< ,< ),
then
lim R k l (ant + bn) = ^(t) for t eCm ,
if and only if
lim [R(ant + bn) ]'n = ^o(t) for t C
where ^0(t) is a non-degenerate reliability function and moreover
^(t) = 1 - [1 - %(t)]k for t e(-<,<).
Proof. The proof may be found in [8].
The types of limit reliability functions of a seriesparallel system depend on the system shape [7], i.e. on the relationships between the number kn of its series subsystems linked in parallel and the number ln of components in its series subsystems. The results based on Lemma 4 and Lemma 5 may be formulated in the form of the following theorem.
Theorem 3. The only non-degenerate limit reliability functions of the homogeneous regular two-state seriesparallel system are:
Case 1. kn = n, | ln - c log n | >> s, s > 0, c > 0. ^:(t) = 1 for t < 0,
0
ft:(t) = 1 - exp[ -1-a ] for t > 0, a > 0; ft2(t) = 1 - exp[- (-t)a] for t < 0, ft2(t) = 0 for t > 0, a > 0; ft3(t) = 1 - exp[-exp[-t]] for t e(-<,<); Case 2. kn = n, ln - c log n » s, s e(-<,<), c > 0. ft(t) = 1 for t < 0,
ftt(t) = 1 - exp[-exp[-ta - s/c]] for t > 0, a > 0;
ft5(t) = 1 - exp[-exp[(-t)a - s/c]] for t < 0,
ft5(t) = 0 for t > 0, a > 0;
ft6(t) = 1 - exp[-exp[b(-t)a - s/c]] for t < 0,
ft6(t) = 1 - exp[-exp[-ta - s/c]] for t > 0, a > 0, b > 0;
ft7(t) = 1 for t < t1,
ft7(t) = 1 - exp[-exp[-s/c]] for tx < t < t2, ft7(t) = 0 for t > t2, ti < t2; Case 3. kn ® k, k > 0, ln ® <. ft8(t) = 1 - [1 - exp[ - (-t)-a ]]k for t < 0, ft8(t) = 0 for t > 0, a > 0; ft?(t) = 1 for t < 0,
ft,(t) = 1 - [1 - exp[-ta]]k for t > 0, a > 0;
ft10(t) = 1 - [1 - exp[-exp t]]k for t e(-<,<).
Proof. The proof may be found in [8].
Using the duality property of parallel-series and seriesparallel systems similar to this given in Lemma 3 for parallel and series systems it is possible to prove that the only limit reliability functions of the homogeneous regular two-state parallel-series system are
ft (t) = 1 - ft;(-t) for t eCft , i = 1,2,...,10.
Applying Lemma 2, it is possible to prove the following fact ([9]).
Corollary 2. If components of the homogeneous two-state parallel system have Weibull reliability functions
R(t) = exp[-b ta ] for t > 0, a > 0, b > 0 and
an = bn/(alog n), bn = (log n/b)1/a, then
ft3(t) = 1 - exp[-exp[-t]], t e(-<,<), is its limit reliability function.
Example 1 (a steel rope, durability). Let us consider a steel rope composed of 36 strands used in ship rope elevator and assume that it is not failed if at least one of its strands is not broken. Under this assumption we may consider the rope as a homogeneous parallel system composed of n = 36 basic components. Further, assuming that the strands have Weibull reliability functions with parameters
a = 2, b = (7.07)-6,
by (5), the rope's exact reliability function takes the form
R36(t) = 1 - [1 - exp[-(7.07)-6t2]36 for t > 0. Thus, according to Corollary 2, assuming
an = (7.07)3/(2>g36), bn = (7.07)^log36
and applying (7), we arrive at the approximate formula for the rope reliability function of the form
R36(t) @ft3((t - bn)/an)
= 1 - exp[-exp [-0.010711 + 7.167]] for t e(-<,<).
The mean value of the rope lifetime T and its standard deviation, in months, calculated on the basis of the above approximate result and according to the formulae
E[T] = Can + bn, s =pan /46, where C @ 0.5772 is Euler's constant, respectively are:
E[T] @ 723, c @ 120.
The values of the exact and approximate reliability functions of the rope are presented in Table 1 and graphically in Figure 4. The differences between them are not large, which means that the mistakes in replacing the exact rope reliability function by its approximate form are practically not significant.
Table 1. The values of the exact and approximate reliability functions of the steel rope
t R36(t) —3(t - b" ) an A = R36 -—3
0 1.000 1.000 0.000
400 1.000 1.000 0.000
500 0.995 0.988 -0.003
550 0.965 0.972 -0.007
600 0.874 0.877 -0.003
650 0.712 0.707 0.005
700 0.513 0.513 0.000
750 0.330 0.344 -0.014
800 0.193 0.218 -0.025
900 0.053 0.081 -0.028
1000 0.012 0.029 -0.017
1100 0.002 0.010 -0.008
1200 0.000 0.003 -0.003
Figure 4. The graphs of the exact and approximate reliability functions of the steel rope
4. Domains of attraction for system limit reliability functions
The problem of domains of attraction for the limit reliability functions of two-state systems solved completely in [11] we will illustrate partly for two-state series homogeneous systems only. From Theorem 1 it follows that the class of limit reliability functions for a homogeneous series system is composed of three functions, ft a(t), i = 1,2,3. Now
we will determine domains of attraction ^ for
ft«
these fixed functions, i.e. we will determine the conditions which the reliability functions R(t) of the
particular components of the homogeneous series system have to satisfy in order that the system limit reliability function is one of the reliability functions
ft>), i = 1,2,3.
Proposition 1. If R(t)is a reliability function of the homogeneous series system components, then
R(t) e D
—1
if and only if
lim 1 " R(r ) = t6 for t > 0. r®-¥ 1 - R(rt)
Proposition 2. If R(t)is a reliability function of the homogeneous series system components, then
R(t) î D— 2
if and only if
(i) 3 y e (-¥, œ) R(y) = 1 and R(y + e) < 1 for e > 0,
(ii) lim 1 - R(rt + y) = t6 for t> 0.
r®0+ 1 - R(r + y)
Proposition 3. If R(t) is a reliability function of the homogeneous series system components, then
R(t) e D—3
if and only if lim n[1 - R(ant + bn )] = e' for t e (-œ, œ)
with
bn = inf{t : R(t + 0) < 1 -1 < R(t - 0)}, n
an = inf{t : R(t (1 + 0) + bn )
< 1 --< R(t (1 - 0) + bn )}.
n
Example 2. If components of the homogeneous series system have reliability functions
R(t) =
1, t < 0 1 -1, 0 < t < 1 0, t > 1,
then
R(t) e D- .
— 2
The results of the analysis on domains of attraction for limit reliability functions of two-state systems may automatically be transmitted to multi-state systems. To do this, it is sufficient to apply theorems about two-state systems such as the ones presented here to each vector co-ordinate of the multi-state reliability functions ([9], [14]).
5. Reliability of large hierarchical systems
Prior to defining the hierarchical systems of any order we once again consider a series-parallel system like a system presented in Figure 3. This system here is called a series-parallel system of order 1. It is made up of components
E. . , i = 1,2,..., k , , = 1,2,...,l. ,
/j ,1 ? 1 5 5 5 ft? J 1 5 5
with the lifetimes respectively
T . , i, = 1,2,..., k , 7, = 1,2,...,l. .
i1 7 j' 1 ' ' ' n 5 1 ' ' ' i,
Its lifetime is given by
T = max{ min {T ,l }}.
1<i1 <kn s<,1 '1j1^
Now we assume that each component
E. . , i = 1,2,..., k , /', = 1,2,...,l. ,
q /1 ? 1 5 5 ' n 5 ./155 ' '
(8)
T .. . , i2 = 1,2,...,k
,...,k ('LJl), j2 = 1,2,...,l
' ' n ' J2 ' ' ' ;
(i1 j1 ) !2
are the lifetimes of the subsystem components Et hh, .
The system defined this way is called a hierarchical series-parallel system of order 2. Its lifetime, from (8) and (9), is given by the formula
T = max{ min [ max ( min T )1},
1<i1 <kn Sij1 <ii1 L1<i2 <kn(i'1 ) 1<j2 <|iJi ) j1'2j2 'J 1
where kn is the number of series systems linked in parallel and composed of series-parallel subsystems E , l are the numbers of series-parallel subsystems
Ein these series systems, kn' are the numbers of
series systems in the series-parallel subsystems E
linked in parallel, and l^'1'1' are the numbers of
components in these series systems of the seriesparallel subsystems E .
In an analogous way it is possible to define two-state parallel-series systems of order 2. Generally, in order to define hierarchical series-parallel and parallel-series systems of any order r, r > 1, we assume that
E.
where
i = 1,2,..., k , j = 1,2,...,l. , i, = 1,2,...,k
1 n 1 i1 2
(i1 J'l)
j = 1,2,..., l(i1jL), ..., i = 1,2,...,k
J 2 5 5 5 i2 5 5 r 55 5,
j = 12 I (ljl..Jr-l/'r-1 )
J r 5 5 * * * 5 ir
and
fa/l".^-1Jr-1)
of the series-parallel system of order 1 is a subsystem composed of components
E
^ , , i, = 1,2,...,k (ilJl), j, = 1,2,..., l
mn J2 ' 2 ' ' ' n ' ^ 2 ill,
(i1 j'l) i2
and has a series-parallel structure.
This means that each subsystem lifetime T^, is given
by
T. ' = max { min (T7 .. . }}, (9)
1/1 <i, ,) 1 (n , ) /2 J J ' v 7
i2
i = 1,2,..., k , / = 1,2,...,l. ,
15 5 5ft? 1 5 5 'i1'
where
('1j1 ) /("1J1 )
n
l (iL./i ...'r-ijr-i) e ^
k , i. , k {'1j1\ VAJl), ..., k n i1 n i2
('1j1..
are two-state components having reliability functions
R (t) = ' ^ ' > t), t e(-¥,¥),
Vl^r/r w v •1J1...1rjr ' ' V ' /'
and random variables
T ' ^ '
i1J1...irjr
where
i = 1,2,..., k , j = 1,2,...,l. , i' = 1,2,...,k
1 n 1 i1 2
(i1 J'l)
r-1 r-1
r
j, = 1,2,..., lii1j1), ..., i = 1,2,...,k
J 2 ' ' ' i2 ' ' r '' 'y
j = 1,2,..., l^1^ -1jr-1), J r 1 1 1 ir '
(i1j1..ir -1 jr-1)
are independent random variables with distribution functions
F . . (t) = P(T . . . < t), t e(-¥,¥),
11 -Wr W V 11 ...irir V '
representing the lifetimes of the components Eiijl..,irjr.
Definition 7. A two-state system is called a seriesparallel system of order r if its lifetime T is given by
T = max { min { max { min )
1<i1<k„ 1£ .1 <li1 1<i2 <kn('1J1) 1< j2 <l W1)
max ( min T )1...}}},
<ir < kS'1 j1..'r-1 jr-1 T 1< jr <1 (i1 j1...ir-1 jr-1) i1 j1..irjr n
.r
k i'i1j1) = = k {nj1.. 'r-1jr-1) = k
where kn is the number of series systems in the seriesparallel subsystems and ln are the numbers of seriesparallel subsystems or respectively the numbers of components in these series systems.
Using mathematical induction it is possible to prove that the reliability function of the homogeneous and regular two-state hierarchical series-parallel system of order r is given by
Rk,kn,in (t) = 1 -[1 -[Rk-i,kn,in (t)]ln ]kn for k = 2,3,...,r and
R1 knin (t) = 1 - [1 - [R(t)]ln ]kn, t e(-¥,¥),
where k , k ), ..., k j2".'r-1jr-1) are the numbers
n ' n ' ' n
of suitable series systems of the system composed of series-parallel subsystems and linked in parallel, li1 ,
), ..., l(j'2J2~.'r-2J-2) are the numbers of suitable
'2 ' ' V-1
series-parallel subsystems in these series systems, and iOui'2j2 Kir-ij>-i) are the numbers of components in the
series systems of the series-parallel subsystems.
Definition 8. A two-state series-parallel system of order r is called homogeneous if its component lifetimes Ti j ... i j have an identical distribution
H--- lrJr
function
F(t) = P(V.Wr < t), t e (-¥, ¥), where
.1 = 1,2,..., kn , .1 = 1,2,...,li1, .2 = l2--k
j, = 1,2,...,l(i'1j1),..., i = 1,2,...,k (i1j1...'r-1Jr-1), 2 i2 r n
j = 1,2,..., l^1^-1jr-1), r ir
(i1 j1)
i.e. if its components E^ have the same reliability
function R(t) = 1 - F(t), t e(-¥,¥).
Definition 9. A two-state series-parallel system of order r is called regular if
where kn and ln are defined in Definition 9.
Corollary 3. If components of the homogeneous and regular two-state hierarchical series-parallel system of order r have an exponential reliability function
R(t) = exp[-1t] for t > 0, l > 0, then its reliability function is given by
Rk,knn (t) = 1 - [1 - [Rk-1kn,„ (t)]ln ]kn for t > 0
for k = 2,3,...,r and R 1,k„,l„ (t) = 1 - [1 - exp[-1l„t]]kn for t > 0.
Theorem 4. If
(i) ft (t) = 1 - exp[-V(t)], t e (-¥, ¥), is a non-
degenerate reliability function,
-
(ii) limlrn-1kn'n = 0 for r > 1,
(iii) lim k'nn +.. +1 [R(aj + bn )]ln = V(t) for t e Cv,
r > 1, t e (-¥, ¥), then
lim R
r,kn,ln (ant + bn) = ft(t) for t e C *, r > 1,
l = l ('1j1) = = l (ij'1..
1)
=l
t e (-¥, ¥).
and
r
Proposition 4. If components of the homogeneous and regular two-state hierarchical series-parallel system of order r have an exponential reliability function
R(t) = exp[-1t] for t > 0, l > 0,
Definition 10. A two-state system is called a parallel-series system of order r if its lifetime T is given by
T = min { max { min { max
1<i1<kn 1< ji<l¡1 1<i2 <kS'1j1 ) 1s h <l(i1J1 )
'2
limlrn-1 knln = 0 for r > 1,
min
(
max
1<ir <k„(i'1 j1 . "r-1 jr-1) 1< jr <,(>1 j1 ...¡r-1 jr-1)
T )] }}}
\ i j ...if if y y '
and
Ou )
1 7 ^ / 1 1 ,
a =-, b = — (— + — +... + —)logk ,
n Ill n l L /2 /:
n n
then
^(t) = 1 - exp[- exp[-t]] for t e (-<, <), (10) is its limit reliability function.
Example 3. A hierarchical regular series-parallel homogeneous system of order r = 2 is such that kn = 200, ln = 3. The system components have identical exponential reliability functions with the failure rate l = 0.01.
Under these assumptions its exact reliability function, according to Corollary 3, is given by
k dumj2...v-1 jr-1 ) are the numbers
where kn , kn n
of suitable parallel systems of the system composed of parallel-series subsystems and linked in series, lh,
l(i1j1', ..., l'j2.Jr-2J-2' are the numbers of suitable
i2 ' ' ir-1
parallel-series subsystems in these parallel systems, and li''J/L..'r-1Jr' are the numbers of components in the parallel systems of the parallel-series subsystems.
Definition 11. A two-state parallel-series system of order r is called homogeneous if its component lifetimes Ti ... i have an identical distribution
11 ... lrJr
function
F(t) = P(T1j,,jr < t):
where
R2,200,3(t) = 1 -[1 -[ 1 -[1 -exp[-0.01-3t]]200 ]3]200 for t > 0.
Next applying Proposition 4 with normalising constants
1
a„ = -
= 11.1,
0.01-9
b = —(^ + —) log 200 = 235.5, n 0.01 3 9
we conclude that the system limit reliability function is given by
^3(t) = 1 - exp[- exp[-t]] for t e (-<, <),
and from (7), the following approximate formula is valid
R 2,200,3 (t) @ ^3 (0.09t - 21.2) = 1 - exp[- exp[-0.09t + 21.2]] for t e (-<, <).
i1 = 1,2,..., kn, j = 1,2,...,l. , i, = 1,2,...,k (i1j1 ), 1 ' ' ' n ' J \ ' ' ' q ' 2 ' ' ' n '
j 2 = 1,2,..., lii1j1 ),..., J 2 5 5 'i2
i = 1,2,...,k (i1j1.Jr-1jr-1 ), j = 1,2,..., l(«..Jr-1 ),
r ' ' ' n 5.7^ 555 i„ 5
i.e. if its components Eiyi . have the same reliability function
R(t) = 1 - F(t), t e(-<,<).
Definition 12. A two-state parallel-series system of order r is called regular if
l = l (i1J1 ) = = l (i1J1...ir-1 jr-1 )= l ¡i * * * i ^ n
1)
and
k (Î1) = = k '¡i'li..Jr-^■j'r-1) = k
where kn is the number of parallel systems in the parallel-series subsystems and ln are the numbers of parallel-series subsystems or, respectively, the numbers of components in these parallel systems.
