CC BY
11
r(t) 1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -
0 -t
02468
Figure 2. The graphs of the steel cover risk function
Using the values given in these Tables 1-4, the formulae (3)-(7) and numerical integration we find:
- the mean values of the cover lifetimes in the reliability state subsets
¥
M(1) = E[T2 24(1)] = J R2,24(t,1)dt @ 4.5634,
0
¥
M(2) = E[T2,24(2)] = J R2,24(t,2)dt @3.2268,
0 ¥
M(3) = E[T2 24 (3)] = J R2,24(t,3)dt @ 2.0408,
0 ¥
M(4) = E[T2,24 (4)] = J R2At,4)dt @ 1.4431,
0
- the second ordinary moments of the cover lifetimes in the reliability state subsets
¥
N(1) = E[r2224(1)] = 2JtR2,24(t,1)dt @ 22.9715,
0 ¥
N(2) = E[T2224 (2)] = 2JtR2,24(t,2)dt @ 11.4879,
0 ¥
N(3) = E[T224 (3)] = 2JtR2,24(t,3)dt @ 4.5944,
0 ¥
N(4) = E[T224 (4)] = 2JtR2,24(t,4)dt @ 2.2967,
0
- the standard deviations of the cover lifetimes in the reliability state subsets
s (1) = VN(1) - [M(1)]2 @ 1.4651, s (2) = 7N(2) - [M(2)]2 @ 1.0370, s (3) = yjN(3) - [M(3)]2 @ 0.6553,
s (4) = ,1N(4) - [M(4)]2 @ 0.4628,
- the mean values of the cover lifetimes in the reliability particular states
M(1) = M(1) -M(2) @4.5634 - 3.2268 = 1.3366,
M(2) = M(2) -M(3) @ 3.2268 - 2.0408 = 1.1860,
M(3) = M(3) -M(4) @ 2.0408 - 1.4431 = 0.5977,
M (4) = M(4) @ 1.4431.
Using the values given in these Tables 5 and the formula (9) we find the approximate value of the moment when the system risk function exceeds an exemplary permitted level 5 = 0.05, namely
t = r-1 (0.05) @ 1.58. 5. Conclusion
Two recurrent formulae for multi-state reliability functions, a general one for non-homogeneous and its simplified form for homogeneous multi-state consecutive " k out of n: F" systems composed of ageing components have been proposed. The formulae for multi-state reliability function of a homogeneous multi-state consecutive " k out of n: F" system has been applied to reliability evaluation of the steel cover composed of ageing components. The considered steel cover was a five-state ageing consecutive "2 out of 24: F" system composed of components with Weibull reliability functions. On the basis of the recurrent formula for steel cover multi-state reliability function the approximate values of its vector components have been calculated and presented in tables and illustrated graphically. On the basis of these vales the mean values and standard deviations of the steel cover lifetimes in the reliability state subsets and the mean values of the steel cover lifetimes in particular reliability states have been estimated. Moreover, the cover risk function and the moment when the risk function exceeds the permitted risk level have been determined.
The input structural and reliability data of the considered steel cover have been assumed arbitrarily and therefore the obtained its reliability characteristics evaluations should be only treated as an illustration of the possibilities of the proposed methods and solutions. The proposed methods and solutions and the software are general and they may be applied to any multi-state consecutive "k out of n: F" system of ageing components.
References
[1] Barlow, R. E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York.
[2] Guze, S. (2007). Wyznaczanie niezawodnosci dwustanowych systemów progowych typu „kolejnych k z n: F". Materiaiy XXXV Szkoiy Niezawodnosci, Szczyrk.
[3] Guze, S. (2007). Numerical approach to reliability evaluation of two-state consecutive „k out of n: F" systems. Proc. Summer Safety and Reliability Seminars, SSARS 2007, Sopot.
[4] Kolowrocki, K. (2001). Asymptotyczne podejscie do analizy niezawodnosci systemów. Monografía. Instytut Badan Systemowych PAN, Warszawa.
[5] Kolowrocki, K. (2004). Reliability of Large Systems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.
[6] Kossow, A. & Preuss, W. (1995). Reliability of linear consecutively connected systems with multistate components. IEEE Transactions on Reliability, 44, 3, 518-522.
[7] Malinowski, J. & Preuss, W. (1995). A recursive algorithm evaluating the exact reliability of a consecutive k-out-of-n: F system. Microelectronics and Reliability, 35, 12, 1461-1465.
[8] Malinowski, J. (2005). Algorytmy wyznaczania niezawodnosci systemów sieciowych o wybranych typach struktur. Wydawnictwo WIT, ISBN 8388311-80-8, Warszawa.
[9] Shanthikumar, J. G. (1987). Reliability of sytems with consecutive minimal cut-sets. IEEE Transactions on Reliability, 26, 5, 546-550.
[10] Xue, J. & Yang, K. (1995). Dynamic reliability analysis of coherent multi-state systems. IEEE Transactions on Reliability, 4, 44, 683-688.
Knopik Leszek
University of Technology and Agriculture, Bydgoszcz, Poland
Some remarks on mean time between failures of repairable systems
Keywords
age replacement, increasing failure rate on average, mean time between failures, mean time between failures or repairs
Abstract
This paper describes a simple technique for approximating the mean time between failures (MTBF) of a system that has periodic maintenance at regular intervals. We propose an approximation of MTBF for vide range of systems than IFRA class of distributions.
1. Introduction
One important research area in reliability engineering is studying various maintenance policies. Maintenance can be classified by two major categories: corrective and preventive. Corrective maintenance is any maintenance that occurs when the system is failed. Some authors refer to corrective maintenance as repair and we will use this approach in this paper. Preventive maintenance is any maintenance that occurs when system is not failed.
A common measure used to describe the reliability characteristics of a repairable system is mean time between failures (MTBF). In most repairable systems, preventive maintenance used to reduce system failure frequency and hence increase the MTBF. It is easy that MTBF is the mean time to a repair service or an age replacement.
The MTBF for a system that has periodic maintenance at a time t can be described by [1], [10]:
i
J R(s)ds
MTBF = -
F(t)
(1)
In paper [1], [10] an approximation to MTBF is provided. Moreover in [10] is proved that
tR(t)- £ MTBF £ t
F(t)
F(ty
and for increasing failure rate on average (IFRA) class of distributions in [1] the following relation by proposed
t £ MTBF £- t
ln(R(t )
F(t)
In this paper, we propose an approximation to MTBF for wide class than IFRA. The equality (1) can be a basis to introduction a new class of ageing distributions (see [7], [8]).
Let us assumption the following notation
M (t) =
1
MTBF
The case when M(t) is monotonic was considered by Barlow and Campo [2], Marshall and Proschan [9], Klefsjo [5] and Knopik [7], [8].
Definition 1. The lifetime T belongs to the class (mean time to failure or replacement) MTFR, if the function M (t) is non-decreasing for tî{t : F(t) > 0} . It has been shown in Barlow [2] and Klefsjo [5] that
IFR c MTFR c NBUE,
where IFR is increasing failure rate class, NBUE is new better than used in expectation class. For absolutely continuous in [2] and for any random variable in [8] it has been proved, that
IFRA c MTFR,
where IFRA is increasing failure rate on average class of distributions. Preservation of life distribution classes under reliability operations has been studied in [7], [8]. The class MTFR is closed under the operations:
(a) formation of a parallel system for absolutely continuous random variables,
(b) formation of series system with identically distributed and absolutely continuous random variables,
(c) weak convergence of distributions,
(d) convolution.
In this paper we propose new approximation and bounds, applicable for a wide range of systems. It is MTFR class of distributions, such that MTBF is non-increasing. The class MTFR contains distributions with unimodal failure rate function. We analyze special case of distribution with MTBF non-increasing as mixture of an exponential distribution and Rayleigh's distributions. This distribution has unimodal failure rate function. However, it is not easy to obtain distribution from mixtures with unimodal failure rate function [12]. The mixtures of two increasing linear failures rate functions were studied in [4]. In [4] showed that a mixture of two distributions with increasing linear failure rate functions does not give distributions with unimodal failure rate function. In section 2, we introduce the proposed model of mixture and we estimate their parameters for an example of lifetime data [11]. In section 3 we propose a simple approximation of to MTBF for lifetime distributions with MTBF non-increasing.
2. Mixture of distributions
We consider a mixture of two lifetimes X1 and X2 with densities f1(t), f2(t), reliability functions R1(t), R2(t), failure rate functions r1(t), r2(t) and weights p and q=1- p, where 0 < p < 1. The mixed density is then written as
f(t) = pfx(t) + (1 - P)f2 (t)
and the mixed reliability function is
R (t) =pRi(t) + (1-p)R2(t).
The failure rate function of the mixture can be written as the mixture
r(t)=rn(t) ri(t)+[1 - rn(t)] r2(t),
where w(t)=pRi(t)/R(t).
Moreover, from [3], we have under some mild conditions, that
lim r(t) = lim min{rl(t),r2(t)} .
In the following propositions, we give some properties for the mixture failure rate function.
Proposition 1. For the first derivative of m (t), we have
m '(t) = rn(t)[1 - m(t)][r2(t) - r()].
Proposition 2. For the first derivative of r(t), we have
r'(t)=[1- œ(t)](- œ(t)(r2(t) - ri(t))2+r'2(t)
+ rn(t)r'i(t).
Proposition 3. If X1 is exponentially distributed with parameter X, then
r'(t)=[1 - œ(t)](- v(t)(r2(t) - X)2+r'2(t)).
We suppose that r2(t) =at, where a>0. Consequently, the reliability function of X2 is the reliability function of Rayleigh's distribution of the form
R2 (t) =exp {- a t2 } for t>0.
Proposition 4. If pX2 < a, then r (t) is unimodal.
Proof. By Proposition 1, we conclude that œ (t) is decreasing for t e (0, ti), where ti = X/a and is increasing for t e (ti, œ). Hence, if t < ti, then œ(t)(r2(t) - X)2 is decreasing from p^2< a to 0, and if t > ti then œ(t)(r2(t) - X)2 is increasing from 0 to <x>. Thus the equation m (t) (r2(t) - X)2 = X has only one solution t2 and r (t) is increasing for t < t2 and decreasing for t > t2.
Proposition 5. If pX2 > a, then there exist t3, and t4, t3 < t1 < t4, such that r (t) decreases in (0, t3), increases in (t3, t4) and decreases in (t4, œ).
Proof. Let h(t)=a - œ(t)(r2(t) - X)2. It is easy to find that h(0)=a - pX2<0, h(tj)= a, h(œ) =-œ. The function h(t) is increasing from h(0) < 0 to h(ti)=a>0, and is decreasing from h(tj)=a>0 to h(x)=-œ.
Thus, there exist t3 and t4, 0<t3<t1<t4 such that r (t) decreases on (0, t3), increases in (t3, t4) and decreases in (t4, œ). This completes the proof.
Example 1. In this example we consider a real life time data from the [11]. We estimate the parameters p, a, X of the model with reliability function
R (t) = p exp (- Xt) + (1 - p) exp (- 0,5at2) for x > 0.
