CC BY
22
r(t) = 1 - Rio,22 (t,2)
= 1 - exp[- exp[(0.2041t - 1^44log10
+ log(50^p log 10)]] for t e (-¥,¥).
The moment when the system risk exceeds the permitted level e.g. 5 = 0.05, according to (11), is
t = r(5) @ 2 years and 11 days.
Comparing the expected values of the elevator lifetimes in the state subset and the elevator mean lifetimes in the particular states in the case when strands failure in dependent and independent way we can conclude that these values are lower in the first case for about 68% percent for exact reliability functions and for about 67% and 69% for approximate reliability functions.
The obtained results illustrate that the increased load of remaining un-failed components causes shortening the lifetime of these components in a significant way. That fact can be interpreted as a decrease of their reliability much faster then for the systems with independent components. Taking into account the presented ship-rope elevator we can notice that the lifetime in the reliability state subset of the elevator under the assumption that strand failure in dependent way is even about 70% shorten then in the case when strands are independent.
5. Conclusion
In the paper the exact reliability analysis and asymptotic approach to the reliability evaluation of homogeneous multi-state parallel-series systems are presented. For these systems the exact and limit reliability functions and other characteristics both in the case when their components are independent and when they are dependent are determined under the assumption that components of systems have exponential reliability functions. Introduced in the paper the method of reliability evaluation of large systems relies on application of some approximate methods based on classical asymptotic approach to this issue. The obtained results are concerned with typical systems with regular structure. Applied in the paper analytical methods are successful rather for not very complex systems. In this background it seems to be justified the extension of this issue for systems with less regular structures and use of any other reliability analysis methods.
References
[1] Blokus, A. (2006). Reliability analysis of large systems with dependent components. International Journal of Reliability, Quality and Safety Engineering 13(1): 1-14.
[2] Blokus-Roszkowska, A. (2006). Reliability analysis of multi-state rope transportation system with dependent components. Proc. International Conference ESREL '06, Safety and Reliability for Managing Risk, A. A. Balkema, Estoril: 1561-1568.
[3] Blokus-Roszkowska, A. (2007). Reliability analysis of homogenous large systems with component dependent failures (in Polish). Ph.D. Thesis, Maritime University, Gdynia - System Research Institute, Warsaw.
[4] Kolowrocki, K. (2004). Reliability of Large Systems. Amsterdam - Boston - Heidelberg -London - New York - Oxford - Paris - San Diego -San Francisco - Singapore - Sydney - Tokyo: Elsevier.
[5] Kolowrocki, K. et al. (2002). Asymptotic approach to reliability analysis and optimisation of complex transport systems (in Polish). Gdynia: Maritime University. Project founded by the Polish Committee for Scientific Research.
[6] Smith, R. L. (1982). The asymptotic distribution of the strength of a series-parallel system with equal load-sharing. Annals of Probability 10: 137-171.
[7] Smith, R. L. (1983). Limit theorems and approximations for the reliability of load sharing systems. Advances in Applied Probability 15: 304-330.
[8] Smith, R. L. & Phoenix, S. L. (1981). Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load-sharing. Journal of Applied Mechanics 48: 75-81.
Appendix
Figure 1. Graphs of the rope elevator exact and approximate reliability functions in the state subset u > 1 a) in the case when components are independent b) in the case when components fail dependently
1,2 -, 1,0 -0,8 0,6 0,4 0,2 0,0
0 2 4 6 8 10 12
-exact reliability function
— - - approximate reliability function 1
- - - ■ approximate reliability function 2
14 16
18
1,2 1 1 -
0.
0,4
b)
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0
-exact reliability function
— - -approximate reliability function 1
- - - ^approximate reliability function 2
0
0
Figure 2. Graphs of the rope elevator exact and approximate reliability functions in the state subset u > 2 a) in the case when components are independent b) in the case when components fail dependently
Figure 3. Graphs of the rope elevator exact and approximate reliability functions in the state subset u = 3 a) in the case when components are independent b) in the case when components fail dependently
Bris Radim
VSB Technical University of Ostrava, Czech Republic
Stochastic ageing models - extensions of the classic renewal theory
Keywords
renewal theory, alternating renewal theory, maintenance processes, ageing, unavailability coefficient Abstract
Exact knowledge of the reliability characteristics as the time dependent unavailability coefficient for example, under influence of different ageing processes as well as under different failure types is very useful to the practitioners who have to find the optimal maintenance policy for their equipment. In this paper found models and their solutions have potential to face the optimisation task under the conflicting issues of safety and economics. Most of the solved models take into account ageing processes. An increasing tendency lately exists to include aging effects into the risk assessment models to evaluate its contribution. We developed different renewal models taking into account different ageing distributions of failures (Weibull, Erlang, log-normal): models with negligible renewal time, models with periodical preventive maintenance, alternating renewal process with lognormal distribution of failure time, and with two types of failures.
1. Introduction
This paper mainly concentrates on the modelling of various types of renewal processes and on the computation of principal characteristics of these processes - the time dependent coefficient of availability, possibly unavailability. The aim is to generate models, most often found in practice, which describe the processes of ageing, further the occurrence of dormant failures that are eliminated by periodical inspections as well as monitored failures which are detectable immediately after their occurrence.
Renewal theory seems to be a feasible option to quantify time-dependent effects on component unavailability due to ageing, periodical inspections, or repairs [1]. Closed form solutions for the asymptotic the failure rate and unavailability can be obtained using Laplace transform. Obtaining the detailed time behaviour may not be a trivial numerical task. Basic information from renewal theory brings Appendix [4], [3]. The following chapter 2 is devoted to models with a negligible renewal time in which a main impact is given on flexible models with the Erlang and Weibull distribution. The solution of these models is received from a Laplace and discrete Fourier transformation. In the following chapter 3 we introduce different models with maintenance. Main attention is paid to models with periodical preventive
maintenance - basic equations for the model are formulated. The solution of a system of equations is demonstrated for the situations with an exponential and Weibull distribution. In the next chapter the alternating model with an inconsiderable renewal time is solved, this is demonstrated for lognormal distribution of time to failure. The final part involves generally formulated alternating models with the occurrence of two types of independent failures. Time-dependent unavailability of components under maintenance and ageing processes can exhibit mathematically complex behaviour [5]. The unavailability may be also dependent on maintenance history. First failure distributions may not be continuous functions. Within this paper we can say that renewal theory provides a feasible approach in selected cases to implement and evaluate interventions given by maintenance and aging processes. A lot of notable asymptotic results on availability analyses are focused on the situation that the components have exponential lifetime distributions. Using so-called phase-type approach, author in [2] shows that the multi-state model also provides a framework for covering other types of distributions, but with limitations - the approach makes use of the fact that a distribution function can be approximated by a mixture of Erlang distributions (with the same scale parameter). Asymptotic analysis of highly
available systems has been carried out by a number of researchers. A survey is given by Gertsbakh [7], with emphasis on results related to the convergence of the distribution of the first system failure to the exponential distribution.
If the lifetimes are distributed arbitrarily, then the system can be described by a semi-Markov process or Markov renewal process. Semi-Markov processes and Markov renewal processes are based on a marriage of renewal processes and Markov chains. Pyke [8] gave a careful definition and discussion of Markov renewal processes in detail. In reliability, these processes are one of the most powerful mathematical techniques for analysing maintenance and random models. A detailed analysis of the non-exponential case (nonregenerative case) is however outside of the scope of the introduction part. Further research is needed to present formally proved results for the general case. Presently, the literature covers only some particular cases, what is also the case of this presentation.
2. Models with a negligible renewal period
In some cases we can take into account a renewal period equal to zero. For example the situation when a time to a renewal is substantially smaller than a time to a failure and its implementation would not influence an expected result. This case was intensively studied in [6]. Basic relationships for Poisson process are derived in [4]:
Renewal function and renewal density are given as follows
»7=1 n\
-Xt
= Xte-'/J
»7=1 1l\
a k=\ a in the expression there is sk e C, which is kth nonzero root of the equation
(s + A)a= Aa sgC, For example for a = 4 nonzero roots are equal to
in
Sj -1) = (-1 + /)A,,
s2=X(e'n - 1) = -2X,
i 371
53 = X(e~Y -1) = (-1 -OA,, and a renewal density
,, x A, lUs,. Slt
hit) — —K V--eSk'
4 k=i 4
= j[l-e-2^ -e-^Sm(Xt)].
0.3 0.25 0.2 0.15 0.1 0.05 0
M
- /m
/ a=I
f 10
Basic definitions from renewal theory see in Appendix. Renewal density is constant
Figure 1. Renewal density for Erlang distribution
Another calculation method was applied for Weibull distribution of time to failure, which has a probability density
f{t)=aX{Xt)a-le-<M)
t> 0,
h(t) = H'(t) = X.
Methodology based on Laplace transforms was dramatically extended in [6] for the case when a time to failure is modelled by the Erlang distribution, which has a probability density function
m=Xm°e~\ t> 0,X- 0, a • 0.
r (a)
After the backward transformation a renewal density is equal to
a > 0 is a parameter of the shape, X > 0 is a parameter of a scale.
A probability density//^, n =2,3...... or a probability
density of time to nth failure can be calculated as a convolution of the function ( /„_/ */). We can express it numerically e.g. with the help of discrete Fourier transformation [6].
By a numerical integration we can determine a vector of a distribution function Fn.
A formula (1) for a calculation of the renewal function
H(t)=£F„(t)
»7=0
(1)
is necessary to substitute by a finite sum of the first K computed terms at the numerical calculation. It can be conducted because these terms converge quickly to a zero at a definite interval [0, 7]. Equally S„ = X/ X2 + ... + X„ has an asymptotic normal distribution N(nu, no2) where ¡J and cr are definite expected value and a dispersion of Xt
Considering that Weibull distribution of time to failure for a > 1 has an increasing failure rate
r(t) = Xa(Xt)a-1
a distribution function F„(t) can be upper estimated by the function
F„(t)< l-Z-^-e-^ =G„(t), t • hu,
i=o /!
where ¡J is an expected value of a time to failure [3].
r(i+—)
We can estimate in this way an error of a finite sum H(t)=£F„(t),
77=0
because a remainder is limited
oo oo
Z ZG„(t), t • «H.
77=Á"+1 77=Á"+1
Exact relationship for the reminder is derived in [6]. The behaviour of the renewal function estimated for different number of members in the above finite sum
we can see in Figure 2.
Figure 2. Renewal function for Weibull distribution
it may be wise to replace it before it has aged too greatly. In this section we shall concentrate on the operating characteristics of some commonly employed replacement policies. A commonly considered replacement policy is the policy based on age (age replacement). Such a policy is in force if a unit is always replaced at the time of failure or zc hours after its installation, whichever occurs first; zc is a constant unless otherwise specified. If zc is a random variable, we shall refer to the policy as a random age replacement policy. Under a policy of block replacement the unit is replaced at times kzc (k = 1,2,...), and at failure. This replacement policy derives its name from the commonly employed practice of replacing a block or group of units in a system at prescribed times kzc(k = 1,2,...) independent of the failure history of the system.
3.1. Replacement based on age
A unit is replaced zc hours after its installation or at failure, whichever occurs first; zc is considered constant. Let R(t) denote the probability that an item does not fail in service before time t. Then
R(t) = R(xc)nR(t-mc),
n e N u {0}: m c < t < (n + l)x c.
The distribution function of a time to failure X is
F(t) = l-R(t) =l-R(Tc)"R(t-mc),
t > 0, n e N u {0}: m c < t < (n + l)x c.
Expected time to failure E(X) is
oo
E(x) = J R{x)dx
o
rj-, (77+l)tc
= E J R(Tc)"R(t-mc)dt
n=0 mc
OO ^c 1 ^c
= ZR(tc)" \R{t)dt =——iR(t)dt,
77=0 (J r[Zr) 0
3. Models with maintenance
In many situations, failure of a unit during actual operation is costly or dangerous. If the unit is characterized by a failure rate that increases with age,
Figure 3. The Weibull distribution function for a unit with the replacement based on age.
When the time to failure is exponentially distributed X ~ exp( 1/p.), then we have
F(t) = l-R(xc)"R(t-mc)
= e-nyac e-myi(t-mc) _ j _ ,
which means that distribution function is independent on replacements. Other words, the unit does not age.
3.2. Block replacement policy
Under a policy of block replacement all components of a given type are replaced simultaneously at times kzc (k = 1,2,...) independent of the failure history of the system. If Xj is time of i-th failure of a unit which has distribution function /■', and probability density fi and R,(i) I- /•', (t), then
Rl(t) = R(xc)nR(t-mc), n e N u {0}: nx c < t < (n + l)x c,
fl(t) = R(xcr^-[l-R(t-mc)l at
Distribution of time to i-th failure for i / \vc can derive on the basis of conditional probability:
x>0, v>0, k = [x/xc], l = [(x + y)/xc], l>k, Pr (X, >v\X,_1=x)
= R((k + l)xc - x)R(xc)(-(i+1))R(x + y-mc). Then
00
Pr(X, > y) =!R((k + iyzc - x)R(xc)('-(k+l) 0
■Rix + v-mjf^ (x)dx
co
= S J R((k + iyzc-x)R(Tc)i-(k+r>
k=0 Hc
■R(x + y-hc)ft._! (x)dx z = x-kxc,n = [{z + y)/xc]
CO TC
= ZR,._1(xc)t¡R(xc-z)R(xcr1
k=0 o
■R(z + y-nxc)f^x(z)dz
0'-1-1-1-1-7-
0 0.5 1 1.5 2 t 2.5
Figure 4. The Weibull distribution function for a unit with block replacement
In Figure 4, we can see time dependencies for R,(i) for Weibull distribution of time to failure.
3.3. Periodical preventive maintenance
May a device goes through a periodical maintenance after a time interval of the operation xc, whose intention is a detection of possible dormant flaws and their possible elimination. The period of device maintenance is id and after this period the device starts operating again. F(t) is here a time distribution to a failure X. Then in the interval [0, xc+ id) there is a probability that the device appears in the not operating state equal to
P(t) = F(t), t ' x c
= l,t> xc.
The state of a failure is considered then both a time to the maintenance after a possible failure and the time when the device is under maintenance. The probability P(t) (also a coefficient of unavailability) for i e 10. x ) is generated by following system of equations:
P(t) = Pn(t\
we7Vu{0}: n{xc +xd)* t<(n + l)(xc +xti),
i> (t) = i>_, (t) - i>_, (/I c )[1 - P0 (t - i(x c +X , ))]
z = 1,2,...,« -1,
P0(t) = F(t), t • 0.
Here /', (t) stands for a probability that a device exists in the failure state provided that before it had gone through i inspections. A term /',_//) - PiA(ix^ represents a probability that a device was all right at the previous inspection and it failed in the interval (it, , t), Pi_\(ixc) P0(t - i(zc + tj)) is a probability that it had failed in the previous inspection and since then it failed again.
3.3.1. Exponential distribution of time to failure
May
F(t) = l-e
-it
For the time t e [0,co) is then a probability P(t) equal
P{t) = F{t-n{xc+xd)\
ie[n(ic +Td),n(xc +Td)+Tc),
P(t) = l,t^[n(xc+xd)+xc,(n + l){xc+xd)),
If Td = 0, the expression for P(t) can be further simplified into the form
P(t) = F(t-mc),
neJVu{0}: n{xc +zd)<f (n + l)(ic +xif).
3.3.2. Weibull distribution of time to failure
Let the intensity of failures of the given distribution of time to failure is not constant, it is then a function of time past since the last renewal. In this case it is necessary for the given t and n, related with it which sets a number of done inspections to solve above mentioned system of n equations and the solution of the given system is not eliminated anyhow as it is at an exponential distribution.
In the Figures 5 and Figure 6 a slightly marked curve draws points of the local extremes in the case of shortening a time to the first inspection.
Figure 5. Coefficient of unavailability for exponential distribution
Figure 6. Coefficient of unavailability for Weibull distribution.
4. Alternating renewal models
Alternating models are those where two of the significantly diverse states appear, between which a model converts from one to another. A faulty device is the example of the alternating model where a time to a repair is compared with a time to failure and it cannot be neglected.
In the case that both a time to failure and a time to a repair follows an exponential distribution, general solution for a calculation of a coefficient of availability can be found in [4].
4.1. Lognormal distribution of a time to failure
If a distribution to a failure has a lognormal distribution, then a probability density is in the form
„ . . 1 ,\nt - A,.
ff(t) = —<p(—-), t ' 0
at o
where (p(x) is standard normal density. In this case a numerical calculation is offered again for the computation of the coefficient of availability. We can compute a probability density of a sum of random quantities X!andXr(X, is an exponential time to a repair) from a discrete Fourier transformation [6], equally as a convolution in the equation
K(t) = Rf (t) + )h(x)Rf (t - x)dx.
o
The calculation of a renewal density is substituted by a finite sum
M0=s/B(0.
«=1
An example: In the following example a calculation for parameter values o= 1/4, /.=80. 1= 1 /2. is done. In the Figure 7 there is a renewal density. The asymptotic value is marked by dots, which is in this case equal to
Figure 7. A renewal density for lognormal distribution
Figure 8 shows a procedure of the coefficient of availability K(t), the asymptotic value is marked by dots again and it is given by the following formula:
Figure 8. Coefficient of availability for a lognormal distribution
5. Alternating renewal models with two types of failures
The following part presents models, which consider an appearance of two different independent failures. These failures can be described by an equal distribution with different parameters or by different distributions.
5.1. Common repair
A device composed of two serial elements can be an example whereas a failure of one of them causes a failure of the whole device. A time to a renewal is common for both the failures and begins immediately after one of them. It is described by an exponential distribution with a mean value 1/x.
A failure occurrence in the renewal time is not taken into account, after the renewal both the parts are considered to be new.
May Xfi and Xf2 are independent random values describing time of failures with probability densities ffl(t) and fr/i). further a time to a repair is with a density fr(t). A probability that no failure occurs in the interval [0,t) is equal to
Rf(t) = P(Xn>tAXf2>t)
= [l-Fn(t)][l-Ff2(t)]=Rfi(t)Rf2(t).
and is a reliability function of the time to failure Xf of the whole device. Then X!has a probability density
With the knowledge f/t) we can calculate the functions describing this alternating process. If the time to failure has an exponential distribution with mean values 1A, and H\i, then
f (t) = = (X + v)e-ani)t.
dt
Then ff{t) has an exponential distribution with a mean value l/(/.+|i) and the coefficient of availability is equal
K(t) =---+ ^ 0.
A, + |o,+x A, + |o,+x
If the analytical procedure is uneasy or impossible, a numerical calculation can be used. For a renewal density computation is desirable instead of the equation
00
h(t) = ff(t) + jh(x)ff(t-x)dx
0
use a renewal equation for a renewal density
00
m = z/n(t)
n=1
and conduct a sum of the only definite number of elements with a fault stated above. f„(t) is a probability density of time to nth failure. Then for the calculation of convolutions is used for example a quick discrete Fourier's transformation. In the Figure 9 there is a graph of a coefficient of availability in the case that a time to a failure Xn and
Xf2 have Weibull distribution. Expected value to the failure EXf is equal to
EXf =
EX f 1EX f 2
EXfl + EX f 2
I 2262 22 + 62
= 736
and that is why an asymptotic coefficient of availability is equal to
Figure 10. Coefficient of availability for independent parts
K =
EXf
EXf + EXr
= 0.79
Figure 9. Coefficient of availability for Weibull distribution
5.2. Two independent parts
Supposing the device consists of two independent parts. The behaviour of each one is described by its alternating model with a given time to a failure and a time to a repair. Maintenance proceeds for both differently and independently. Equally, the failure of one of them can appear regardless of the state of the other part, even in the state of a failure. Let us consider the whole device to be in the state of a failure when at least one of the parts is in the state of a failure. Ka(t) is a coefficient of availability of the first part and Kbt) is a coefficient of availability of the second one. For the whole device K(t) is equal to
K(t) = Ka (t)Kb (t).
May dormant faults occur in the first part, with Weibull's distribution and with an expected value EX = 2 and a parameter of the form a = 2 which are eliminated by periodical inspections with a period xc = 2 (See Models with periodical preventive maintenance) and the second part is equal as in the previous model. The course of the coefficient of availability as the product of already computed partial ones is designed in the Figure 10.
6. Conclusion
In this paper a few types of renewal processes, which differentiate in a renewal course and a type of probability distribution of a time to failure, were described. These processes were mathematically modelled by the means of a renewal theory and these models were subsequently solved. In the cases, when the solving of integral equations was not analytically feasible, numerical computations were successfully applied. It was known from the theory that the cases with the exponential probability distribution are analytically easy to solve. With the gained results and gathered experience it would be possible to continue in modelling and solving more complex mathematical models which would precisely describe real problems. For example by the involvement of certain relations which would specify the emergence, or a possible renewal of individual types of failures which in reality do not have to be independent on each other. Equally, it would be practically efficient to continue towards the calculation of optimal maintenance strategies with the set costs connected with failures, exchanges and inspections of individual components of the system and determination of the expected number of these events at a given time interval.
Acknowledgements
This research is supported by The Ministry of Education, Youth and Sports of the Czech Republic. Project CEZ MSM6198910007.
References
[1] Aldemir, T. (2004). Characterization of time dependent effects due to ageing. Proc. of the Kick off meeting, JRC Network on Incorporating Ageing Effects into PSA Applications, European Commission, DG JRC - Institute for Energy, Nuclear Safety Unit, Probabilistic Risk & Availability Assessment Sector. Edited by M. Patrik and C. Kirchsteiger.
[2] Aven, T. & Jensen, U. (1999). Stochastic models in reliability. Springer-Verlag , ISBN 0-387-98633-2.
[3] Barlow, R. & Proschan, F. (1965). Mathematical Theory of Reliability. Wiley.
