Řešení vybraných modelů s obnovou

1. Řešení vybraných modelů s obnovou Radim Briš VŠB - Technical University of Ostrava (TUO), Ostrava, The Czech Republic radim.bris@vsb.cz

2. Introduction Contents • Renewal process • Alternating renewal process • Models with a negligible renewal period • Models with periodical preventive maintenance • Alternating renewal models • Alternating renewal models with two types of failures • Conclusions

3. Introduction • This paper mainly concentrates on the modeling of various types of renewal processes and on the computation of principal characteristics of these processes – the coefficient of availability, resp.unavailability. • The aim is to generate stochasticageing models, most often found in practice, which describe the occurrence of dormant failures that are eliminated by periodical inspections as well as monitored failures which are detectable immediately after their occurrence. • Mostly numerical mathematical skills were applied in the cases when analytical solutions were not feasible.

4. Renewal process Random process is called renewal process. Let we call Nta number of renewals in the interval [0, t] for a firm t ≥ 0, it means From this we also get that SNt ≤ t < SNt+1 is called renewal function

5. Renewal process Renewal equation An asymptotic behaviour of a renewal of renewal function: function h(t) that is defined asrenewal density. is renewal equation for a renewal density

6. Alternating renewal process X1,X2…resp. Y1,Y2…are independent non-negative random variables with a distr. function F(t) resp. G(t). A random process {S1, T1, S2, T2…..} is then an alternating renewal process. Coefficient of availability K(t) (or also A(t) - availability) is h(x) is a renewal process density of a renewal {Tn}n=0∞, F(t) is a distribution function of the time to a failure, resp. 1 – F(t) = R(t) is reliability function. andasymptotic coefficient of availability is

7. Models with a negligible renewal period Poisson process: Gamma distribution of a time to failure: Using Laplace integral transformation we obtain: is kth nonzero root of the equation(s + λ)a = λa For example for a = 4 nonzero roots are equal to: For example for a = 4 nonzero roots are equal to: and a renewal density

8. Models with a negligible renewal period Weibull distribution of time to failure: α > 0 is a shape parameter, λ > 0 is a scale parameter Using discrete Fourier transformation: where μ is an expected value of a time to failure: We can estimate in this way an error of a finite sum because a remainder is limited

9. Models with a negligible renewal period Weibull distribution:

10. Models with periodical preventive maintenance May a device goes through a periodical maintenance. Interval of the operation τC, (detection and elimination of possible dormant flaws). The period of a device maintenance …τd F(t) is here a time distribution to a failure X. In the interval [0, τc+ τd) there is a probability that the device appears in the not operating state The probability P(t) (coefficient of unavailability) for

11. Models with periodical preventive maintenance Exponential distribution of time to failure, τd = 0: Coefficient of unavailability for Exponential distribution.

12. Models with periodical preventive maintenance Weibull distribution of time to failure, τd = 0: 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. Coefficient of unavailability for Weibull distribution.

13. Alternating renewal models Lognormal distribution of a time to failure: We use discrete Fourier transformation for: 1. pdf of a sum Xf+Xr(Xris an exponential time to a repair), as well as for 2. convolution in the following equation: renewal density can be estimated by a finite sum

14. Alternating renewal models An example:In the following example a calculation for parameter values σ=1/4, λ=8σ, τ=1/2, is done. A renewal density for lognormal distribution Coefficient of availability for lognormal distribution

15. Alternating renewal models with two types of failures Two different independent failures. These failures can be described by an equal distribution with different parameters or by different distributions. Common repair: A time to a renewal is common for both the failures and begins immediately after one of them. It is described by an exp.distribution, with a mean 1/τ. For a renewal density we have In case of non-exponential distribution we use and we estimate the function by a sum of the finite number of elements with a fault stated above. fn(t) is a probability density of time to n-th failure. For the calculation of convolutions we can use a quick discrete Fourier’s transformation.

16. Alternating renewal models with two types of failures Example: have Weibull distributions Coefficient of availability for Weibull distribution

17. Conclusions • Selected ageing processes were mathematically modelled by the means of a • renewal theory and these models were subsequently solved. • Mostly in ageing models the solving of integral equations was not analytically feasible. In this case 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 occurance, or a possible renewal of individual types of failures which in reality do not have to be independent. • 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.

18. Thank you foryour attention.

19. RISK, QUALITY AND RELIABILITYhttp://www.am.vsb.cz/RQR07/September 20-21, 2007International conferenceTechnical University of Ostrava, Czech Republic • Call for papers •  Risk assessment and management • Stochastic reliability modeling of systems and devices • Maintenance modeling and optimization • Dynamic reliability models • Reliability data collection and analysis • Flaw detection • Quality management • Implementation of statistical methods into quality control in the manufacturing companies and services • Industrial and business applications of RQR e.g., Quality systems and safety • Risk in medical applications

20. RISK, QUALITY AND RELIABILITYhttp://www.am.vsb.cz/RQR07/September 20-21, 2007International conferenceTechnical University of Ostrava, Czech Republic Inivited keynote lecturesKrzysztof Kolowrocki: Reliability, Availability and Risk Evaluation of Large Systems Enrico Zio: Advanced Computational Methods for the Assessment and Optimization of Network Systems and Infrastructures Sava Medonos: Overview of QRA Methods in Process Industry. Time Dependencies of Risk and Emergency Response in Process Industry. Marko Cepin: Applications of probabilistic safety assessment Eric Châtelet: will be completed later