Řešení vybraných modelů s obnovou

Radim BrišVŠB - Technical University of Ostrava (TUO), Ostrava,

The Czech Republicradim.bris@vsb.cz

Contents

• Introduction

• Renewal process

• Alternating renewal process

• Models with periodical preventive maintenance

• Models with a negligible renewal period

• Alternating renewal models

• Conclusions

• Alternating renewal models with two types of failures

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 stochastic ageing 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.

Renewal process

Random process is called renewal process.

Let we call Nt a number of renewals in the interval [0, t] for a firm t ≥ 0, it means

From this we also get that SNt ≤ t < S Nt+1

is called renewal function

,,,01

0

n

iin NnXSS

0nnS

tSnN nt :max

)(tFtSPnNP nnt

)()()(1)( 111 tFtFtFtFtStSPnNP nnnnnnt

0,)( tENtH t

0 111 )()()()()(

n nn

nnnt tFtFtFnnNnPtH

Renewal process

t

duuFutHtFtH0

.)()()()(

1)(

lim t

tHt

htHhtH

t

)]()([lim

Renewal equation

An asymptotic behaviour of a renewal of renewal function:

)()()(

lim)(0

tHt

ttHtHth

t

function h(t) that is defined as renewal density.

is renewal equation for a renewal density t

duufuthtfth0

,)()()()(

Alternating renewal process,... 1111 nnnn XYXYXS

.... 1111 nnnnn YXYXYXT

A random process {S1, T1, S2, T2…..} is then an alternating renewal process.

X1,X2…resp. Y1,Y2…are independent non-negative random variables with a distr. function F(t) resp. G(t).

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.

and asymptotic coefficient of availability is

).(lim tKKt

Models with a negligible renewal period

.!

)(

!

)()(

11

tn

tte

n

etnENtH

n

nt

n

tn

t

)()( tHth

.0,0)( tetf t

Poisson process:

Gamma distribution of a time to failure: .0,0,0,)(

)()(

1

ata

ettf

ta

1

1

,0,)(a

k

tsk tea

s

ath k

Using Laplace integral transformation we obtain:

,Csk is kth nonzero root of the equation (s + λ)a = λa

For example for a = 4 nonzero roots are equal to:

)1()1(

,2)1(

,)1()1(

2

3

3

2

21

ies

es

ies

i

i

i

and a renewal density

)sin(14

)( 2 teeth tt

For example for a = 4 nonzero roots are equal to:

Models with a negligible renewal period0,)()( )(1 tettf t

1

)()(n

n tFtH

,),(!

)(1)(

1

0

nttGei

t

tF nt

n

i

i

n

Weibull distribution of time to failure:

Using discrete Fourier transformation:

where μ is an expected value of a time to failure:

.)1

1(

We can estimate in this way an error of a finite sum

α > 0 is a shape parameter,

λ > 0 is a scale parameter

K

nn tFtH

1

)()(

because a remainder is limited nttGtFKn

nKn

n ,)()(11

Models with a negligible renewal period

Weibull distribution:

Models with periodical preventive maintenance

c

c

t

ttFtP

,1

),()(

.0),()(

...

,1,...,2,1))((1)()()(

...

),)(1()(:0),()(

0

011

ttFtP

niitPiPtPtP

ntnNntPtP

dcciii

dcdcn

The probability P(t) (coefficient of unavailability) for ,0t

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

Models with periodical preventive maintenance

Exponential distribution of time to failure, τd = 0:

).)(1()(:0),()( dcdcc ntnNnntFtP

Coefficient of unavailability for Exponential distribution.

Models with periodical preventive maintenance

Weibull distribution of time to failure, τd = 0:

Coefficient of unavailability for Weibull distribution.

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.

Alternating renewal models

0),ln(

1)( t

t

ttf f

t

ff dxxtRxhtRtK0

.)()()()(

Lognormal distribution of a time to failure:

We use discrete Fourier transformation for: 1. pdf of a sum Xf + Xr (Xr is 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

N

nn tfth

1

).()(

Alternating renewal models

123.01

)(lim

rf

t

EXEX

th

An example: In the following example a calculation for parameter values σ=1/4, λ=8σ, τ=1/2, is done.

A renewal densityfor lognormal distribution

Coefficient of availabilityfor lognormal distribution

938.0

)(lim

rf

f

t

EXEX

EX

tK

Alternating renewal models with two types of failures

).()()(1)(1)()( 212121 tRtRtFtFtXtXPtR fffffff

).()( tRdt

dtf ff

t

ff dxxtfxhtfth0

)()()()(

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 .)()(1

n

n tfth

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.

Alternating renewal models with two types of failures

6.362

6222

22

22

21

22

21

ff

fff EXEX

EXEXEX 79.0

rf

f

EXEX

EXK

tXtX ff 21have Weibull distributions

Example:

Coefficient of availability for Weibull distribution

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.

Thank you for your attention.

RISK, QUALITY AND RELIABILITY http://www.am.vsb.cz/RQR07/  

September 20-21, 2007

International conference

Technical 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

RISK, QUALITY AND RELIABILITY http://www.am.vsb.cz/RQR07/  

September 20-21, 2007

International conference

Technical University of Ostrava, Czech Republic

Inivited keynote lectures  

Krzysztof 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