Transcript Prezentace aplikace PowerPoint

Řešení vybraných modelů
s obnovou
Radim Briš
VŠB - Technical University of Ostrava (TUO), Ostrava,
The Czech Republic
[email protected]
Contents
• Introduction
• 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
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
n
S0  0, S n   X i , n  N ,
i 1
Random process
Sn n0
is called renewal process.
Let we call Nt a number of renewals in the interval [0, t] for a firm t ≥ 0, it means
Nt  maxn : Sn  t
From this we also get that SNt ≤ t < S Nt+1
PNt  n  PSn  t  Fn (t )
PNt  n  PSn  t  Sn1  t  Fn (t )1  Fn1 (t )  Fn (t )  Fn1 (t )
H (t )  ENt , t  0 is called renewal function



n 0
n 1
n 1
H (t )   nP( N t  n)   nFn (t )  Fn1 (t )  Fn (t )
Renewal process
t
Renewal equation
H (t )  F (t )   H (t  u ) F (u )du.
0
An asymptotic behaviour of a renewal of renewal function:
lim
t 
H (t ) 1

t

lim[ H (t  h)  H (t )] 
t 
h

H (t )  H (t  t )
 H (t )
t 0 
t
h(t )  lim
function h(t) that is defined as renewal density.
t
h(t )  f (t )   h(t  u ) f (u )du, is renewal equation for a renewal density
0
Alternating renewal process
Sn  X1  Y1  ... X n1  Yn1  X n ,
Tn  X1  Y1  ... X n1  Yn1  X n  Yn .
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
t
t
K (t )  1  F (t )   h( x)1  F (t  x)dx  R(t )   h( x) R(t  x)dx,
0
0
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
K  lim K (t ).
t 
(4)
Models with a negligible renewal period
Poisson process:
f (t )  e  t
t  0,   0.

(t ) n e t
(t ) n
t
H (t )  ENt   n
 te 
 t.
n!
n!
n 1
n 1

h(t )  H (t )  
Gamma distribution of a time to failure: f (t ) 
Using Laplace integral transformation we obtain:
sk  C , is
kth
nonzero root of the equation (s +
λ)a
 (t ) a 1 et
(a)
h(t ) 
=
a
a 1

  sk
k 1
a
e sk t ,
t  0,
λa
For example for a = 4 nonzero roots are equal to:
For example for a = 4 nonzero roots are equal to:

t  0,   0, a  0.
,
s1   (e
i
2
 1)  (1  i ) ,
s2   (e i  1)  2 ,
s3   (e
i 3
2
 1)  (1  i )
and a renewal density
h(t ) 

1  e
4
 2 t
 e t sin( t )

Models with a negligible renewal period

Weibull distribution of time to failure: f (t )  (t ) 1 e(t ) , t  0
α > 0 is a shape parameter,

H (t )   Fn (t )
Using discrete Fourier transformation:
λ > 0 is a scale parameter
n 1
n 1
t
( )i
i 0
i!
Fn (t )  1  

e  t  Gn (t ),
t  n ,

where μ is an expected value of a time to failure:
(1 

1
)
 .
K
H (t )   Fn (t )
We can estimate in this way an error of a finite sum
n 1

because a remainder is limited

 F (t )   G (t ),
n  K 1
n
n  K 1
n
t  n
Models with a negligible renewal period
Weibull
distribution:
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
P(t )  F (t ),
 1,
t c
t c
The probability P(t) (coefficient of unavailability) for
P(t )  Pn (t ),
n  N  0: n( c   d )  t  (n  1)( c   d ),
...
Pi (t )  Pi 1 (t )  Pi 1 (i c )1  P0 (t  i( c   d ))
...
P0 (t )  F (t ),
t  0.
i  1,2,...,n  1,
t  0,  
Models with periodical preventive
maintenance
Exponential distribution of time to failure, τd = 0:
P(t )  F (t  n c ),
n  N  0: n( c   d )  t  (n  1)( c   d ).
Coefficient of unavailability for Exponential distribution.
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.
Alternating renewal models
Lognormal distribution of a time to failure:
f f (t ) 
1 ln t  
(
),
t

t0
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:
t
K (t )  R f (t )   h( x) R f (t  x)dx.
0
renewal density can be estimated by a finite sum
N
h(t )   f n (t ).
n 1
Alternating renewal models
An example: In the following example a calculation for parameter values σ=1/4,
λ=8σ, τ=1/2, is done.
lim h(t ) 
t 
1
 0.123
EX f  EX r
A renewal density
for lognormal distribution
lim K (t ) 
t 
EX f
EX f  EX r
 0.938
Coefficient of availability
for lognormal distribution
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.



R f (t )  P( X f 1  t  X f 2  t )  1  Ff 1 (t ) 1  Ff 2 (t )  R f 1 (t ) R f 2 (t ).
f f (t )  
d
R f (t ).
dt
For a renewal density we have
t
h(t )  f f (t )   h( x) f f (t  x)dx
0

In case of non-exponential distribution we use
h(t )   f n (t ).
n 1
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
Example:
X f1  t  X f 2  t
have Weibull distributions
Coefficient of availability for Weibull distribution
EX f 
EX 2f 1 EX 2f 2
EX 2f 1  EX 2f 2
2262

 3.6
22  62
K
EX f
EX f  EX r
 0.79
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