Markovian Method of Finding the Availability of Repairable Systems

8/6/2019 Markovian Method of Finding the Availability of Repairable Systems

http://slidepdf.com/reader/full/markovian-method-of-finding-the-availability-of-repairable-systems 1/8

Volume 2 No.5, MAY 2011 ISSN 2079-8407

Journal of Emerging Trends in Computing and Information Sciences

©2010-11 CIS Journal. All rights reserved.

http://www.cisjournal.org

243

Markovian Method of Finding the Availability of Repairable Systems

M. Y. AshkarDepartment of Mathematics,

University of Bahrain, PO Box 32038,Kingdom of Bahrain.

[email protected]

ABSTRACT

In this paper the transition probability matrix is used to tackle the problem of the availability of repairable systems. The

method is applicable to a system governed by a two-parameter distribution for the life-time and repair-time. It will be

concentrated on the Weibull and Gamma distributions without and/or with singularities. The method is involving matrices

multiplication of massive size. For this purpose an arithmetic method is applied to carry out this multiplication that, in turn,

represents the availability of a repairable system.

Keywords: Point of Availability, Markovian Method, Transition Matrix 1. INTRODUCTION

The Availability or point-availability term is

insufficient to know about how many times a device has

been replaced or repaired upon failure. However we areinterested in the condition of such item, working or failure

so it is under repair. In fact the device may be replaced

on daily base or more than once a day or it could have

never been replaced at all. Other measurement is still

important and needed, such as the device's reliability. So

the relationship between reliability, maintainability and/or

availability is very strong and important, Ascher [1],

Ashkar [2], Barlow [3], Kapur [8] and Leemis [15].For a repairable system, the life-time cycle can be

described by a sequence of up and down states, i.e.

working and under repair. So the system operates until it

fails, then it is repaired (or replaced, if its replacement cost

less and save more time) and is returned to its original

operating state, this process will continue alternating

between these two states for sufficient time and this

process is called a renewal process. It is a sequence of

independent random variables, Ashkar [2] and Cox [4]. In

this case, the random variables are the time-to-failure and

the time-to-repair. Each time a device fails and is repaired,

a renewal is said to have occurred. This type of renewal

process is known as an Alternating Renewal Process because the state of the device alternates between a

functioning (up-state) and under repair (down-state), Cox

[4]. This process is illustrated as shown in figure1

Ashkar[2].

Figure 1 illustrates the nature of alternating of a system

between up-time (operating) and down-time (under repair)

over the time.

The assumptions in the renewal theory is that the

failure items are replaced or repaired with new ones so

they are "as good as new" hence the name 'renewal' isobtained, Kijima [12], [13], Elsayed [6] and Leemis [15].

The availability, as a definition, is somewhat

flexible and is largely based on what types of down-times,

or under repair, one chooses to consider in the analysis.

There are number of different classifications of

availability, such as, Instantaneous (or Point) Availability,

Average up-time Availability and Steady State

(Asymptotic) Availability Kececioglu [10], [11].

The instantaneous availability, or point

availability, is defined as the probability that a system will

be in operation at any random time, t . This is very similar

to the reliability function where it gives a probability that asystem will function at the given time, t . Unlike reliability,

the instantaneous availability measure incorporates

maintainability information. At any given time, t , the

system will be operational if the item is either functioning

properly from 0 to t with probability R(t)=1-F(t), or it is

functioning properly since the last repair at time u, 0 < u <

t , with probability ∫ −t

duumut R0

)()( , where m(u) is

the renewal density function of the system Elsayed [6],

Kececioglu [10]. Hence, the point availability is given as

the summation of these two probabilities,

i.e. ∫ −+=t

duumut Rt Rt A0

)()()()( , Cox [4].

The above sum of probabilities represents

Volterra's integral equation of the second kind. It can be

solved numerically to find A(t), and it can be applied in

systems without singularities Fox [7].

Ashkar [2] has shown that this estimation of point

availability is possible through a mixture distribution and

complicated calculation .However, the used method is

limited to non singular two-parameter distributions.

In this paper, the method used is invented to cope

with a very wide class of distributions regardless of their

scale and position parameters. It could be applied to

Gamma, Weibull, Lognormal and most of two parameters

distributions. Also, it can be used to find the pointavailability of systems of Bath-tub shaped hazard rate. But

http://www.cisjournal.org/


http://[email protected]/










244

here, the method will be applied strictly to systems of

Weibull and Gamma distribution. In this article, a direct

method is illustrated in section 2. The transition

probability matrix is obtained in section 3. Finally point

availabilities of systems of weibull and gamma are

obtained in section 4 and 5.

2. THE DIRECT METHOD

This method, in principle, finds out the point

availability of the system directly from the hazard

function. We found that the hazard function ),( x x xh δ +

can be expressed with reference to the definition of the

conditional probability as

}Pr{),( x X x x X x x x xh >+≤<=+ δ δ

)(},Pr{

x X P x X x x X x

>>+≤<= δ

)(1

)()(

xF

xF x xF

−

−+=

δ

(1))(

)(1

)(

)()(

x R

x x R

x R

x x R x R δ δ +−=

+−=

The above formula (1) gives the failure

probability function for any life distribution function.

Consider the limit Z(x), which may exist and be finite. If it

does exist, then

)(

)()()(

)(

1lim)(

0 x R

x f

x

x x R x R

x R x Z

x=

+−=

→ δ

δ

δ (2)

The above Z(x) is called the hazard rate function

and then

)()()(

)(1

)(

)()( x x Z

x R

x x R

x R

x x R x Rδ

δ δ ≅

+−=

+−(3)

The term x x Z δ )( is approximately the failure

probability in the interval x)xx,( δ + .The

quantity x x Z δ )( is used to approximate, where it is

appropriate, the failure as well as the repair time

distributions. When there are singularities, the original

formula (1) is used; otherwise (2) will be used.

However, it must be mentioned that the hazard functiondepends, where it is an up time, on the current age of the

component; and on how long it has been down, if it is a

down time. So we need only to take account of the age of the

system to see what will happen next. Thus, if we use the

current age as a state, the system is Markovian and,

therefore, we can describe it by an initial state vector and atransition matrix, Ross [19].

It is clear that the hazard function ),( x x xh δ + can be

calculated from R(x). The term ),( x x xh δ + measures

the mortality near the point x. Meanwhile, the term

),(1 x x xh δ +− is the probability of survival from x to

)( x x δ + .

The method is applicable to both discrete and

continuous distributions if it can be approximated bydiscrete ones.

In the continuous case the subdivision between

consecutive steps is made very small and is 0.01 in ourcalculations. So the continuous distribution can be

approximated by a discrete one. All we need to deal with

such a problem is to find the elements of the transition

matrix shown in figure 2. These elements are obtained

merely from the hazard function.

3. THE TRANSITION MATRIX

The nature of state transition matrix of size

[(2n+2)x(2n+2)] is shown in figure 2. The transition

matrix is divided into four regions to allow transition

between up states and down states. Each of the up times

and the down times is specified with n+l states. This is tocover the range of the study, and n is large enough to

fulfill the requirements of the problem.

We made the transition matrix finite by

truncating the life time and the repair time distributions

suitably at a very large n to make it manageable. This can

be done by making the hazard rate at the state n equal to

unity. The same thing is true for the corresponding

functions for the down time. For the up time, the hazard

function ),( x x xh δ + will be represented by p(x). Thesurvival rate will be represented by l-p(x). The same thing

is true for the down time but with q(x) and l-q(x) instead.

For instance, the term23

1 p− , denotes the probability of

the system surviving from state i=2 to the state i=3. If the

system fails to survive at the state i=2 then it goes, instead,

to the down state region with probability 23 p . From up

state at t=0, the system can go either to state i=l with

probability ( 011 p− ) staying still in an up state, or fails

and goes to the down state i=l with probability 01 p . A

system may stay in the up state and go to states i=2, 3, ..

,n-l with probabilities

)1(.,),........1(),1( ,12312 nn p p p −−−− without failure.

If the system has survived this long, it will surely

be transferred into the down state. So the system will fail

according to its hazard function. By then the system will

be transferred to the down state, where it can go through

similar steps. So we can describe the method as the

following: The transition matrix is divided into four

regions. In a clockwise order, the first region is the top

left. A system in the up position starts within the first

region. Upon failure it goes to the second region where it

can be transferred to the third region where the system is

under repair and stays there for a certain time. Byrepairing the system it will be transferred to region four

where it can be transferred to the first region to start again.










245

This circulation would continue as long as the system is in

use.

Up-Time Down-Time

)4321001234....( nni =

.

0...0000010000...0

1...000000000...0

0........0

0...000000...0

0...010000000...0

0...0010000000

0...000100000...0

......

...00000100...0

0...000010...0

0...00001...0

0...00000000...0

......................

0...000000000...1

0....0000100000...0

1

2

1

0

0

1

2

1

,1,1

233,2

122,1

011,0

0101

1212

2323

,1,1

nnnn

nnnn

qq

qq

qq

qq

p p

p p

p p

p p

n

n

n

n

−−

−−

−

−

−

−

−

−

−

−

−

−

Figure2 illustrate the nature of the Transition

Matrix of size (2n+2)x( 2n+2)Between states

Up-Time Down-Time

nn 4321001234....

)00000001000....0()0( = AV

Figure 3 illustrates the (2n+2) initial availability row vector

3.1 Initial availability and availability row

vector

A new system is put into service starts at t=0. Theavailability at this point state is "One". And non-

availability is "Zero". Thus the initial vector is the first

essential in calculating the availability at any state time.

Given the probability of being up or under repair as

illustrated in the transition matrix, we can deduce the point

availability, or the vector's availability for the next point.

This row vector will replace the initial one for the next

step. This can be done by multiplying the vector by the

transition matrix. The resultant row vector will show not

only the probability the system is up or down, but the

probabilities of the different times that it has been up or

down. Also it will be used again as the initial availability

row vector for the next step.To obtain the next point availability, we repeat

the foregoing process. Every repeated multiplication givesthe instant point availability. This process will be repeated

until the system reaches an asymptotic point availability

which, as a check, should be very close to the theoretical

one. At every state time the initial row vector will contain

(2n+2) elements (probabilities). These elements should

sum up to unity. These elements are probability values. A

set of n+1 elements of these refer to up state, let us denote

it by (Ui), The other n+1 refer to down state, let us denote

it by (Di). The point availability, simply is ∑+

=

1

1

n

i

iU and the

point of non-availability is ∑+

+

22

2

n

n

i D where

11

1

22

2

=+∑ ∑+

=

+

+

n

i

n

n

ii DU

3.2 Problem of a very large matrixmultiplication

The idea of finding the point availability looks

very simple especially when the size of the transitionmatrix is reasonably small. More formally, let us call the

transition matrix TM[(2n+2)x(2n+2)] and the current row

vector as AV(i)[[1]x(2n+2)]. At the commencement

AV(0)(2n+2) contains the point availability at t=0 where

PA(0)=1. The process to calculate AV(i) can be found as

the following :

,...,2.1,0)]22(1)[1()]22()22[(

)(

)]22(1)[( =+×+=+×+×+× ini AV nn

xTM

ni AV

i

Very small steps are used (such as width is

δt=0.01). Yet our aim is to study the system availability upto t=20. This makes the matrix TM of size 4002x4002, for

which a huge computer memory is required. And more

memory needed when the system availability is studied up

to t=50 or more.

It is notable, however, that most the elements of TM are

zeros. We will take advantage of this phenomenon and

calculate the availability vector with an arithmetical

method which condenses the matrix multiplicationmethod.

More formally, let us denote the elements of AV(i) by

I.e. AV(i) now is ( a1 a2 a3 . . . an+1 an+2 . . . a2n+1 a2n+2 )

And the elements of the resultant availability vector, AV(i)

appears as

( c1 c2 c3 . . . cn+1 cn+2 . . . c2n+1 c2n+2 )

This ci's, i=1,2,3,… are calculated as the following;

c1 = a2(1-pn-1,n )

c2= a3(1-pn-2,n-1)

c3= a4(1-pn-3,n+2)










246

…………………

…………………

cn =an+1 (1-p01 )

cn+1 = an+2 q01 + an+3 q12+ an+4 q23 + … + a2n+2 qn-1,n +

a2n+2 cn+2 = a1 + a2 Pn-1,n+ a3 Pn-2,n-1 + a4 Pn-3,n-2+ … + an

P12 + an+1 P01

cn+3 =an+2 (1-q01 )

cn+4 =an+3 (1-q12 )

………………..

……………….

c2n+2 =a2n+1 (1-qn-1,n ). (4)

Having obtained all the elements of the

availability vector, we now proceed to find the point

availability and the non-availability of the system at the

point i which are given, respectively, as

)(1

1

∑+

=

=n

j

jciPA

(5)

∑+

+=

=22

2

)(Nn

n j

jciPA (6)

By getting the point availability from (5), we

proceed to the next point. This can be done by repeating

the same procedure after replacing a j by c j .The method

and its operation are illustrated in figure 4.

Figure 4 flow-chart illustrates the structure of Arithmetical

calculation of PA

The time interval between consecutive points is

δt=0.01 the calculation will continue until the systemreaches the asymptotic state. This method accepts any

distribution for the up-time and/or down-time and gives

results similar to those distributions if studied using the

integral equation method. Weibull and Gamma

distributions have been treated by this technique.

4. WEIBULL DISTRIBUTION

The Weibull distribution was introduced by W.

Weibull [21]. Its use in reliability theory was discussed by

Weibull [22]. The distribution is widely used in reliability

theory. It includes monotone increasing, monotone

decreasing and constant hazard rates. It is noted that the

Weibull distribution has many different algebraic forms

depending on the way in which the parameters are defined.

Although all forms are essentially the same in regard to

the important distributional properties, there is no

generally accepted algebraic form for the Weibull

distribution. The probability density function, thecumulative distribution and the reliability functions of the

Weibull family are given respectively.

0,0,1)(1

)1

(

≥>−>=+

+−

xk mekx x f m

xm

k

m (7)

0,0,11)(1

)1

(

≥>−>−=

+

+−

xk me xF

m x

m

k

(8)

0,0,1)(

1)

1

(

≥>−>=

+

+

−

xk me x R

m x

m

k

(9)

The mean and the variance of Weibllll

distribution are given respectively as

)1

2()

1()( 1

1

+

+Γ

+= +

−

m

m

m

k X E m (10)

1

2

2)

1()]

1

2()

1

3[()( +

−

+×

+

+Γ−

+

+= m

m

k

m

m

m

m X Var (11)

where (*)Γ is the complete Gamma function , i.e.

dxe x x−

∞−

∫=Γ0

1)( α α . For the purpose of the study we

define the parameters m and k in order that each member of

the family has mean equal to unity. This can be found by

putting E(X)=1 in (10) and solving the resulting equation

for the given values of m. In table 1 below, we display the

values of m and its corresponding values of k that are used

in our calculation.

Table 1 Weibull parameters

m k Mean-0.75 0.5533414 1.0

-0.50 0.7071068 1.0

-0.25 0.8548628 1.0

0.0 1.0000000 1.0

1.0 1.5707064 1.0

2.0 2.1362189 1.0

3.0 2.6998792 1.0

4.0 3.2627404 1.0

The reliability function will play the main role in

this method. Using the reliability function of a Weibulldistribution (9), the hazard function can be given, by usingthe formula (1) as










247

1

1

)1

(

))(1

(

1),(+

+

+−

++

−

−=+m

m

xm

k

x xm

k

e

e x x xh

δ

δ (12)

The hazard function (12) will be used for given

values of m and k which make the MTTF and the MTTR

as required.

We now obtain the elements of the transition

matrix as illustrated in figure 2. This matrix consists

merely of those values obtained from the hazard function

(12). We will call the probability of up-time (l-pij) for the

system to survive and pij for the system to fail. The (l-qij)

are corresponding probabilities for down-time. (l-qij)

means that the system is still under repair q ij means that

the system goes from being under repair to the up-state,and in all cases j=i+1. After any point time, t=i, the system

is only in one position, either in the up-state withprobability (1-pij ), or down-state with probability pij .At

any point time the system is in one well defined state. We

come now to finalize the results of the point-availability of

a system of Weibull. Once the hazard function is essential

to the aim of the calculation, two hazard functions of the

form (12) will be stated. One is for the up-time and the

other for the down-time. It is assumed that the mean time

to failure and the mean time to repair are identical and

equal to unity, i.e

1)1

2()

1( 1

1

=+

+Γ

+== +

−

m

m

m

k MTTR MTTF m (13)

Where k and m are the parameters of Weibull

distribution and )1

2(

+

+Γ

m

mis the complete gamma

function. For every given value of m we can define only

one value of k. This can be found as

1)}1

2(){1()1( +

+

+Γ+= m

m

mmK (14)

8.01

1)(lim

41

=+

=∞→

t PAt

(15)

It means that the point-availability of the system

on the long run is 0.8, and mild oscillations can be caused.

The results are shown in figure 6

The parameter k(l) of MTTF=l is defined from

the formula (14), while the parameter k(1/4) where

MTTR=1/4 of the repair time can be given as:

,4 / 1)()1

(121

1

=Γ+

= +++

−

mmm

m

k MTTR (16)

1

)1

2

(4)1()4 / 1(

+

+

+

Γ+=

m

m

m

mK (17)

5. GAMMA DISTRIBUTION FUNCTION

The probability density function, the cumulative

density function and the reliability function are given

respectively by

β α

α β α

x

e x x f −

−

Γ= 1

)(

1)( (18)

dses xF

s x

β α

α β α

−−

Γ= ∫

1

0)(

1)( (19)

dses x R

s x

β α

α β α

−−

Γ−= ∫

1

0)(

11)( (20)

Thus the hazard function will be evaluated from

(1) by using the reliability function formula (20), which is,

itself, evaluated using numerical integration. Thus

)(

)(1),(

x R

x x R x x xh

δ δ

+−=+ (21)

It is clear from (20) and (21) that the calculation

of the cumulative distribution function involves the

integration of the singular integrand. However, the

Gamma distribution is defined to have mean unity, this

would make the density function of the Gamma

distribution written as a function of only one parameterα .

Since the mean =αβ =1 then α β / 1= and (18) can be

rewritten as

xe x x f α α

α

α

α −−

Γ= 1

)()( (22)

And in a similar manner (19) can be expressed in

(23). This step is taken to make the Gamma calculations

comparable with the Weibull calculations later on.

dses xF s

x

α α α

α

α −−

Γ= ∫ 1

0)(

)( (23)

The point availability of a system in which the

failure time, and the repair time are distributed each with

Gamma distribution is our main interest. The methodwhich was elaborated previously will be applied. Similar

procedures to those with the Weibull are carried out. The

many numbers involved in the numerical results are

converted into graphs which are shown in figure 7. It can

be noted from these graphs, a mild oscillation occurs and

then settles down to the asymptotic position, during a

relatively short time of operation. Several values of the

parameter are considered. These values are α =0.25,

0.50, 0.75, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0. The values of

)(α Γ which are widely tabulated are used in the

calculation. Note that, since failures and repairs are

identically distributed, the asymptotic result gives that;










248

5.0)(lim =+

=∞→ MTTR MTTF

MTTF t PA

t

(24)

To make further generalization of failures and

repairs of Gamma, we assume that 1= β in (18), (19)

and (20) then the Gamma distribution can be expressed in

an easy way. The mean of the distribution will be a

function of its shape parameterα , the point availability of

the system can be considered in a similar way. In the

following lines several cases of the Gamma distributionare considered to show the system behavior. In all cases

the parameter 1= β is taken for the up-time and the down

time. Different21 ,α α Are used and they are illustrated

in the table 2 below.

Table 2 illustrates the different values of

parameters21

,α α for the up-time and down

time together with its asymptotic results

Up-time

Mean=

1α

Down-

time

Mean=

2α

Asympt

tic

results

Down-

time

ean=

2α

Asymptotic

results

Down-

time

Mean

2α

Asymptot

results

0.25 0.5 0.3333 1.0 0.200 2.0 0.111

0.50 0.5 0.5000 1.0 0.333 2.0 0.200

0.75 0.5 0.6000 1.0 0.429 2.0 0.273

1.00 0.5 0.6667 1.0 0.5.00 2.0 0.333

2.00 0.5 0.8000 1.0 0.667 2.0 0.500

3.00 0.5 0.8571 1.0 0.750 2.0 0.600

4.00 0.5 0.8889 1.0 0.800 2.0 0.667

5.00 0.5 0.9091 1.0 0.8.33 2.0 0.715

In table 2 the asymptotic values of the point

availability are shown. More details about the point-

availability which are specified in table 2 can be obtained

by considering figure 8.The point-availability values for

small values of t have been calculated. Note that theasymptotic values are indicated as in table 2 for each case.

6. CONCLUSIONS

At the end it can be concluded that the method

looks comprehensive, simple and easy to use. It can be

used despite the nature of the used distribution, with and

without singularities. Beside, it has wide range of use.

The numerical results obtained in this article are turned

into graphs shown in figures 5,6,7 and 8. The behaviors of point availability of the considered systems look natural in

term of its oscillation. The only difficulty encountered is

that of finding the transition matrix and carrying out themultiplication. Otherwise it is very efficient and reliable

for those systems of two-parameter distribution.

Figure 5 shows point availability of a system of Weibull

distribution where MTTF= MTTR = 1 Where m=0.25, 0.50,0.75,1, 2,3 ,4 ,5 , 6 and 7.










249

Figure 6 shows the point availability of a system of Weibull

distribution where MTTF= 1, MTTR = 0.25 Where m=0.25,

0.50, 0.75,1, 2,3 ,4 ,5 , 6 and 7

Figure 7 shows group of the point availability of a system of

Gamma distribution where MTTF= MTTR with mean

Equal unity Where m=0.25, 0.50, 0.75,1, 2,3 ,4 ,5 , 6 and 7

Figure 8 shows point availability of a system of

Gamma distribution given in table 2 where

β=1. α=2 ---,β=1. α=2 –-–-–- , β=1. α=2 ────










250

REFERENCES

[1] Ascher, H. and Feingold, H. "The Repairable

System Reliability", Marcel Dekker, New York,

1984.

[2] Ashkar, M.Y. " Point Availability of a Mixture

Distribution", IST Transaction of applied

Mathematics- Modeling and Simulation, Vol.1

No.1 (2),pp.1-9.2010.

[3] Barlow, R. and L. Hunter, " Reliability Analysis

of a One unit System", Operations Research, Vol.

9, pp. 200-2008, 1961.

[4] Cox, D. R. "Renewal Theory", Methuen, London,

1962.

[5] Davis, D.J., " An Analysis of Some Failure Data",

J. Am. Stat. Assoc., Vol. 47, 1952.

[6] Elsayed, E., "Reliability Engineering", Addison

Wesley, Reading, MA, 1996.

[7] Fox. L., and Goodwin, E. T., "The Numerical

Solution of Non-Singular Integral Equation",

Phil. Trans. Roy. Soc. A245,PP 501-534, 1953

.

[8] Kanatzavelos, A., " Reliability of Simple System

Under Different Assumption of Break down and

Repair Times", Msc. University of Essex,England 1985.

[9] Kapur, K.C. and L.R. Lamberson, " Reliability

in Engineering Design", John Wiley & Sons,Inc., New York, 1977.

[10] Kececioglu, D., "Reliability Engineering

Handbook", Volume 1, Prentice Hall, Inc., New

Jersey, 1991.

[11] Kececioglu, D., "Maintainability, Availability, &

Operational Readiness Engineering", Volume 1,

Prentice Hall PTR, New Jersey, 1995.

[12] Kijima, M. and Sumita, N., " A Useful

Generalization of Renewal Theory: counting

process governed by nonnegative Markovian

increments", Journal of Applied Probability, 23,71-88, 1986.

[13] Kijima, M., "Some results for repairable systems

with general repair ", Journal of Applied

Probability, 20, pp.851-859, 1989.

[14] L'Ecuyer, P., Proceedings of the 2001 Winter

Simulation Conference, pp.95-105, 2001.

[15] Leemis, L.M., "Reliability - Probabilistic Models

and Statistical Methods", Prentice Hall, Inc.,

Englewood Cliffs, New Jersey, 1995.

[16] Mettas, A., " Reliability Allocation and

Optimization for Complex Systems", Proceedings

of the Annual Reliability & Maintainability

Symposium, 2000.

[17] Piessens, R. Mertens,I. and Branders, M.Automatic Integration of Functions Having

Algebraic End Point Singularity', Angewandte

Informatic, 16, pp.65-68., 1974.

[18] Racicot,R. L. " Numerical Solution and Inference

for Interval-Reliability of Repairable

Components", IEEE, Transaction on reliability,

Vol. R-24, No. 1 1975.

[19] Ross, S. M.,"Stochastic Processes", Second

edition ,John Wiley & Sons, Newyork,1996.

[20] Tillman, F.A., " Numerical Evaluation of Instantaneous Availability", IEEE, Transactions

on Reliability, Vol. R32, No.1, 1983.

[21] Weibull, W., " A Statistical theory of Strength of

Materials" Ingeniors Vetenskaps Akademien

Handlingar, No 151, The phenomenon of Rupture

in Solids, Ibid No.153, 1939.

[22] Weibull, W., " A Statistical Distribution Function

of Wide Applicability", Journal of Applied

Mechanics, Vol. 18, pp. 293-297, 1951




Date post:	07-Apr-2018
Category:	Documents
Upload:	saeedullah81
View:	218 times
Download:	0 times

Markovian Method of Finding the Availability of Repairable Systems

Documents