A study of asymptotic availability modeling for a failure and a repair rates following a Weibull distribution Текст научной статьи по специальности «Математика»

A STUDY OF ASYMPTOTIC AVAILABILITY MODELING FOR A FAILURE AND A REPAIR RATES FOLLOWING A WEIBULL

DISTRIBUTION

Salem Bahri a, Fethi Ghribi b, Habib Ben Bacha a,c •

a Electro Mechanical systems laboratory (LASEM), Department of Mechanical Engineering-ENIS

e-mail: Salem.BenBahri@enis.rnu.tn b Department of Mathematical and Computer Science National Engineering School of Sfax (ENIS), University of Sfax BP W, Sfax, 3038, Tunisia e-mail: fethi.ghribi@,enis.rnu.tn c King Saud University- College of Engineering in Alkharj-P.O Box 655, Elkharj11942, Kingdom of Saudi Arabia e-mail: hbacha@ksu.edu.sa

Abstract:

The overall objective of the maintenance process is to increase the profitability of the operation and optimize the availability. However, the availability of a system is described according to lifetime and downtime. It is often assumed that these durations follow the exponential distribution. The work presented in this paper deals with the problem of availability modeling when the failure and repair rates are variable. The lifetime and downtime were both governed by models of Weibull (the exponential model is a particular case). The differential equation of the availability was formulated and solved to determine the availability function. An analytical model of the asymptotic availability was established as a theorem and proved. As results deduced from this study, a new approach of modeling of the asymptotic availability was presented. The developed model allowed an easy evaluation of the asymptotic availability. The existence of three states of availability for a system has been confirmed by this evaluation. Finally, these states can be estimated by comparing the shape parameters of the Weibull model for the failure and repair rates.

Keywords: Availability function, asymptotic availability, failure rate, repair rate, Weibull distribution

1. Introduction

The last two decades witnessed major progress in the development of new maintenance strategies [1]. The primary objectives of these strategies are to reduce equipment downtime, also increase reliability and availability of the equipment which at the same time optimizes the life-cycle costs [2]. The need for high reliability and availability is not just restricted to safety-critical systems [3]. In general, current technology has ensured that the equipments for industrial application, for example, telephone switches, airline reservation systems, process and production control, stock trading system, computerized banking etc. all require very high availability [2]. Reliability is generally described in terms of the failure rate or mean time between failures (MTBF), while availability is normally associated with total downtime [2]. There is some research on increasing system availability [4]. Goel and Soenjoto proposed a generalized model [4]. Markov models are also implemented to analyze the system availability, which combines both software and hardware failures and maintenance processes [4]. Khan and Haddara [1] proposed a methodology for risk-based maintenance to increase availability of a heating, ventilation and air-conditioning (HVAC) system. Garg S. et al. [3] developed a model for a transactions based software system which

employs preventive maintenance to maximize availability, minimize probability of loss, minimize response time or optimize a combined measure. The steady state availability can be modelled using standard formulae from Markov regenerative process (MRGP) theory. The Service rate and failure rate are assumed to be functions of real time (Weibull distribution) [3]. The failure and repair rates are supposed constant (X and p respectively), so that system availability can be modeled using a Markov chain in Refs. [5,7]. But, Khan and Haddara [1] considered that the Weibull model is more robust than the other models. Dai et al [4] studied the availability of the centralized heterogeneous distributed system (CHDS) and developed a general model for the analysis. The repair time was exponentially distributed. For the failure intensity function (failure rate), the G.O model presented by Goel and Okumoto was used [4]. Some other research considered that the availability depends on both reliability and maintainability and is defined as the ratio of requested service time to practical service time [6, 7]

Nomenclature

A(t) Availability function

A<x> Asymptotic availability

Mt) Failure rate

^(t) Repair rate

p Shape parameter of Weibull distribution for Failure rate

n Scale parameter of Weibull distribution for Failure rate

a Shape parameter of Weibull distribution for repair rate

0 Scale parameter of Weibull distribution for repair rate

Review of the literature indicates that there is a new trend to use availability and reliability modeling as a criterion to plan maintenance tasks. However, most of the previous studies assumed the failure and/or repair rates are constant. It seems that there is a need for a more generalized methodology that can be applied for variable rates. The present study adopts a new fundamental approach for the asymptotic availability modeling where the failure and repair rates were governed by the Weibull distribution.

This paper is organized as follows. In Section 2, the differential equation of the availability is established. Section 3 is dedicated to the resolution of the differential equation to determine the instantaneous availability. The model of the asymptotic availability is developed in Section 4. Finally, in Section 5, the conclusions along with future research directions are presented.

2. The mathematical formulation of the availability differential equation

According to the standard "Association Française de Normalisation - AFNOR X 06-503" [8, 9] , in order to have a system available at time t+dt, there are two possibilities:

• the first is that the system is available at time t and does not have breakdown between t and t+dt

• the second is the system is unavailable at time t but it is repaired between t and t+d. These expressions are transformed by the following probabilities:

> A(t+dt): The probability that the system is available at time (t+dt),

> A(t): the probability that the system is available at time t,

> 1-D(t)dt : The probability that the system does not have breakdown between t and t+dt, knowing that it had already functioned until the time t,

> 1-A(t): The probability that the system is unavailable at time t

> D(t)dt: The probability that the system is repaired between t and (t+dt), knowing that it was already failing until the time t.

With:

- D(t) : Instantaneous failure rate

- D(t) : Instantaneous repair rate

Fig. 1 shows the state diagram of the system.

Mt)

l(t)

Fig. 1. State transition diagram

A(t+dt)= probabilities (that the system is up at t and is no break down between t and (t+dt))+ probabilities (the system to be down at time t and it is repaired between t and (t+dt)) [8, 9].

=p® - [m+lmm co

Then:

(4)

This expression represents the differential equation of first order of the availability. [4,8, 9].

3. The availability function

For t > 0, the failure and repair rates, which are modeled using a Weibull distribution, are

given by :

- m - : (: (5)

^ ' , v CM

" i'W=f(t) <«)

- Eq. (4) can be solved by "Mathematica software", by taking account of the initial conditions A(0)=0 if the system is in the failure state and A(0)=1 if the system is in the functioning state and can be obtained the following solutions:

• If A(0) = 0then,

A(t) = AoCO = a V W Lrfay Vrt

(7)

If A(0) = 1 then

_rT'r_fnV - fiii'Wiitf \

d<£} = A^i) = a V ^ ( 1 + a*"01 Jja V 4W u^du J : deduced that: Al® = « -M0

(8)

It can be deduced that:

ta£t) (9)

There are four parameters in the availability functions (7), (8), p, rj, a, and 0. The sensitivity of different parameters is described in Figures 2, 3, and 4.

Fig. 2. The availability A(t) for p =0.5, a=1.5 (P < a)

Fig. 3. The availability A(t) for p = a=1.5

Fig 4. The availability A(t) for p =1.5, a= 0.5 (P > a)

4. The Asymptotic availability

4.1. Theorem

Demonstration

It can be assumed that:

Where:

KO -*ffi> - if.A(t)=Ao(t)

And

rm = Vk®= if A(t)=A1(t)

It may be necessary to prove that:

ik* ¡m^m-i.

Ms titi' ' '

There are four intermediate results can be used to explain this. > 1st result:

Proof:

From Eq. (12), this can be obtained by substituting Ai(t) by Eq.(9)

Pa CO = ^rpf *^ -Az(0) = ^AM

(tit?

»tit?

An analogy with Eq. (9) can be deduced:

titj

№

Therefore, from Eq. (17), it is necessary, to verify the Is result, to prove.

issas*^= 0

llm

e ■ x ^(¡3

f>r f Vs

(10)

(11) (12)

(13)

(14)

(15)

(16)

(17)

(18) (19)

Referring to Eqs. (5) and (6): ¿.-r.r-1- gnfriP

ttiti arn*"1 SJfn" a is?

Then, Eq. (19) will become:

v $££>1 s if«1

The limit study of Eq. 18 gives:

(20)

(21)

Figure 1:

(22) And

fT f-"ia

lim e" " =0

7 f ■■

.-(3-£S

lim " = 0

So, lim rtXJ = lim if,(t I = 11m and the 1st result is verified.

—----^-»-t (E

(23)

> 2nd result :

L^ 3 ft\a

dv

Proof:

From Eqs. 16 and 17, the T^t) function is written as:

According to Eqs. (7) and (20), The Eq. (25) will become as follow:

A change of variables is applied in Eq. (26): u = t ■ v => du = £dt?

r0co = (i+f g) (|) )• V w «(£} ¿w ^ v*-1*?

(24)

(25)

(26)

(27)

(28)

and the 2nd Result is

Therefore, = [^(rj " « (jj )«

verified.

> 3rd result :

According to the shape parameters P and a, the r0(t) function should satisfy the two following

inequalities:

a) If p< a, then

-if-rs

1 — a V W

Or

b) If p> a then

-fif-flf

Proof:

a) For p< a

-trf-f1!"

1 _ a v w

ivijj

i-r^-®1

(29)

(30)

(31)

And ((0 - Qyy £ (0 (32)

By referring to the second result (24) and the two above mentioned inequalities (31) and (32), so, the tfltXi function can be put under the form of the following inequality:

The calculations of exponential integral allow to express the inequality (33) as follow:

-a P -pi

■ (J

rji) 23

¡^Tj ^ K

JWL,

(33)

(34)

(35)

f f-.'3i-.'3-ll r ¡-(tf-itfil

So,

satisfied. b) For p> a

A similar development and demonstration is used for this case also

e ]tM], ¿? gf i^"1 + » (£}* £ (V g/ + a (ff ) v

the first inequality (29) is

(36)

And

(0V + (i)V^((if + (|}>

(37)

According to the second result (24) and the two above mentioned inequalities (36) and (37), so, the r0(t) function can be put under the form of the following inequality:

e „

f1

- V

.05 £

The calculations of exponential integral allow to express the inequality (38) as follow:

(38)

(39)

(40)

(41)

So,

J I J

satisfied.

> 4th result :

lim rnÇif) = 1

, the second inequality (30) is

(42)

Proof: a) p< a

By referring to the third result "inequality (29)", to prove the fourth result, it can be sufficient to show that the limits:

JM^iJ-tsUff)

= i

(43)

And

lllEl

f—' "I flri

Then Jim fc®

1 - a

- 1

(44)

J af cm

«ÎL

- 49*@1

' (45)

L-.Hr-r)1

lim

= [1- B] = 1

(46) And

(47)

So, linj^Cfcl = lifp<a b) p > a

In the same way as explained in the previous case, according to inequality (30), to prove the fourth result, it can be sufficient to show that the limits:

i-fl u 1

fr—iu,T ft

And

-1

lim -

J

Then

lim

And

Jim

[a_.Hr-r)i

= [l- a] = i

4B*(B1

(48)

(49)

(50)

(51)

(52)

So, Jim ra(il = 1 and the 4 result (42) is proven also for a< |3 Finally, the theorem (10) ensues therefore of results 1 and 4

The availability A(t) is plotted together with The i / function in figure 5 for p<a, figure 6 for P=a and figure 7 for p>a.

The three figures show that the availability A(t) " with its two solution Ao(t) and Ai(t) " and the function have tendency to converge towards the same limit when the time t is more

important.

Fig.5. The availability A(t) and

M ( t )

/ (t )+X (t )

for p =0.5, a=1.5 (P < a)

A (t) ,

Mt)

y. (t) +A (t)

1

0.8

□ . 6

□ .4

0.2

Aj.(t) for 77 = 1000 and 6=2000

A[i(t) for 77 = 1000 and 8=2000

^ (t)

V (t) +A (t)

for 77 = 1000 and 6=2000

5000

10000

15000

20000

Fig.6. The availability A(t) and

M (t )

/ (t )+À (t )

for A(t) for p = a=1.5

A(t) ,

^ (t) pi (t) +A (t) 1

o.s

0.4

5000

Ai(t) fou 77 = 1000 and 0=2000

A0(t) foe 77 = 1000 and 0=2000

--^ (t)- £ou „ = iooo and 6=2000

pt (t) + A (t)

10000

15000

20000

Fig.7. The availability A(t) and

M ( t )

/ (t )+À (t )

for A(t) for p =1.5, a=0.5 (P > a)

4.2. Asymptotic availability evaluation

According to (10), the asymptotic availability is defined by: = llimAt9= lim

A^ = lim

(53) (54)

f^f." r—-x s. T

1SB

The study of the limit of the function will be done according to three following cases: 1st case: p<a

Uni;__

Then,

= lin —77T = 7^= 1

S-H-« 14— HC

The converge of the L function, when the time t is more important, to the A®= 1 with the

sensitivity of the scale parameters (r|<9, r|=9 or r|>9) is shown in Fig. 8.

(55) (56)

^ (t) +A (fc)

(t)

(t) + A (t)

(t)

H (t) -+- X (t)

(t)

V (te ) + A (te)

for 77 = lOOO and ©=2000 for 77 = 1500 and ©=1500 for 77=2000 and © = 1000

Fig.8. The

M ( t )

/ (t )+À (t )

function limit studies for P=0.5, a=1.5 (P < a).

2nd case: p=a

un, i,™ =

r—'- K ft1,t/ r —t-ED« Va/

In this case, the asymptotic availability is defined to be equal to i i j-i

4 = - *

Particular cases:

P=a=l: the exponential models

¿a ■ * -j

i

if r|=0, then

a -i

(57)

(58)

(59)

(60)

Fig. 9 shows the asymptotic availability plotted with □ = □ with the sensitivity of the scale parameters (n<0, or n>9).

-^ ; '--for 77 = 1000 and 6 = 2000

(sL (t) +A (t)

fJ (t)

-i—i—i--for 77 = 1500 and 6 = 1500

f Ct) +A (t)

---foe 77 = 2000 and 6 = 1000

Mt)+A(t)

o.e -

0.4 -

0.2 -

V (t) M (t) +A (t)

1 r

0.8 -

5000

10000

15000

20000

Fig.9. The function

M ( t )

M (t )+À (t )

if p= a=1

3rd case: p>a

11» ¿g- lim —= "OD

— « V 1

ltK

- Q

(61) (62)

The converge of the 777^-—function, when the time t is more important, to the A.,= 0 with the sensitivity of the scale parameters (n<0, n=9 or n>9) is shown in Fig.10.

V- (t)

0.8

□ . 6

0.4

0.2

(t)

¡-i (t) + A (t)

(t)

(t) + A (t)

(t)

t* (t) + A (t)

foe 77 = 1000 and e=2000 for 77 = 1500 and e = 1500 for 77 = 2000 and 8 = 1000

Fig.10. The

5000

P ( t )

10000

15000

20000

fl (t )+À (t )

function limit studies for P=1.5, a=0.5 (P>a).

5. Conclusion

In this paper, the presented work extended the classic availability model to a new asymptotic availability model when the failure and repair rates are distributed according to the Weibull model. The analysis of asymptotic behavior of the system according to the developed model allowed to extract the following result:

The asymptotic availability depends only on the shape parameters of the Weibull models P and a. The scale parameters n and 9 do not have an influence in the limit of the availability.

- If b < a then, the system is fully available

- If b > a the system resides in the down state, then, it is unavailable

- Ifb = a , in this case, the asymptotic behavior of the system is analogous to a system

governed by the exponential model.

Thus, the future plan includes the research on a novel approach, which will be the combination of two different models (Weibull, Gamma,) or (Weibull, lognormal).

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