STUDIES ON RELIABILITY AND AVAILABILITY OF AREPAIRABLE SYTEM WITH MULTIPLE DEGGRADATIONS M.A.El-Damcese

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50

STUDIES ON RELIABILITY AND AVAILABILITY OF AREPAIRABLE SYTEM WITH MULTIPLE DEGGRADATIONS

M.A.El-Damcese¹ & M.S. Shama²

1 Department of Mathematics, Faculty of science^, Tanta University, Egypt

2 Department of Basic Science, Preparatory Year, King Saud University, Saudi Arabia

ABSTRACT

This paper deals a repairable system consist of one unit with (n1) degraded states and one failed state. Let degraded rates, failure rate and repair rates of the system are assumed to be exponentially distributed. The expressions of a steady-state availability and reliability function are derived. We used special case of the system (one state degradable system) to analysis the availability function and the reliability function and compare the results to another studied model.

Keywords: Availability, Reliability, Steady-state availability.

1. INTRODUCTION

Redundancy is a technique widely used to improve System reliability by minimizing the Interruptions caused by machine failures. Any system becomes unreliable due to various reasons .Researchers [1,2,3] define different types of failures .There are three basic types of failure wear out failure, random failure, and infant mortalities .In the traditional systems, the units of the system have only two states up and down. However, in many situations the units of the system can have finite number of states.

In [4] stochastic analysis of a repairable system with three units and two repair facilities was introduced. In [5], reliability characteristic of cold-standby redundant system was introduced. In [6], some reliability parameters of a three state repairable system with environmental failure were evaluated. In [7], human error and common-cause failure modeling was established for a two-unit multiple system. In [8], reliability modeling of 2-out-of-3 redundant system is introduced subject to degradation after repair. Recently Reliability and availability analysis of a standby repairable system with degradation facility were studied in [9]. In. [10] Reliability measures of a degradable system with standby switching failures and reboot delay were introduced.

In this paper, we consider a system consists of (n1) degraded state and one failed state. We develop the expressions for steady-state availability function and reliability function using Laplace transform techniques then we study a special case of the system (one state degradable system) in details.

1.1. The Notations



i: The degradable rate of statei,i1,2,....,n1..

n: The failure rate of state

n  1

.

i: The repair rate of state

i , i  1 , 2 ,..., n

. )

(t

P_i : Probability that the system is in state

i

. Pi: The steady-state probability of state

i

.

) (s

P_i^ : Laplace transform ofP_i(t).

A: The steady-state availability of the system.

Apar: The steady-state availability of parallel system.

) (t

A_s : Availability of a one state degradable system.

A : The steady-state availability a one state degradable system. S

) (t

R : Reliability function of the system.

) (s

R^ : Laplace transform of R(t). Laplace Transform ofP_i(t) are defined as:

(2)

51 .

,....

2 , 1 , 0 , ) ( )

(

0

n i

dt t P e s

P_i^ 



^ ^^st _i  1.2. Model Description

We consider a system consists of a one unit with (n1)independent degradable rates and one failure rate_n. Let degradable rates and failure rate are constants denoted by_i, i1,2,...,n1and_nrespectively. This means that degradable rates and failure rate follow the exponential lifetime distribution .At time 0, the system is in working state .As soon as a first degradation occurs, and the system goes into the second working state. We assume that no more degradation rates will occur once the system is degraded and no failure rate only from state(n1) .During the time the system transit from degraded state to next degraded state till reach degraded state

) 1

(n .Let



_i,i1,2,...,n represent the repair rate of state

i

. Because of the memoryless property of the exponential distribution, a repaired system is as good as new. In this paper various states probabilities have been evaluated in the form of Laplace transform. The expressions of availability and reliability characteristics will be obtained in addition to we illustrate a particular case of this system in details.

Figure 1. System configuration diagram.

1.3. Mathematical Formulation of the Model

Based on the state transition diagram in Figure1, we can derive the following differential equations:

) 1 ( )

( )

) ( (

1 0 1

0









 ⁿ

i i

iP t

t dt P

t

dP  

 

⁽⁾ ⁽⁾ ^, ¹ ¹ ⁽²⁾

) (

1

1    





 _ P t P_ t i n

dt t dP

i i i i i

i   

) 3 ( )

( )

) ( (

1 t P t dt P

t dP

n n n n n

 



  

Initial conditions:



 

 otherwise

i where P_i

, 0

0

, ) 1 0 (

Taking the Laplace transform of equations(1)(3)and applying initial conditions, we have

 

( ) ( ) 1 (4)

1 0

1  









 n

i i

iP s

s P

s

 



s



_i



_i_₁



P_i^(s)



_iP_i_₁^(s)0 , 1in1 (5)



s



_n



P_n^⁽s⁾



_nP_n_1^⁽s⁾⁰ ⁽⁶⁾

Further simplification of these Laplace transforms is difficult or impossible without specifying the relationships among_i fori1,2,...,n and among_ifori1,2,...,n, we derive steady state probability for each state

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52

 

     

⁽⁷⁾

) ( lim

1 1

1

1 1

1

1 0

0

    







 





























 















 





 n

i i n

i

k k n

i j

j j n n

i

i i n

n

i

i i n

t P t

P













 

     

^,¹ ¹ ⁽⁸⁾

) ( lim

1 1

1

1 1

1









 















 





    







 

























 Pt i n

P n

i i n

i

k k n

i j

j j n n

i

i i n

n

i j

j j n t i

i













     

⁽⁹⁾

) ( lim

1 1

1

1 1

1



  







 

























 















 





 n

i i n

i

k k n

i j

j j n n

i

i i n

n

i i t n

n P t

P











1.4. Steady-State Availability Analysis of the System

From equations (7)(9)we find that the steady-state availability of one-unit with

( n  1 )

independent degradable rates and one failure rate is

     

⁽¹⁰⁾

1 1

1

1 1

1

1 1

1

1 1

1



  



  







 

















 



















 















 





 















 



 n

i i n

i

k k n

i j

j j n n

i

i i n

n

i

k k n

i j

j j n n

i

i i n

A

















Special case: In this case, we substitute



_i



and  _i  , we have

     

 

⁽¹¹⁾

1

1 1

1

1 1

1

1 1

n n

i

i i n n

i

i i n

N n

i

i i n n

n

i

i i n n

A





















































 



 















As we know the steady-state availability of the parallel system (n units) can be obtained from this equation

 

⁽¹²⁾

1 1

1 _n

n n n

n

Apar





















 



 





 

 



 





 



 Since,

     

(13)

1

1 n

n

i

i i n

n     



 



  



 

 

⁽¹⁴⁾

1

1 1

1

n n

i

i i n n

i

i i n

Apar

























 



 

From equation (11) and (14), we find one unit that has (n-1) independent degradable rates with one failure rate and equal without repair has the same steady-state availability on (n) units in parallel structure with a single repair facility.

1.5. Reliability Analysis of the System

To study the reliability function for this model, There must be at least state (n) is absorbing state and the transition rate from this state equal to zero. We study the reliability function when all degradable states are absorbing. In this case we substitute_i 0in equations(1)(3) and taking the Laplace transform, we have

 

⁰ ⁽¹⁵⁾

)

( ₁ n

s s

P i

i

i 



  _





(4)

53

 

⁽¹⁶⁾

) (

1

0



^ 1







 ⁿ 

i

i i

s s

R



An inverse Laplace transform results in

) 17

! ( )

(

1

0

t n

i i i

i e t t

R



^



 

 







This is the same of steady-state availability of (n) components that are i.i.d with constant failure rate in parallel structure without repair.

2. ILLUSTRATED EXAMPLE (ONE STATE DEGRADABLE SYSTEM)

To find closed expression of the availability function and reliability function of this system, we consider the system has two failure rates₁, ₂and two repair rates₂,₂. From equations (1)(3)and puttingn2, we drive the following differential equations

) 18 ( )

( ) ( ) ) (

(

2 2 1 1 0 1

0 P t P t P t

dt t

dP   

) 19 ( )

( ) ( ) ) (

(

0 1 1 1 1

1 P t P t

dt t

dP   

) 20 ( )

( ) ( ) ) (

(

1 2 2 2

2 P t P t

dt t

dP   

With initial condition P₀(0)1andP₁(0)P₂(0)0.Using the Laplace transform technique, the solutions ofP_i^(s), i=0, 1, 2 are given

  

 

⁽²¹⁾

) (

1 2 2 2 2 1 2 1 2 1 2 1 2

2 2 1

0

           









 



s s s s s s

s s s

P

 

⁽²²⁾

) (

1 2 2 2 2 1 2 1 2 1 2 1 2

2 1

1

           







 



s s s s s s s s P

 

⁽²³⁾

) (

1 2 2 2 2 1 2 1 2 1 2 1 2

2 1

2

           





 



s s s s s s s P

The steady-state probability becomes:

 

) 24 ( )

(

1 2 2 2 2 1 2 1

2 1 2 0

0

       







 



 P P

) 25 ( )

(

1 2 2 2 2 1 2 1

2 1 1

1

       







 



 P P

) 26 ( )

(

1 2 2 2 2 1 2 1

2 1 2

2

       





 



 P P

Thus, the steady-state availability of a two independent degradable rates is

 

) 27 (

1 2 2 2 2 1 2 1

1 2 1 2 1

0

       









 



P P A_S

Special case: In this case, we assume that failure rates are equal and repair rates are equal, so we substitute₁₂ and ₂₁ into equations (21)(23)we have

 

      

⁽²⁸⁾

)

0 (

   











 

 



 



s s

s s s s P

 

     

( )

  

⁽²⁹⁾

)

( ₂

2 2

1

   

























 



 

 



 



s s

s s s P

          

⁽³⁰⁾

)

( ₂

2 2

2

   





















 



 

 



 



s s

s s P

(5)

54 Inverse Laplace transforms of these equations yield

   

^ ^ ⁽³¹⁾

)

0(

e t

t

P ^ ^









 

 

 

   

^ ^

 

^ ^ ⁽³²⁾

) (

2 2

2 1

t

t t e

e t

P ^ ^ ^^

















   

 



 

 

   

^ ^

 

^ ^ ⁽³³⁾

) (

2 2

t

t t e

e t

P ^ ^ ^^















   



 



 

 

The availability function of the system can be written as

 

¹ ⁽³⁴⁾

) 2 ( ) ( )

( ₂

2 1

0 

 





 

 



 





 



 



 ^ ^





 











^^

e t t

P t P t A

t s

The steady –state availability can be obtained from this equation

 

⁽³⁵⁾

) 2 (

lim ₂

2









 

 A t

A _s

S t

3. CONCLUSIONS

In this paper, we studied a system of one unit with (n1)degraded state and one failed state. Results indicate that a one unit that has (n-1) independent degradable rates with one failure rate and equal without repair has the same steady-state availability of (n) units in parallel structure with a single repair facility. We also found the system without repair has the same steady-state availability of (n) components that are i.i.d with constant failure rate in parallel structure without repair.

4. REFERENCES

[1] Gupta, R., Tyagi, P.K. and Kishan, R. A two-unit system with correlated failures and repairs, and random appearance and disappearance of repairman. International Journal of Systems Sciences, 27(6), 561-566 (1996).

[2] Lam, Y. Calculating the rate of occurrence of failures for continuous time Markov chains with application to a two-component parallel system. Journal of Operational Research Society, 46, 528-536 (1995).

[3] Sim, S.H. and Endrehyi, J. A failure repair model with minimal and major maintenance. Journal of IEEE Trans. on Reliability, 55(1), 134-140 (2006).

[4] Wei, L., Attahiru, S.A. and Yiqiang, Q.Z. Stochastic analysis of a repairable system with three units and two repair facilities. Journal of Microelectronics Reliability, 38(4), 585-595 (1998).

[5] Agarwal, S.C., Mamta, S. and Shikha, B. Reliability characteristic of cold-standby redundant system.

International Journal of Research and Reviews in Applied Sciences, 3(2), 193-199 (2010).

[6] Sachin, K. and Anand, T. Evaluation of some reliability parameters of a three state repairable system with environmental failure. International Journal of Research and Reviews in Applied Sciences, 2(1), 96-103 (2009).

[7] El-Damcese, M.A. Human error and common-cause failure modeling of a two-unit multiple system. Journal of Theoretical and Applied Fracture Mechanics,26, 117-127 (1997).

[8] Chander, S. and Singh, M. Reliability modeling of 2-out-of-3 redundant system subject to degradation after repair. Journal of Reliability and Statistical Studies, 2(2), 91-104 (2009).

[9] El-Damcese, M.A., Shama,M.S. Reliability and availability analysis of a standby repairable system with degradation facility . International Journal of Research and Reviews in Applied Sciences, 16(3), (2013).

[10] El-Damcese, M.A., Shama,M.S. Reliability measures of a degradable system with standby switching failures and reboot delay. Applied Mathematics in Engineering, Management and Technology, 1(4), 1–16 (2013).