J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
Reliability Analysis of k-out-of-n: G Redundant System in the Presence of Common Cause Shock Failures and with Imperfect Fault Coverage K. MALLIKARJUNUDU1, A. A. CHARI2, S. MAKBUL HUSSAIN3, M. RAVI KUMAR4, K. SUBHASHINI5 and V. RAGHAVENDRA PRASAD6 1
Sr. Assistant Professor of Mathematics, G. P. R. Engg. College, Kurnool, A.P., India. 2 Professor of OR & SQC, Rayalaseema University, Kurnool, A.P., India. 3 Dept. of Mathematics, Osmania Degree College, Kurnool. A.P., India. 4,5 Assoc. Professor of Mathematics, G. P. R. Engg. College, Kurnool, A.P., India. 6 Assistant Prof. of Mathematics, G. P. R. Engg. College, Kurnool, A.P., India. ABSTRACT This paper deals with the Reliability analysis of k-out-of-n: G redundant system in the presence of Common Cause Shock Failures (CCF) along with imperfect fault coverage. S. Akhtar (1994) has discussed the reliability analysis of k-out-of-n: G redundant system with Perfect and imperfect fault coverage and derived Reliability measures namely Rs (t), As (t), MTTF, MTBF. This paper attempts to extend the above model further with the system under the impact of CCF along with imperfect FC. The model derives the Reliability measures such as Rs (t), As (t), MTTF, MTBF. Also Numerical results are provided to highlight the impact of CCF in support of the model. Keywords: Reliability, Availability, MTTF, MTBF, Repairability, Imperfect Fault Coverage, Common Cause Shock Failures (CCF). Reader Aids: General Purpose: Solve some system models. Special math needed for derivations: Probability theory, Markov process, Laplace Transforms, Matrices. Special math needed to use results: Probability. Results useful to: System reliability analysis.
1. INTRODUCTION Redundancy is the key factor to improve the system reliability. However computer based systems intended for critical
application are vulnerable to uncovered failures1 and uncovered component failures cause immediate system failure even in the presence of adequate redundancy1,2. Also components in the system may not fail
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
754
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
independently of each other, but there are many factors such as common cause shock failures (CCF) will cause multiple component failures in the system3,4,5,6,9. Therefore an accurate analysis for complex system structure considering the CCF is quiet essential in real world applications. This paper considers both the aspect of uncovered fault and CCF and their impact on the k–out–of– n : G redundant system. This paper derive reliability measures of kout–of–n : G redundant system in the above conditions. The influence of these two sources of failures on the system are independent of each other and requires the attention for analysis of system to arrive at the accurate assessment of reliability measures of the system.
5. If the failed components are not repairable, i.e the fault is not covered and chance of fault coverage is Pc and fault uncovered is with probability (1-Pc) 6. Uncovered fault of components causes immediate system failure, in spite of adequate redundancy1. 7. There are single & multiple repair facilities. Single failure repair is attended in FIFO basis, if there are more than one failure. In case of multiple repair facilities each unit has it’s own repair facility and no waiting time. 8. Repaired unit is as new as the original. The repair rate is µ for a single repair facility and i.µ for a multiple repair facility for any state i .
2. MODEL FOR RELIABILITY AND AVAILABILITY
NOTATION: i
2.1 Assumptions and Notation 1. The system is k-out-of-n: G redundant configuration with s–Independent and identical components. 2. The components fail individually with a constant rate λ. 3. Also the components in the system are under the impact of non-lethal CCF which follow binomial failure model3, 4 i.e., the non-lethal CCF failures occur at a rate of β. Ps is the probability of failure of a component with the occurrence of non-lethal shock. So, probability of a specific set of k components fail simultaneously due to non-lethal CCF is βPsk 3,4. 4. The failed units are repairable. Also the fault is covered i.e repair is perfect.
λi µi β Ps ρ Pc 1-Pc Pi(t) Pi*(s) Ωs Ωf Ωc
: Number of operational unit (label for a state) : Intrinsic failure rate, i = 0,1,2………….n-k. : repair rate in state ‘i’, i = 1,2,…………..,n-k. : Constant rate of occurrence of non-lethal CCF. : Probability of a shock intensity upon occurrence of CCF. : λ / µ (Rho) : Probability of fault coverage Pr {system recovers / fault occurred} : Probability of fault uncovered : Probability of the system in state i at time t : Laplace transform of Pi (t) : Set of all successful states : Set of all failure states : Set of all states
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
P,R,A
Rs(t) Ass(t) Asm(t) MTBFs MTBFm MTTF
X
Di
: Steady state (t→ ∞) Probability of the state of system, reliability of & Availability of the system. : Reliability of the system. : Availability of the system with single repair : Availability of the system with unlimited (multiple) repair facility. : Mean time between failure of a system when there is single repair facility : Mean time between failure of a system when there is multiple repair facility. : Mean time to failure of a system. : Det. of coefficient matrix pertaining to Lap lace transformations of Markov equations. : Det. after ith column is replaced by initial vector of matrix X.
n -k
Ass(t )=
n -k i=0
(1)
Reliability of system = Pr {system is operational over a period of time} for µ i = 0
≠ 0 & µ i = i.µ
(4)
∞
∫R
(t)dt
(5)
(t)dt
(6)
MTBFm = A sm (t)dt
(7)
MTTF =
s
0 ∞
MTBFs =
∫A
ss
0 ∞
∫ 0
(ii)
i
i
(3)
When µ i = 0, Rs(t) = Ass(t) = Asm(t)
Ri(t) = exp (-λt)
i=0
i
0& µi=µ
Where Pi(t) is Probability of state i of a system at time t and is a solution of the Markov equations.
(i)
∑ P ( t) ,
∑ P ( t) for µ
Asm(t) =
System behaviour can be modeled with Markov approach as follows: The reliability of each unit
n -k
i≠
i
i=0
2.3 Fault – Coverage an Common Cause Shoch Failures:
2.2. Markov Model
Rs(t) =
∑ P ( t) , for µ
755
(2)
Availability of system = Prob {system is operational at a specified time}
The failures in the system are covered or uncovered. Each unit fault is covered with probability Pc, 0<Pc<1 The components in the system may fail synchronising (multiple) owing to occurrence of CCF failures2,3,4,5. The CCF failures are assumed to be non-lethal type and obey Binomial failure rate model2,3,4 occurrence of CCF lead to an absorbing failure rate i.e for k –out –of –n : G system this would be n-k+1. Therefore, model gives two types of states.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
756 •
0 < i≤ n-k (Ωs : System success states) • n-k <i ≤ n (Ωf : System failure states) & Ωc = Ωf ∪ Ωs : The non-lethal CCF occurs at a constant rate, say β (Occurrence of nonlethal CCS obeys exponential law). Ps is the probability of a component fails under the impact of non-lethal CCS and rate of k components fail simultaneously under the impact of non-lethal CCS is βPsk Under the above assumptions Markove model is discussed here to analyse the K- out – of n: G system if fault is covered (see fig-1). There is a transition to an absorbing state with probability 1-Pc which will hold in every state. Referring to the fig-1 , the Markovian equations of the present model are:
{
}
P0' (t) = − P0 (t) λ 0 + βPsn − k +1 + P1 (t)µ 1 (8)
{
Pi' (t) = − Pi −1 (t)Pc λ i −1 − Pi (t) λ i + µ i + βPsn +1− (k +i) + Pi +1µ i +1
}
(9)
(10)
+ Pn − k −1(t)Pc λ n − k −1
{
n − k −1
+ ∑ Pi (t) (1 − Pc )λ i + βPsn +1− (k + i) i =0
}
&P0 (t) + P1(t) + P2 (t) + − − − − +
(11) (12)
Pn − k +1(t) = 1
and Pi(0) =1, Pi(t)=0 for t ≠0. where λi = (n-i)λ , µi = i. µ for multiple repair facility and µ i = µ for single repair facility 2.4. Reliabilty Function – Transient Analysis: Using Laplace Cramer’s Rule,
Transforms
and
( )
n
∑ Di (s ) at s = r j . exp r j . t
Pi(t) =
j =1
for 0 ≤ i ≤ n
∏ (r j − rz ) n
(13)
z =1 z≠ j
The rj&rz,0<j,z ≤n are the roots of X = 0
P n − k +1 = 1 −
i = 1,2, − − − − − − −, n − k − 1 .
Pn' − k (t) = − Pn − k (t){µ n − k + λ n − k + βPs }
Pn' − k +1 (t) = Pn − k (t){λ n − k + βPs }
n−k
∑ P (t) i=0
i
when µ = 0, Equation (13) reduces to P 0 (t) = exp( − A 1 .t), where
A 1 = λ 0 + βP s3
i +1 i i +1 Pi (t) = (Pc λ)i π (n − j + 1) . ∑ exp(−A p .t) / ∏ (A p − A q ), where A r = λ r −1 + βPsn +1− (k + r −1) j=1 p =1 q =1
(14)
p≠ q
i =1,2…………n-k.& r = 1,2……………..,i+1. n −k
n −k
i =0
i =0
Pn − k +1 = 1 − ∑ Pi (t) and R s (t) = ∑ Pi (t) Average time to failure of the system is
MTTF
=
∞
∫R
s
(t)dt
0
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
(15)
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
757
And Mean time between failure of the system with single repair facility is
2.4.1. Availability Analysis For repairable system (single and unlimited)
P n − k +1 = 1 −
∞
n−k
∑
i= 0
MTBFs = ∫ A ss (t)dt
P i (t) and
(18)
0
n −k
A s (t) = ∑ Pi (t) , when µ ≠ 0
(16)
i=0
here Pi(t) are given in equation (13) Single repair facility: When a single repair facility is provided for the k-out-of-n: G system the Availability of the system is
Unlimited repair facility: If unlimited repair facility is allowed, i.e., µi = i . µ Availability of the system is n−k
As (t) = ∑ Pi (t), µ ≠ 0, whenµ i = iµ
(19)
i=0
n−k
and Mean time before Failure of the system is
i=0
MTBFm = ∫ A sm (t) dt
Ass (t) = ∑Pi (t), µ ≠ 0, whenµ i = µ . (17)
∞
(20)
0
Table 1: Reliability and Availability values of the system for λ=10-5 /10-5, µ=10-5 /104, Ps= 0.95, Pc= 0.95 (single repair facility). Time(X104)
Reliability
β=0
Availability
β =10
-5
β=0
β =10-5
0
1
1
1
1
1
0.98505
0.90356
0.98515
0.90415
2
0.96795
0.81349
0.96976
0.81747
3
0.94670
0.72828
0.95394
0.73911
4
0.92064
0.64777
0.93804
0.66826
5
0.88991
0.57233
0.92225
0.60420
6
0.85509
0.50241
0.90666
0.54628
7
0.81699
0.43834
0.89131
0.49391
8
0.77648
0.38029
0.87620
0.44656
9
0.73442
0.32823
0.86134
0.40375
10
0.69159
0.28198
0.84674
0.36505
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
758
Table2: MTTF and MTBF values of the system for µ=2x10-4, Ps=0.9,Pc=0.95 (Single repair facility). MTTF(X105) β=0 β =10-5 4.2771 1.1001 2.1385 0.89228 1.4257 0.74432 1.0693 0.63663 0.55548 0.85542 0.71285 0.49237 0.61101 0.44199 0.53464 0.40088 0.47523 0.36672 0.42771 0.33789
Rho 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Pcλ0
Pcλ1
0
Pcλ2
1
µ1
T0
Pcλn-k-2
2
µ2
T1
MTBF(X105) β=0 β =10-5 1.2619 16.778 8.2678 1.1663 5.3481 1.0792 3.8514 0.99824 2.9401 0.92254 2.3310 0.85193 0.78647 1.8994 1.5811 0.72622 1.3393 0.67111 1.1513 0.62097 Pcλn-k-1
n-k-1
. . . .
µ3
µn-k-1
T2
T n-k-1
n-k
µn-k
T n-k
n–k+1
T0 T1 T2
= = =
(1-Pc) λ0+βPsn-k+1 (1-Pc) λ1+βPsn-k (1-Pc) λ2+βPsn-k-1
T n-k-1 T n-k
= =
(1-Pc) λn-k-1+βPs2 λn-k+βPs
Fig.1: Markov Graph-K-Out-of-N-:G Redundant System In The Presence of Ccf with Imperfect Fault Coverage Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
2.5 Steady State Behaviour of the System Assuming that the system is observed for a longer period i.e.,as t → ∞ steady-state availability of the system is
n −k A (t) = lim s lim ∑ P (t) t→∞ t→∞ i=0 i and therefore
A ss (∞) = lim A ss (t) where µ i = µ , t →∞ A sm (∞ ) = lim A sm (t) where µ i = i . µ t →∞
and MTBF of the system are also significantly reduced even for small values of non lethal CCS failure rates of β = 10-5 / 104 time units (see Tables (1) and (2). Thus, the model ascertains that k-out-of-n: G system is greatly affected by occurrence of CCS along with imperfect FC. This model indicates that redundant configuration like kout-of-n: G system will be greatly influenced by the occurrence of non-lethal CCS and needs to think about the strategies to strengthen and to achieve the goal of reliability indices by suitable optimal sparing redundancy policy. REFERENCES 1.
3. EXAMPLE AND DISCUSSION Rs(t), As(t), MTTF, MTBF for single repair facility are computed for individual failure rate λ = 10-4/104 time units, various CCF failure rates β = (0,10-5/104 time units) & for Ps =0.95, Pc =0.95& µ=10-3/104, for a given example of 1 – out – of - 3 system of the model. The values are given in Table.1 and Table.2.
2.
3.
4. CONCLUSIONS The present paper discusses the reliability analysis of k-out-of-n : G redundant system in the presence of common cause shock failures (CCF) along with fault coverage. The result of this model indicate that occurrence of CCF failures along with FC will reduce reliability & availability function values significantly (i.e., almost 70% reduction) and even MTTF
759
4.
5.
Amari S.V, Dugan J.B and Misra R.B: Optimal Reliability of system’s subject toimperfect fault-coverage, IEEE Transactions on Reliability, Volume 48. No.3, Sept., (1990). Akhtar S: Reliability of K-out-of-n: G systems with imperfect-faultcoverage, IEEE Transactions on Reliability, Vol. 43 (1):PP. 101-106 March (1994). Atwood C.L: Estimations for the Binomial failure rate common cause Model US Nuclear regulatory commission report NUREG/CR-1401, EGG-EA-5112 Published: (1980). Atwood C.L,Suitt W.J: users guide to BFR-A computer code based on the Binomial failure Rate Common cause Model, US Nuclear Regulatory commission report, NUREG/CR2729, ECG-EA-5502. (1983). Billinton.R.B and Allan R.N: Reliability evaluation of Engineering
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
760
6.
7.
K. Mallikarjunudu, et al., J. Comp. & Math. Sci. Vol.2 (5), 753-760 (2011)
systems, concepts and techniques, Plenum press, New York (1983). Chari A.A: optimal redundancy of Kout-of-n: G system with two kinds of CCFs, Micro Electronics and Reliability Vol. 34 (6). PP, 1137-1139 (1994). Chari. A. A, Sastry M.P and Varma.: Reliability analysis in the presence of chance common-cause shock failures, Micro Electronics, Reliability Vol. (31.15) (1991).
8.
9.
10.
Dugan J.B: fault tree and imperfect coverage IEEE transactions on Reliability, Vol. 38, PP:775 â&#x20AC;&#x201C; 787 June, (1989). Gangloff. W.C: common-cause failure analysis IEEE Transactions on Power Apparatus and Systems, Vol.94; PP. 27 â&#x20AC;&#x201C; 30. Jan (1975). Kuo. W and Zuo M.J: Optimal Reliability modelling: Principles and applications, John Wiley Inc., (2003).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)