MULTISTATE COHERENT SYSTEMS WITH MULTIPLE STATE

TRANSITION AT A TIME

V. M. Chacko

Department of Statistics, University of Calicut, Kerala -673635, India

chackovm@gmail.com •

M. Manoharan

Department of Statistics, University of Calicut, Kerala-673635, India

Abstract

A key requirement in defining a multistate coherent system (MCS) is the relevance condition of its components. A new class of MCSs is introduced with a new component relevance condition. Also we introduce a more general relevance condition. They are compared with some existing component relevance conditions. Based on the two new relevance conditions, two component importance measures for MCSs are defined. They are most appropriate for comparing components when certain type of system improvement is sought. We introduce new joint importance measures for two or more components with respect to the proposed relevance conditions. The new MCS classes include several existing MCSs as special case. An illustrative example of the proposed MCSs is also provided.

Keywords: Reliability, MCS, relevance condition, component importance, joint importance.

1. Introduction

Let us consider a coherent system with n components C=(1,2,...,n}. Furthermore, suppose that each component can be in one of M+1 states, {0,1,2,...,M}, where '0' is the failed state and 'M' is the maximal or "perfect" state. To describe such a multistate system (MSS), a general theory has been developed in the literature.5,9,11 A binary state system (BSS) of n components can be described by a structure function ^:{0,1}n ^ {0,1}, which presents the state of system as a function

of states of its n components.4 A binary system is statistically coherent if it satisfies the following conditions;4

(i)^(x)is non-decreasing in each argument, where x = (x1,x2,...,xn)

with x e {0,1}, and

(ii) for each i, there exist a vector (..,x), such that ^(1.,x) > 0(0i,x),

where (.,., x) = (x1, x2,..., *,_!,„ x+1,..., xn ).

Note that the condition (i) and (ii) gives, 0(j) = j, j = 0,1 where j = (j, j,..., j).

In practice, a system and its components often have more than two states of performance.5 The structure function of the MSS is 0:S" ^ S, S = {0,1,...,M}, which relates the level of performance of system to level of performance of each of its components. There are various approaches which extends the structure function from the binary case to the multistate case.5,9,10,11 The effort resulted in, extension of the requirement of non-decreasing binary structure function to MSS structure function. Also note the condition ^(j) = j, j = 0,1 of the binary coherent system (BCS) is extended to

the MSS requiring = j, j 6(0,1,..., M }.n The condition (ii) of relevancy in BCS is extended in various different ways. Some extensions can be seen in Refs. 1, 2 and 12.

In this paper, we extend the relevance condition to the MSS case in a general way, which includes several existing relevance conditions as special cases. Section 2 introduces the new class of multistate coherent system(MCS)s and its generalization by introducing a reasonable component relevance condition. The two new classes are compared with the some existing classes. Section 3 introduces two new component importance and joint importance measures for the proposed MCSs. Section 4 provides an example of an offshore electrical power generation system. Discussion and conclusion are given in section 5.

2. Component relevancy and the new classes of MCSs

In this section we discuss the new relevance condition and its generalization on which two new classes of MCSs are defined. Consider the following component relevance conditions.

NAT13: For every component i and level j>0, there exist (..,x) such that

<(j, x) > j and <((j - 1)i, x) < j.

GRI.111: For every component i and level j>0, there exist (..,x) such that <(j .,x) ><((j-1),,x).

GRI.211: For every component i, there exist (.,, x) such that <(0,, x) <<M ,, x).

and EP9: For every component i and level j> 1, there exist (..,x) such that

<(j,x) ><(0i,x).

NAT and GRI.1 indicate degree of relevance of each component to every level of performance; while GRI.2 merely states that < is not a constant in any of its arguments.

Now consider a situation in which some component is not relevant to every level of performances, i.e., the system degrades from state j to j-1 or j-2 etc when the component degrades only from state j to j-2 or j-3 etc. In order to degrade the system, component must degrade more than one level of performance. For example,14 let s = {0,1,2,3,4}, and the component can take 0, 2, and 4 when the system can take 0, 1, 2, 3 and 4. Consider the structure function <2 having 5 components in Ref.14. From the minimal path vectors of <2, we have, <2(41,42,23,44,25) = 4 ><2(41,42,23,24,25) = 3, when the 4th component degrades from state 4 to state 2, the system degrades from state 4 to state 3. Now consider the structure function < with three components in Ref.14. We have, <1(41,02,43) = 4 ><1(41,02,23) = 2, when the third component degrades

from state 4 to state 2, the system degrades from state 4 to state 2. Here fourth component must degrade from state 4 to state 2 for the system to degrade from state 4 to state 3 with respect to <2. The third component must degrade from state 4 to state 2 for the system to degrade from state 4 with respect to <.

We define a new component relevance condition as, degrading a component from state j to state j-2 can cause system failure or degradation while degradation of the component from state j to j-1 cannot cause system failure or degradation.

Now the new class of MCSs can be defined as follows.

Definition.1.: A multistate system of n components with structure function < belonging to class CM.1 if < is non-decreasing, <(j) = j, and for each component, there exist (.., x) such that

<( j,x) > ((j - 2)i,x).

Now consider the generalization of the new relevance condition, one or more than one level of degradation of the component can cause the system degradation, i.e., when the component 'i' degrades from state j to state j'£{j-l,j-2, j-3,...., 1, 0), the system degrades from state j to any lower state. Thus we define the generalized class of MCSs with this relevance condition.

Definition 2.: A multistate system of n components with structure function < belonging to class CM.2 if < is non-decreasing, </>(]) = j, and for each component, there exist (..,x) such that

<(j,X)>(]',.,x) where j'a{j-l,]-2]-3,....,l,0}.

In the following section we introduce the component importance and joint importance measures to the new classes of MCSs.

3. Component importance and Joint importance measures

We consider the problem of measuring the reliability importance and structural importance of individual components, and the joint reliability importance and joint structural importance of two or more components in the new classes of the MCSs. The main advantage of defining a new relevance condition is to obtain the importance measures.6 At the reliability design phase, the joint importance can improve system designer's understanding of the relationship between the components and the system, and among the components,3 which is quite desirable. Birnubaum measure provides the importance of a component in the BSS.6 It is further extended to the MSS.7,15 Now we consider x = (X1,X2,...,Xn) as a random vector with component states x t as random

variables and p = Pr{X. = j} where j e S = {0,1,..., M}. For the BSS with structure function <, the Birnubaum reliability importance6 of component i is

I,.(B) = P(<(1,.,x)-<(0„x) = 1) = h(1,.,p) -h(0i,p).

where h(p), p = (p1, p2,..., pn) and v. pt = (x,. = 1), is the reliability function of the BCS,

h(p) = ph(1„ p) + (1 - p,. )h(0,., p) = pI(B) + h(0,., p).

Therefore,

^ = I. (B).

dp.

We propose the following component importance measures for the two classes of new

MCSs.

1 I..(CM. 1) = P(<(] , x) > <((] - 2) x)).

2. I,(CM.2) = P(<(],x) ><(]',.,x)), j'e {j -1, j - 2.....1,0}.

Let the distribution of x .. be described by p. = (p.0, p(1,..., pM). The reliability function of the MCS with minimum satisfactory system level j, is p(<( x) > j) = ^ p<<( j., x) > j), since

jsS

p.0 + p n + p.2 +... + p m = 1. Now we prove the following theorems.

Theorem 1. For the CM.1 class, i.(CM .1) is the rate of improvement of P(<(x) > j) with respect to p j.

Proof. Clearly,

P(<(x) > j) = X pj [P(<(j,., x > j) - P(<((j - 2)., x) > j)] + P(<((j - 2) .., x) > j),

S\{ j-2)

since 1 -pj_2 = p.0 + pn +... + pj-3 + p..-1 +... + p M. Differentiating p(<(x) > j) partially with respect to p j , we get

I . (CM. 1) = ap(<(x) > j) = p(<(] , x) > j) - P(<((j - 2)., x) > j) = P(<((j - 2)., x) < <(i, x)).

Theorem 2. For the CM.2 class, i.(CM 2)is the rate of improvement of P(<(x) > j) with respect to p .

Proof. Clearly,

P(<(x) > j) = X p ij [P(<( j, x) > j) - P(<j\, x) > j)] + P(<Kj\, x) > j),

s\{ j}

where 1 - p.., = p.0 + p +... + pif x + pif+i +... + pM and differentiating the P(<(x) > j) partially with respect to pwe get,

I i (CM .2) = aP(<(x) > j) = P(<(ji , x) < <(j, x)), j'6 {j -1, j - 2,...,1,0}. dp,j

Now define the structural definition of the component importance (when reliabilities of components are not given) with respect to the new relevance conditions.

Consider <(x) = 1 if <(x) > j and 0 otherwise. We define the structural importance of a component as follows.

Definition.3.: Let <:Sn ^ S be the MCS structure function in CM.1 class. Then < is said to have the following measures of structural importance for the level j of component i:

Ij (CM .1) = 1 XMax{0,Kj,, x)-<((j - 2)i, x)}.

(M + i.) (x:x, = j}

Definition.4.: Let <:Sn ^ S be the MCS structure function in CM.2 class. Then < is said to have the following measures of structural importance for the level j of component i:

I j (CM .2) = 1 EM^R^O-,. , x) - j x)}, j'6 {j -1, j - 2,...,1,0}.

(M + 1) {x:x , = j}

In order to define the joint importance measures for two or more components in the new classes of MCSs, we recall the joint structural importance measure(JSIM)s8 and joint reliability importance measure(JRIM)s8 for the MSS with relevance condition in GRI.1. The JSIM (ij) for two components i and j with the new relevance conditions can be obtained by replacing M with m-2

or m'6{m -1,m -2,...,2,1,0}in,

MM „

JSIM(i,I) = YX(SIM(i,l;m,k) -SIM(i,l;m,€)}

m=1k =1

j ^ — EEx(<(mi, ki, x,i)=<(Mi, ki, x,i) = j - q)

where SIM(,,l;m,k)= q=1_.

(M + 1)n-2

Here x(true)=1 and x(false)=0, and <(m,.,kl,xa) = j,<(M.t,kl,xa) = j - q where x. = (x1,...,mt,...,kl,...,xn)

determines the critical path vector to the level j with state m of component i. The JSIM (i,j,k) for three components can be obtained as, for € = n - 2 or €6{n -1, n - 2,...,2,1,0},

M M M

JSIM(i,l,r) = XXX{JSIM(i,l,r;m,k,n) - JSIM(i,l,r;m,k,«)},

k=1 n=1 m=1

where JSIM(i,l, r; m,k,n) = JSIM(i,l,r;m,k,n) - JSIM(i, l, r; m,i,n).

Thus we can find JSIM of any number of components w. r. t. both relevance conditions in the MCS classes CM.1 and CM.2. Thus JSIM8 holds with new relevance conditions by replacing M with suitable m-2 or m'6{m -1,m - 2,...,2,1,0} .

Now we consider the JRIM for k components in the MSS.8 The JRIM of state ¿1of component a1, state b2 of component a2,..., state bk of the component ak (k < n) of the MSS is

Chacko V.M., Manoharan M. - MULTISTATE COHEREHT SYSTEMS WITH MULTIPLE STATE TRANSITION AT A TIME # ° P5

[Vol.lJ 2°12, June

d kE

JRIM(a, ,...,a,;b,,...,b,) =-—^-, k = 2,...,n,

' k 1 R b^ b2...9Ra,bk

where e P(0(x) >y)is the expected system performance and rabt is the reliability function with

j=0

respect to level bt of component ai.

Here m=m-1 in the expansion of es. 8 The results also holds true with the other values of M=m', m'<= {m-1,m-2,m-3,...,0}. Hence the JRIM with the new MCSs can be obtained with appropriate m' values for m, i.e., m'=m-2 or m'e{m-1,m-2,...,2,1,0}.

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Now consider some implications based on the new relevancy definitions. In fact one can easily prove the following implications.

Theorem 3. CM.1=>EP, CM.2=>NAT=>GRI.1=>GRI.2=>EP, CM.2=>GRI.1, CM.2=>GRI.2 and CM.2=>EP.

It is clear that all the existing relevance conditions are special cases of CM.2 (or CM.1) relevance condition. Hence the existing MCSs are special cases of the proposed MCSs.

4. Example

Ref.14 considered an offshore electrical power generation system, which supply two nearby oilrings with electrical power. Both oilrings have their own main generation, represented by equivalent generators a1 and a3 each having capacity of 50MW. In addition the oilrings has a

standby generator a2 that is switched into the network in case of outage of a, or a3 , or may be used in extreme load situations in either of the oilrings. The a2 also has capacity 50MW. The control unit, u , continuously supervises the supply from each of the generators with automatic control of the switches. If for instance the supply from A3 to oilring 2 is not sufficient, whereas the supply from a, to oilring 1 is sufficient, U can activate a2 to supply oilring 2 with electrical power through the subsea cables L . The components have states {0, 2, 4} and the system have states {0, 1, 2, 3, 4}, where 0, 1, 2, 3 and 4 represents the states of the system at capacities 0MW, 12.5MW, 25MW, 37.5MW, and 50MW respectively.

Table I .Minimal path vectors of ^

Levels U A a2

2 2 2 0

2 4 0 2

4 2 4 0

4 4 0 4

4 4 2 2

Table II .Minimal path vectors of

Levels U A L A A3

1 4 4 2 2 0

1,2 2 0 0 0 2

2 4 4 2 4 0

2 4 4 4 2 0

3 4 4 2 2 2

3,4 2 0 0 0 4

4 4 4 2 4 2

4 4 4 4 2 2

4 4 4 4 4 0

The minimal path vectors to the levels are given in table 1 and table II, of the structure functions,

^1(U,A1,A2) = I(U > 0)min(A1 + A2I(U = 4),4), the amount of power that can be supplied to platform 1, and

^2(U, A1,L,A2,A3) = I(U > 0) min(A3 + A21(U = 4)I(A1 = 4)L/4,4), the amount of power that can be supplied to platform 2, when I (.) is the indicator function. One may easily verify from the tables that the new relevance condition of CM.1 and CM.2 are found to be holding good w. r. t. the structure functions considered in the example. As done for JSIM and JRIM for two components11 and JSIM for three components,8 we can compute the concerned joint importance measures for any number of components in the power generation system.

5. Discussion and Conclusion

The theory of MSS reliability models has been developed to cope with many real-life situations. The present paper introduced two classes of MCSs with a new relevance condition and its generalization. It is shown that many MCSs introduced earlier in the literature are included in the new classes. Structural definitions of importance and joint structural importance measures are given, and new reliability importance and joint reliability importance measures are introduced. In system engineering, a practical and difficult problem is the identification of those groups of components that mostly influence the system behavior with respect to safety and reliability. In this respect, the main advantage of our importance and joint importance measure with respect to the new MCS models is the information provided by them for the reliability theoreticians and design analysts. It gives useful information for safe and efficient operation of the system, where existing importance measures gives information about individual component importance and joint importance of components with some limited number of relevance conditions.

Acknowledgments

Authors are thankful to the financial support from CSIR1 and UGC2, India.

References

1. A. M. Abouammoh, and Al-kadi, On measures of importance for components in multistate coherent system, Microel. Reliab., 31(2) (1991) 109-122.

2. A. M. Abouammoh, and Al-kadi, Multistate coherent system of order k, Microel. Reliab., 35(11)(1995)1415-1421.

3. M. J. Armstrong, Joint reliability importance of components, IEEE Trans. Reliab., 44(3) (1995) 408-412.

4. R. E. Barlow, and F. Proschan, Statistical theory of reliability and Life testing, (Holt, Rinheart & Winston, NewYork, 1975).

5. R. E. Barlow, and A. Wu, Coherent system with multistate components, Math. Oper. Res., 3 (1978) 275-281.

6. Z. W. Birnbaum, On the importance of different components in a multi-component system, in Multivariate analysis II, ed. P. R. Krishnaiah (NewYork, Academic Press, 1969), p. 591592.

7. V. C. Bueno, On the importance of components for multistate monotone systems, Stat. Prob. Lett, 7 (1989) 51-59.

8. V. M. Chacko and M. Manoharan, Joint importance measures for the multistate system, Advances in Performance and Safety of Complex systems, eds. A. K., Verma, P. K, Kapur, S. G., Ghadge, (Macmillan, New Delhi, 2008), p. 308-314.

9. E. El-Neweihi and F. Proschan, Degradable systems; a survey of multistate system theory, Commu. Statist. Theory and Methods, 13 (1984) 405-432.

10. E. El-Neweihi, F. Proschan, and J. Sethuraman, Multistate coherent systems, J. Appl. Prob, 15 (1978) 675-688.

11. W. S. Griffith, Multistate reliability models, J. Appl. Prob., 17(1980) 735-744.

12. F. C. Meng, Component relevancy and characterization results in multistate systems, IEEE Trans. Reliab., 42 (1993) 478-483.

13. B. Natvig, Two suggestions of how to define a multistate coherent system, Adv. Appl. Prob, 14 (1982) 434-455.

14. B. Natvig, S. Sormo, A. T. Holen, and G. Hogasen, Multistate reliability theory -a case study, Adv. Appl. Prob, 18(1986) 921- 932.

15. S. Wu, Joint importance of multistate systems, Comp. Indust. Eng., 49(2005) 63-75.