reliability of linear consecutively-connected systems with multistate components

518 IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 3, 1995 SEPTEMBER

Reliability of Linear Consecutively-Connected Systems with Multistate Components

Andreas Kossow Hochschule fiir Technik & Wirtschaft Wismar, Wismar

Wolfgang Preuss Hochschule fiir Technik & Wirtschaft Dresden, Dresden

Key Words - Consecutive-k-out-of-n:F system, Linear consecutivelyconnected system, Multistate components

Abstract - A linear consecutively-connected system with multistate components (LCCSMC) consists of n+2 linear ordered statistically independent multistate components Ci, i E [O,n], and the sink C,,, (which is absolutely reliable in a certain sense). System failure is caused by the Ci. If Ci is in state 0 then it is failed, if it is in the state j (1 s j l kj for a given kj) then there are paths from Ci to the next minu, n -i+l) components. The system fails iff there is no path from CO to C,,,. This system generalizes the linear consecutive-k-out-of-n:F system and the consecutively- connected system of Shanthikumar (1987). The paper gives recursive algorithms for determining the LCCSMC reliability.

Acronyms &

CWn:F LCCSMC

ShCCS

Notation

1. INTRODUCTION

Abbreviations

consecutive-k-out-of-n:F system linear consecutively-connected system with multistate components Shanthikumar consecutively-connected system [2].

1 index for component Ci, i E [O,n] unless otherwise stated C;, C,,, [component i , sink] n, k; parameters of the LCCSMC k {ki: i E [O,n]} j index for component state, j € [O,kJ p i j probability that Ci is in state j pj probability that a component is in state j (for pi = p j ) p i , qi [reliability, unreliability] of Ci for CWn:F and ShCCS

R (k,n) LCCSMC reliability with parameters n, k 9( .) indicator function: S(True) = 1, S(Fa1se) =O.

Other, standard notation is given in "Information for Readers & Authors" at the rear of each issue.

Assumptions for LCCSMC 1. The system has consecutively ordered components Ci,

i E [O,n], and the sink C,, 1, as sketched in figure 1. CO is the source.

(qi +Pi 3 1)

2. State Zi of Ci is a discrete r.v. with pmf

i Zi = j implies a path from Ci to each of Ci+l, Ci+*, ...,

c. . . r+mmO,n-i+ 1).

Zi = 0 implies the failure state of C;.

3. The Zi are s-independent. 4. Sink C,,, 1, as the target of paths, is absolutely reliable

5 . The system is failed iff there is no path from CO to 4

to receive paths.

C,,+,, ie, unreliability is caused by Ci (see figure 1).

Z , = j

(source) 1 . . (sink) +----. c, c, ... c, C,+l ... c,., c, a . . c,, ... C" C"+I

I J Y

inner system

Figure 1. LCCSMC

Example The components of the inner system are a set of relay sta-

tions; the source is a transmitter station; and the sink is an absolutely reliable receiver. Each of the n + 1 stations Ci consists of ki amplifiers.

If all the amplifiers of a Ci are operating, then a signal from Ci reaches the next min(ki, n - i + 1) stations. The failure of one amplifier reduces the range of C; by 1 station, ie, a signal from that station reaches the next k;- 1 stations, and so on. If all the amplifiers fail s-independently and have the same reliability p (O<p< l), then we obtain (if the amplifiers are the only unreliable elements in a station):

p i j = binmu; p , k ; ) 4

Generalization of ShCCS & Ck/n:F LCCSMC generalizes the ShCCS & CWn:F.

A. Let,

p i j = 0, for j = 1, ... , k i - l ,

p . . = p . for j = ki, 1J 1

Pi,o = 4i.

Then this restricted LCCSMC corresponds to ShCCS. Thus LCCSMC generalizes ShCCS.

0018-9529/95/$4.00 01995 IEEE

519 KOSSOW/PREUSS: RELIABILITY OF LINEAR CONSECUTIVELY-CONNECTED SYSTEMS

B. Let,

15k,=k,

po = 1 (the source is absolutely reliable).

Then this restricted LCCSMC fails if at least k consecutive (in-

We give recursive algorithms to evaluate the LCCSMC ner) components fail, and is thus a linear Ckln:F. for 3 s v I k. +c)

Corollary 2 shows a special case. reliability.

Corollary 2 . In corollary 1 let, 2. RECURSION FORMULAS

p i j = 0, f o r j = l , ..., k i - l ,

pij = pi for j = k;,

Theorem 1. Let 1 s ki = k. R ( k , n ) can be evaluated recur- sively as follows:

for I 5 n < k; (2) Then the LCCSMC corresponds to ShCCS (see section 1).

k

(3) R ( k , n ) = I , for 0 5 n e ko;

R(k,n) = a , ( n ) . R ( k , n - v ) , for n L k; (9)

Corollary 1 invokes the most interesting case. Corollary 3 . In corollary 2 let,

1 5 kj = k. Corollary 1. Let ki be the maximum range of C;, and let k = max(ki: 01i~n).

R(k-0 ) = 1 - po,o, (5 ) Then the LCCSMC corresponds to a linear Ckln:F with non- i.i.d. component states.

R ( k , n ) = 1, for 0 5 n < k;

n

R ( k , n ) = a , ( n ) . R ( k , n - v ) + % + I @ ) , (13) u = l

(6) k for 1 5 n e k,; min(k,n) ~ ( k , n ) = 1 - qr, for n = k ;

r = l R(k ,n ) = a , ( n ) . R ( k , n - v ) , forn L b; (7) u = l

520 IEEE TRANSACTIONS ON RELIABILITY. VOL. 44. NO. 3. 1995 SEPTEMBER

Corollary 4. Let the system be linear CWn:F with i.i.d. components @i = p).

R ( k , n ) = 1, for 0 5 n < k; (16)

R ( k , n ) = 1 - qk, for n=k; (17)

R ( k , n ) = R ( k , n - 1 ) - p.qk .R(k ,n-k- l ) , for n > k.

+( 18)

Remark: Eq (13) - (18) correspond to the recursive formulas in [l].

Corollary 5 . Let,

1 5 k j = k

P i j = Pj.

Then R ( k , n ) can be obtained from (1) with,

= po), (2) & (3)

The a , ( n ) are independent of n. + Theorem 1 and some corollaries are proved in the

appendix.

3. EXAMPLES

3.1 Example 1

An LCCSMC has CO, Cl, C2, C3, C4 (C4 is the sink) with system parameters:

n = 3 , k = 3 ;

Use corollary 1.

KOSSOWIPREUSS: RWJASILITY SF 1 .JNEAR CONSECUTIVELY-CONNECTED SYSTEMS

APPENDIX

A.l Proof of Theorem 1

R(k,O) = 1 - po,o is obvious, because the system con-

Let n 1 1. sists only of the source CO and the sink C1.

Notation

A

m min ( k , n ) .

Thus,

the LCCSMC is good (functioning) Ci is in any state j , 0 I r I j I s I k

Pr{A} = R ( k , n ) ,

m r m - 1 r 1

m - 1

.Pr{AI 0 A,'!!:)}. r=O

If n < k , ie, m = n, then

n-1 k

~

521

r=O j = n + l

because there is no path from the inner system to-the sink, ie, the complete system can operate only if CO is in any statej 2 n + l . Then the final expression in (A-1) is ~ , + ~ ( n ) and (2) holds.

If n L k , ie, m = k , then

k - 1

Pr{AI n A,'!!:)} = 0, r=O

because there is no path from Cn-k+l , . . . , C,, (consisting of k components) to the sink, and the maximum range of the sub- system CO, ..., Cn-k is k , ie, from here, no path goes to the sink too. Therefore (3) holds. Q. E. D.

A.2 Proof of Corollary 1

know, The proof follows directly from theorem 1. For it, we

~ , + ~ ( n ) = 0, if n 2 ko;

p i j = 0, if j > ki.


Corollary 2 comes from corollary 1. Eq (9) is obvious - it follows also from (5) & (6); and (1 1) can be obtained from (8). Q. E. D.

Q. E. D.


For a linear Ckln:F.

p i j = 0, f o r j = 1 , ... , k-1,

po = 1 ,

Then (13) follows from (9). The coefficients a,(n) are,

522 IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 3, 1995 SEPTEMBER

hold; on the other hand, we obtain, = 1 - qk + qk’(1-qk-1) + qk’

qn*ak(n- l ) = Pn-k‘qn’qn-1‘ ( 1 -qk-v’tl)

k

= 1 - n qr r = 1

Thus (14) holds. ie, (15) holds. Q. E. D.

Let n L k + l then, A S Proofs of Corollary 4 & 5

R(k,n) = R ( k , n - l ) - q n * R ( k , n - l )

k

+ a , ( n ) * R ( k , n - v ) u = 2

k

= R ( k , n - l ) - qn* a , ( n - l ) - R ( k , n - l - U ) u = l

Corollary 4 follows directly from corollary 3. Corollary 5 is a special case of theorem 1 . Q.E.D.

REFERENCES

[l] J.G. Shanthikumar, “A recursive algorithm to evaluate the reliability of a consecutive-k-out-of-n:F system”, IEEE Trans. Reliability, vol R-31, 1982 Dec, pp 442-443. J.G. Shanthikumar, “Reliability of systems with consecutive minimal cutsets”, IEEE Trans. Reliability, vol R-36, 1987 Dec, pp 546-550.

[2]

AUTHORS

k Dr. Andreas Kossow; Hochschule fiir Technik & Wirtschaft Wismar; Fachbereich Maschinenbau; Ph.-Muller-Str.; 23966 Wismar; Fed. Rep.

Andreas Kmsow (born 1954) is a Professor of Mathematics at Hochschule + tav(n) - q n . a , - l ( n - l ) ] . R ( k , n - U ) GERMANY.

v = 2

fiir Technik & Wirtschaft Wismar. He received the Dr. (1981) in Mathematics from the University of Rostock and the Dr. habil. (1988) in Engineering Science

On the one hand, for the coefficients under the sum, from the Technological University of Wismar. He deals with combinatorics, mathematical cybernetics, and the application of the mathematical reliability theory in engineering.

Dr. Wolfgang Preuss; Hochschule fiir Technik & Wirtschaft Dresden; Fachbereich Informatik Mathematik; Fr.-List-Platz 1; 01069 Dresden; Fed. Rep. GERMANY.

Wolfgang Preus~ (born 1944) is a Professor of mathematics at Hochschule f i r Technik & Wirtschaft Dresden. Before, he was a Professor at Technical University Wismar and at University of California Santa Barbara. He received the Dr. (1971) and Dr. habil. (1976) both in Mathematics, from the Faculty of Mathematics and Science of the Mining Academy, Freiberg. He deals with integral transformations, generalized functions, and reliability theory.

a2 ( n ) - 4 n ’a1 ( n - 1 = q n ‘Pn- 1 - q n * ( 1 - q n - 1 = 0

and for 3 I U I k,

au(n) - qn-av - l (n -1 ) = q n * [ g %-.I ‘Pn-u+l

U-3

Manuscript received 1994 February 2.

IEEE Log Number 94-12140 4 T R b

Unavailability Analysis of Periodically Tested Standby Components (Continued from page 5 17)

AUTHOR Techn. and Dr. of Science degrees in Nuclear Physics from the Helsinki Univ. of Technology. His research interests are in reliability & risk assessment,

100 papers on these subjects.

Manuscript received 1994 January 3.

Dr. J. K. Vaurio; W t r a n Voima Q; bvi i sa power Station; ~ O B O X 23; 07901 Loviisa, FINLAND.

J.K. Vaurio is a Risk Assessment Program Manager and Head of Training at Imatran Voima Oy. He was a Research Program Manager on Reliability, Risk Assessment, and Man-Machine Interaction projects for US DOE Fast Reac-

Stochastic processes, estimation, and information theory. He has published over

tor Safety Program. He holds Dipl.Eng (MASc) in Engineering Physics, Lic. IEEE Log Number 94-1 1825 4TRb

reliability of linear consecutively-connected systems with multistate components

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