# Reliability theory for large linear systems with helping neighbors

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<ul><li><p>IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 3, 1992 SEPTEMBER 343 </p><p>Reliability Theory for Large Linear Systems with Helping Neighbors </p><p>Winfrid G. Schneeweiss, Senior Member IEEE FernUniversit&, Hagen </p><p>Key Words - Fault-tree, Helping neighbor, Consecutive k- out-of-n:F system, Shannon expansion, Boolean algebra </p><p>Reader Aids - Purpose: Widen state of the art Special math needed for explanations: Boolean algebra, </p><p>Special math needed to use results: None Results useful to: Reliability analysts </p><p>elementary probability </p><p>Abstract - This paper shows ways to model large systems, whose redundancy consists of the ability of neighbors to help (replace faulty units), at least for a degraded mode of operation. A general approach of determining and evaluating a fault-tree for such systems is given. Then the bulk of the paper is concerned with 1-dimensional (linear) arrays of components, which leads the way to linear consecutive quasi-3-out-of-n:F systems and to circular con- secutive 3-out-of-n:F systems. In all cases, explicit formulas - most of them recursive - are given for system unavailability and for mean system-failure frequency for non-identical s-independent com- ponents. As to methodology, the good adaptation of the Shannon decomposition to finding recursive results is amply demonstrated. </p><p>1. INTRODUCTION </p><p>With the ever increasing complexity of modern engineer- ing & social systems, the need for adequate and not overly com- plicated reliability analyses grows continually. Often in large multi-component systems, helpful redundancy for faulty com- ponents is confined to helping neighbors. In other words: Often a large system can tolerate many isolated faults but no lumped groups of faulty components. This idea is visible behind the con- cept of the consecutive k-out-of-n:F system, be they linear or circular [l-3,7]. This paper assumes that only direct neighbors can help. </p><p>After a brief description of a general approach, the bulk of the material is concerned with 1-dimensional (linear) arrays of components. In the circular case a special version of the theory of consecutive 3-out-of-n:F systems is presented, using a fault-tree approach, which appears to be new in this field. In the open-line case a modified consecutive 3-out-of-n:F systems theory is presented. This modification is necessary since at both ends of an open line, not only triples but pairs of failing components cause system failure. The fault-trees are correct also for the case of s-dependent components. But no examples of sdependance are discussed here. The formulas for availability ( A ) can easily be transformed to formulas for reliability; simply replace A by R ( t ) . </p><p>Many results are compared with corresponding results from the literature. Details of common reliability algebra are documented in the appendix. The reader who is not familiar with the Shannon decomposition should study appendix A.l first. </p><p>System availability is derived in the cited literature in various - partially ingenious - ways in closed form and via recursive formulas. The recursive fault-tree based approach presented here appears to be new. </p><p>Section 2 outlines the general approach of how to find and analyze fault trees for systems with helping neighbors, and gives examples that hint at the possibility of finding recursive for- mulas for fault-tree functions. The main sections 3 & 4 show the derivation of, and outline the analysis of a) recursive fault- trees for linear (open-ended), and b) circular arrays of components. Section 5 compares various existing results, and section 6 summarizes this whole investigation of recursive fault- tree functions. </p><p>Notation </p><p>a,b Boolean quantities, typically literals or terms A availability * B Boolean n-space, B = (0, l } ck component k; k E {l , ..., n}- U unavailability * : U= 1 -A. U= A X fault-tree indicator variable *: X = 1 for bad state, X=O </p><p>for good state. 9 Boolean function describing the fault-tree </p><p>h failure rate * : h = 1 /MTTF P repair rate *: p= 1/MTTR MTTF mean time-to-failure (more accurate name than MTBF) MTTR mean time-to-repair A, V Boolean conjunction (AND), disjunction (OR) </p><p>* Index Implies </p><p>i j S system C,i,j </p><p>L,iJ L,i,j </p><p>L ,iJ </p><p>U mean failure frequency * : v = 1 / (MTTF + MTTR) </p><p>- U Boolean negation (NOT a ) : ?i= 1 - a </p><p>k ck edge ij (between nodes i & j) </p><p>circular 3-out-of- (j - i + 1 ) :F system ranging from Ci to 5, inclusive; linear 3-out-of- (j- i + 1 ) :F system [L,ij]-type system that is also bad when only Ci & Ci-] (and not also Ci+2) are bad [L,iJ]-type system that is also bad when only & 5 (and not also qW2) are bad </p><p>Other, standard notation is given in Information for Readers & Authors at the rear of each issue. </p><p>0018-9529/92$03.00 01992 IEEE </p></li><li><p>344 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 3, 1992 SEPTEMBER </p><p>General Assumption Since unavailability is (for any index): </p><p>1. Components have 2-states and are mutually ~ Pr{X=ll = E{Xl, </p><p>s-independent . ( 5 ) 2. Links do not fail. (Assumed for ease of analysis) </p><p>one has at once, for s-independent X 1 , . . .&, system unavailability U, on replacing in the properly processed fault- </p><p>2. GENERAL APPROACH tree function X by U [6]. Hence, by (4), for few multiplica- tions [8], </p><p>In large multi-component systems helpful redundancy for faulty components is often confined to helping neighbors. The system fails if any node and all its neighbors are faulty. </p><p>U, = U, { u4 + A ~ [ u ~ ( U, + A ~ u ~ u ~ ) + A , u ~ u ~ ] ) </p><p>+ ASUiU2U6. (6) Example 1 . 1"WlarlY ~Onnected 6-cOmponent System. As is easily verified [8 ] , from U, (even when given in non- </p><p>polynomial form as in (6)), the mean frequency of system failure is: Figure 1 is the graph showing the components with their </p><p>connections (eg, communication lines or power lines). The associated part of a Boolean fault-tree function is, for ideal links and fallible nodes: vs = u5{u4(p5 + p4) + A4[Ul(U2(p5 - A4 + pl -k p2) </p><p>By Boolean absorption (see appendix A. 1) the last two terms are superfluous: </p><p>Figure 1. Graph of a 6-Component System [Shows the components as nodes and the various connections via edges. Neighbors are adjacent nodes (at graph-theoretical distance l)] </p><p>Aiming at reliability characteristics, primarily at unavailability, this type of function is readily analyzed via the Shannon expansion [5,6] (see appendix A. 1). For instance, ex- pand with respect to X,, then simplify/decompose: </p><p>If failures of links also have to be modeled, the fault-tree becomes more complex but nothing substantially changes. On- ly the indicator variable 4 of neighbor j of the node i under consideration is replaced by Xu%, where Xi is the indicator variable of link (edge) ij. Cases where links are bidirectional are easily modeled by setting Xi=Xji, </p><p>As is well known, from U, or A,, and v, one can easily determine - for the usually given s-coherent systems - MTTF & MTTR. In fact [4], for any appropriate index (see defini- tions on Notation list) - </p><p>MTTF = 1/X = A/v, MTTR 1/p = U/v. (8) </p><p>Details are not discussed in this paper, ie, any detailed analysis usually stops when v, is determined. </p><p>No doubt, for large n the above analysis of a fault-tree becomes more tedious. Hence one looks for simplifications for regular system topologies. In this paper 1-dimensional arrays of components are investigated with topology graphs as in figure 2. </p><p>Example 2. Linear Open-Ended 6-Component System </p><p>Since components C2 & C5 are the only neighbors of C1 & c6, respectively, a plausible form of the fault-tree function, showing the minimal cut-sets is: </p><p>= x5{x4 + %[xl(x2 + %&&) + xlx&I) By the Shannon expansion, and subsequent Boolean absorption: </p></li><li><p>SCHNEEWEISS: RELIABILITY THEORY FOR LARGE LINEAR SYSTEMS WITH HELPING NEIGHBORS 345 </p><p>1 2 3 4 * * n-3 n-2 </p><p>(single) helping direct neighbor. Then by figure 3 the correspon- ding Boolean fault-tree function 4 is: y - - - - - - y @-@- - - - - - -@ </p><p>a) b) XL",l,, , X1X2 V X2X& v...V Xfl-$,,-2X,,- 1 V X,,- lXw ( 16) </p><p>Figure 2. Figure 2. Linear Arrays of Components [a. Open Ended, b. Closed] The first & last terms have length 2, whereas all the others have </p><p>length 3. Hence this is not the "classical" consecutive 3-out- of-n:F system [2]. </p><p>+ %5(x1x2 x2x3x4 v x&J5) n-1 n </p><p>(10) Figure 3. Linear, Open-Ended Spatial-Arrangement of n Components </p><p>Using the definition for XL,,l,n this is shortened to yield the first typical recursive result: </p><p>&",1,6 = x6(xL',1,4 </p><p>To determine reliability characteristics such as system- availability, -reliability, -MTTF, and -MTTR, eq (16) has to be transformed. From the numerous ways of doing so, I choose the Shannon expansion (decomposition) [5,6]. An easy-to-handle </p><p>x5) + %&',1,5* (' ') </p><p>In Order to get rid Of operators, by avb=a+ab recursive relation for XLn,l,n is now derived. By (A-1) for i = n : for indicators a and b, the last OR (V) can be deleted: </p><p>+ X,,(X,X2 v ... v x,,-3xfl-2xfl- 1) . (16') For further processing one can insert XLt,l,4 from (A-13) and XL,,1,5 from (A-15). 0 </p><p>Example 3. Circular Closed 6-Component System </p><p>By the Boolean absorption law (A-2), the term X,,-3X,,-2X,,- 1 is absorbed by X,, - 1. Apply (A-1) again with i = n - 1 to the shortened first term of the r.h.s. of (16'): </p><p>A Boolean fault-tree description is: y = XlX2 v ... v X,,-Jfl-3Xfl-2 v x,,-1; the result is: </p><p>xC,1,6 = x1x2x3 v x2x& x3xJ5 x4x&6 </p><p>The Shannon approach yields: </p><p>xC,1,6 = x6(x2x&4 x4x5 x&l x1x2) + x 6 x L , 1 , 5 (for sufficiently large n, yet to be determined): Use the definition for XLr,ij; then a recursive form of (16') is </p><p>= %[X5(X4 + %Xl) + X;X2(Xl + XlX&4)1 X L " , l , n = X,,(Xfl-1 + % - l x L , , l , f l - 2 ) + ~ n X L ' , l , n - l . (17) </p><p>+ %xL,1,59 (14) By definition, and figure 3: </p><p>As usual, for s-independence, system unavailability is found on replacing X by U. </p><p>x~ , ,~ , , , = x,,(xlx2 v ... v X,,-~X,,-~X,,-~ v X,,.~X,,-~) </p><p>+ ~ & L ' , l , n - I . (19) 3. </p><p>3.1 Fault-Tree </p><p>ARRAYS OF By (A-2) the next to last term of the expression in parentheses is absorbed by the last one. Hence by (16), the recursive for- mula is: </p><p>A linear system consists of a linearly arranged set of com- ponents 1 - n; it fails whenever a faulty component has no X ~ , , i , n = X&~",i,n-i -%-%',i,n-i. (20) </p></li><li><p>346 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 3, 1992 SEPTEMBER </p><p>To get a recursion for XLt,l,n as a function of other XL,,l,i's on- ly, just replace XL",l,n-l in (20) according to (17) with n </p><p>multiplications) : </p><p>V L , ~ , ~ = UnU~",l,,,-l ( P n + P L " , l , n - l ) replaced by n - 1 . The result is (for a minimal number of + AnUL' , l ,n - l ( - A n + P L , , I , ~ - I ) . (25) </p><p>The minimal values of n, for (17) & (20) respectively to hold true, viz 4 & 3, are determined in appendix A.2. </p><p>3.2 Unavailability </p><p>For s-independent components, (17) immediately (on replacing X by U) yields for n 1 4 : </p><p>Likewise (20) yields for n 1 3 : </p><p>A closed linear system consists of a circular arranged (closed) set of n components; it fails whenever a faulty compo- nent has no helping direct neighbor. Then by figure 2b a cor- responding fault-tree can be described algebraically by: </p><p>XC,l,n = X,X,X, v...v xn-,xn- ,xn v xn- lXJ, </p><p>This is a fault-tree description of the classical circular 3-out- of-n:F system [l-3,7]. </p><p>As in section 4, the Shannon decomposition procedure is applied repeatedly. By (A-1) for i = n (26) transforms to: </p><p>The resulting algorithm (Alg#l) is conceptually slightly simpler than the one using (22). </p><p>Alg#l: In order to determine UL-,l,n: </p><p>XlX2X3 v . . . v X, -2Xn - v Xn - ,X1 V XIXz = X2X3X4 v ... v Xn-4Xn-3Xn-2 V & - & - I V Xn-lXl v XlX,. </p><p>1. Compute all the UL,,l,k up to and including UL, , l ,n- l . 2. Use UL'1,n-l and UL', l ,n-2 to compute UL", l ,n via Expanding this expression with respect to Xn-l yields: </p><p>(21). 0 XC,l,n = xn[xn-l (x2x3x4 v *.- v xn-5&-4&-3 </p><p>v xn-2 v X I ) + Xn-1XL',l,n-21 + X J L , l , n - l - Details are in section 6. Simplifications for equal components are obvious. </p><p>(28) </p><p>A continued expansion of the expression in the parentheses 3.3. Mean Failure-Frequency </p><p>Since systems which allow for fault-tree functions being disjunctive normal forms (disjunctions of conjunction terms) without negated variables are s-coherent [5], (21) together with XC,l,n = Xn(Xn-l[Xn-2 Z I - ~ ( X I X I X L , ~ , ~ - ~ ) ] (22) or (23) can be used to determine mean system-failure fre- </p><p>yield: </p><p>in the recursion: </p><p>quency v, via very simple rules [8]. Specifically, (21) & (22) + X n - J L r , 1 , n - 2 1 + X J L , l , n - I . (29) </p><p>For practical work with (29) we need a similar recursive result for XL,ij, which was not derived in section 3. By definition: </p><p>vL",l ,n = Un[Un-l(Pn + Pn-1) </p><p>+ A n - l U ~ ' , l , n - 2 ( ~ U n - An-1 + PL', l ,n-2)1 xL,iJ = X&+,Xi,, v ... v xj-,x/-1q (30) </p></li><li><p>SCHNEEWEISS: RELIABILITY THEORY FOR LARGE LINEAR SYSTEMS WITH HELPING NEIGHBORS 347 </p><p>X L , i j = Xj (X iX i+1Xi+2 V ... V 4 - 4 X j - 3 3 - 2 - </p><p>v + & I ) + 3 X L , i j - l = l g$ . . 1 (XJ j+1Xi+2 v ... v xj-+Jj-3 vxj-2) + X,_1XL, i j -2 ] + 3 X L j j - l . </p><p>Finally, after another application of (A-1), the recursion is: </p><p>- X,,jj = q X i - l ( x j - 2 + &XL,j j -3) + $-1XL,iJ-21 </p><p>+ 3 X L , j j - l . (31) Specifically, for (29), wherej=n, and i = 1 & 2, respectively: </p><p>~ L , l , n = X n [ X n - l ( X n - Z + %-2xL , l , n -3 ) + Z t - l x L , l , n - ~ I </p><p>+ % X L , l , n - l , (31) </p><p>xL,2,n = x ~ x n - l ( x n - 2 + Xn-2xL , z ,n -3 ) + x n - ~ L , ~ , n - 2 1 </p><p>+ ~ I J L , Z , n - l . (31 ) </p><p>In order to get a more homogeneous recursion than (29), ie, one without XL, , l , n -2 , expand X L j , l with respect to X1 & X2. By (18), using (A-1) again: </p><p>x L ! , l J = X l X 2 v X2X3X4 v ... v 3 - 2 4 - 1 4 = X , ( X , v x 3 x 4 x 5 v ... v Xj-*xJ-13, + XIXL ,* j = X l ( x 2 + X 2 x L , 3 j ) + X l X L , Z j . (32) </p><p>Inserting this ( for j=n-2) in (29) yields: </p><p>x c , l , n = X n { X n - l [ X n - 2 + % - 2 ( X 1 + XlxL ,2 ,n -3 )1 </p><p>+ % - l [ X l ( x 2 + x2xL,3,n-2) + xlXL,2,n-21} </p><p>+ X I J ~ , ~ , ~ - ~ , for n 1 7 . (33) Hence XC,l,n can be determined via recursion for XL,l,n, X L , ~ , ~ , XL,3,n, all 3 of them according to (31). Appendix A.2 shows that (33) holds true for n 17. </p><p>4.2. Unavailability </p><p>For s-independent Xs (29) yields immediately (on replac- ing X by U): </p><p>U ~ , l , n = u n { u n - l [ u n - 2 + A n - Z ( U 1 + A l U ~ , ~ , n - 3 ) 1 </p><p>+ An- lUL , l , n -2 ) + AnUL, l ,n - l - (34) </p><p>Alternatively (33) yields: </p><p>As an auxiliary result we find from (31): </p><p>(35) </p><p>4.3. Mean Failure-Frequency </p><p>From (35) there follows immediately [8]: </p><p>5. COMPARISON WITH EXISTING RESULTS </p><p>The following comparisons are typical rather than ex- haustive. They provide for a nice bunch of practically useful secondary results. Eq (36) & (35) are the same as [2, (la) & ( 5 ) ] , respectively; see appendix A.3. </p><p>the following recursion is derived in appendix A.3, which is equivalent to [3, Theorem] for k=3: </p><p>For A c , </p><p>J3q (A-36) corresponds to [ l ,...</p></li></ul>

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