EIR-BerichtNr.522 Eidg. Institut fur Reaktorforschung Wurenlingen Schweiz Graph theoretical calculation of systems reliability with semi-Markov processes U.Widmer Wr WurenMngen, Juni 1984 EIR-BERICHT NR. 522 GRAPH THEORETICAL CALCULATION OF SYSTEMS RELIABILITY WITH SEMI-MARKOV PROCESSES U, WIDMER - 1 - Abstract The determination of the state probabilities and related quantities of a system characterized by an SMP (or a homogeneous MP) can be performed by means of graph-theoretical methods. The calculation procedures for semi-Markov processes based on signal flow graphs are reviewed. Some methods from electrotechnics are adapted in order to obtain a representation of the state probabilities by means of trees. From this some formulas are derived for the asymptotic state probabilities and for the mean life-time in reliability considerations. - 2 - Table of contents Page 1. Introduction 3 2. Graphs k 3. Semi-Markov processes 9 h. Topological formulas based on cycles Ik 5. Topological formulas based on trees 17 6. Topological formulas for reliability determination 23 7. An example 32 References 36 - 3 - 1. Introduction The evaluation of the reliability of technical systems is frequently based on the theory of Markov Processes (MP). More recently also semi- Markov processes (SMP) have been considered as a possible extension of the MP /l - h/. In several cases, they allow a more realistic description of the systems evolution in time. Important quantities characterizing the reliability of a system following an SMP are the state probabilities (as functions of time), their asymptotic values and the mean life-time. With each given SMP a directed graph is defined with points and arrows representing the states and the transition probabilities (densities) re­ spectively. We want to present several graph-theoretical methods which enable us to find the above quantities, as functionals of the transition probabilities, directly form an inspection of the graph using ways and subgraphs. First calculations on SMPs with the help of graphs were suggested in papers /5>6,7/. The actual topological methods, which yield the "topological formulas", have been developed however in electrotechnics where they have found a large field for applications. A complete des­ cription of the topological methods is given in the book of Chen /8/ to which we will frequently refer. We will translate the theory formulated in the language of electrotechnics to SMPs. We will also achieve, in this way, an extension and a deeper understanding of the results in /9/. In the chapters 2 and 3 we give brief surveys on the basic notions of graph theory and of the SMP as far as needed for the following discussions. A topological formula for the state probabilities based on cycles is introduced in chipter k. We then develop in chapter 5 another topological formula based or. trees. These methods are then transferred and applied to reliability theory in chapter 6. In chapter 7 an example illustrates our considerations. The author wishes to thank to Dr. H. Hirschmann for motivating this work and for his vaLuable help and advice. - u - 2. Graphs In this chapter we introduce some notations and properties of directed graphs, in which every arrow has been assigned a weight. Two elementary theorems on certain sets of ways between two points are formulated. A (finite) graph G consists of a finite number of points and a finite nuiber of arrows (directed edges) connecting the points. The points are assigned the numbers i = 1, 2, 3, ... and the arrows from point i to 1 2 point j are labelled b. ., b Note that there may exist more than J ^ ij» lj* Q one arrow from i to j. For i = j a special arrow b.. is defined for all i. A weight f of G assigns every arrow b. a complex number or a complex k ^ function f(b..). By definition we set f(b°.) - 1 for all i. IJ n k A multiset m of arrows b.. is a family of arrows in which an arrow may also appear more than once: k, k 1 2 m { b. , b. , ... } (1) Vl 12J2 The domain of definition of f is extended to every multiset m by k k f(m) = f(l).1. ) f(b.2. ) ... (2) Vl V2 In the following the multisets will be built up by the arrows contained in a way w or a subgraph s of G. We will write f(w) or f(s) for the expression (2) of the corresponding multiset m. In this paper we will only consider sums I f(«) (3) m eM - 5 - in which M contains with m all other multisets which result if the numbers k. > 0 of (l) are replaced by other possible integers k! > 0. 1 xl 2 A new "reduced" graph G' arises form G if the family of arrows b.,., b..}, is replaced by a simple arrow b. .. For i ^ j G' contains at most one i J arrow from i to j and for i » j there remains , in addition, the arrow o , . b.. (see Fig . 1)x . If we define k k f(b..) = f(b.^) + f(b.?) + .... CO ij IJ IJ a straightforward calculation shows E f(m) I f(m«} (5) meM m'cM' for certain multisets m' of the reduced graph G*. As we are only interested in sums like (5) we will always assume that G is reduced. Figure 1 : A graph G and its reduced graph G* For the SMP we have to consider ways on the graph G. A way is a sequence of the kind w • b. b. (6) 1 1 V2 V3 n--i1 n ^ is called the start and i the end of w and w is called "a way from i1 to in". If w is a way from i to k and v is a way from k to j - 6 - w and v can be combined to a way wv from i to j. The sets W of ways considered here will always consist of ways with a common start and a common end, which are called the start and the end of W. For sets of ways W and V with the same start and the same end we define the sum as the union W + V = {w, v / w c W, v E V } (7) If the end of W coincides with the start of V we define WV = {wv / w e W, v e V } (8) If the start and the end of W coincide we define /10,ll/ W* = {b?.} + V + WW + . (9) n Every arrow b.. defines also a set of ways {b. .} , which by a slight abuse of notation will be denoted again by b... The basic sets of ways from i to j are W. = lw = b b. / i, ^ j for k = 2, ... n} (10) _ en W. = {w • b b. .} (11) W.. contains all ways from i to j and W,. contains the ways which meet j only once. We need the following matrices with sets of ways as elements : B - ( b.. ) , V * ( W. ) (12) With this definitions the following theorem is valid /12/ : Theorem 1 ; W. • W. W«. , W « B + B2 + U IJ JJ If we apply f to Theores 1 we obtain - 7 - Z_ f(w) = Z f(w) (1 - Z f(w)) X (13) weW. weW. weW.. ij U JJ because from (2), (8) and (9) we derive Z f(w) = Z f(w) Z f(w) (lU) WEW. .W., weW. veW., IJ jk IJ jk Z f(w) - (1 - I f(w) f1 (15) WEW*. weW.. n n The application of f to the second formula of Theorem 1 yields Z f(w) = f(b.. • Zb.A. * Z VW^.j •... ) weW.. K K»K = f(b. .) + Z f(b.v) f(b, .) + ... (.16) ij -^ 1K *J Thus one obtains the equation of matrices ( Z f(w) ) = ( f(b..) ) + ( f(b..) )2 + ... weW ij = ( f(b..) )(1- ( f(b..) )) 1 (17) Another method to achieve a representation of W.. by the arrows of G is the method of elimination; Here the points of the graph are successively eliminated by using the following algorithm /13/ : - The elimination of a point n is performed by cancelling n and replacing the arrows b.. by the sets of ways b.. + b. b* b . for i.j # n. IJ IJ in nn nj If several points are eliminated the arrows are replaced by more and more complex sets of ways. If all points, except one, are eliminated a single set of ways remains. This is used in - 8 - Theorem 2 : W.. is obtained if all points, except i, are eliminated. b..W. is obtained if all arrows b.. , k f i, are dropped Ji ij Jk and all points, except j, are eliminated. Proof : According to the definitions (7), (8) and (9) we have W. = b. + b. b« b . for the graph of Fig. 2. IJ IJ in nn nj Figure 2 Figure 3 Some reflection shows that after the elimination of the points k e K the arrows b.., i,j i. K, are replaced by the set of ways i J w = b b. , i . ... i e K (18) ii- I j ' 2' n 2 n 3o W.. is obtained if all points k # i are eliminated. After elimination of all points k # i,j the graph of Fig.3 remains, with B.. (B..) the set of ways froir. i to j (i) not meeting i and j. From Fig.3 one derives W.. « B*.B.. and thus b..W.. = b..B*.B... This expression is also the ij ii ij ji ij Ji n ij result of the additional elimination of i; thus the second statement of Theorem 2 is proved.