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. r-i - 9 -
3. Semi-Markov processes
We define now the semi-Markov process (SMP) and its representation by graphs.
Semi-Markov processes have been considered as a nat'iral generalization of the homogeneous Markov process (MP) and have been thoroughly explored /l,lU/. In the following we confine ourselves to processes with finite state space. Then the SMP is a stochastic process with a finite number N of states j, 1< j< N< The sequence of states occupied by a semi-Markovian system is described by a sequence of random variables X (n = 0, 1, 2, ... ) with n values j. Thus X = j means that after the n-th change of state the n state of the system is j. The "holding time" spent in sta&e j after the n-th change is described by a random variable V (n = 0, 1, 2, ... ) with values in (O," ). The SMP is defined by /!/
x x = n-1 (t) = w \ i *<*) max { n : Es V < t } (19) ~\t)+ k 0 n if the condition
P(X V t P(X = t/X ) ( n+l " J> n « ' V •" V Vl' •'• V * n+l *' V n is fulfilled and independent of n. Because of equation (15) a transition probability
(t) = P(X t/X i (t)= G (t) 21 °ij n+l ' *' V n " > • «ij ij < > is defined for each direct transition from state i to ,j. We call g..(t) "the transition func+ion from i to j". Independency of n of the function G. .(t) implies that the system, after having entered a state,always evolves in accordance to the same law which is independent of the total number of states occupied earlier.
We define the following probabilities -10-
rn D. .(t) = P(X = j for exactly one integer m >1 with Z V,
We define the (conditional) state probabilities of the SMP by
Y..(t) * P(X(t) = j / X(0) = i ) (2^0
For arbitrary start conditions at t • 0 the system evolves, according to equation (2k), as
P(X(t) - j) = Z Y..(t) P(X(0) « i) (25) i 1J
An expression for the state probabilities, equation (2^), in terms of the d..(t), equation (23), can be found in the following way; ^ J The probability that the system stays for a time interval longer than t in state j is
1 - * G,.(t) (26) k JK Coupling this with the probability d..(t) dt that the system enters j at time t we have, for the start condition
P(X(0) - i) « 1 (27) - 11 -
Y..(t) = 6..(1- Z G., (t)) + /* d..(t-s)(l - Z G.Js)) ds IJ IJ k .lit o IJ k jk
= 6..(1- Z G.,(t)) + d. . « (1 - Z G.) (28) ij k jk ij k jk where 6.. is the Kronecker-delta and "»" denotes the convolution operation.
The Laplace transform of Y..(t) can be expressed in a closed analytic form by the transforms of d.. and g.. :
With the Laplace operator defined by
At Lg (X) = o/" e~ g(t) dt (29) we have
L(g*g') = Lg Lg' ; L 1 = VX i y* g(s) ds = Lg/X (30)
Application of equation (22) to the expression of equation (28), yields
1 " ZkLgik LY.. = (6.. + Ld .) T-* &• (31) U U iJ X
Connection to the graph formalism :
A representation of the SMP by directed and weighted graphs was introduced already by Howard and by Sittler /5»7/»
- the points of the graph are the states - the arrows are the (nontrivial) direct transitions - the weights are defined by
f(b..) = Lg..(A) (32)
The probability density for an indirect transition from state i to j with an intermediate state k is given by 111 : - 12 -
P(X X k W t+dt/X i} n*2 = J- n+1 " ' VW n ' =
dt / g.k(t-s) ^.(s) ds = (g.k. gy) dt (33)
Equation (33) holds since the time interval t spent for the i - j transition is the sum of the time intervals for the transitions i- k and k- j.
The Laplace transform of the r.h.s. of equation (33) reads
Lgik^cj " f(bikV {3k)
If the path from i to j involves m > 1 intermediate states the probability corresponding to equation (33) contains m convolution operations.
The probability of a general transition, equations (22) and (23), takes into account all possible paths from i to j. This is expressed by the following theorem 11/ :
Theorem 3 : Ld. . = I f(w) ; Ld. . = 1 f(w) 1J weW. . 1J weW. .
Using theorems 1 and 3 and equations (Ik) and (15) we may write
_ Ld.. Ld.. Ld.. = i*1— ; Ld. . = 3^3- (35) J 1 - Ld.. 1J 1 - Ld.. n n
To calculate Ld. . or Ld. . from the f(w), according to theorem 3, the ij ij elimination method of theorem 2 can be used for a successive replacement of
f(b.)f(b ) f(b. ) by *S SSL- (36) ** 1 - f(b ) nn for all n ^ i,j. Because of equation (31) we can finally calculate LY.. from Ld. .. ij - 13 -
From theorems 1 and 3 we obtain also the well known renewal equation for the state probabilities Y.. /1,2/ : We think all LY.., Ld.. and Lg.. ordered in the three matrices
v Lg.. ) = ( f(b..) ) , ( lid. . ) and ( LY. . ) (37) U U U ij According to equation (17) we may write
1 + ( Ld.. ) = 1 -! ( I_f(w) ) = (1 - ( Lg..))"1 (38) 1J w eW. . 1J Using equation (31) we obtain
-1 X " ZkLgik Theorem h : ( LY. . ) = (1 - ( Lg. . )) ( 6. . , lfc ) IJ lj lj X
From theorem U we derive an interesting relation of the weights f(b..)
Since Z Y..(t) = 1 and I LY..(X) • 1/X (39) we have for all i
1(6.. + l_ f(w) )( 1 - I f(b..) ) = 1 (1+0) j 1J weW. . k Jlt - Ik -
h. Topological formulas based on cycles
In the theory of electrical networks there have been developed methods which permit to calculate the solutions of linear systems of equations directly from an asscociated graph. We use the book of Chen /8/ as reference. It is seen from theorem k that the LY. . are the solutions of a linear system of equations with f(b..) = Lg.. as coefficients. In this and the next chapter we will apply the graph-theoretical methods in /8/ for the calculation of LY.. as J. J functions of f(b..). U
The set of paths is a special set of ways defined by
P.. = {w / w = b b. . , i, i , ... i all different} (kl) iJ n2 ifcJ 2 k For two paths
p = b b. . , q = b. . b. . (1+2) V2 1k-l1k J1J2 "• Jm-lJ- P/%q =
The elements of U P.. are called cycles. The set C has as elements the unions 1 11 of disjunct cycles :
C = tc « c + ... +c / eel/ P , C.AC. •
r(c) = n denotes the number of disjunct cycles c. of c. See chapter 7 for an example of the sets P.. and C.
The following formula of Mason is of fundamental importance /8,13/ :
Theorem 5 : det(l -• ( f(b..) )) - 1 + Z (-l)r*c)f(c) 1,1 ceC - 15 -
From this formula the topological formulas for SMPs are derived :
Theorem 6 : Z f(p) (l + Z (-l)^C'f(c) ) peP. . ceC, CAP = $ i f(w) = —-" ^ (uuj 1J ceC, c~j = $
I f(p) (1 + Z (-Dr(c)f(c) ) peP. . ceC, CAP = i| l_ f(w) = —^ ^ (1*5) WEW.. 1 + I (-l)rlC;f(c) 1J ceC
Proof : If the outgoing arrows from j are cancelled j becomes absorbing. But in this case W.. coincides with W. . and (kk) coincides with (1*5). Thus it is sufficient to prove (U5).
In (IT) we derived
( Z_ f(w) ) = (1 - ( f(b ) ))~1 - 1 (1.6) W£W.. J U It is well known that . A. . (1 - ( f(b..) ))_i = y (1.7) 1J det(l - ( f(b..) )) with A, . the cofactor s of 1 - ( f(b. .) )• Apart from a sign they are the x j i j determinants of 1 - ( f(b. .) )* in which the j-th row and the i-th column i« have been dropped.
From /8, Th. 3.8/ we have
A. . = Z f(p) (1 + I (-l)r(c)f(c) ) for i * j (U8) peP. . CEC, CAP = $ IJ * r(c) A.. - 1 + E (-l) f(c) (k9) ceC, c«i * ^
From (U6) and (1.8) the equation (1*5) is proved for i f j. For i • j we obtain - 16 -
A.. z_ f(w) = a _ x weW.. det(l - ( f(b..) )) 11 ij 1 + Z (-l)r(c)f(c) I (-l)r(c)f(c) ccC e i C£C C/Ni » ^ " • - 1 = - ' ^ * n (50) 1 + I (-l)rU,f(c) 1 + I (-l)rlC,f(c) ceC ceC This coincides again with (1*5). LJ
The topological formula for LY.. is ; ov easily derived from theorems 3 and 6 and equation (31). - 17 -
5. Topological formulas based on trees
In this chapter the connection between the SMP and the MP is studied more thoroughly. A topological expression of the state probabilities is derived, which is valid for both MP and SMP.
The MP are a subclass of the SMP and thus all the topological formulas of the the last chapter are also valid for MP. If the MP has the transition rates p.. the weight of the last chapter will be
fd,..) = L„..e-V = r^r . U. - I U-. (5D i J
The fractions of theorem 6 are expressed in terms of f(b..). For MP, f(b..) U U itself is now a fraction and thus the fractions of theorem 6 may be simplified. However this simplification procedure can be rather involved. We therefore use another topological formula which examines the trees of the graph. This formula will be given in theorem 8. First we show that there is a strong connection between the state proabilities of an MP and an SMP:
Theorem 7 : Suppose an MP with transition rates u.• and an SMP with transition functions g.. have the same graph of transitions. Let LY!.(A, p. , l Then L&, (A) ALY (A ij > Sfc.. l Proof : It is well known that for the matrices M " ( Wij > ' Mii " " "i (52) ( LY!.(A) ) . (A - M)"1 » A_1(l - A^M)"1 (53) - 18 - Thus A LY!.(A, u..) = LY! .(l, u. ./\) (51*) From the definition '29) of the Laplace transform it is easily seen that for ervery A'> 0 0 « Lg.-(A') « I Lg..(A«) < E Lg..(0)« 1 (55) IJ . lj - lj J J Thus we may define Lgi.(X') 0 < y-.(x') = X' u < - (56) 1J 1 " £ Lg,k(X') For fixed A'> 0 the u..(X') may be considered as transition rates of U an MP. If u--(X') is inserted in (51) the corresponding transfer function Lg!.(A) satisfies ij Lg!.(X') = Lg..(A') (57) Prom theorem k we see that for two different SMPs which fulfill (57) for a fixed A' > 0 one also has LYI.(X') = LY..(A') (58) The theorem is proved if (58) and (5*0 are combined. I—» Prom theorem 7 a second proof of theorem k is obtained if (56) is inserted in (53) : Lg-,(X) X LY..(A) = ( 1 - id )_1 (59) 1J 1 " Z Lg..(A) This is identical with theorem k. In the following we choose for the MT the weight f(b..) * v.. (60) - 19 - and for the SMP Lg. .(X) (61) f(h.,) l.J 1 - I Lg.. U) which corresponds to (60) according to (51) or (56). For the topological, formula of theorem 8 we need the directed trees of the graph G. A directed tree with "reference point" i (/8/) is a subgraph of G without cycles in which all points, except i, have exactly one outgoing arrow and i has no outgoing arrow. The set T. of directed trees is T. = {t / t is a directed tree of G with reference point i} (62) For every point j in G we define the following graph : G. arises from G by adding to every point i f* j an arrow b. from i to j with the weight f(b..) (63) An example of a set of trees is given in chapter 7. For an example of a graph G. see Fig. h. We further define J T. = {t / t is a directed tree of G. with reference point j} (6k) J j T.. = {t/ t eT. and in t is a way from i to j with arrows from G} (65) J* J with the convention T..=T.. t e T. - T.. means t £ T., t i T... JJ J J Ji J Ji Figure h : A graph G and its associated graph G. - 20 - The topological formula for the state probabilities based on trees is given by Theorem 8 : For every SMP with the weights (6l), (63) LY. . is given by t&T.. LX..U) = U 1J X I_ f(t) + I w Z_ _ f(t) teT. k J teT.-T., Proof : By theorem 7 it is sufficient to prove theorem 8 for MP because every SMP can be considered as an MP with variable u..(X) according to (6l) and (56). The proof for MP will be derived from a corresponding result in Chen /8/. By (53) we have to calculate A. . ( LY ) = (X - M)"1 = ( ±* ) (66) 1J det(X - M) where A.. are the cofactors of A - M. We take the (i,j)-element of the transposed matrix (X - M) to define the weight of the arrows b.. of a graph G*. G arises from G if all directions of the arrows are inverted and at every point i an arrow b.. is added. 11 The weights are f(b .) = - W , f(b.) = X (67) Theorem 3.12 in /8/ states det(x - M)T = I (-l)\l(r)qlr;f(r. ) (68) reR where R is the set of semifactors of G; R = {r / r is a subgraph of G which contains all points of G. r con tains no cycles exexcepc t the b.. and to every point is pointing exactly one arrow} A component of r is a maximal set of points r in which every two points - 21 - are connected by a sequence of arrows. q(r) is defined as the number of components with an even number of points (see Figure 5) • If we replace the weight (66) by f(b..) p.. , f(b..) = A (70) ji ii equation (68) changes to d<.l(X - M) E f(r) (71) reR For the transformation of the semifactors into trees t eT. we have to consider two cases : If r contains b.. we replace for i 4 j the arrows b.. by arrows b. JJ ii i from i to j with the same weight f(b.) = X . If all other arrows are inverted and b.. is dropped we obtain the graph G. and r is trans- formed to a tree t eT. (see Fig. 5). Every t eT. arises exactly once from J J an r ER and we have (72) Figure 5 : A semifactor and its associated tree t e T. If r does not contain b.. the point j lies in a component with a loop n 4 j. There is exactly one way w from n to j. Let now be k the nn' point with b . the last arrow of w. We drop b . and again we replace the b.. by the b. and invert all other arrows (see Fig. 6). By this process we obtain a tree t cT. - T. and every t e T. - T. arises exactly once from an r e R. We have - 22 - f(r) = p., f(t) (73) Thus I f(r) = A Z_ f(t) + Z v Z_ _ f(t) (7M reR teT. k ° teT.-T., Figure 6 : A semifactor and its associated tree teT. From equations (66), (71) and (jh) we obtain the denominator of theorem 8. For the nominator we start with theorem 3.13 in Chen /8/ : A.. = Z (-l)q(r,f(r) (75) 1J reR.. with the weight (67), R. . is defined by R.. = {r/rvb..cR and there is a way in r from i to j} (76) and q(r) again is the number of components of r with an even number of points. If we change the weight (67) to (70) we get A. . T, f(r) (77) reR. . U With similar arguments as for (71) the proof is completed by deriving Z f(r) l_ fit) (78) reR. teT ij ij - 23 - 6. Topological formulas for reliability determination In reliability considerations the strongest interest lies on several time independent quantities like the asymptotic values Y..(°°) of the state probabilities or the system lifetime. In the following we will apply the topological formulas developed in chapters h and 5 to the determination of these quantities. For the calculation of the asymptotic values Y..(«°) we assume that there U are no absorbing states and that I of g At) dt = Z Lg..(o) = 1 (79) j j Throughout this chapter we assume further v.. = /"t g..(t) dt < » , v. = I v.. (80) U o °ij I . IJ It is known /2,p,198/ that with these assumptions the asymptotic values of the state probabilities lim Y..(t) = Y.(-) (81) t-x» 1J J exist independent of the initial state i. Y.(°°) may also be calculated J with the Laplac? transform by lim Y .(X) = Y.(-) (82) In this chapter we use the weight f(bij} " o/Wgij(t) dt (83) which is the value for A * 0 of the weight (_U) of chapter 3 : f(b .) - Lg..(A) (8U) 1J i J - 2k - Because of (79) the sum over all columns of the matrix 1 - ( Lg-.(O) ) (85) vanishes. Therefore the determinant of this matrix is zero and thus, according to theorem 5» 1 + I (-l)r(c)f(c) = 0 (86) ceC From theorem 6 we derive the following topological formula for Y.(») : J r(c) Theorem 9 : v. I f(p) (1 + E (-1) f(c)) peP. . ceC, Cr»p = Proof : From the derivative of equation (29) with respect to X 1 - E. Lg..(X) v = E --^Lg..(X) = lim ^ ^ (£7) : dX 1J j X*; X and from (31) and (82) we conclude Y.(~) = lim X v.(6. . + LI. .) » v. lim X Ld .(X) (P3) J ^ J 10 U J x^0 ij By theorem 3 we replace Ld.. by the sum of f(w) over w e w. . and by I j . I j __ theorem 6 , equation (1*5), we obtain the topological formula for Ld. .. By taking the limit in (88) we see that the limit lim 7 (1 + £ (-l)r(c)f(c)) = a (89) X-K> A ceC exists in accordance to (86). Equation (88) yields v Y (») = ^ E f(p) (1 + E (-l)r(c)f(c)) (90) peP. . ceC, CAP = <{> From E Y.(») - 1 (91, j J the denominator is determined. This completes the proof. • - 25 - The l.h.s. of (90) is independent of i. Thus for all i the r.h.s. of (90) must be equal with the r.h.s. of (90) for i = j : E f(p) (1 + E (-l)r(c)f(c)) = - E (-l)r(c)f(c) (92) peP. . CEC, C/»p = $ ceC, c/\j # $ The l.h.s. of (92) is the nominator of the r.h.s. of (UU) while the r.h.s. of (92) is equal to the denominator of the r.h.s. of (UU) because of (86). Thus with theroem 3 we obtain Ld..(0) = /*d..(t) d4. = 1 (93) U o IJ whenever the expression (92) does not vanish. Equation (93) expresses the fact that every system at state i will arrive at state j after a certain time. Another topological formula for Y.(«) is derived with the help of theorem 8 J and the trees defined in equation (62) : Theorem 10 : v. E f(t) J teT. Y.(-) = J J E v. I f(t) j J teT J Proof : In theorem 8 we used the weight of equation (6l) : X Lg..(X) f(b..) » ^ (9k) 1J 1 - ELg..(A) k 1K By (87) f(b..) lim f(b ) - —^- (95) A-K) J i and thus, for every tree t e T., we obtain n v. lim f'(t) * v. f(t) (96) i X A-K> J It is easily seen that on the r.h.s. fo theorem 8 a factor A may be split - 26 - off in the denominator. By (71) and ilk) the denominator ist det(* - M). Let c be c » lira j det(A - M) (97) X-KI We insert theorem 8 in (82) to obtain Y.(») = -lim Z_ f»(t) (98) 3 C A-»o teT.. 01 n which together with (9b) and (91) yields tne desired result. LI There is a tight connection between theorems 9 and 10 : From theorem 5 and (89) we see a - lim 7 det(l - ( Lg..(A) )) (99) A-o 1J If we insert the definition of M from (52), (60) and (6l) we derive from (97) and (87) 1 A(5.. - Lg..) 6.. - Lg.. c - lim ± ( iJ U- ) - lim j det( -^ "• ) (100) X+O 1 - Z Lg., \+0 V. k ^ J Thus _ ,..,, a • c II v. (101) i This means that the denominators of the fractures in theorem 9 and 10 are equal and so do the nominators. With (92) this leads to Z (-l)r(c)f(c) = Z f(t) (102) ceC, CAJ j» 1^ teT. /9/ uses theorem 9 for the special case of the MP for which the weight (83) and v. from equation (80) assume the values 1 V- • f(b .) - /\i.,e~Vdt - -^ (103) v. - Z /" p.. t e" V dt • -7 (10U) 1 • 0 ij y. J 1 - 27 - As is immediately seen the weight f(b..) f(b..) = ^ = u. . (105) U vi IJ reduces theorem 10 to Z f(t) teT. Y.(») = J-x (106) J Z f(t) teT where T is the set of all trees of G. The same simplification will be possible in theorem 11 derived below. As a second application of the topological formulas of the last chapter we derive formulas for the system life-time. For this we make the following assumptions : The state i represents the entirely intakt system and the absorbing state j represents the irreparably failed system. Between this extreme states we may have partially failed states. We consider two cases : 1) all partially failed states are operating states 2) the partially failed states are considered as non- operating but repairable states The probability of the system to be in an operating state is E Y..(t) = 1 - Y..(t) in the first case (107) k i j 1J 1J Y..(t) in the second case (108) In the first case the life-time u.. is the mean time for a transition from i to the absorbing state j, while in the second case the life-time u. . i3 the time spent in the operating state i : u.. = /" (1 - Y..(t)) dt = L(1-Y..)(0) (109) IJ o IJ u u.. « ,/"*..(t) dt - LY..(0) (110) - 28 - For the topological formulas of these quantities we need the 2-trees with reference points i and j /8, p.2UU/ : T. . = {t / there are no outgoing arrows from i and j and exactly one form every other state, t contains no cycles} (ill) T., . = {t /t eT. . and there is a way in t from k to i} (112) ik.j i,j ^ Where we set T.. . = T. .. For an example of 2-trees see Fig. 7. ii.J i.J T. . = {b .b .b, ., b .b .b ., b .b, b . i,j ni mi ki m ki ny nj km mi b .b, b ., b .b, .b ., b .b .b, . ni km mi nj ki mi nj mi ki b .b, b ., b .b, b .} in ni km m. nj km nyJ T.,, . = ft .b. b ., b .b. b .} jk,i ni km mj nj km ny Figure 7 : A graph and its 2-trees T. . and T. . i »J Jk,i From theorem 8 we derive Theorem 11 £ v £ f(t) u. . £ f(t) teT. J v. £ f(t) 1 teT. u. . l .j U £ f(t) teT. J Proof : The state j is absorbing, i.e. y. « 0 for all k. Thus from Jk theorem 8, LI * 1/A and f from (9*0, f'(b.) * A , we obtain - 29 - Z_ _ f'(t) teT.-T.. f - LY. . • J—^ (113) X ij A Z_ f'(t) teT. Every t eT.-T.. contains at least one b.. Thus the X in the denominator of J Ji J- (113) is compensated by the f'(b ) = X in the nominator . In the limit It X -*• 0 only the f'(t) for which t contains exactly one b remains in the nominator. This arrow b lies in t on the way from i to j. If b is cancelled a 2-tree teT, . . arises. Every such t is built exactly once. ki,j Hence the formula for u.. is proved by using (95). By (71) and (lh) the denominator of the fraction of theorem 8 is independent of j. Thus we replace it by the denominator for Y.. used in (113) : E_ f»(t) teT. Y.. = 1 (iiU) 11 x z_ f (t) teT. J As u.. = 0 for all 2". the only outgoing arrow at j in a tree t with non- vanishing f'(t) is b.. Thus every non-vanishing term of the nominator of (llU) contains the factor f'(b.) = X which is cancelled by the X in the denominator. For X •* 0 we obtain for u.. the same denominator as for u.. _ iJ U and in the nominator of u.. all trees teT.. appear. _ Finally we give an argument why the topological formulas developed in electrical network theory are equally applicable to SMPs. For this we no longer assume that j is absorbing in general. Only for the determination of u.. the state j has to be made absorbing. - 30 - Theorem 12 : The graph of an MP with rates u.. = u.. for all i,j is considered as an electrical network with resistances 1/u... Then u is the total resistance between any points n and m. nm Proof : We chose an enumeration in which n = 1 is the first and m is the las. state of the MP. Let U. be the electrical potential from i to m, I the current flowing if a source is applied between 1 and m and R. = U./I. We want to prove R, (115) u.l m Kirchhoff's current law asserts that the sum of the currents vanishes in the nodes of the network : I V-. IU. - U.) for i i l,m (116) • ij i J O £ u. . (U. - U.) = ± I +1 for i=l, -I for i=m (117) . IJ i J J The equation for i = m is redundant because the sum of all equations vanishes. Let M' be the matrix M in which the last line and the last column are cancelled. Then (116) and (117) may be written as 'Rl * ll\ - M' (118) 1V1 7 and hence _1 - M» (119) m-1 For the calculation of u the state m is made absorbing. Let Y..(t) be the state probabilities of this new MP. Let ( LY.. )' be "the matrix i J ( LY.. ) in which the last line and the last column have been cancelled. U With (53) and the determination of matrix inverses with the help of cofactors one easily derives ( LY.. )' U - M»)- 1 (120) - 31 - Hence by (110) we obtain With the same arguments as in theorem 12 it may be proved : If for all points i an additional resitsance l/A is inserted between each i and m the total resistance between 1 and m becomes LY ,(A). Equation (121) is then obtained as the special case A = 0. - 32 - 7. An example As an illustration of the topological formulas we calculate for the SMP given in Fig. 8 the quantities LY . and Y^09) for i = 1, 2, 3, k and ulk and ulU- *d0 4 Figure 8 The set of cycles Z and the set C from equation (H3) are b b b } Z = { b22, b33, b12b21. \h\2*21, *23*3k\2> l3 3^U2 21 C Z { b +b b +b b b +b b } " ^ 33 22' 33 l2 21» 33 lU\2 21 The sets of paths from equation (Ul) are 11 { b12b21» V^l' b13b3U\2b21} P12 = {b12'blA2»b13b3^2} (b , b b , b^b^b^} P13S 13 12 23 Ik {bU*' bl3V b12b23b3U} We uso theorems 3 and 6 to calculate R. , Ld. . • f(w) (122) weW. U with R. . z f(p) (l + Z <-i)r(c)f(c)) (123) PEP ceC, c/>p • $ ij - 33 - 1 + T. (-l)r(c)f(c) (12U) ct.C w^'th the notation h. . = Lg.. = f(b..) (125) ij 1J iJ we obtain N = 1 - h22 - h33 - h21h12 - hlUhU2h21 - h23h3UhU2 - h13h3l;hU2h2i + h h + h h h + h h h h 22 33 33 12 21 33 lU U2 21 R = h h + h h h + h h h h h h h h h h h ll 12 21 lU l*2 21 13 3U Ul 21 " 33 12 21 " 33 lU U2 21 R12 = h12 + hlUhl»2 + h13h3UhU2 " h33h12 " h33hU*hU2 R13 = h13 + h12h23 + hlUhU2h23 ' h22h13 \k = \h + hl3h3U + h12h23h3U ' hlUh33 " hH*h22 ~ h22h13h3>* + hlUh22h33 Finally, by (31), we obtain LY.. N + R R LYn = —— (1 - h12 - h13 - hlU) , LY12 = — (1 - h21 - h22 - h23) AN AN LY = -^ (1 - h - hk) , LYU= -ilL(i-h ) 15 XN 3i " ±4 XN ^ and by theorem 9 for all i : v. R..(0) Y.(-) = J—iJ (126) J E v. R..(0) We could calculate Y.(«>) also with theorem 10 and the trees of equation (62) J Tl " *b3UV>21> T { b b b b b b 2 ' 13 3U l*2» 3l» U2 12> T3 * { blUbU2b23' bl»2b21b13» bl3bU2b23} TU " < b21blUb3U» b21b13b3^' b23b3^bH*» b12b23b3U» b13b23b3^} - 3k - Hence S. = E f(t) (127) 1 tcT. I 51 = h3H\2h21 52 = h13h3lA2 + h3Uhlt2h12 3 = hlUhl»2h23 + hl»2h21hl3 + h13hU2h23 Si* = h21hlUh3U + h21h13h3U + h23h3Uhll* + h12h23h3U + hI3h23h3U and by theorem 10 : V. S.(0) Y.(-) = J [1 (128) i Z v. S.(0) i J J We verified in (192) and (102) that for all i S.(0) = R..(0) (129) We see that R . (0) and S,(0) look quite different. But if we make use of the identities (79) which yield h12(0) + h13(0) + hlU(0) - 1 , h21(0) + h22(0) + h23(0) = 1 h23(0) + h3U(0) = 1 , hU2(0) = 1 we can eliminate h (0), h „(0), h,-(0) and h (0) from the expressions of ^(0) and S,(0). This gives us RlU(0) = SU(C) = h3U(0)(h21(0) + h23(0) - h21(0)h12(0)) * h3U(0)(l - h22(0) - h12(0)h21(0)) (130) which could have been obtained directly from R , (0). Finally for u . and u , we need the 2-trees (see equations (ill), (112)) = T {b b b V 11.U " 23 3^ 2lV T {b T {b b b b } 21.U" l2V • 31,U ' 21 13» 13 23 - 35 - Now theorem 11 yields the desired result if we insert in (127) for S,(0) n = n the last expression of (130) and if we vise h (0) • Po(°) 1 - P?(°) v.d-h^o)) U1U i - h22(o) - h12(o)h21(o) v1h3lt(0)(l - h22(Q)) + v2h12(0)h3U(0) + v3h13(0)(l - h22(0)) Ulk h3U(o)(i - h22(o) - h12(o)h21(o)) - 36 - References 1. V. Nollau : Semi-Markovsche Prozesse. WTB Band 260, Akademie Verlag, Berlin, 1980 2. K.W. Gaede : Zuverlassigkeit. Mathematische Modelle. Carl Hanser Verlag, Munchen, 1977 3. C. Singh and R. Billinton : System reliability, modelling and evaluation. Hutchinson of London, 1977 h. H. Hirschmann : The semi-Markov process. Generalizations and calculation rules for application in the analysis of systems. EIR-Bericht Nr. U90, Wurenlingen, 1983 5. R. W. Sittler : Systems analysis of dicrete Markov processes. IRE Trans, on Circuit Th. 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