STUDY OF THE STRUCTURE OF SYSTEMS: STRUCTURE FUNCTIONS
Theorem 17.11. Let A, be the number of links having k components and Bk the number of cuts having k components. Then
19 Reliability Networks
72 Ill S T R U C T U R E F U N C T I O N S A N D R E L I A B I L I T Y N E T W O R K S
(1) an r-fold graph G = (S, U) without loops and in which two vertices 0 E S and Z E S are distinguished and called, respectively, the origin and end, and
(2) a mapping A : U +w A e such that
W j ) = ( S , Sk) 3 Q(u,.) = (Si, S,)
*
& j ) # A h , . ) ,where 52 is the mapping that corresponds to each arc the pair of its ends (cf. Section 18).
The mapping A corresponds a component to each arc of the graph.
Several arcs may correspond to the same component,6 and it may happen that there is no arc corresponding to a given component. Figure 19.1 gives an example of a reliability network where
(19.1) (19.2) (19.3)
e = { e l 9 e2, e3, e4
I,
s
= f 0,z,
A , B,c > ,
u
= { (0, A 12, (0, A 13, (0, B ) , (4 B ) ,(4 Z ) , (B, C ) , (B, Z ) , ( C , B ) , ( Z , B )
I. .
The mapping A is indicated by the e , attached to the arcs.
To simplify the figures, if two symmetric arcs (Xi, X j ) and ( X i , Xi) concern the same component, it will be convenient to replace these two sym- metric arcs with a single arc with two opposing arrowheads (Fig. 19.2). To further simplify the presentation, the names of the vertices may be omitted when this will introduce no confusion.
C
FIG. 19.1. FIG. 19.2.
Links of a Reliability Network. To any subset e, c e of components one may correspond the partial graph G,(e,) of the graph G, obtained by
This is subject to the condition that they d o not have the same end points. Arcs having the same end points, like the two arcs from 0 to A in Fig. 19.1, will be marked with the index of the component to which it corresponds through the mapping d.
I 9 R E L I A B I L I T Y N E T W O R K S 73 retaining only the arcs of G to which correspond a component belonging to e, :
(19.4) G,(eJ = ( S , U,(e,))
with (19.5)
that there exists in the graph G,(a) a path from 0 to Z . U,(e,) = { u E U
I
d(u) E el.
A link of a network 32. will refer to a subset a c e of components such
Example. { e l , e 2 , e, } is a link of the network in Fig. 19.2, as may be seen in Fig. 19.3 which represents the corresponding partial graph. To each link there corresponds one or more paths of the graph. Inversely, to a path p = ( u , , u 2 ,
...
, u,) from 0 to Z there corresponds a link a formed by the images of the arcs u l , u 2 , ...,
uI through the mapping A.FIG. 19.3.
Example. To the path p = ((OA),, AB, BZ) there corresponds the link a = { e3 9 e4
1.
Cuts of a Reliability Network. A cut of a reliability network 32. will refer to a subset b c e of components such that the subset of arcs U,(b) defined by (19.5) contains a cut of the graph G relative to a subset of vertices including 0 but excluding Z.
Example.
the complementary subset { e l , e 2 , e3 } is a cut of the network (and at the same time a link); { e4 } is neither a cut nor a link.
To any cut b of a network there thus corresponds one or more cuts of the graph, included in U,(b). In the example above, the set of arcs { AZ, BZ } is a cut of the graph that does not include the arcs OB, (04,
,
and (OA),.
Note that the concepts of cut in a graph and of cut in a reliability net- work are not identical. It would no doubt be preferable to use two different
74 Ill S T R U C T U R E F U N C T I O N S A N D R E L I A B I L I T Y N E T W O R K S
terms as we have done in the case of paths and links,7 but we have not found a satisfactory equivalent for the descriptive term “ cut.”
System Defined by a Reliability Network. Given a reliability network 32, consider the system S having e for its set of components and such that for each state of the set of components in which the components of the subset el are in a good state and those of the complementary subset
<
have failed, the system functions if el is a link of the reliability network X, and it fails in the contrary case. This system S is defined without ambiguity from the network X. Its structure function* cp(x,, x 2 , ...,
x,) takes the value 1 for any state x = (xl, x 2 ,.. . ,
x,) such that the subset of components for which xi = 1 is a link of the network; this then is also a link of the structure function. If the subset of components is not a link of the network, cp takes the value 0, and the subset is not a link of the structure function. It is easy to see, similarly, that the network and the structure function have the same cuts.Example. Figure 19.4 represents the table of values of the structure function cp of the system defined by the network of Fig. 19.2. For the state (1, 1, 1, 0) of the set of components, one has cp = 1 since { e , , e,
,
e3 } is a link; on the other hand, cp(0, 0, I, 0) = 0 since { e3 } is not a link. One may equivalently reason in terms of cuts: q(0, 0, 0, 1) = 0 since { e l , e, , e3 } is a cut. (Note.Y
,
0 0 0 0 0 0 0 0 I 1 I I I I 1 I __
- -y 2
0 0 0 0 1 I I 1 0 0 0 0 I I I I -
- .\- 3
0 0 I 1 0 0 I 0 1 0 1 0 1 0 1 1 FIG. 19.4.
- s4
0 I 0 I 0 I 0 0 1 0 I 1 0 I 0 1 -
- ‘P
0 0 0 0 1 0 0 0 I 0 I I I I I 1 -
-
’
Hansel [28] uses the termpath tc designate what we have called a link; then to a link* See Section 15.
of the network corresponds a partial graph of the graph.
19 R E L I A B I L I T Y N E T W O R K S 55 that it is the subset of components for which x i = 0 that must be considered.) On the other hand, cp(0, 0, 1, 1) = 1 since { e l , e2 } is not a cut.
The identity between the links and cuts of a reliability network and those of the structure function defined by this network permits one to apply to reliability networks the notions defined in Sections 16 and 17. We shall now briefly review these notions using the language of reliability networks.
Minimal Cuts and Minimal Links. A link a is minimal if no subset a' c c a is a link of the network. A cut b is minimal if no subset b' c c b is a cut of the network.
Examples. In the network of Fig. 19.2, { e l , e 2 , e3 } is not a minimal link;
{ e l , e3 } is one and is also a minimal cut. { e l , e4 } is another example of a minimal cut.
Complementarity Relations. Recall properties (16.22) and (1 6.23) : (19.6) a is a link o zi is not a cut,
(1 9.7) b is a cut o 6 is not a link.
Degenerate Networks. A network is degenerate if:
(1) it possesses no link (the system never functions), or
(2) it possesses no cut (the system functions whatever the state of its components); the extremities 0 and Z are then identical.
Example (Fig. 19.5) The networks XI (Fig. 19.5a) and X2 (Fig. 19.5b) are degenerate.
B C
FIG. 19.5.
Fundamental Property of Reliability Networks. Adding one or more arci to a graph cannot suppress a path existing between the origin 0 and the end Z . The repair of a broken component thus could not entail the failure of
76 1 1 1 S T R U C T U R E F U N C T I O N S A N D R E L I A B I L I T Y N E T W O R K S
a system representable by a reliability network (this hypothesis is reasonable but not absolute for real systems).
This property may be written in the following form : (19.8) a‘ c a, (a’ is a link) (a is a link).
One has the same property for cuts:
(19.9) b’ c b, (b’ is a cut)
=-
(b is a cut).In effect, by adding arcs to a cut in a graph, one still obtains a cut. Properties (19.8) and (19.9) may be expressed in the following theorem.
Theorem 19.1. In a reliability network, a subset of components including a link is also a link; a subset of components including a cut is also a cut.
In more concrete terms, one may say that the systems defined by reli- ability networks are such that repair of a broken component cannot entail failure of the system, and that the failure of a component cannot entail the functioning of a failed system. We have already remarked at the end of Sec- tion 16 that properties (16.24) and (16.25) are not satisfied by all structure functions. We shall return to this point in Section 21.
Theorem 19.II. A cut contains at least one component from each link, and a link contains at least one component from each cut. In other words, any link and any cut have at least one component in common.
In fact, let a be a link and b be a cut; suppose they have no component in common, that is, a c 6. Then, according to the preceding theorem, 6 is a link, and according to the property of complementarity (19.6), b is not a cut, which is contradictory to the hypotheses.