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6 NETWORKED SYSTEMS RELIABILITY Reliability of Computer Systems and Networks: Fault Tolerance, Analysis, and Design Martin L. Shooman Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0 - 471 - 29342 - 3 (Hardback); 0 - 471 - 22460 -X (Electronic) 283 6 . 1 INTRODUCTION Many physical problems (e.g., computer networks, piping systems, and power grids) can be modeled by a network. In the context of this chapter, the word network means a physical problem that can be modeled as a mathematical graph composed of nodes and links (directed or undirected) where the branches have associated physical parameters such as flow per minute, bandwidth, or megawatts. In many such systems, the physical problem has sources and sinks or inputs and outputs, and the proper operation is based on connection between inputs and outputs. Systems such as computer or communication networks have many nodes representing the users or resources that desire to communicate and also have several links providing a number of interconnected pathways. These many interconnections make for high reliability and considerable complexity. Because many users are connected to such a network, a failure affects many people; thus the reliability goals must be set at a high level. This chapter focuses on computer networks. It begins by discussing the sev- eral techniques that allow one to analyze the reliability of a given network, after which the more difficult problem of optimum network design is introduced. The chapter concludes with a brief introduction to one of the most difficult cases to analyze—where links can be disabled because of two factors: (a) link congestion (a situation in which flow demand exceeds flow capacity and a link is blocked or an excessive queue builds up at a node), and (b) failures from broken links. A new approach to reliability in interconnected networks is called surviv- ability analysis [Jia and Wing, 2001 ]. The concept is based on the design of 284 NETWORKED SYSTEMS RELIABILITY a network so it is robust in the face of abnormal events—the system must survive and not crash. Recent research in this area is listed on Jeannette M. Wing’s Web site [Wing, 2001 ]. The mathematical techniques used in this chapter are properties of mathe- matical graphs, tie sets, and cut sets. A summary of the relevant concepts is given in Section B 2 . 7 , and there is a brief discussion of some aspects of graph theory in Section 5 . 3 . 5 ; other concepts will be developed in the body of the chapter. The reader should be familiar with these concepts before continuing with this chapter. For more details on graph theory, the reader is referred to Shooman [ 1983 , Appendix C]. There are of course other approaches to net- work reliability; for these, the reader is referred to the following references: Frank [ 1971 ], Van Slyke [ 1972 , 1975 ], and Colbourn [ 1987 , 1993 , 1995 ]. It should be mentioned that the cut-set and tie-set methods used in this chapter apply to reliability analyses in general and are employed throughout reliabil- ity engineering; they are essentially a theoretical generalization of the block diagram methods discussed in Section B 2 . Another major approach is the use of fault trees, introduced in Section B 5 and covered in detail in Dugan [ 1996 ]. In the development of network reliability and availability we will repeat for clarity some of the concepts that are developed in other chapters of this book, and we ask for the reader’s patience. 6 . 2 GRAPH MODELS We focus our analytical techniques on the reliability of a communication net- work, although such techniques also hold for other network models. Suppose that the network is composed of computers and communication links. We rep- resent the system by a mathematical graph composed of nodes representing the computers and edges representing the communications links. The terms used to describe graphs are not unique; oftentimes, notations used in the mathematical theory of graphs and those common in the application fields are interchange- able. Thus a mathematics textbook may talk of vertices and arcs; an electrical- engineering book, of nodes and branches; and a communications book, of sites and interconnections or links. In general, these terms are synonymous and used interchangeably. In the most general model, both the nodes and the links can fail, but here we will deal with a simplified model in which only the links can fail and the nodes are considered perfect. In some situations, communication can go only in one direction between a node pair; the link is represented by a directed edge (an arrowhead is added to the edge), and one or more directed edges in a graph result in a directed graph (digraph). If communication can occur in both direc- tions between two nodes, the edge is nondirected, and a graph without any directed nodes is an ordinary graph (i.e., nondirected, not a digraph). We will consider both directed and nondirected graphs. (Sometimes, it is useful to view DEFINITION OF NETWORK RELIABILITY 285 ab1 42 dc3 5 6 Figure 6 . 1 A four-node graph representing a computer or communication network. a nondirected graph as a special case of a directed graph in which each link is represented by two identical parallel links, with opposite link directions.) When we deal with nondirected graphs composed of E edges and N nodes, the notation G(N, E) will be used. A particular node will be denoted as n i and a particular edge denoted as e j . We can also identify an edge by naming the nodes that it connects; thus, if edge j is between nodes s and t, we may write e j  (n s , n t )  e(s, t). One also can say that edge j is incident on nodes s and t. As an example, consider the graph of Fig. 6 . 1 , where G(N  4 , E  6 ). The nodes n 1 , n 2 , n 3 , and n 4 are a, b, c, and d. Edge 1 is denoted by e 1  e(n 1 , n 2 )  (a, b), edge 2 by e 2  e(n 2 , n 3 )  (b, c), and so forth. The example of a network graph shown in Fig. 6 . 1 has four nodes (a, b, c, d) and six edges ( 1 , 2 , 3 , 4 , 5 , 6 ). The edges are undirected (directed edges have arrowheads to show the direction), and since in this particular example all possible edges between the four nodes are shown, it is called a complete graph. The total number of edges in a graph with n nodes is the number of combinations of n things taken two at a time  n! / [( 2 !)(n − 2 )!]. In the example of Fig. 6 . 1 , the total number of edges in 4 ! / [( 2 !)( 4 − 2 )!]  6 . In formulating the network model, we will assume that each link is either good or bad and that there are no intermediate states. Also, independence of link failures is assumed, and no repair or replacement of failed links is con- sidered. In general, the links have a high reliability, and because of all the multiple (redundant) paths, the network has a very high reliability. This large number of parallel paths makes for high complexity; the efficient calculation of network reliability is a major problem in the analysis, design, or synthesis of a computer communication network. 6 . 3 DEFINITION OF NETWORK RELIABILITY In general, the definition of reliability is the probability that the system oper- ates successfully for a given period of time under environmental conditions (see Appendix B). We assume that the systems being modeled operate con- tinuously and that the time in question is the clock time since the last failure 286 NETWORKED SYSTEMS RELIABILITY or restart of the system. The environmental conditions include not only tem- perature, atmosphere, and weather, but also system load or traffic. The term successful operation can have many interpretations. The two primary ones are related to how many of the n nodes can communicate with each other. We assume that as time increases, a number of the m links fail. If we focus on communication between a pair of nodes where s is the source node and t is the target node, then successful operation is defined as the presence of one or more operating paths between s and t. This is called the two-terminal problem, and the probability of successful communication between s and t is called two- terminal reliability. If successful operation is defined as all nodes being able to communicate, we have the all-terminal problem, for which it can be stated that node s must be able to communicate with all the other n − 1 nodes, since communication between any one node s and all others nodes, t 1 , t 2 , . . . , t n − 1 , is equivalent to communication between all nodes. The probability of success- ful communication between node s and nodes t 1 , t 2 , . . . , t n − 1 is called the all- terminal reliability. In more formal terms, we can state that the all-terminal reliability is the probability that node n i can communicate with node n j for all pairs n i n j (where i ϶ j ). We wish to show that this is equivalent to the proposition that node s can communicate with all other nodes t 1  n 2 , t 2  n 3 , . . . , t n − 1  n n . Choose any other node n x (where x ϶ 1 ). By assumption, n x can communicate with s because s can communicate with all nodes and communication is in both direc- tions. However, once n x reaches s, it can then reach all other nodes because s is connected to all nodes. Thus all-terminal connectivity for x  1 results in all-terminal connectivity for x ϶ 1 , and the proposition is proved. In general, reliability, R, is the probability of successful operation. In the case of networks, we are interested in all-terminal reliability, R all : R all  P(that all n nodes are connected) ( 6 . 1 ) or the two-terminal reliability: R st  P(that nodes s and t are connected) ( 6 . 2 ) Similarly, k-terminal reliability is the probability that a subset of k nodes 2 ≤ k ≤ n) are connected. Thus we must specify what type of reliability we are discussing when we begin a problem. We stated previously that repairs were not included in the analysis of net- work reliability. This is not strictly true; for simplicity, no repair was assumed. In actuality, when a node-switching computer or a telephone communications line goes down, each is promptly repaired. The metric used to describe a repairable system is availability, which is defined as the probabilty that at any instant of time t, the system is up and available. Remember that in the case of reliability, there were no failures in the interval 0 to t. The notation is A(t), and availability and reliability are related as follows by the union of events: DEFINITION OF NETWORK RELIABILITY 287 A(t)  P(no failure in interval 0 to t + 1 failure and 1 repair in interval 0 to t + 2 failures and 2 repairs in interval 0 to t + · · ·) ( 6 . 3 ) The events in Eq. ( 6 . 3 ) are all mutually exclusive; thus Eq. ( 6 . 3 ) can be expanded as a sum of probabilities: A(t)  P(no failure in interval 0 to t) + P( 1 failure and 1 repair in interval 0 to t) + P( 2 failures and 2 repairs in interval 0 to t)+·· · ( 6 . 4 ) Clearly, • The first term in Eq. ( 6 . 4 ) is the reliability, R(t) • A(t)  R(t)  1 at t  0 • For t > 0 , A(t) > R(t) • R(t)  0 as t  ∞ • It is shown in Appendix B that A(t)  A ss as t  ∞ and, as long as repair is present, A ss > 0 Availability is generally derived using Markov probability models (see Appendix B and Shooman [ 1990 ]). The result of availability derivations for a single element with various failure and repair probability distributions can become quite complex. In general, the derivations are simplified by assuming exponential probability distributions for the failure and repair times (equiv- alent to constant-failure rate, l, and constant-repair rate, m). Sometimes, the mean time to failure (MTTF) and the mean time to repair (MTTR) are used to describe the repair process and availability. In many cases, the terms mean time between failure (MTBF) and mean time between repair (MTBR) are used instead of MTTF and MTTR. For constant-failure and -repair rates, the mean times become MTBF  1 / l and MTBR  1 / m. The solution for A(t) has an exponentially decaying transient term and a constant steady-state term. After a few failure repair cycles, the transient term dies out and the availability can be represented by the simpler steady-state term. For the case of constant-failure and -repair rates for a single item, the steady-state availability is given by the equation that follows (see Appendix B). A ss  m / (l + m)  MTBF / (MTBF + MTBR) ( 6 . 5 ) Since the MTBF >> MTBR in any well-designed system, A ss is close to unity. Also, alternate definitions for MTTF and MTTR lead to slightly different but equivalent forms for Eq. ( 6 . 5 ) (see Kershenbaum [ 1993 ].) Another derivation of availability can be done in terms of system uptime, U(t), and system downtime, D(t), resulting in the following different formula for availability: 288 NETWORKED SYSTEMS RELIABILITY A ss  U(t) / [U(t) + D(t)] ( 6 . 6 ) The formulation given in Eq. ( 6 . 6 ) is more convenient than that of Eq. ( 6 . 5 ) if we wish to estimate A ss based on collected field data. In the case of a com- puter network, the availability computations can become quite complex if the repairs of the various elements are coupled, in which case a single repairman might be responsible for maintaining, say, two nodes and five lines. If sev- eral failures occur in a short period of time, a queue of failed items wait- ing for repairs might build up and the downtime is lengthened, and the term “repairman-coupled” is used. In the ideal case, if we assume that each element in the system has its own dedicated repairman, we can guarantee that the ele- ments are decoupled and that the steady-state availabilities can be substituted into probability expressions in the same way as reliabilities are. In a practi- cal case, we do not have individual repairmen, but if the repair rate is much larger than the failure rate of the several components for which the repairman supports, then approximate decoupling is a good assumption. Thus, in most network reliability analyses there will be no distinction made between reli- ability and availability; the two terms are used interchangeably in the network field in a loose sense. Thus a reliability analyst would make a combinatorial model of a network and insert reliability values for the components to calculate system reliability. Because decoupling holds, he or she would substitute com- ponent availabilities in the same model and calculate the system availability; however, a network analyst would perform the same availability computation and refer to it colloquially as “system reliability.” For a complete discussion of availability, see Shooman [ 1990 ]. 6 . 4 TWO-TERMINAL RELIABILITY The evaluation of network reliability is a difficult problem, but there are several approaches. For any practical problem of significant size, one must use a com- putational program. Thus all the techniques we discuss that use a “pencil-paper- and-calculator” analysis are preludes to understanding how to write algorithms and programs for network reliability computation. Also, it is always valuable to have an analytical solution of simpler problems for use to test reliability com- putation programs until the user becomes comfortable with such a program. Since two-terminal reliability is a bit simpler than all-terminal reliability, we will discuss it first and treat all-terminal reliability in the following section. 6 . 4 . 1 State-Space Enumeration Conceptually, the simplest means of evaluating the two-terminal reliability of a network is to enumerate all possible combinations where each of the e edges can be good or bad, resulting in 2 e combinations. Each of these combinations of good and bad edges can be treated as an event E i . These events are all mutually TWO-TERMINAL RELIABILITY 289 exclusive (disjoint), and the reliability expression is simply the probability of the union of the subset of these events that contain a path between s and t. R st  P(E 1 + E 2 + E 3 ·· ·) ( 6 . 7 ) Since each of these events is mutually exclusive, the probability of the union becomes the sum of the individual event probabilities. R st  P(E 1 ) + P(E 2 ) + P(E 3 ) + · · · ( 6 . 8 ) [Note that in Eq. ( 6 . 7 ) the symbol + stands for union ( U ), whereas in Eq. ( 6 . 8 ), the + represents addition. Also throughout this chapter, the intersection of x and y (x U y) is denoted by x . y, or just xy.] As an example, consider the graph of a complete four-node communication network that is shown in Fig. 6 . 1 . We are interested in the two-terminal reli- ability for node pair a and b; thus s  a and t  b. Since there are six edges, there are 2 6  64 events associated with this graph, all of which are presented in Table 6 . 1 . The following definitions are used in constructing Table 6 . 1 : E i  the event i j  the success of edge j j ′  the failure of edge j The term good means that there is at least one path from a to b for the given combination of good and failed edges. The term bad, on the other hand, means that there are no paths from a to b for the given combination of good and failed edges. The result—good or bad—is determined by inspection of the graph. Note that in constructing Table 6 . 1 , the following observations prove help- ful: Any combination where edge 1 is good represents a connection, and at least three edges must fail (edge 1 plus two others) for any event to be bad. Substitution of the good events from Table 6 . 1 into Eq. ( 6 . 8 ) yields the two-terminal reliability from a to b: R ab  [P(E 1 )] + [P(E 2 ) +·· ·+P(E 7 )] + [P(E 8 ) + P(E 9 ) + · · · + P(E 22 )] + [P(E 23 )+P(E 24 ) + · · · + P(E 34 ) + P(E 37 ) + · · · + P(E 42 )] + [P(E 43 )+P(E 44 ) + · · · + P(E 47 ) + P(E 50 ) + P(E 56 )] + [P(E 58 )] ( 6 . 9 ) The first bracket in Eq. ( 6 . 9 ) has one term where all the edges must be good, and if all edges are identical and independent, and they have a probability of success of p, then the probability of event E 1 is p 6 . Similarly, for the second bracket, there are six events of probability qp 5 where the probability of failure q  1 − p, etc. Substitution in Eq. ( 6 . 9 ) yields a polynomial in p and q: R ab  p 6 + 6 qp 5 + 15 q 2 p 4 + 18 q 3 p 3 + 7 q 4 p 2 + q 5 p ( 6 . 10 ) 290 NETWORKED SYSTEMS RELIABILITY TABLE 6 . 1 The Event-Space for the Graph of Fig. 6 . 1 (s  a, t  b) No failures: ΂ 6 0 ΃  6 ! 0 ! 6 !  1 E 1  123456 Good One failure: ΂ 6 1 ΃  6 ! 1 ! 5 !  6 E 2  1 ′ 23456 Good E 3  12 ′ 3456 Good E 4  123 ′ 456 Good E 5  1234 ′ 56 Good E 6  12345 ′ 6 Good E 7  123456 ′ Good Two failures: ΂ 6 2 ΃  6 ! 2 ! 4 !  15 E 8  1 ′ 2 ′ 3456 Good E 9  1 ′ 23 ′ 456 Good E 10  1 ′ 234 ′ 56 Good E 11  1 ′ 2345 ′ 6 Good E 12  1 ′ 23456 ′ Good E 13  12 ′ 3 ′ 456 Good E 14  12 ′ 34 ′ 56 Good E 15  12 ′ 345 ′ 6 Good E 16  12 ′ 3456 ′ Good E 17  123 ′ 4 ′ 56 Good E 18  123 ′ 45 ′ 6 Good E 19  123 ′ 456 ′ Good E 20  1234 ′ 5 ′ 6 Good E 21  1234 ′ 56 ′ Good E 22  12345 ′ 6 ′ Good Continued . . . Three failures: ΂ 6 3 ΃  6 ! 3 ! 3 !  20 E 23  1234 ′ 5 ′ 6 ′ Good E 24  123 ′ 45 ′ 6 ′ Good E 25  123 ′ 4 ′ 56 ′ Good E 26  123 ′ 4 ′ 5 ′ 6 Good E 27  12 ′ 345 ′ 6 ′ Good E 28  12 ′ 34 ′ 56 ′ Good E 29  12 ′ 34 ′ 5 ′ 6 Good E 30  12 ′ 3 ′ 456 ′ Good E 31  12 ′ 3 ′ 45 ′ 6 Good E 32  12 ′ 3 ′ 4 ′ 56 Good TWO-TERMINAL RELIABILITY 291 TABLE 6 . 1 (Continued) E 33  1 ′ 2345 ′ 6 ′ Good E 34  1 ′ 234 ′ 56 ′ Good E 35  1 ′ 234 ′ 5 ′ 6 ′ Bad E 36  1 ′ 2 ′ 3456 ′ Bad E 37  1 ′ 2 ′ 345 ′ 6 Good E 38  1 ′ 2 ′ 34 ′ 56 Good E 39  1 ′ 23 ′ 456 ′ Good E 40  1 ′ 23 ′ 45 ′ 6 Good E 41  1 ′ 23 ′ 4 ′ 56 Good E 42  1 ′ 2 ′ 3 ′ 456 Good Four failures: ΂ 6 4 ΃  6 ! 4 ! 2 !  15 E 43  123 ′ 4 ′ 5 ′ 6 ′ Good E 44  12 ′ 34 ′ 5 ′ 6 ′ Good E 45  12 ′ 3 ′ 45 ′ 6 ′ Good E 46  12 ′ 3 ′ 4 ′ 56 ′ Good E 47  12 ′ 3 ′ 4 ′ 5 ′ 6 Good E 48  1 ′ 234 ′ 5 ′ 6 ′ Bad E 49  1 ′ 23 ′ 45 ′ 6 ′ Bad E 50  1 ′ 23 ′ 4 ′ 56 ′ Good E 51  1 ′ 23 ′ 4 ′ 5 ′ 6 Bad E 52  1 ′ 2 ′ 345 ′ 6 ′ Bad E 53  1 ′ 2 ′ 34 ′ 56 ′ Bad E 54  1 ′ 2 ′ 34 ′ 5 ′ 6 Bad E 55  1 ′ 2 ′ 3 ′ 456 ′ Bad E 56  1 ′ 2 ′ 3 ′ 45 ′ 6 Good E 57  1 ′ 2 ′ 3 ′ 4 ′ 56 Bad Continued . . . Five failures: ΂ 6 5 ΃  6 ! 5 ! 1 !  6 E 58  12 ′ 3 ′ 4 ′ 5 ′ 6 ′ Good E 59  1 ′ 23 ′ 4 ′ 5 ′ 6 ′ Bad E 60  1 ′ 2 ′ 34 ′ 5 ′ 6 ′ Bad E 61  1 ′ 2 ′ 3 ′ 45 ′ 6 ′ Bad E 62  1 ′ 2 ′ 3 ′ 4 ′ 56 ′ Bad E 63  1 ′ 2 ′ 3 ′ 4 ′ 5 ′ 6 Bad Six failures: ΂ 6 6 ΃  6 ! 6 ! 0 !  1 E 64  1 ′ 2 ′ 3 ′ 4 ′ 5 ′ 6 ′ Bad Substitutions such as those in Eq. ( 6 . 10 ) are prone to algebraic mistakes; as a necessary (but not sufficient) check, we evaluate the polynomial for p  1 and q  0 , which should yield a reliability of unity. Similarly, evaluating the 292 NETWORKED SYSTEMS RELIABILITY polynomial for p  0 and q  1 should yield a reliability of 0 . (Any network has a reliability of unity regardless of its topology if all edges are perfect; it has a reliability of 0 if all its edges have failed.) Numerical evaluation of the polynomial for p  0 . 9 and q  0 . 1 yields R ab  0 . 9 6 + 6 ( 0 . 1 )( 0 . 9 ) 5 + 15 ( 0 . 1 ) 2 ( 0 . 9 ) 4 + 18 ( 0 . 1 ) 3 ( 0 . 9 ) 3 + 7 ( 0 . 1 ) 4 ( 0 . 9 ) 2 + ( 0 . 1 ) 5 ( 0 . 9 )( 6 . 11 a) R ab  0 . 5314 + 0 . 35427 + 0 . 0984 + 0 . 0131 + 5 . 67 × 10 − 4 + 9 × 10 − 6 ( 6 . 11 b) R ab  0 . 997848 ( 6 . 11 c) Usually, event-space-reliability calculations require much effort and time even though the procedure is clear. The number of events builds up exponentially as 2 e . For e  10 , we have 1 , 024 terms, and if we double the e, there are over a million terms. However, we seek easier methods. 6 . 4 . 2 Cut-Set and Tie-Set Methods One can reduce the amount of work in a network reliability analysis below the 2 e complexity required for the event-space method if one focuses on the min- imal cut sets and minimal tie sets of the graph (see Appendix B and Shooman [ 1990 , Section 3 . 6 . 5 ]). The tie sets are the groups of edges that form a path between s and t. The term minimal implies that no node or edge is traversed more than once, but another way of defining this is that minimal tie sets have no subsets of edges that are a tie set. If there are i tie sets between s and t, then the reliability expression is given by the expansion of R st  P(T 1 + T 2 + · · · + T i )( 6 . 12 ) Similarly, one can focus on the minimal cut sets of a graph. A cut set is a group of edges that break all paths between s and t when they are removed from the graph. If a cut set is minimal, no subset is also a cut set. The reliability expression in terms of the j cut sets is given by the expansion of R st  1 − P(C 1 + C 2 +·· ·+C j )( 6 . 13 ) We now apply the above theory to the example given in Fig. 6 . 1 . The min- imal cut sets and tie sets are found by inspection for s  a and t  b and are given in Table 6 . 2 . Since there are fewer cut sets, it is easier to use Eq. ( 6 . 13 ) rather than Eq. ( 6 . 12 ); however, there is no general rule for when j < i or vice versa. [...]... (6.47) involves 125 intersections followed by complex calculations involving expansion of the union of the resulting events (inclusion–exclusion); clearly, hand computations are starting to become intractable A similar set of equations can be written in terms of cut sets In this case, interrupting path ab, ad, or ac is sufficient to generate all-terminal cut sets Pall Pall 1 − P([no path ab] + [no path... (6.14c), we get a closer approximation at the expense of j computing [ j + ( 2 )] [ j( j − 1)/ 2] terms Rab ≤ 1 − [2q3 + 2q4 ] + [5q5 + q6 ] 0.9978 + 5 × 0.15 + 0.16 0.997851 (6.18) Equation (6.18) is not only an approximation but an upper bound In fact, as more terms are included in the inclusion–exclusion formula, we obtain a set of alternating bounds (see Shooman [1990, Section 3.6.5]) Note that Eq... involving more and more terms in the expansion of Eq (6.13) at each stage An equation similar to Eq (6.24) would be used for the last two terms in the computation to determine when to stop computing alternating bounds The process truncates when the error approximation yields an estimated value that has a smaller error bound that that of the required error We should take note that the complexity of... ad Term √ √ √ √ √ √ √ √ √ p6 √ — √ — √ — q 3 p3 q 3 p3 , q 4 p2 , q 3 p1 , q 6 qp5 q 2 p4 42 Rall-terminal Rall Α i 1 i ϶ 25, 31, 35, 36 P(E i ) (6.44) Note that events 25, 31, 35, and 36 represent the only failure events with three edge failures These four cases involve isolation of each of the four vertices All other failure events involve four or more failures Substituting the terms from Table 6.3... order of f or “big O of f.” For example, if f 5x 3 + 10x 2 + 12, the order of f would be the dominating term in f as x becomes large, which is 5x 3 Since the constant 5 is a multiplier independent of the size of x, it is ignored, so O(5x 3 + 10x 2 + 12) x 3 (see Rosen [1999, p 105]) In both cases, the dominating complexity is that of expansion for the inclusion–exclusion algorithm for Eqs (6.12) and (6.13),... vertex (for Rst where neither s nor t is b) This is the crux of the matter, since we must still include node b in the all-terminal computation Of course, eliminating node b does not invalidate the transmission between nodes a and c If we continue to use Eq (6.46) to define all-terminal reliability, the transformations given in Table 6.2 are correct; however, we must evaluate all the events in the brackets... is too large for the capacity of the line, if congestion ensues, and if a queue of waiting messages forms that causes unacceptable delays in transmission 2 If messages do not go through because of interrupted transmission paths or excessive delays, information is fed back to various nodes An algorithm, called the routing algorithm, is stored at one or more nodes, and alternate routes are generally invoked... a network with n nodes and e edges, an adjacency matrix is an n × n matrix where the rows and columns are labeled with the node numbers The entries are either a zero, indicating no connection between the nodes, or a one, indicating an arc between the nodes An adjacency matrix for the graph of Fig DESIGN APPROACHES 313 Nodes a c a′ b′ 0 1 1 0 0 a Nodes b c a′ b′ b 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1... 20 60 units The reliability of this network can be easily calculated as the product of the edge reliabilities: 0.94 × 0.96 × 0.94 × 0.94 × 0.93 0.71415 The resulting network is shown in Fig 6.11(a) Now, we repeat the design procedure by calculating a maximum-reliability 316 NETWORKED SYSTEMS RELIABILITY Prim’s Algorithm for Minimum Cost a TABLE 6.5 a See Cost 1–2 1–4 2–3 2–4 3–6 4–6 1–5 2–6 3–4 4–5 1–3... edge failures These four cases involve isolation of each of the four vertices All other failure events involve four or more failures Substituting the terms from Table 6.3 into Eq (6.44) yields Rall Rall p6 + 6qp5 + 15q2 p4 + 16q3 p3 0.96 + 6 × 0.1 × 0.95 + 15 × 0.12 × 0.94 + 16 × 0.13 × 0.99 0.531441 + 0.354294 + 0.098415 + 0.011664 (6.45a) 0.995814 (6.45b) Of course, the all-terminal reliability is lower . system by a mathematical graph composed of nodes representing the computers and edges representing the communications links. The terms used to describe. evaluating the two-terminal reliability of a network is to enumerate all possible combinations where each of the e edges can be good or bad, resulting in