Using minimal cuts to evaluate the system reliability of a stochastic-flow network with failures at nodes and arcs

Reliability Engineering and System Safety 75 (2002) 41±46

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Using minimal cuts to evaluate the system reliability of a stochastic-¯ow network with failures at nodes and arcs Yi-Kuei Lin Department of Information Management, Van Nung Institute of Technology, Chung-Li, Tao-Yuan, Taiwan 320, Republic of China Received 19 June 2001; accepted 17 August 2001

Abstract This paper deals with a stochastic-¯ow network in which each node and arc has a designated capacity, which will have different lower levels due to various partial and complete failures. We try to evaluate the system reliability that the maximum ¯ow of the network is not less than a demand (d 1 1). A simple algorithm in terms of minimal cuts is ®rst proposed to generate all upper boundary points for d, and then the system reliability can be calculated in terms of such points. The upper boundary point for d is a maximal vector, which represents the capacity of each component (arc or node), such that the maximum ¯ow of the network is d. A computer example is shown to illustrate the solution procedure. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Node failure; Minimal cut; Upper boundary point for d; Stochastic-¯ow network; System reliability

1. Introduction Reliability evaluation of a network has attracted many researchers' attention in recent decades. In a binary-state network (with no ¯ow in it), the network and its arcs are being in normal or failed states [1,3,10,18]. The system unreliability can be evaluated in terms of minimal cuts (MCs) since MCs are directly related to the modes of system failure. The system reliability (i.e. 1 Ð system unreliability) can then be obtained. If we remove some arcs from the network and thus the source s disconnects the sink t, then the set of such arcs is called a cut. A MC is a cut, which will not be a cut after removing any arc from it. Aggarwal et al. [2] extended the binary-state network to the case of node failure. They proposed a concept that the failure of a node implies the failure of arcs incident from it. Then the original network with node failure can be modi®ed to a conventional network with perfect nodes. Fig. 1 simply shows such a concept. Let Paj and Pni be the probabilities that aj and ni are successful in the original network, respectively. Note that such a concept is only adaptable to a binary-state network. Several authors [4,14,15] proposed algorithms to calculate the system reliability for a binary-state ¯ow network by assuming perfect nodes and that each arc has a designated capacity which will have zero level only due to any failure. E-mail address: [email protected] (Y.-K. Lin).

Rueger [19] extended the network to the case that nodes as well as arcs all have a positive-integer capacity and may fail. A branching tree is built in which the tree-nodes are the disjoint terms of a symbolic reliability expression. In a stochastic-¯ow network, each arc has several possible capacities and may fail. Several authors proposed methods to calculate the system reliability of such a ¯ow network by assuming perfect nodes [6,7,9,13,16,17,20,21]. The system reliability, the probability that the maximum ¯ow of the network is not less than (d 1 1), can be calculated in terms of upper boundary points for d (i.e. d-MCs [13,16,21] or maximum lower vectors for level d [20]). Each upper boundary point for d is denoted by a vector which represents the capacity level of each arc such that the maximum ¯ow of the network under such level is d. It is in fact a maximal vector, which means the maximum ¯ow will be larger than d after any improvement in capacity level of an arc. Extending the stochastic-¯ow network to the general case that nodes as well as arcs all have several capacities and may fail, Lin [17] used MPs to evaluate the system reliability. This paper mainly uses the MCs to solve the same problem again. A simple algorithm is proposed to generate all upper boundary points for d and then the system reliability can be computed in terms of such points. The proposed solution procedure can be applied to evaluate the system reliability, one performance index, for realworld systems such as computer systems, telecommunication systems, transportation systems, electric-power

0951-8320/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0951-832 0(01)00110-7

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Y.-K. Lin / Reliability Engineering and System Safety 75 (2002) 41±46

Fig. 1. The concept of Aggarwal et al. [2] for a binary-state network with node failure: (a) original network (node failure), (b) modi®ed network (perfect node).

transmission systems and logistics systems, etc. An example is shown in Section 5 to illustrate the solution procedure. The storage and computational complexities are analyzed in Section 6.

in {XuV…X† ˆ d}: Hence Pr{XuV…X† # d} ˆ Pr{XuX # Xi for an upper boundary point Xi for d} 8 9 < = [ ˆ Pr {XuX # Xi } : : all upper boundary points X for d ;

2. Assumptions Let G ˆ …A; N; M† be a stochastic-¯ow network where A ˆ {ai u1 # i # n} is the set of arcs, N ˆ {ai un 1 1 # i # n 1 p} the set of nodes, and M ˆ …M1 ; M2 ; ¼; Mn 1 p† with Mi the maximal capacity of ai for i ˆ 1; 2; ¼; n 1 p: A cut is a set of nodes and arcs, which disconnects the source s and the sink t. A minimal cut is a cut, and it is not a cut if any component (arc or node) is removed away. Furthermore, G is assumed to satisfy the following assumptions: 1. The capacity of each ai is an integer-valued random variable which takes integer values [ {0,1,2,¼,Mi} according to a given distribution. 2. The capacities of different components are statistically independent. 3. The ¯ow in G must satisfy the ¯ow-conservation law [11]. 3. A stochastic-¯ow network with node failure 3.1. De®nition of upper boundary points for d Let X ˆ …x1 ; x2 ; ¼; xn1p †; where xi denotes the (current) capacity of ai, be a capacity vector, and V(X) be the maximum ¯ow of the network (from s to t) under X. Suppose there are m MCs: K1, KP2,¼,Km. The capacity of Kj under X is de®ned by CKj …X† ; ai [Kj xi : By max-¯ow min-cut theorem [11], V…X† ˆ min1#j#m CKj …X†: A capacity vector X is called an upper boundary point for d if and only if: (i) V…X† ˆ d and (ii) V(X 1 ei) . d for each ai which is unsaturated under X (i.e. xi , Mi) where ei is a (n 1 p)-tuple with 1 in ith position and 0 others. In particular, if X is an upper boundary point for d, then V(Y) . d for each Y . X (where Y $ X if and only if yi $ xi for i ˆ 1; 2; ¼; n 1 p and Y . X if and only if Y $ X and yi . xi for at least one i). 3.2. System reliability The upper boundary points for d are those maximal ones

i

It is known that the system reliability Rd11 ; Pr{XuV…X† $ d 1 1} ˆ 1 2 Pr{XuV…X† # d} for a given demand (d 1 1). If {X1 ; X2 ; ¼; Xr } are the set of upper boundary points for d, let Bi ˆ {XuX # Xi } for i ˆ 1; 2; ¼; r: Thus, Pr{XuV…X† # d} ˆ Pr{B1 < B2 < ¼ < Br }: Methods such as inclusion±exclusion [12,16,17], disjoint subset [1,8], or state-space decomposition [4,6,13] can then be applied to compute Pr{B1 < B2 < ¼ < Br }: 3.3. Theory for upper boundary points for d For each upper boundary point X for d, by max-¯ow mincut theorem [11], there exists at least one MC Kj such that CKj …X† ˆ d and CKw …X† $ d for w ± j: The following is a necessary condition for an upper boundary point for d. Lemma 1. If X is an upper boundary point for d, then there exists a MC Kj such that CKj …X† ˆ d

and

xi ˆ Mi for each ai Ó Kj :

Proof. Suppose to the contrary that there exists an ai Ó Kj such that xi , Mi : Set Y ˆ X 1 ei : Then Y . X; Ckj …Y† ˆ d and CKw …Y† $ CKw …X† $ d ;w ± j: This means that V…Y† ˆ d which contradicts to that X is an upper boundary point for d. Hence, xi ˆ Mi ;ai Ó Kj : A Each capacity vector X which satis®es CKj …X† ˆ d and xi ˆ Mi for each ai Ó Kj is said to be generated by Kj. For convenience, let C d ˆ {Xu'Kj such that CKj …X† ˆ d and xi ˆ Mi for each ai Ó Kj }: Hence, V…X† # d for each X [ C d since there exists a Kj such that CKj …X† ˆ d: Remove those non-maximal ones from C d to obtain C d,max ; {XuX is maximal w.r.t. $ in C d}. The following lemmas further show that C d,max is the set of upper boundary points for d.

Y.-K. Lin / Reliability Engineering and System Safety 75 (2002) 41±46

Lemma 2.

If X [ C d,max, then V…X† ˆ d:

Proof. Suppose X [ C d,max but V(X) , d. Without loss of generality, we assume V…X† ˆ d 2 1: Hence, there exists a Kw such that C Kw …X† ˆ d 2 1: Choose an ai [ Kw such that xi , Mi (this ai can be found, otherwise that xi ˆ Mi ;ai [ Kw implies that CKw …X† ˆ CKw …M† , d and so V(M) , d which is a contradiction). Set Y ˆ X 1 ei : Then Y . X and CKw …Y† ˆ CKw …X† 1 1 ˆ d: Hence, Y [ C d which contradicts to that X [ C d,max. We conclude that V…X† ˆ d: A Lemma 3.

C d,max ˆ {upper boundary points for d}.

Proof. Firstly, we claim that each upper boundary point X for d will satis®es X [ C d,max. Suppose to the contrary that X Ó C d,max. As X [ C d by Lemma 1, there exists a Y [ C d,max such that Y . X and V…Y† ˆ d: This contradicts to that X is an upper boundary point for d. Hence X [ C d,max. Conversely, we claim that each X [ C d,max is an upper boundary point for d. Suppose to the contrary that X is not an upper boundary point for d. Then there exists an upper boundary point Y for d such that Y . X and so Y [ C d,max. This contradicts to that X [ C d,max. Hence, each X [ C d,max is an upper boundary point for d. A After generating C d, we use the comparison concept to obtain all upper boundary points for d. However, the comparison method will be unnecessary if G is series-parallel. Lemma 4. Cd:

If the network is series-parallel, then C d;max ˆ

Proof. Such a network can be considered as the series of its MCs K1, K2,¼,Km. Suppose X [ C d but X Ó C d,max. Without loss of generality, we may assume that X is generated by K1, i.e. C K1 …X† ˆ d and xi ˆ Mi ;ai Ó K1 : Then there exists a Y [ C d,max such that Y . X and so C K1 …Y† . CK1 …X† ˆ d and CKw …Y† ˆ CKw …X† ˆ CKw …M† . d ;w ± 1: Hence, V(Y) . d which contradicts to that Y [ C d,max. Thus, X [ C d,max and so C d ˆ C d;max : A

negative integer solutions of xr1 1 xr2 1 ¼ 1 xrnr ˆ d and xi # Mi for each i ˆ r1 ; r2 ; ¼; rnr ; and (1.2) setting xi ˆ Mi for each ai Ó Kr. Step 2. (Use comparison method to obtain C d,max) Suppose C d ˆ {X1 ; X2 ; ¼; Xu }: If the network is series±parallel, then C d is the set of upper boundary points for d. Otherwise, use the comparison method to remove those non-maximal ones from C d to obtain all upper boundary points for d as follows: (2.1) I à {f } (2.2) FOR i à 1 TO u and i Ó I (2.3) FOR j à i 1 1 TO u 2 1 and j Ó I (2.4) IF Xi , Xj THEN Xi is not an upper boundary point for d. I à I < {i} and goto step 2.7). ELSEIF Xi $ Xj THEN I à I < {j}. (2.5) j à j 1 1 (2.6) Xi is an upper boundary point for d (2.7) i à i 1 1 (2.8) END 5. An illustrative example An illustrative network is given in Fig. 2, which shows a simple computer network in which each arc represents a transmission line (it consists several physical transmission lines, e.g. T3 cable, E1 cable, optical ®ber) and each node represents a computer center (it consists several switches). The capacities of arcs a1 to a6 all take values from {0, 1, 2, 3, 4} with probability {0.01, 0.02, 0.02, 0.05, 0.9}, and the capacities of nodes a7 and a8 both take values from {0, 1, 2, 3, 4, 5, 6} with probability {0.01, 0.01, 0.02, 0.02, 0.02, 0.02, 0.9}. Therefore, M ˆ …M 1 ; M2 ; M3 ; M4 ; M5 ; M6 ; M7 ; M8 † ˆ …4; 4; 4; 4; 4; 4; 6; 6†: There are eight MCs: K1 ˆ {a1 ; a5 }; K2 ˆ {a1 ; a7 }; K3 ˆ {a5 ; a8 }; K4 ˆ {a2 ; a3 ; a5 }; K5 ˆ {a1 ; a4 ; a6 }; K6 ˆ {a2 ; a6 }; K7 ˆ {a2 ; a7 }; K8 ˆ {a6 ; a8 } and K9 ˆ {a7 ; a8 }: The supervisor would like to know the system reliability that the computer system can transmit at least eight messages from s to t simultaneously (i.e. R8). We ®rst generate all upper boundary points for 7 as follows: Step 1. ² Generate all X ˆ …x1 ; x2 ; ¼; x8 † [ C 7 by K1: (1.1) All feasible non-negative integer solutions (x1,x5)

4. Algorithm to generate all upper boundary points for d As those approaches of Refs. [13,18,20,21], we suppose all MCs, K1 ; K2 ; ¼; Km ; have been pre-computed. In particular, all MCs can be ef®ciently derived from those algorithms [5,10]. We can determine all upper boundary points for d by the following steps: Step 1. (Generate C d) With respect to each K r ˆ {ar1 ; ar2 ; ¼; amr }; r ˆ 1; 2; ¼; m; generate all X ˆ …x1 ; x2 ; ¼; xn1p † [ C d by (1.1) applying the implicit enumeration to ®nd all non-

43

Fig. 2. A simple stochastic-¯ow network.

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Y.-K. Lin / Reliability Engineering and System Safety 75 (2002) 41±46

Table 1 The result of the illustrated example

Table 1 (continued)

MC

X [ C d (step 1)

Is an upper boundary point for 7?

K1

X1 ˆ (4,4,4,4,3,4,6,6) X2 ˆ (3,4,4,4,4,4,6,6)

YES YES

K2

X3 ˆ (4,4,4,4,4,4,3,6) X4 ˆ (3,4,4,4,4,4,4,6) X5 ˆ (2,4,4,4,4,4,5,6) X6 ˆ (1,4,4,4,4,4,6,6)

YES NO NO NO

K3

X7 ˆ (4,4,4,4,4,4,6,3) X8 ˆ (4,4,4,4,3,4,6,4) X9 ˆ (4,4,4,4,2,4,6,5) X10 ˆ (4,4,4,4,1,4,6,6)

YES NO NO NO

K4

X11 ˆ (4,4,3,4,0,4,6,6) X12 ˆ (4,4,2,4,1,4,6,6) X13 ˆ (4,4,1,4,2,4,6,6) X14 ˆ (4,4,0,4,3,4,6,6) X15 ˆ (4,3,4,4,0,4,6,6) X16 ˆ (4,3,3,4,1,4,6,6) X17 ˆ (4,3,2,4,2,4,6,6) X18 ˆ (4,3,1,4,3,4,6,6) X19 ˆ (4,3,0,4,4,4,6,6) X20 ˆ (4,2,4,4,1,4,6,6) X21 ˆ (4,2,3,4,2,4,6,6) X22 ˆ (4,2,2,4,3,4,6,6) X23 ˆ (4,2,1,4,4,4,6,6) X24 ˆ (4,1,4,4,2,4,6,6) X25 ˆ (4,1,3,4,3,4,6,6) X26 ˆ (4,1,2,4,4,4,6,6) X27 ˆ (4,0,4,4,3,4,6,6) X28 ˆ (4,0,3,4,4,4,6,6)

NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO

K5

X29 ˆ (4,4,4,3,4,0,6,6) X30 ˆ (4,4,4,2,4,1,6,6) X31 ˆ (4,4,1,1,4,2,6,6) X32 ˆ (4,4,4,0,4,3,6,6) X33 ˆ (3,4,4,4,4,0,6,6) X34 ˆ (3,4,4,3,4,1,6,6) X35 ˆ (3,4,4,2,4,2,6,6) X36 ˆ (3,4,4,1,4,3,6,6) X37 ˆ (3,4,4,0,4,4,6,6) X38 ˆ (2,4,4,4,4,1,6,6) X39 ˆ (2,4,4,3,4,2,6,6) X40 ˆ (2,4,4,2,4,3,6,6) X41 ˆ (2,4,4,1,4,4,6,6) X42 ˆ (1,4,4,4,4,2,6,6) X43 ˆ (1,4,4,3,4,3,6,6) X44 ˆ (1,4,4,2,4,4,6,6) X45 ˆ (0,4,4,4,4,3,6,6) X46 ˆ (0,4,4,3,4,4,6,6)

NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO

K6

X47 ˆ (4,3,4,4,4,4,6,6) X48 ˆ (4,4,4,4,4,3,6,6)

YES YES

K7

X49 ˆ (4,4,4,4,4,4,3,6) X50 ˆ (4,3,4,4,4,4,4,6) X51 ˆ (4,2,4,4,4,4,5,6) X52 ˆ (4,1,4,4,4,4,6,6)

NO NO NO NO

K8

X53 ˆ (4,4,4,4,4,4,6,3) X54 ˆ (4,4,4,4,4,3,6,4) X55 ˆ (4,4,4,4,4,2,6,5) X56 ˆ (4,4,4,4,4,1,6,6)

NO NO NO NO

K9

X57 ˆ (4,4,4,4,4,4,6,1) X58 ˆ (4,4,4,4,4,4,5,2)

NO NO

MC

X [ C d (step 1)

Is an upper boundary point for 7?

X59 ˆ (4,4,4,4,4,4,4,3) X60 ˆ (4,4,4,4,4,4,3,4) X61 ˆ (4,4,4,4,4,4,2,5) X62 ˆ (4,4,4,4,4,4,1,6)

NO NO NO NO

such that x1 1 x5 ˆ 7 and (x1,x5) # (4,4) are (4,3) and (3,4). (1.2) Set …x2 ; x3 ; x4 ; x6 ; x7 ; x8 † ˆ …M2 ; M3 ; M4 ; M6 ; M7 ; M8 † ˆ …4; 4; 4; 4; 6; 6†: Two Xs are generated: X1 ˆ …4; 4; 4; 4; 3; 4; 6; 6† and X2 ˆ …3; 4; 4; 4; 4; 4; 6; 6†: ² Generate all X ˆ …x1 ; x2 ; ¼; x8 † [ C 7 by K2 : (1.1) All feasible non-negative integer solutions (x1,x7) such that x1 1 x7 ˆ 7 and (x1,x7) # (4,6) are (4,3), (3,4), (2,5) and (1,6). (1.2) Set …x2 ; x3 ; x4 ; x5 ; x6 ; x8 † ˆ …M2 ; M3 ; M4 ; M5 ; M6 ; M8 † ˆ …4; 4; 4; 4; 4; 6†: Four Xs are generated: X3 ˆ …4; 4; 4; 4; 4; 4; 3; 6†; X4 ˆ …3; 4; 4; 4; 4; 4; 4; 6†; X5 ˆ …2; 4; 4; 4; 4; 4; 5; 6† and X6 ˆ …1;.4; 4; 4; 4; 4; 6; 6†: .. All Xs are listed in Table 1. Step 2. (2.1) I ˆ {f} (2.2) Test whether X1 is an upper boundary point for 7. (2.3) j ˆ 2 (2.4) X1 ˆ …4; 4; 4; 4; 3; 4; 6; 6† ñ X2 ˆ …3; 4; 4; 4; 4; 4; 6; 6† and X1 à X3 : I ˆ {f } (2.3) j ˆ 3 (2.4) X1 ˆ …4; 4; 4; 4; 3; 4; 6; 6† ñ X3 ˆ …4; 4; 4; 4; 4; 4; 3; ..6† and X1 à X3. I ˆ {f } . (2.3) j ˆ 8 (2.4) X1 ˆ …4; 4; 4; 4; 3; 4; 6; 6† . X8 ˆ …4; 4; 4; 4; 3; 4; 6; 4†: .. I ˆ {8} . (2.6) X1 is an upper boundary point for 7. (2.2) Test whether X2 is an upper boundary point for 7. .. . 2.8) END. The ®nal result is concluded in Table 1. So X1 ˆ …4; 4; 4; 4; 3; 4; 6; 6†; X2 ˆ …3; 4; 4; 4; 4; 4; 6; 6†; X3 ˆ …4; 4; 4; 4; 4; 4; 3; 6†; X7 ˆ …4; 4; 4; 4; 4; 4; 6; 3†; X47 ˆ …4; 3; 4; 4; 4; 4; 6; 6† and X48 ˆ …4; 4; 4; 4; 4; 3; 6; 6† are all upper boundary points for 7. In order to compute Pr{XuV(X) # 7}, we let B1 ˆ {XuX $ X1 }; B2 ˆ {XuX $ X2 }; B3 ˆ {XuX $ X3 }; B4 ˆ {XuX $ X7 }; B5 ˆ {XuX $ X47 } and B6 ˆ {XuX $ X48 }: Then

Y.-K. Lin / Reliability Engineering and System Safety 75 (2002) 41±46

Pr {XuV…X† # 7} ˆ Pr{B1 < B 2 < B3 < B4 < B5 < B6 } ˆ 0:4027004 after calculating. So the system reliability R8 ˆ 1 2 Pr{XuV…X† # 7} ˆ 0:57972996 for demand 8. 6. Complexity analysis The number of non-negative solution X satisfying xr1 1 xr2 1 ¼ 1 xrnr ˆ d and …xr1 ; xr2 ; ¼; xrnr † # …Mr1 ; Mr2 ; ¼Mrnr † ! nr Y nr 1 d 2 1 and …Mri 1 1†; d iˆ1

are

respectively. Thus, the number of Xs generated by Kr is bounded by ! n ( ) r nr 1 d 2 1 Y min ; …Mri 1 1† : d iˆ1 Hence, the number of X [ C d is bounded by ! ( ) m m X nr 1 d 2 1 Y ; V; min …Mri 1 1† : d rˆ1 rˆ1 The proposed algorithm needs O((n 1 p)´V ) storage space in the worst case. In the worst case, step 1 generates C d in time O(V ). For each X [ C d, it further takes O((n 1 p)´V ) time to compare with other Xs in the worst case. Hence, step 2 to obtain C d,max needs O((n 1 p)´V 2) time in the worst case. In sum, the algorithm needs O((n 1 p)´V 2) time in the worst case. 7. Conclusions Based on minimal cuts, this paper proposes a simple algorithm to generate all upper boundary points for a given demand d for a stochastic-¯ow network in which each arc and node has several capacities and may fail. The upper boundary points for d can be used to calculate the system unreliability, i.e. Pr{XuV(X) # d}. If we try to evaluate the system reliability Pr{XuV(X) $ d}. Then we can generate all upper boundary points for (d 2 1) and calculate Pr{XuV…X† $ d} ˆ 1 2 Pr{XuV…X† P # d 2 1}: The mean ¯ow of the network is thus V…M† dˆ1 …d £ Pr{XuV…X† ˆ d}†; where Pr{XuV…X† ˆ d} ˆ Pr{XuV…X† # d} 2 Pr{XuV…X† # d 2 1}:

For the perfect-node case, the best existing algorithm by Yeh [21] tests whether the maximum ¯ow of the network is d under X for each X [ C d. Yeh's algorithm totally takes O(n 2´V(M)´V ) time in which the maximum ¯ow algorithm needs O(n´V(M)) time. Our algorithm only uses the comparison concept to directly generate all upper boundary points for d without applying the maximum ¯ow algorithm.

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However, the proposed algorithm is simpler to understand and to program on a digital computer. In this paper, we only consider the case of single commodity. Future research can develop a process to evaluate the system reliability for the networks that allow multiple commodities to be transmitted from source s to sink t. In the multiple commodities case, different commodity consumes the capacity of a component at different rates. Acknowledgements The author would like to thank the anonymous referees for helpful and constructive comments. This work was supported in part by the National Science Council, Taiwan, Republic of China, under Grant No. NSC 89-2213-E-238008. References [1] Abraham JA. An improved algorithm for network reliability. IEEE Trans Reliab 1979;28:58±61. [2] Aggarwal KK, Gupta JS, Misra KB. A simple method for reliability evaluation of a communication system. IEEE Trans Commun 1975;23:563±5. [3] Aggarwal KK, Rai S. Symbolic reliability evaluation using logical signal relat ions. IEEE Trans Reliab 1978;R-27:202±5. [4] Aggarwal KK, Chopra YC, Bajwa JS. Capacity consideration in reliability analysis of communication systems. IEEE Trans Reliab 1982;31:177±80. [5] Al-Ghanim AM. A heuristic for generating minimal path and cutsets of a general network. Comput Ind Engng 1999;36:45±55. [6] Clancy DP, Gross G, Wu FF. Probability ¯ows for reliability evaluation of multiarea power system interconnections. Electr Power Energy Syst 1983;5:100±14. [7] Doulliez P, Jamoulle J. Transportation networks with random arc capacities. RAIRO, Rech Oper Res 1972;3:45±60. [8] El-Neweihi E, Proschan F, Sethuraman J. Multistate coherent systems. J Appl Probab 1978;15:675±88. [9] Evans JR. Maximal ¯ow in probabilistic graphs Ð the discrete case. Networks 1976;6:161±83. [10] Fard NS, Lee TH. Cutset enumeration of network systems with link and node failures. Reliab Engng Syst Safety 1999;65(2):141±6. [11] Ford LR, Fulkerson DR. Flows in networks. NJ: Princeton University Press, 1962. [12] Grif®th WS. Multistate reliability models. J Appl Probab 1980;17:735±44. [13] Jane CC, Lin JS, Yuan J. On reliability evaluation of a limited-¯ow network in terms of minimal cutsets. IEEE Trans Reliab 1993;42(3):354±61. [14] Lee SH. Reliability evaluation of a ¯ow network. IEEE Trans Reliab 1980;29:24±6. [15] Lee DW, Yum BJ. Determination of minimal upper paths for reliability analysis of planar ¯ow networks. Reliabil Engng Syst Safety 1993;39:1±10. [16] Lin YK. On Reliability evaluation of a stochastic-¯ow network in terms of minimal cuts. J Chin Inst Ind Engrs 2001;18(3):49± 54. [17] Lin YK. A simple algorithm for reliability evaluation of a stochastic¯ow network with node failure. Comput Oper Res 2001;28(13):1277±85. [18] Malinowski J, Preuss W. A parallel algorithm evaluating the

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Y.-K. Lin / Reliability Engineering and System Safety 75 (2002) 41±46

reliability of a system with known minimal cuts (paths). Microelectron Reliab 1997;37(2):255±65. [19] Rueger WJ. Reliability analysis of networks with capacity-constraints and failures at branches and nodes. IEEE Trans Reliab 1986;35:523±8.

[20] Xue J. On multistate system analysis. IEEE Trans Reliab 1985;34:329±37. [21] Yeh WC. A simple approach to search for all d-MCs of a limited-¯ow network. Reliab Engng Syst Safety 2001;71(1):15±19.