Reliability of a computer network in case capacity weight varying with arcs, nodes and types of commodity

ARTICLE IN PRESS

Reliability Engineering and System Safety 92 (2007) 646–652 www.elsevier.com/locate/ress

Reliability of a computer network in case capacity weight varying with arcs, nodes and types of commodity Yi-Kuei Lin Department of Information Management, Vanung University, Chung-Li city, Tao-Yuan, Taiwan 320, ROC Received 9 January 2006; received in revised form 28 February 2006; accepted 28 February 2006 Available online 18 April 2006

Abstract The computer network can be modeled as a capacitated-flow network. This paper concentrates on a two-commodity capacitated-flow network with three characters: (1) nodes as well as arcs have multiple possible capacities and may fail, (2) each component (arc/node) has both capacity and cost attributes; and (3) the capacity weight varies with arcs, nodes and types of commodity (or named file). We study the possibility that a given quantity of two types of files can be transmitted through this network simultaneously under the budget constraint. Such a possibility is named the system reliability which is a performance index to measure the quality level of supply demand systems such as computer, telecommunication, electric-power transmission and transportation systems. The approach of minimal paths is applied to describe the relationship among flow assignments and capacity vectors. A simple algorithm in terms of minimal paths is proposed to evaluate the system reliability. r 2006 Elsevier Ltd. All rights reserved. Keywords: Computer network; System reliability; Capacity weight; Unreliable nodes; Cost attribute; Minimal paths

1. Introduction From the quality and service management view point, it is an important issue to measure the transmission ability to meet the customers’ demand for a computer network. Network analysis is a crucial approach to solve the network problems. In a binary-state network (without flow through it), each arc has good/bad states. The system reliability, the probability that the source s communicates with the sink t, can be computed in terms of minimal paths (MPs). A MP is an ordered sequence of arcs from s to t that has no cycle. Note that a minimal path is different from the so-called minimum path. The latter is a path with minimum cost whenever the arc containing the cost attribute. The system reliability can be treated as a performance index to measure the quality level for a reallife system with supply demand character. Extending to a binary-state flow network [1], the capacity of each arc (the maximum flow passing the arc per unit time) has two levels, Tel.: +886 3 4515811; fax: +886 3 4621348.

E-mail address: [email protected]. 0951-8320/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2006.02.005

0 and a positive integer. The system reliability is the probability that the maximum flow from the source to the sink is not less than the demand. In a capacitated-flow network composed of multistate arcs, the maximum flow from s to t is not a fixed number. Hence, such a network is also multistate. Without cost attribute of each arc, several authors [2–7] had presented algorithms to generate all lower boundary points for the demand d in order to evaluate the system reliability. The lower boundary point for d is a minimal system state meeting the demand constraint. Note that only singlecommodity is considered in this network. However, many real-world flow networks allow multicommodity (multiple types of commodity) to be transmitted through it. Such a network is called a multicommodity capacitated-flow network. In a broadband computer network, multicommodity (audio files, video files, multimedia, etc.) share the bandwidth (arc capacity) simultaneously. In the past few decades, many authors [8–12] discussed the multicommodity minimum cost flow problem to minimize the total cost of multicommodity by assuming the arc is deterministic (i.e., the

ARTICLE IN PRESS Y.-K. Lin / Reliability Engineering and System Safety 92 (2007) 646–652 Table 1 The total flow for two networks

Flow of commodity 1 Flow of commodity 2 Total flow

Network 1

Network 2

5 4 9

2 6 8

capacity of each arc is a constant). Besides, several authors [13–18] have solved the multicommodity maximal flow problem to find the maximal total flow by also assuming the arc is deterministic. However, the maximal total flow is not an appropriate performance index especially in case different type of commodity consumes the arc capacity differently. For example, as the data shown in Table 1, the total flow in network 1 is larger than that in network 2. But this fact does not imply that network 1 provides a better transmission ability if commodity 2 consumes the quantity of capacity more than commodity 1. This paper tries to evaluate the system reliability for a multicommodity capacitated-flow network under the budget constraint with three characters: (1) each node as well as the arc has several possible capacities and may fail; (2) each component (arc/node) has both capacity and cost attributes; and (3) the capacity weight varies with arcs, nodes and types of commodity. For convenience, we concentrate on a two-commodity capacitated-flow network (named TCFN throughout this paper). This work is organized as follows. In Section 3, we discuss the relationship among flow assignments and capacity vectors. And we define the lower boundary point for (d1,d2;B) where dk the required demand of commodity k, k ¼ 1, 2 and B the transmission budget. The system reliability, the probability that the given demand (d1,d2) can be transmitted through the TCFN under budget B, can be computed in terms of all lower boundary points for (d1,d2;B). In Section 4, an algorithm based on MPs is proposed to generate all lower boundary points for (d1,d2;B). The storage and computational time complexity are also analyzed. A simple example is shown in Section 5 to illustrate the proposed algorithm and how the system reliability be computed. Finally, Section 6 shows that the reliability problem in twocommodity case cannot be simplified to single-commodity case. 2. Assumptions and nomenclature Let G(A, Q, M, C, W) be a TCFN where A ¼ fai j1pipng the set of arcs, Q ¼ fai jn þ 1pipn þ qg the set of nodes, M(M1, M2,y,Mn+q) with Mi the maximal capacity of ai, C ¼ fcki ji ¼ 1; 2; . . . ; n þ q; k ¼ 1; 2g with cki the transmission cost through ai per commodity k and W ¼ fwki ji ¼ 1; 2; . . . ; n þ q; k ¼ 1; 2g with wki the capacity weight denoting the consumed capacity on ai per commodity k. For instance, if one GIF-type file (commodity 1) in a computer network loads 10 packets exactly and if the

647

capacity in counted in terms of the number of packets, then w1i ¼ 10. The current capacity of the component ai is denoted by xi and the vector X  ðx1 ; x2 ; . . . ; xnþq Þ is called the capacity vector representing the system state. Without loss of generality for capacity weight wki , we assume that w2i Xw1i X1 for each component ai. The network G is required to satisfy the following assumptions: 1. The arcs and nodes all have several possible capacities and may fail. 2. Two types of commodity are transmitted from s to t. 3. The current capacity xi takes values from {0, 1, 2,y,Mi}, i ¼ 1; 2; . . . ; n þ q. 4. The capacities of different components are statistically independent. 5. Flow of each type of commodity must satisfy the flow conservation [19]. 2.1. Nomenclature dxe X pY

the smallest integer such that dxeXx ðx1 ; x2 ; . . . ; xnþq Þpðy1 ; y2 ; . . . ; ynþq Þ: xi pyi for i ¼ 1; 2; . . . ; n þ q X oY ðx1 ; x2 ; . . . ; xnþq Þoðy1 ; y2 ; . . . ; ynþq Þ: X pY and xi oyi for at least one i ðd 1 ; d 2 Þpðd 01 ; d 02 Þ: d k pd 0k for k ¼ 1, 2 ðd 1 ; d 2 Þoðd 01 ; d 02 Þ: ðd 1 ; d 2 Þpðd 01 ; d 02 Þ & d k od 0k for at least one k ðd 1 ; d 2 Þ þ ðd 01 ; d 02 Þ: (d1+d10 ,d2+d20 ) 3. Problem formulation Since the nodes are unreliable, we redefine a path qas an ordered sequence of arcs and nodes which connects s and t. The MP is also redefined as an ordered sequence of arcs and nodes whose proper subsets are no longer paths. Suppose that P1 ; P2 ; . . . ; Pm are MPs form s to t. The TCFN is described in terms of the capacity vector X ¼ ðx1 ; x2 ; . . . ; xnþq Þ and the flow assignment (F1,F2) where F 1 ¼ ðf 11 ; f 12 ; . . . ; f 1m Þ and F 2 ¼ ðf 21 ; f 22 ; . . . ; f 2m Þ with f kj the flow (integer-value) of commodity k through Pj, j ¼ 1; 2; . . . ; m, k ¼ 1; 2. Such a flow assignment is feasible under X if it satisfies the following condition, 2 3 X X 1 2 6w1 f j þ w2i fj7 6 i 7pxi for i ¼ 1; 2; . . . ; n þ q. (1) 6 ai 2Pj 7 ai 2Pj For convenience, let fX denote the set of (F1, F2) feasible under X. The flow assignment (F1,F2) is said to meet the demand and budget constraints if it satisfies constraints (2)–(4), m X j¼1

f kj ¼ d k ; k ¼ 1; 2,

(2)

ARTICLE IN PRESS Y.-K. Lin / Reliability Engineering and System Safety 92 (2007) 646–652

648

0 1 n X X X @ c1 f 1j þ c2i f 2j ApB, i i¼1

2 6w1 6 i 6

ai 2Pj

X

f 1j

(3)

ai 2Pj

þ

w2i

ai 2Pj

X ai 2Pj

2

3 f 2j 7 7pM i 7

for

i ¼ 1; 2; . . . ; n þ q.

Let F be the set of all flow assignments meeting the demand and budget constraints. The capacity vector X is said to meet the demand and budget constraints if there exists a ðF 1 ; F 2 Þ 2 fX meeting the demand and budget constraints. Let O be the set of such X. The system reliability Rd 1 ;d 2 ;B is thus 

¼ PrfOg ¼

meets the demand and budget constraints

X



ðF 1 ; F 2 Þ 2 F,

constraint

(1)

have

3

X

for i ¼ 1; 2; . . . ; n þ q.

loss of generality for ai, suppose that l Without m P P 1 2 1 2 w1 a1 2Pj f j þ w1 a1 2Pj f j ox1 for a1, then (F1,F2) is feasible under the capacity vector (Xe1) where e1 is a (n+q)-tuple vector with 1 at position 1 and 0 at others. It means that ðX 2e1 Þ 2 Ol which contradicts the factm that X is P P minimal in O. Hence, w1i ai 2Pj f 1j þ w2i ai 2Pj f 2j ¼ xi for i ¼ 1; 2; . . . ; n þ q. & For each ðF 1 ; F 2 Þ 2 F, generate the corresponding capacity vector Z F 1 ;F 2 ¼ ðz1 ; z2 ; . . . ; znþq Þ via zi ¼ l P m P w1i ai 2Pj f 1j þ w2i ai 2Pj f 2j for i ¼ 1; 2; . . . ; n þ q. In fact,

PrfX g;

X 2O

where PrfX g ¼ Prfx1 g  Prfx2 g      Prfxnþq g by Assumption 4 (note that Pr{xi} is the probability that the capacity of ai is exactly xi). We name the minimal one in O as a lower boundary point for (d1,d2;B). Equivalently, X is a lower boundary point for (d1,d2;B) if and only if (i) X 2 O and (ii) Y eO for any capacity vector Y such that Y oX . Thus we have the following result: Lemma 1. For each X 2 O, there exists at least one lower boundary point Y for (d1,d2;B) such that Y pX . Suppose there are r lower boundary points for (d1,d2;B): X 1 ; X 2 ; . . . ; X r . Let subset S i  fX jX XX i g, i ¼ 1; 2; . . . ; r. The system reliability can be formulated as follows: Rd 1 ;d 2 ;B ¼ PrfX jX XX i for a lower boundary point X i forðd 1 ; d 2 ; BÞg ¼ PrfS 1 [ S2 \ . . . \ S r g.

X

each

6w1 f 1j þ w2i f 2j 7 6 i 7pxi 6 ai 2Pj 7 ai 2Pj (4)

Rd 1 ;d 2 ;B ¼ Pr X jX

Proof. For said that

ð5Þ

It can be calculated by applying several methods such as inclusion-exclusion method [3,7,20–25], disjoint subsets [4,22] and state-space decomposition [2,26,27]. Note that PrfY XX g ¼ Prfy1 Xx1 g  Prfy2 Xx2 g      Prfynþq Xxnþq g if Y ¼ ðy1 ; y2 ; . . . ; ynþq Þ. The remainder work is how to search for all lower boundary points for (d1,d2;B). The following Theorem shows a necessary condition for a lower boundary point for (d1,d2;B). Theorem 1. Let X be a lower boundary point for (d1,d2;B). Then there exists an ðF 1 ; F 2 Þ 2 F such that 2 3 X X 1 xi ¼ 6 f 1j þ w2i f 2j 7 6wi 7 for i ¼ 1; 2; . . . ; n þ q. 6 ai 2Pj 7 ai 2Pj (6)

the Z F 1 ;F 2 meets the demand and budget constraints since (F1,F2) is feasible under Z F 1 ;F 2 . We call such Z F 1 ;F 2 a candidate of lower boundary point for (d1,d2;B). Then the following result arises. Lemma 2. Each lower boundary point for (d1,d2;B) is a candidate of lower boundary point for (d1,d2;B). For convenience, let C ¼ fZ F 1 ;F 2 jðF 1 ; F 2 Þ 2 F g be the set of all candidates of lower boundary point for (d1,d2;B). The following Theorem further shows that Cmin{X|X is minimal in C} is the set of lower boundary points for (d1,d2;B). Theorem 2. {X|X (d1,d2;B)} ¼ Cmin.

is

a

lower

boundary

point

for

Proof. Firstly, suppose that X is a lower boundary point for (d1,d2;B) (note that X 2 C by Lemma 2) but X eCmin i.e., there exist a Y 2 C such that Y oX . Then Y 2 O, which contradicts that X is a lower boundary points for (d1,d2;B). Hence, X 2 Cmin . & Conversely, suppose that X 2 Cmin (note that X 2 O) but it is not a lower boundary point for (d1,d2;B). Then there exists a lower boundary point Y for (d1,d2;B) such that Y oX . By Lemma 2, Y 2 C that contradicts that X 2 Cmin . Hence, X is a lower boundary point for (d1,d2;B). 4. Algorithm As those approaches taken in [2–7,24,28] we suppose all MPs have been pre-computed. Virtually, MPs can be efficiently derived from those algorithms discussed in [29–31]. Step 1: v 0, I f, C f, Cmax f.

ARTICLE IN PRESS Y.-K. Lin / Reliability Engineering and System Safety 92 (2007) 646–652

Step 2 (To generate F): Obtain all flow assignments (F1,F2) satisfying the demand and budget constraints, m X

f kj ¼ d k ;

k ¼ 1; 2,

n X

@c1

X

f 1j þ c2i

i

(7)

(8)

storage space in the worst case. Each solution O(m)m time to test whether l Pof Eq. (7) needs P 1 1 2 it satisfies wi ai 2Pj f j þ wi ai 2Pj f 2j pM i for each i and

ai 2Pj

i¼1

2 6w1 6 i 6

X

f 1j

þ

ai 2Pj

1

X

f 2j ApB,

ai 2Pj

X

w2i

ai 2Pj

3 f 2j 7 7pM i 7

fori ¼ 1; 2; . . . ; n þ q.

(9)

Step 3 (To obtain C): Transform each (F1,F2) into the candidate X ¼ ðx1 ; x2 ; . . . ; xnþq Þ according to Eq. (10), 2 3 X X 1 xi ¼ 6 f 1j þ w2i f 2j 7 6wi 7 for i ¼ 1; 2; . . . ; n þ q, 6 ai 2Pj 7 ai 2Pj (10) X and C C [ X v. set v v þ 1, X v Step 4 (To obtain Cmin): C ¼ fX 1 ; X 2 ; . . . ; X v g. (4.1) (4.2) (4.3) (4.4) (4.5) Cmax (4.6) (4.7)

For i ¼ 1 to v and ieI For j ¼ i þ 1 to v with jeI If X i XX j , I ¼ I [ fig and go to step (4.6) Elseif X j 4X i , I ¼ I [ fjg j ¼jþ1 Xi is a lower boundary point for (d1,d2;B) and Cmax [ X i i ¼iþ1 End.

4.1. Complexity analysis The number of feasible solutions of Eq. (7) are   m þ d1  1 m

Pm

1 j¼1 f j

¼ d 1 and

and    m þ d1  1 m þ d2  1 , m m respectively. The number of solutions of constraints (7)–(9) is bounded by    m þ d1  1 m þ d2  1 . m m Similarly, the number of X transformed according to Eq. (10) is bounded by 

m þ d1  1 m



 m þ d2  1 . m

Hence, the algorithm needs     m þ d2  1 m þ d1  1 O ðn þ qÞ m m

j¼1

0

649

O(m(n+q)) time for all i. Hence, it takes     m þ d2  1 m þ d1  1 O mðn þ qÞ m m time to obtain all solutions ofl constraints (7)–(9) in the m P P worst case. Since the number w1i ai 2Pj f 1j þ w2i ai 2Pj f 2j has been processed in step 2, it does not need any time to transform (F1,F2) into X via Eq. (10). In the worst case, the number of elements of O is    m þ d2  1 m þ d1  1 m m and so it takes     m þ d1  1 m þ d2  1 O ðn þ qÞ m m time to test an element of O whether it is minimal in O and     ! m þ d1  1 2 m þ d2  1 2 O ðn þ qÞ m m time for all elements. Hence, the computational time complexity of the algorithm in the worst case is 0 !2 !2 1 m þ d2  1 m þ d1  1 A O@ðn þ qÞ m m ! !! m þ d1  1 m þ d2  1 ¼ O mðn þ qÞ m m 0 !2 !2 1 m þ d2  1 m þ d1  1 A. þ O@ðn þ qÞ m m    m þ d2  1 m þ d1  1 Note that m is less than . m m 5. A numerical example We use the benchmark [32,33] shown in Fig. 1 to illustrate the proposed solution process. There are seven MPs: P1 ¼ fa13 ; a1 ; a9 ; a3 ; a11 ; a7 ; a14 g, P2 ¼ fa13 ; a1 ; a9 ; a3 ; a11 ; a6 ; a12 ; a8 ; a14 g, P3 ¼ fa13 ; a1 ; a9 ; a4 ; a12 ; a8 ; a14 g, P4 ¼ fa13 ; a1 ; a9 ; a4 ; a12 ; a6 ; a11 ; a7 ; a14 g, P5 ¼ fa13 ; a2 ; a10 ; a5 ; a12 ; a8 ; a14 g, P6 ¼ fa13 ; a2 ; a10 ; a5 ; a12 ; a6 ; a11 ; a7 ; a14 g and P7 ¼ fa13 ; a2 ; a10 ; a5 ; a12 ; a4 ; a9 ; a3 ; a11 ; a7 ; a14 g. The data of arcs and nodes are shown in Table 2. We assume the source and the sink both have infinite capacity and are perfect. If the demand (d1, d2) is set to be (3,3) and C ¼ $2450, then the

ARTICLE IN PRESS Y.-K. Lin / Reliability Engineering and System Safety 92 (2007) 646–652

650

a9

a11

a3

Table 2 The data of arcs and nodes

a7

a1 a4

s = a13

a6

t = a14

Component

Capacity

Probability

w1i

w2i

c1i a

a1

0b 1 2 3 4 5

.01 .01 .01 .01 .02 .94

1

2

30

60

a2

0 1 2 3 4 5

.01 .01 .01 .02 .02 .93

1

2

60

90

a3

0 1 2 3 4 5

.01 .01 .01 .01 .01 .95

1

2

90

120

a4

0 1 2 3 4 5

.01 .01 .02 .03 .03 .90

1

2

60

120

a5

0 1 2 3 4 5

.01 .01 .01 .01 .02 .94

1

2

30

60

a6

0 1 2 3 4 5

.01 .01 .02 .02 .03 .91

1

2

60

90

a7

0 1 2 3 4 5

.01 .01 .01 .01 .01 .95

1

2

90

120

a8

0 1 2 3 4 5

.01 .01 .01 .03 .03 .91

1

2

60

120

a9–a12

0 1 3 5 7 9

.01 .01 .01 .01 .02 .94

1

2

60

90

a8

a2 a10

a5

a12

Fig. 1. A benchmark [32,33].

network reliability R3,3;2450 can be calculated by the following steps: Step 1: v ¼ 0, I ¼ f, C ¼ f, Cmax ¼ f. Step 2: Obtain all F 1 ¼ ðf 11 ; f 12 ; f 13 ; f 14 ; f 15 ; f 16 ; f 17 Þ and F 2 ¼ 2 2 2 2 2 2 2 ðf 1 ; f 2 ; f 3 ; f 4 ; f 5 ; f 6 ; f 7 Þ of the following constraints: f 11 þ f 12 þ f 13 þ f 14 þ f 15 þ f 16 þ f 17 ¼ 3, f 21 þ f 22 þ f 23 þ f 24 þ f 25 þ f 26 þ f 27 ¼ 3,

ð11Þ

fð30ðf 11 þ f 12 þ f 13 þ f 14 Þ þ 60ðf 21 þ f 22 þ f 23 þ f 24 Þg þ f60ðf 15 þ f 16 þ f 17 Þ þ 90ðf 25 þ f 26 þ f 27 Þg þ f90ðf 11 þ f 12 þ f 17 Þ þ 120ðf 21 þ f 22 þ f 27 Þg þ f60ðf 13 þ f 14 þ f 17 Þ þ 120ðf 23 þ f 24 þ f 27 Þg þ f30ðf 15 þ f 16 þ f 17 Þ þ 60ðf 25 þ f 26 þ f 27 Þg þ f60ðf 12 þ f 14 þ f 16 Þ þ 90ðf 22 þ f 24 þ f 26 Þg þ f90ðf 11 þ f 14 þ f 16 þ f 17 Þ þ 120ðf 21 þ f 24 þ f 26 þ f 27 Þg þ f60ðf 12 þ f 13 þ f 15 Þ þ 120ðf 22 þ f 23 þ f 25 Þg þ f60ðf 11 þ f 12 þ f 13 þ f 14 þ f 17 Þ þ 90ðf 21 þ f 22 þ f 23 þ f 24 þ f 27 Þg þ f60ðf 15 þ f 16 þ f 17 Þ þ 90ðf 25 þ f 26 þ f 27 Þg þ f60ðf 11 þ f 12 þ f 14 þ f 16 þ f 17 Þ þ 90ðf 21 þ f 22 þ f 24 þ f 26 þ f 27 Þg þ f60ðf 12 þ f 13 þ    þ f 17 Þ þ 90ðf 22 þ f 23 þ    þ f 27 Þgp2450, a1 : df 11 þ f 12 þ f 13 þ f 14 þ 2f 21 þ 2f 22 þ 2f 23 þ 2f 24 ep5, a2 : df 15 þ f 16 þ f 17 þ 2f 25 þ 2f 26 þ 2f 27 ep5,

ð12Þ

a3 : df 11 þ f 12 þ f 17 þ 2f 21 þ 2f 22 þ 2f 27 ep5, a4 : df 13 þ f 14 þ f 17 þ 2f 23 þ 2f 24 þ 2f 27 ep5, a5 : df 15 þ f 16 þ f 17 þ 2f 25 þ 2f 26 þ 2f 27 ep5, a6 : df 12 þ f 14 þ f 16 þ 2f 22 þ 2f 24 þ 2f 26 ep5, a7 : df 11 þ f 14 þ f 16 þ f 17 þ 2f 21 þ 2f 24 þ 2f 26 þ 2f 27 ep5, a8 : df 12 þ f 13 þ f 15 þ 2f 22 þ 2f 23 þ 2f 25 ep5, a9 : df 11 þ f 12 þ f 13 þ f 14 þ f 17 þ 2f 21 þ 2f 22 þ 2f 23 þ 2f 24 þ 2f 27 ep9, a10 : df 15 þ f 16 þ f 17 þ 2f 25 þ 2f 26 þ 2f 27 ep9,

a

US dollars. Pr{the capacity of a1 is 0} ¼ 0.01.

b

c2i

ARTICLE IN PRESS Y.-K. Lin / Reliability Engineering and System Safety 92 (2007) 646–652

a11 : df 11 þ f 12 þ f 14 þ f 16 þ f 17 þ 2f 21 þ 2f 22 þ 2f 24 þ 2f 26 þ 2f 27 ep9, a12 : df 12 þ f 13 þ    þ f 17 þ 2f 22 þ 2f 23 þ    þ 2f 27 ep9. ð13Þ Seven (F1,F2) are obtained: (3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0), (2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0), (2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0), (1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0), (0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 1, 0, 0) and (0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 1, 0, 0). The corresponding total costs are 2370, 2370, 2340, 2340, 2340, 2310 and 2430, respectively. Step 3: Transform each (F1,F2) into X ¼ ðx1 ; x2 ; . . . ; x12 Þ according to 2 3 X X 1 xi ¼ 6 f 1j þ w2i f 2j 7 6wi 7 for i ¼ 1; 2; . . . ; 12 (14) 6 ai 2Pj 7 ai 2Pj We obtain X 1 ¼ ð5; 4; 5; 0; 4; 0; 5; 4; 5; 4; 5; 4Þ, X 2 ¼ ð5; 4; 4; 1; 4; 0; 4; 5; 5; 4; 4; 5Þ, X 3 ¼ ð4; 5; 4; 0; 5; 0; 4; 5; 4; 5; 4; 5Þ, X 4 ¼ ð5; 4; 5; 0; 4; 0; 5; 4; 5; 4; 5; 4Þ, X 5 ¼ ð5; 4; 4; 1; 4; 0; 4; 5; 5; 4; 4; 5Þ, X 6 ¼ ð4; 5; 4; 0; 5; 0; 4; 5; 4; 5; 4; 5Þ and X 7 ¼ ð4; 5; 4; 0; 5; 1; 5; 4; 4; 5; 5; 5Þ. Step 4: C ¼ fX 1 ; X 2 ; X 3 ; X 4 ; X 5 ; X 6 ; X 7 g. Check each Xi whether it is a lower boundary point for (3,3;2450) or not. (4.1) i ¼ 1 (4.2) j¼2 (4.3) X 1  X 2 and X 2 4X 1 . I ¼ f. (4.2) j¼3 (4.3) X 1  X 3 and X 3 4X 1 . I ¼ f. (4.2) j¼4 (4.3) X 1 XX 4 . I ¼ f1g. (4.5) X1 is a lower boundary point for (3,3;2450) and Cmax ¼ fX 1 g (4.1) i ¼ 2 ^ After further checking, Cmax ¼ fX 4 ; X 5 ; X 6 ; X 7 g are all lower boundary points for (3,3;2450). Let S 1 ¼ fX jX XX 4 g, S2 ¼ fX jX XX 5 g, S3 ¼ fX jX XX 6 g and S 4 ¼ fX jX XX 7 g. Hence, the system reliability R3;3;2450 ¼ PrfS 1 [ S2 [ S 3 [ S4 g ¼ 0:676618532 can be computed by the inclusion-exclusion method. 6. Discussion The method discussed by Lin [3] studied the system reliability problem for single-commodity and failure nodes case without cost attribute. If we let wki be constant for i, d ¼ w1i d 1 þ w2i d 2 , B be unlimited and treat the original problem (in TCFN) as a single-commodity case, then the approach of Lin [3] can be applied to evaluate Rd 1 ;d 2 ;B (i.e., Rd,0;N). However, the following illustration indicates that the TCFN model cannot be simplified to a singlecommodity model. We use the network of Fig. 1 to

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illustrate the difference between them. If w1i ¼ 1, w2i ¼ 3 and ðd 1 ; d 2 Þ ¼ ð1; 1Þ, then d ¼ 4. The capacity vector X ¼ ð2; 2; 2; 0; 2; 0; 2; 2; 2; 2; 2; 2; 4; 4Þ permits the flow d ¼ 4 pass through P1 and P5. But it is obvious that the same capacity vector X cannot permit ðd 1 ; d 2 Þ ¼ ð1; 1Þ since the second type of commodity can pass through neither P1 nor P5. Hence, the reliability problem in two-commodity case cannot be simplified to single-commodity case. Acknowledgements This work was supported in part by the National Science Council, Taiwan, Republic of China, under Grant no. NSC 94-2213-E-238-001. References [1] Aggarwal KK, Chopra YC, Bajwa JS. Capacity consideration in reliability analysis of communication systems. IEEE Trans Reliab 1982;31:177–80. [2] Lin JS, Jane CC, Yuan J. On reliability evaluation of a capacitatedflow network in terms of minimal pathsets. Networks 1995;25:131–8. [3] Lin YK. A simple algorithm for reliability evaluation of a stochasticflow network with node failure. Comput Oper Res 2001;28:1277–85. [4] Xue J. On multistate system analysis. IEEE Trans Reliab 1985;34:329–37. [5] Yeh WC. A revised layered-network algorithm to search for all d-minpaths of a limited-flow acyclic network. IEEE Trans Reliab 1998;47:436–42. [6] Yeh WC. A simple algorithm to search for all d-MPs with unreliable nodes. Reliab Eng Syst Safety 2001;73:49–54. [7] Yeh WC. A new approach to evaluate reliability of multistate networks under the cost constraint. Omega 2005;33:203–9. [8] Cremeans JE, Smith RA, Tyndall GR. Optimal multicommodity network flows with resource allocation. Naval Res Logistics Q 1970;17:269–79. [9] Evans JR. A combinatorial equivalence between a class of multicommodity flow problems and the capacitated transportation problem. Math Programming 1976;10:401–4. [10] Kennington JL. Solving multicommodity transportation problems using a primal partitioning simplex technique. Naval Res Logistics Q 1977;24:309–25. [11] Tomlin JA. Minimum-cost multicommodity network flows. Oper Res 1966;14:45–51. [12] Weigel HS, Cremeans JE. The multicommodity network flow model revised to include vehicle per time period and node constraints. Naval Res Logistics Q 1972;19:77–89. [13] Assad AA. Multicommodity network flows-a survey. Networks 1978;8:37–91. [14] Ford LR, Fulkerson DR. A suggested computation for maximal multicommodity network flows. Manage Sci 1974;20:822–44. [15] Held M, Wolfe P, Crowder HP. Validation of subgradient optimization. Math Programming 1974;6:62–88. [16] Hu TC. Multi-commodity network flows. Oper Res 1963;11:344–60. [17] Jarvis JJ. On the equivalence between the node-arc and arc-chain formulations for the multi-commodity maximal flow problem. Naval Res Logistics Q 1969;16:525–9. [18] Rothechild B, Whinston A. On two commodity network flows. Oper Res 1966;4:377–87. [19] Ford LR, Fulkerson DR. Flows in networks. NJ: Princeton University Press; 1962. [20] Griffith WS. Multistate reliability models. J Appl Probab 1980; 17:735–44. [21] Hudson JC, Kapur KC. Reliability bounds for multistate systems with multistate components. Oper Res 1985;33:153–60.

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