Overall-terminal reliability of a stochastic capacitated-flow network

MATHEMATICAL COMPUTER MODELLING PERGAMON

Mathematical

and

Computer

Modelling

36

(2002)

173-181 www.elsevier.com/locate/mcm

Overall-Terminal Reliability of A Stochastic Capacitated-Flow Network YI-KUEI Department

of Information

LIN

Management,

Chung-Li,

Tao-Yuan,

Van Nung Institute Taiwan

of Technology

320, R.O.C.

yklinQcc.vit.edu.tw

(Received November

2001; accepted December 2001)

Abstract-For a stochastic and directed capacitated-flow network in which the capacity of each arc has several possible values, this article generalizes the system reliability problem from single source node and single sink node cases to an overall-terminal case. Given the demand for each node pair simultaneously, a simple algorithm is proposed first to generate all lower boundary points for such demands in terms of minimal paths. The lower boundary point is a vector denoting the current capacity of each arc. The system reliability, the probability that the system satisfies the demands simultaneously, can be calculated in terms of such lower boundary points by applying the inclusion-exclusion method. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords-Network

flow, Reliability, Overall-terminal,

Capacity, Minimal path.

NOMENCLATURE N

set of nodes

n

number of arcs

ai

arc i

A

{ai 1 1 5 i 5 n}:

M

(Ml, M-J,. , Mn): maximal capacity vector where Mi (a positive integer) denotes the maximal capacity of arc ai for each i= 1,2,...,n

( q,z2,

.

(i,_i)

set of arcs

, zn): a (current)

capacity

(source node, sink node): pair

a node

D

a prespecified set of node pairs

E(i, j)

set of MPs from i to j

I.1

number of elements in a set

m

total number of MPs in all node pairs of D, i.e., m = C(r,jlED kth

lE(i,j)l

MP for Ic = 1,2,.

,m

flow through Pk

vector where xi E (0, 1,2,. , Mi} denotes the (current) capacity of ai for each i = 1,2,. ,n

total flow from node i to node j,

(N, A, M): stochastic and directed capacitated-flow network

demand for node pair (i, j)

This work was supported 2213-E-238-002.

(fl, f2, , fm): a flow pattern or flow assignment i.e., Fi>i = CP~&q(,j)

in part by the National Science Council,

Taiwan, R.O.C.,

0895-7177/02/S - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(02)00113-9

fk

under Grant No. NSC 89-

Typeset

by An,is-Tl$

Y.-K.

174

LIN

set of demands for D, i.e., the set of di,j for (i,j) E D

dD

Y
(Yl>YZ,...,Y?z) < (11,22,..,,3%): and Yi < xi for at least one i Y < X

system reliability

RD

Y
(Yl,Y2,.‘.,Y7%)

I

probability

Pd.1

(w,22,...,G%):

YiIziforalli=1,2,...,n

1. INTRODUCTION Traditionally,

the maximum

flow problem

was addressed

to a directed

capacitated-flow

network

in which the flow is from the single source node s to the single sink node t and the capacity of each arc is deterministic. Such a problem is to find the maximum flow F,,, from s to t within

arc-capacity

constraints.

Ford and Fulkerson [l]. transportation networks,

The most classic method

is the labeling

However, in many real networks and electric power transmission

algorithm

proposed

by

such as telecommunication networks, networks, the capacity of each arc is

stochastic due to maintenance, failure, etc. Such a network is named a stochastic capacitatedflow network in this article. Then the maximum flow is stochastic and so the problem is to study R,,t that the the distribution of the maximum flow or equivalently to evaluate the probability is named system reliability maximum flow is not less than a given demand d,,t. This probability in [2-41. In order to evaluate lower boundary minimal

paths

points (MPs),

it is no longer a path This paper extends

it, Xue [5] and Lin et al. [3] proposed an algorithm to generate all for d,,t (named d,,t-MP in [3]) in terms of modular decomposition and

respectively.

A MP is a path whose remainder

after removing

any arc in

[6]. the system

reliability

problem

to the overall-terminal

case in which demand

for each node pair is given. Thus, the system reliability is the probability that the flow Fi,j is not less than the given demand di,j for each node pair (i, j). Let D denote some specified node pairs. For simplicity, we first consider the case that there are demands only for D. We will define the lower boundary points for the demands and then propose a simple algorithm to generate them in terms of MPs. Such lower boundary points are vectors denoting the current capacity of each arc. The system reliability can then be calculated in terms of such points by applying the inclusion-exclusion proposed

of the proposed whenever

method.

algorithm

A numerical

example

and how the system

reliability

algorithm

is analyzed.

in various

The system

cases is presented

can be computed. reliability

to illustrate

The computational

is the overall-terminal

the time

reliability

D is the set of all node pairs.

2. ASSUMPTIONS ASSUMPTION 1. Each node is perfectly

reliable.

ASSUMPTION 2. The capacity of each arc ai is an integer-valued random distribution. values from (0,1,2,. . . , I&} according to a given probability ASSUMPTION 3. The capacities

of different

arcs are statistically

ASSUMPTION 4. The Aows in G satisfy the flow conservation will disappear or be created during transmission.

3. MODEL 3.1.

Flow

Patterns

The flow pattern i.e.,

and

Capacity

variable

which takes

independent. law 111. This means

that

no Aow

BUILDING

Vectors

F = (_fl, fi, . . . , fm) is feasible

under

M if and only if it does not violate

M,

m c{.fk k=l

I ai E h}

I Mi,

foreachi=1,2

,...,

n,

(1)

Overall-Terminal Reliability

whereC~~“=,Lfrc I ai E x=

(Zl,ZZ,..

Pk} is the total

flow through

175

F. Similarly,

ai under

F is feasible

under

if and only if F satisfies

. , z,)

m c

I ai E h)

{d

foreachi=1,2

5 xi,

,...,

n.

(2)

k=l

let Ux = {F 1F is feasible under X}. The flow pattern F is said to satisfy the system demand

For convenience,

do if and only if

for each (i, j) E D.

(3)

The capacity vector X is said to satisfy the system demand dD if and only if there exists an X satisfies do if and only if there exists F E Ux satisfying the system demand dD. Equivalently, an F E UX such that 3.2.

System

Fi,j = di,j for each (i, j) E D. (See Lemma

Reliability

1 and its proof in Appendix

A.)

Evaluation

Let R = {X 1 X satisfies the system demand dD}. The system reliability RD, the probability that the system satisfies the system demand dD, is thus, Pr{R}. Let flmi, z {X 1 X is minimal

in a}

= {X 1 X E Q and Y # R .for any capacity

X E flmi, is called a lower boundary point for dD. In particular, X* E Qmi, such that X > X*. Hence, RD = Pr{X

1 X > X’ for a lower boundary

vector

Y with Y < X}.

Each

for each X E a, there exists

point

an

X* for dD}.

Therefore, in order to evaluate RD, one straightforward method is to search for all lower boundary points for do first. Then as in evaluating R,)t case, either rule such as inclusion-exclusion rule [4,5,7-111, disjoint subset [9,12,13], or state-space decomposition [2,3,14,15] can be applied. 3.3.

Theory

A necessary condition in the following theorem

for a capacity vector X to be a lower boundary (see its proof in Appendix B).

THEOREM 1. Let X be a lower boundary point for do.

each F E Ux such that Fi,j = dij, Given

each

V (i,j)

F such

flow pattern

that

such that Xi = c&{fk (Xl, 22,. . . ,x,) ity vector as 2i E (0, 1,2, . . . , Mi} for each that p contains all lower boundary points orem 2 that Pmin = {X I X is minimal in Appendix C.)

Then

Xi =

point

~~zl

{fk

for dD is shown

1 ai E pk},

Vi,

for

E D. Fi,j = di,j for each

(i, j)

E D, the

vector

XF

=

I Ui E Pk} for i = 1,2,. . . , n is obviously a capaci. Let p be the set of such XF. Theorem 1 implies for do (i.e., Rmin c p). We will further see in Thep} is the set of lower boundary points for do. (See

4. ALGORITHM As those in [3-5,10,11,14],

the proposed

algorithm

generates

all lower boundary

points

for dD

in terms of MPs which are set to be known in advance. List and arrange all MPs between each (i, j) E D orderly. All 1ower boundary points for dD are generated from the following steps. STEP 1. Generate

all flow patterns

F = (f~, f2, . . . , fm) of the following

constraints:

m c

{f/c

1 ai E pk)

5

Mi,

foreachi=1,2

,...,

n,

(4)

k=l C

SEW&j)

fk

= di,j,

for each (i, j) E D.

(5)

Y.-K.

176

STEP 2. Transform

LIN

F = (fi, f2,. . . , fm) into XF = (xi, x2,. . . ,x,)

each feasible zi=g{ficiaitPk},

foreachi=1,2

,...,

via

n.

(6)

k=l

STEP

3.

Let

p =

a lower boundary nonminimal

. . , X”} be the set of such generated

{X1,X2,. point

for do in the following

(i.e., use the comparison

I = 4 (I is the stack which stores the index of nonlower I = 4.)

(3.2)

FORi=lTOuwithi$I

(3.3) FORj=iflTOu,withj$I (3.4) IF Xi _> Xj, THEN

Xi is not a lower boundary

Step (3.7). ELSEIF Xj > Xi, THEN (3.5) j+j+l (3.6) X” is a lower boundary (3.3)

method

each Xi is

to delete those

ones).

(3.1)

(3.7)

XFS. Check whether

for do.

point

Xj is not a lower boundary

point

boundary

point

points

for do.

I +- I U {i}

Initially,

and GOT0

for do. I +- I U {j}.

for do

ici+l END

5. We use the telecommunication

A NUMERICAL network

EXAMPLE

of Figure

1 to illustrate

the proposed

algorithm.

In

this network, each node denotes a city (or a switch center) and each arc represents a transmission line composed of Tl cables., A Tl cable supplies 24 phone calls. Three Tl cables are installed

2 al

02

a3

ah

a5 10

3

a4

4

Figure 1. A telecommunication

network

Table 1. The arc data of the example.

Unit: Tl cable or 24 phone-calls.

Overall-Terminal

Reliability

177

Table 2. Three cases on system demand of the example.

Case 1 Case2 Case 3: (Overall-terminal.) D {(G’),

(1,3), (194)) (2,3), (2,4), (3,4), (211)) (3,1), (4,1), (3>2), (4,2), (4,3))

dl,z

&,3

&,4

&,3

dz.4

&,4

dz,l

&,I

&,I

d3,z

&,z

d4,3

1

1

1

2

1

0

0

0

0

0

1

2

Unit: 24 phone-calls

from City 1 to City 2. Hence, the capacity distribution.

The detail

for each arc is listed in Table

considered in Table 2. In this example, it is known lower boundary CASE

al,

1.

do

2) = {Pl,

points =

Hence,

that

List all MPs between

where

m = 8. The approach

demand

all (i,j)

and computed,

are All

respectively.

E D in the following

order:

P2 = {Q,,u~},

P3={alja2},

p4 =

where P7 = {&j},

(P7r PB),

cases on the system

n = 6 and (Adi, n/12, Ma, MJ, Ms, AIs) = (3,3,2,3,2,2).

where Pi = {al},

p21,

1. Three

for do and RD in three cases are generated

{d 1,2,d1,3,d~}.

Ejb3)={P3,fi,P5,P6}>

‘9(4,3) =

of al has 0, 1, 2, or 3 Tl cables with a given probability

{al,a3,a6},

p5={a5ja6},

p6={a5,a4,a2},

P8 = (U4,Q).

to search for all lower boundary

points

for do can be described

as follows. STEP

1. Generate

all flow patterns

F = (fl,

f2, . . . , f8)

of the constraints

f1+f3+f4<3, f3+fS+.fSi3, f4

5

fi+fS+f8

2,

(7)

13,

fi+f5+f6<2, f4+f5+f712, fl

+ f2

=

1,

f3+f4+f5+f6=3,

(8)

f7 + f8.=

Totally,

17 F are generated:

2,2,0), (0,1,2,~,0,0,1,~),

(0,1,3,0,0,0,2,0),

(1,0,2,0,1,0,1,1),

(1,0,2,0,0,1,2,0),

(0,1,2,OJ,OJJ),

0,1,1,1,1),

2.

(1,0,1J,OJ,IJ),

(l,O,O,l,O,2,1,1), (0,1,1,2,0,0~0,2), and (1,0,0,1,1,1,0,2). (1,0,0,2,0,1,0,2),

(~,OJJJ,O~O,2)~

STEP

into

2.

Transform

each F = (f~ , f2, . . . ,fB) Xl

XF

(0,1,2,0,0,1,2,0),

=

(OJ,l,~,O,1,1,~),

(O,IJ,I,l,O,O,2),

(x1,~2t~3r~4yx5,x6)

(l,O,l,O,O, (l,O,l,

(I,O,I,O,2,0,0,2), via

=f1+f3+f4,

22=f3+f6+f8r x3 =

f4,

x4=.f2+f6+f8, x5=f2+f5+.f6, z6=f4+f5+f7.

(9)

178

Y.-K.

Totally,

five different

XF

are obtained:

(3,3,1,2,1,2),

X4 = (2,3,1,3,2,2),

STEP

the minimal

3. Find

LIN

X ’ = (3,3,0,1,1,2), = (3,3,2,3,1,2).

and’X5

set in {X1,X2,X3,

X4,X5}

X2 = (2,3,0,2,2,2),

to obtain

all lower boundary

X3

=

points

for dD. (3.1) I = 4. (3.2) i = 1. (3.3) j = 2. (3.4) X’=(3,3,0,1,1,2)~X2=(2,3,0,2,2,2)andX2~X1, (3.5) j = 3. (3.4) X3 = (3,3,1,2,1,2)

I=$.

> X’ = (3,3,0,1,1,2).

X3 is not a lower boundary

point

for dD.

I = (3). (3.5) j = 4.

point for do.

(3.4) X1 is a lower boundary (3.7)

i = 2.

The final result

is listed in Table 3. Table 3. Lower boundary

point for dD in Case 1 of the example. Is Xz a lower boundary

XZ X’ = (3,3,0,1,1,2)

YES

X2 = (2,3,0,2,2,2)

YES

x3 = (3,3,1,2,1,2)

NO

x4 = (2,3,1,3,2,2)

NO

X5 = (3,3,2,3,1,2)

NO

By the inclusion-exclusion

rule, RD = Pr{(X

point for dD?

> X1) U (X 2 X2)} = Pr{X

2 X1} + Pr{X

>

X2} - Pr{(X 2 Xl) n (X 2 X2)} = Pr{X 2 (3,3,0,1,1,2)} +Pr{X 2 (2,3,0,2,2,2)} - Pr{X 2 (3,3,0,2,2,2)} = (0.8 x 0.75 x 1 x 0.95 x 0.95 x 0.85) + (0.90 x 0.75 x 1 x 0.9 x 0.9 x 0.85) (0.8 x 0.75 x 1 x 0.9 x 0.9 x 0.85) = 0.5119125. CASE 2. No lower boundary cannot

satisfy

point

for do can be generated

(dl,z, d1,3, d4,3) = (1,4,2)

CASE 3. The final result

terminal

reliability

simultaneously.

of Case 3 by the proposed RD is 0.538249219.

Table 4. Lower boundary

and so RD = 0, i.e., the system

algorithm

is listed in Table 4 and the overall-

points for do in Case 3 of the example. 1s a lower boundary point for do?

x5 = (3,3,2,1,0,2)

YES

Overall-Terminal

6. COMPUTATIONAL Computational

Time Complexity

Reliability

TIME

179

COMPLEXITY

of Step 1 in the Worst Case

For each node pair (i, j) E D, the number of feasible solutions which satisfies CPkEE(z,jj fit = di,j is at least IE(i,j)l

and is bounded by

IE(i,j)l + di,j - 1 & > ’ ( Hence, the total number of feasible solutions F = (fl, fi, . . . , fm) which satisfies equation (5) is bounded by IE(i,j)l C( (i,j)ED

+ di,j - I di,j 1 ’

Each solution needs O(m) time to test whether it satisfies ~~?“=,{f~ 1 ai E Pk} _< AL%for each arc ai and O(m +n) time for all arcs in the worst case. Hence, it takes IE(i,j)l

+ di,j - 1 di,j

time to obtain all solutions of Step 1 in the worst case. Computational

Time

Complexity

of Step 2 in the Worst Case

Each solution in Step 1 needs O(m . n) time to be transformed to the capacity vector in the worst case. Thus, it takes

time to obtain all XF of Step 2 in the worst case. Computational

Time

Complexity

of Step 3 in the Worst

Case

It further needs IE(i>j)l + di,j - 1 4,j time to test each solution in Step 2 whether it is minimal and 2 IE(i,.i)l

+ 4,j

-

1

disj )))

time to test all solutions in Step 2 in the worst case. Hence, the proposed algorithm needs

time in the worst case. Note that

and so.

IE(i,j)I + d,,j - 1 di,j

>

2

C (i,g)ED

IE(i,j)l = m.

180

Y.-K.

7. CONCLUSIONS For a directed this article

capacitated-flow

evaluates

simultaneously.

the system

A simple

network reliability

algorithm

LIN

AND

DISCUSSION

in which the capacity that the system

is proposed

satisfies

first to generate

system demands in terms of MPs. Then the system reliability such lower boundary points by applying the inclusion-exclusion is, thus, algorithm

the overall-terminal

reliability

of each arc has several the demands

for all (i, j) E D

all lower boundary

points

for

can be calculated in terms of method. The system reliability

D is the set of all node pairs.

whenever

values,

The proposed

takes

IE(i, j)l + d,,j -

1

di,j time in the worst case which means that the computational

time in the worst case is bounded

by

k.n.

for a real number Another

k.

straight

method for evaluating the same system reliability is first to apply the algorithm by either Xue [2] or Lin et al. [4] for each (i, j) E D to obtain Ri,j totally in IDI times, where R,,, = Pr{X ( there exists an F E Ux such that Ft,j > d,,j}. And then let RD s n(i,jJED Ri,j to approximate

RD.

It is obvious

that

Pr{X

( there exists an F E Ux such that

Fi,j 2 d+,

v (il A E Dl f rI(t,j)ELJPr{X 1there exists an F E UX such that Fi,j 2 d,,j}. However, see in Table 5 that the values of RD are over exaggerated in all cases of our example. Note that in Case 1, RD = R1,2 x R1,3 x R4,3 = 0.995125 x 0.85637625

x

0.971625

we can

= 0.8280202.

Table 5. The values RD vs.RD in the example

Case 1

RD

RD

RD - RD (error)

0.8280202

0.5119125

0.3161077

Case 2

0.6453629

0

0.6453629

Case 3

0.8491870

0.5382492

0.3019378

APPENDIX LEMMA 1. Let X be a capacity vector. V (i, j) E D, then there exists a AOW pattern

A

If the AOW pattern F E l_Jx satisfies Fi,j 2 di,j, HE U,y with H < F such that Hi,j =di3j, ‘d (i, j) E D.

PROOF. For any (i, j) E D such that Fi,j > di,j, we may assume FQ = di,j + 1 without loss of generality. Let F = (f;, fi, . . . , f;n) = (_f~,f2,. . . ,fw-lr fw - 1, fw+l,. . . , fm) for a P, E E(i,j) with fW > 0. Then F < F and pi,j = Fi,3 - 1 = di,j (note that Fi,J = Fi,j 2 di>j for any other (i, j) E D). Th is implies that we can repeat the above procedure in finite times to deduce F to an H E Ux such that Hi,j = di,j for each (i,j) E D. I

APPENDIX PROOF

B

OF THEOREM 1. Suppose to the contrary that there exists an arc ai such that C& = (21,~~ ,...) ~i_l,~ci-l,~i+l,..., z:,). {jk ) ai E Pk}
Overall-Terminal

Reliability

APPENDIX THEOREM

2.

Note PROOF. Theorem 1). Y < X. Then Conversely, Y < X. Then that

181

C

Pmin = 0min.

that X E p implies that Suppose that X E Pmin, Y E p, which contradicts suppose that X E flmin, Y E 0, which contradicts

X E R but X to that but X to that

and that X E Rmin implies that X E p (due to $ Rmin, i.e., there exists a Y E SZrnin such that X E Pmin. Hence, Pmin C Qmin. # Pmin, i.e., there exist a Y E PInin such that X E Rrnir,. Hence, Ornin & Pmi~~and we conclude

Prnin = Gin.

I

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