374 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 4, DECEMBER 2001

An Efficient Method for EvaluatingNetwork-Reliability With Variable Link-Capacities

Seung Min Lee and Dong Ho Park, Member, IEEE

Abstract—An efficient method is proposed to evaluate the net-work reliability with variable link capacities when the simple pathsof the network are known. Most of the evaluation techniques pro-posed in the literature so far are based on enumerating the-com-posite paths, each of which is a union of simple paths. Althoughsome of those methods lead to correct results, the redundancy isstill quite large and may occur repeatedly each time the higherorder composite paths are generated. This paper proposes a newmethod based on the concepts of additivity and eligibility proper-ties defined in the text. We identify a composite path as a subnet-work which in general contains more simple paths than those in-volved in composition, and add only a minimal set of links at eachstep which gives maximal increase on the maximum capacity flowof the subnetwork. Thereby we reduce the possible occurrence ofredundancy significantly. The number of composite paths consid-ered for the capacity computation is also greatly reduced. Further-more, it is not necessary to keep the information on how many andwhich simple paths have been used in each composite path. Somenumerical examples illustrate the efficiency of the method.

Index Terms—Additivity, composite path, eligibility, maximumcapacity flow, simple path.

ACRONYMS1

simple pathcomposite pathminimal success : minimal set of links which en-sures successful operation of the network

ELGSP set of reserved for checking eligibilityFSP set of available failureLIST set of success found in the processLP-EM evaluation method in this paperMCF maximum capacity flow

NOTATION

aasubpath of onset of all which are equivalent to onset of all for the given networkset of all for the subnetwork induced byMCF of the subnetwork induced byrequired amount of capacity flow for the network

Manuscript received May 8, 1999; revised March 10, 2000; July 1, 2000; andAugust 17, 2000. This work was supported in part by The Hallym Academy ofSciences, Hallym University, Korea.

The authors are with the Department of Statistics, Hallym University,Chunchon 200-702 Rep. of Korea (e-mail: SMLee1@sun.hallym.ac.kr;DHPark@sun.hallym.ac.kr).

Publisher Item Identifier S 0018-9529(01)11343-6.

1The singular and plural of an acronym are always spelled the same.

I. INTRODUCTION

EXACT evaluation of network reliability for a networkwith variable link capacities has attracted much attention

in the literature. The network is modeled as a probabilisticgraph , which consists of a set of nodes and a setof links. Each link of the network can have different capacity,and the network is required to transmit a specified amount offlow from the source node to the terminal node, e.g., a computercommunication network which allows only a fixed amount ofdata exchange among terminals of various computer centers, atransport system of a large town with limited maximum trafficon various roads, or a hydraulic system which carries gas orfluid through a pipeline network with capacity limitations.In these cases, successful operation of the network is notnecessarily characterized by connectedness only, but by theMCF that can be transmitted through the network. The networkreliability is measured as the probability of transmitting therequired amount of flow successfully from the source node tothe terminal node.

Many algorithms have been suggested to obtain the capacityrelated reliability for a network with variable link-capacities.The method in [5] is based on the concept of lexicographic or-dering, and uses a labeling scheme to route the flow throughthe network. Another approach [7] generates valid groups bysuccessive replacement of links. All the resulting valid groups,which are mutually disjoint, are used for evaluating the networkreliability. Recently, more algorithms have been proposed toevaluate various measures for capacity-related reliability whenthe of the network are known. Some of these algorithmsare in [1], [2], [4], [6], [8], [9], [10]. Reference [6] uses a listof failure to generate higher order . However, [8] arguesthat this method does not generate enoughand fails to givecorrect results in general. Reference [2] also proposes a tech-nique to evaluate the network reliability using; [1] suggestsa method for evaluating a performance index of the network,which is essentially the same as the-expected value of MCFof the network. But, both [1], [2] have some drawbacks in com-puting the capacity and give incorrect results. Reference [10]expands the method of [1] to allow the links to have multiplestates of capacities with corresponding probabilities. Reference[9] presents a counter example to show that the method of [10]fails in certain cases. To complement and correct the drawbacksof the preceding results, [8] proposes a method based on the

enumeration technique using the idea of clique and key_cutto reduce the redundancy. However, the redundancy is still quitelarge and can occur repeatedly each time the higher orderaregenerated. Reference [4] suggests a technique for computing the

capacity using , whereas minimal cuts are used in [8].

0018–9529/01$10.00 © 2001 IEEE

LEE AND PARK: EVALUATING NETWORK-RELIABILITY WITH VARIABLE LINK-CAPACITIES 375

The LP-EM in this paper treats a as a union of allcontained in the subnetwork induced by it, rather than as a unionof only those involved in composition, and reflects the structureoriginated from the given network. Then, only a minimal setof links is added at each step, which gives maximal increaseon the maximum capacity flow of the subnetwork by iterativelyperforming -or- operations.

Section II describes the basic procedure, and defines the prop-erties of additivity and eligibility.

Section III describes in detail the LP-EM methodology andalgorithm using the properties; and provides a numerical ex-ample to exemplify the use of algorithm.

Section IV discusses the LP-EM time complexity and com-pares LP-EM with other existing methods. Numerical examplesillustrate the efficiency of LP-EM.

II. A DDITIVITY AND ELIGIBILITY PROPERTIES

Assumptions:

1) The nodes are perfect. No node has a capacity limit.2) The links are -independent. Each link either functions or

fails with known probabilities.3) All the links are undirected. Each link flow is bounded by

the link capacity.4) No flow can be transmitted through a failed link.5) The network is good iff a specified amount of flow can be

transmitted from the source node to the terminal node.6) The of the network, considering connectivity only, is

known.A is a minimal set of links connecting the source node

and the terminal node; a is a union of . A itself canbe considered as a . The is defined as the MCF of thesubnetwork induced by ; the network is in a functioning stateif a specified amount of flow, , can be transmitted fromthe source node to the terminal node. If , then

is a success ; otherwise, is a failure . A successwhich contains no other successas a proper subset is a .Under assumption #6, the network reliability can be obtained bysolving the following three subproblems.

1) Generate enough efficiently to find all possible .2) Compute MCF for each generated in subproblem #1.3) Evaluate the network reliability with all found.Subproblem #2 is essentially the same as computing MCF of

a network; it can be computed by applying well known methodssuch as the Ford and Fulkerson algorithm [3]. Subproblem #3has been discussed extensively in the literature, e.g., [1], [9],[10]. Thus, this paper is focused on subproblem #1. Here, re-dundancy implies both duplication and absorption. A duplica-tion occurs when a is generated more than once; an absorp-tion implies that a contains other as subsets.

A. Basic Procedure

The basic idea is to add, each time, a choice of minimal set oflinks which gives maximal increase on MCF of the subnetworkinduced by a failure . Such a choice is made from a set offailure and is referred to as a qualified. Each iteration hasthe contents [ ] and [FSP]; a qualified is to be taken fromFSP.

From the set of , it is straightforward to check the successand the failure . Begin with:

FSP set of all failure

Then, the qualified is which has the maximum MCFamong the in FSP, because it adds, to the initial ,a choice of minimal set of links which gives maximal increaseon MCF among all possible choices in FSP. Next, set

FSP FSP

and select the qualified in FSP. If is a success ,then record it, and repeat the process with

FSP FSP

otherwise, repeat the process with

FSP FSP

With the contents and [FSP], each time the qualifiedis taken from FSP, set FSPFSP . There are three cases.

1) : as a success inLIST. Then restart the process with [] and [FSP].

2) : in composition: restartthe process with [ ] and [FSP].

3) There is no choice: to the step where the lastwas added to generate, at which time, (last

) for some . Restart the process with [] and [FSPcorresponding to ].

The whole process is stopped when it is necessary to retreatfurther at . The set of is obtained from LIST byremoving the possible redundancy.

B. Additivity and Eligibility

This subsection introduces the new concepts of additivity andeligibility which are used in the process to check and remove thepossible redundancy effectively. Given: 1 and 2 and

, then and are if .We assume that : the set of links added to by

is nonempty.Definition 1: A has if

.Let have the additivity on , and let there exist awhich adds fewer links to than . Then, belongs to

, but is not in ; this contradicts the additivityof on . Thus, a having the additivity assures itselfto add only a minimal set of new links to when andcombine to generate a new.

Definition 2: For given , a hasif .

A possessing the eligibility on in ensures that noother than those in would be newly introduced to the sub-

network induced by . Suppose that, currently, we have:[ ] and [FSP]. For FSP, would be a redundancyif contains a not in FSP. The reason is that a

would not be in FSP at a given stage, because either it is al-ready a success , or all containing it have been checkedpreviously in the process. With current , generate new

376 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 4, DECEMBER 2001

Fig. 1. Bridge network.

by adding only the possessing both eligibility and additivity.For a , is a . Using set theory, it can be shownthat lemmas #1–#3 hold and each is used in the process as achecking procedure for corresponding property.

Lemma 1: Two and are equivalent on if.

Lemma 2: A has additivity on if there is nosuch that is a proper subset of .

For eligibility checking, remove the having no eligibilityfrom FSP, instead of selecting the with eligibility. ELGSP isintroduced for this purpose. At each iteration, assume that theprocess has: [] [FSP] [ELGSP]. Initially, [ELGSP the set ofall success ]. Select the qualified for ; set FSP FSP

; then find all success starting with on . In doingthis, do not generate the which contain any one of thosein ELGSP. When the process retreats back to, update ELGSPby adding and restart the process.

Lemma 3: A has no eligibility on in FSP if thereexist in ELGSP such that .

Example: Consider the bridge network in Fig. 1. There are 4: (2,3,4), (2,5), (1,4), (1,3,5). Let:

FSP ELGSP

then would not be generated, because (1,3,5) hasno additivity. If:

FSP ELGSP

then two new with are generated, because both (1,4) and(1,3,5) in FSP have eligibility and additivity. If:

FSP

ELGSP

then no new are generated with , because (1,3,5) has noeligibility.

III. A LGORITHM

This section presents the algorithm and exemplifies its useby solving the bridge network of Fig. 1. In checking the prop-erties of additivity and eligibility, one needs only the informa-tion on the links in subpath , not itself, with current

. In the process, keep each in FSP and ELGSP in theform of subpath for saving time and memory. A is an

on if has the eligibility on (in FSP)and the additivity on . The qualified subpath is the one whichgives the maximal increase on MCF among the available ad-ditive subpaths. Here, denotes the number of compositions

forming the current and is used as a bench mark for re-treating.

Specific Notation for Algorithm:levelcurrent at level

ELGSP( ) ELGSP for checking eligibility at levelFSP( ) FSP at level

number of additive subpaths oncurrent additive subpaths on ;set of all containing line

({all links of the network})

A. Algorithm

1. /* Initialization */Set , , FSP(0) {set of all

}, and ELGSP(0) .If , then STOP. Go to 3.

2. /* Check Eligibility and Additivity */Check Eligibility: Set FSP( )FSP( ) for ELGSP( ) .If FSP , then go to 5.Check Equivalence & Additivity: For each

FSP( ), remove all subpaths whichare the same as , but not itself.Then, is an additive subpath if thereis no in FSP( ) which is a propersubset of .

3. /* Compute MCF with Additive Choices*/Compute for each additive sub-path .Arrange all additive subpaths in de-scending order of , and recordthem as s along with its corre-sponding , where . Set

.4. Set FSP( ) FSP( ) .

/* Record Success */If , then [recordin LIST. Go to 6.]/* Advance */If FSP , then go to 5.If , then [if , thengo to 5.]Set ,FSP k ,ELGSP k ,

. Go to 2.5. /* Retreat */

Set . If , then STOP.6. /* Update ELGSP */

If , then go to 5.Remove nonadditive subpaths fromFSP( ), if .Set ELGSP(k) ELGSP(k) , and

. Go to 4.

LEE AND PARK: EVALUATING NETWORK-RELIABILITY WITH VARIABLE LINK-CAPACITIES 377

TABLE IPROCESS FORFIG. 1

TABLE IIPROCESS FORFIG. 1 (CONTINUED)

TABLE IIIPROCESS FORFIG. 1 (CONTINUED)

B. Use of Algorithm

To exemplify the use of LP-EM, consider the bridge networkin Fig. 1. Let [2,6,4,5,3], and . Atinitialization, all are additive. Arrange the additive subpathsin descending order of the magnitude of MCF; then set

FSP nonadditive subpaths

The process is summarized in Tables I–III. The number incolumn #1 indicates the algorithm number in Section III-A;when there is no change, the cell is left unfilled. If the successcolumn indicates , then the corresponding is recordedin LIST. For , only those links in are listed in thetables.

After finishing the processes in Table I, go to #6. Because, go to 5 and set . Retreat to the point where

, and restart the process.After Table II, because FSP(1) , go to 5 and set .

Retreat to the point at and restart the process.After Table III, because , check . Because

, go to 5 and set. Because , the whole process STOPs. The final

list of success contains (2,3,4,5), (1,2,3,4), (1,2,4,5) with noredundancy; thus, all of them are . For the bridge network

of Fig. 1, can be evaluated using the existing methods, forexample [10], as:

When for all , then .

C. Detecting a Failure A Priori

To reduce the number of MCF computations, [8] introducesthe concepts of key_cut and cross_link. They construct thepath_groups by key_cut and detect those- generated withina path_group as failure using cross_link. The cross_linkis the set of links common to the forming , andis detected as a failure if ;

is the capacity of link . However, the idea of cross_linkdoes not work in general. Consider the bridge network inFig. 1. Let [2,6,4,5,3], and . For

, is the cross_link , and isobviously a success 2-, although [8] identifies it as a failure

. Similar concepts are proposed, in conjunction with LP-EM,which always work.

Definition 3: A link is a common_link on if is in forall .

Proposition 1: A is a failure if there exists acommon_link on such that .

Proposition 1 holds, since all in the subnetwork induced byare considered, not just thoseused in composition. On the

378 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 4, DECEMBER 2001

other hand, a subnetwork if ,where .

Definition 4: Link is a complete_link if constitutes asubnetwork.

Proposition 2: A generated within is a failureif is a complete_link such that .

The links which are connected directly to the source node orto the terminal node are complete_links. If () is a minimal cut ofthe network, then is also a complete_link. For a complete_link, let be a obtained by compositions within . Then,

is a common_link on , and Proposition 2 holds. Propositions1 and 2 help detect a failure without computing its MCF andcan be used any time during the process without affecting thelogical flow of the algorithm. For the bridge network (Fig. 1),links 1, 2, 4, 5 are complete_links and among them, 1 and 5 havesmaller capacities than . Then, instead of constructingpath_groups as in [8], check during the process only if link 1 (or5, respectively) is in each of forming .

IV. DISCUSSION

Notation:number of failure of the network;number of subpaths in ELGSP();number of subpaths in FSP();number of subpaths remaining in FSP();number of additive subpaths in FSP().

A. Computational Time Complexity

The performance of LP-EM depends largely on the numberof generated during the process, which in turn relies not onlyon the structure of the network, but also on the link capacitiesand the magnitude of for the given network. To discussthe algorithm’s time complexity, let be fixed, and be thetotal number of for the network.

Let: [ ] [ ] [FSP( )] [ELGSP( )]. Then

because all equivalent and the having no eligibility whichwas checked before reaching have been removed fromFSP( ). Observe the following time complexity for each step.

• If , then retreat and generate no.Otherwise, check the eligibility of each subpath; the timecomplexity for eligibility-checking is .

• Remove the subpaths not having eligibility from FSP().The time complexity for equivalence and additivitychecking is .

• The time complexity involved in computing MCF is. This computing time can be saved by applying

propositions 1 and 2 appropriately.• Arrangement of additive subpaths in descending

order of its MCF requires the time complexity:.

Observe that , and that decreasesfaster as the process moves on. The total number ofgenerated

in the process will never reach , because ischecked.

Remark: In the process, the current additive subpath is thequalified one, and gives maximal increase on MCF by addingto the current . However, there could be more than 1 additivesubpath which gives the same amount of maximal increase onMCF. The following tie-breaking rule can determine the quali-fied subpath among the available additive subpaths.

1) Choose the subpath which gives the maximal increase onMCF.

2) If tied in #1, compare the smallest link capacity ()of each of the tied additive subpaths. The one with thegreatest is chosen as the qualified subpath. If theyare tied again, then compare the next of each subpath,and so on.

If the numbers of links in the tied additive subpaths are dif-ferent, then the one with the smaller number of links is chosen.Such arrangement is not essential, but is helpful in further re-ducing redundant contained in LIST at the completion of theprocedure.

B. Comparison With Other Methods

The existing composite path enumeration techniques in [2],[6], [8] are based on the concept of forming- , .At each order , they generate all- possible, check for suc-cess, remove the redundant, and then generate -from the set of failure - . As a result, the redundancy oc-curs not only within the same order but also between orders,and the redundancy between orders possibly causes the redun-dancy among the success throughout the process. Conse-quently, many redundant are generated in the process evenwhen the network has moderate number of links. The four ad-vantages of LP-EM over other existing methods are given here,and the methods numerically compared bythree examples.

Advantages:

1) The information on how many and which have beenused in each are not needed; hence, a routine just forthe generation of new is not necessary.

2) Generate new using only additive subpaths which pos-sess the eligibility and additivity, and only when there isa possibility of with the current by checking

. Thereby, a much smaller number of aregenerated.

3) Compute MCF of after removing the possible cause ofboth duplications and absorptions; hence, the number ofMCF computations is further reduced.

4) The process of checking the redundancy is essentially thesame for LP-EM and the existing methods, but LP-EMdeals with smaller sets, each element of which containsfewer links. An additional job needed in LP-EM is to sortthe additive subpaths in descending order of MCF; it canbe lightened by recording the successin LIST whilecomputing MCF.

The three examples: a) do not check at initialization,and b) apply the tie-breaking rule of Section IV-A only oncewhen the given are arranged right after initialization.

LEE AND PARK: EVALUATING NETWORK-RELIABILITY WITH VARIABLE LINK-CAPACITIES 379

TABLE IVNUMBER OFcp FOR COMPUTING MCF

Example 1: For the bridge network of Fig. 1, the are:, , , , the superscript denotes its

MCF. When , the number of generated duringLP-EM is 6, including 2 for checking . The 2-(2,3,4) (1,3,5) (1,2,3,4,5) would not be generated at allduring the process; but other existing methods generate it, com-pute its MCF and remove it as a redundant success 2-, andthe same would be generated again as 3-.

Example 2: Consider a parallel network of 10 links, witheach link capacity 1. For , LP-EM generatesand computes MCF for 53 , including 8 for checking

, whereas other methods generate and compute MCFfor at least 1014 .

Example 3: Table IV compares LP-EM with that of [8] interms of the total number of , of which the MCF needs to becomputed for the 5-node, 8-link network, in [8: Fig. 4]. The linkcapacities of the network are

linkslink capacity

For the network, the are: (2,3,7,8) (2,5) (2,4,8) (2,3,6)(1,7,8) (1,3,5) (1,3,4,8) (1,4,5,7) (1,6) . For this compar-ison, LP-EM uses Propositions 1 and 2 in Section III-C and [8]applies the concepts of key_cut to reduce the number of MCFcomputations. The number in parenthesis in the first column ofTable IV denotes the number of resulting for the givenvalue of .

When , each of (2,3,7,8), (2,5), (2,4,8), (2,3,6) isa success at ; thus, check

. Because each of the remaining failurecontainslink 1 which is a complete_link such that , concludethat by Proposition 2 without computing;thus, retreat and stop the process. Therefore, there is no compu-tation of MCF in this case. As Table IV shows, LP-EM greatly

reduces the number of MCF computations, even when there ex-ists a key_cut.

ACKNOWLEDGMENT

The authors are pleased to thank the referees and AssociateEditor, Dr. Chien, for their valuable comments on an earlier ver-sion of this paper.

REFERENCES

[1] K. K. Aggarwal, “A fast algorithm for the performance index of atelecommunication network,”IEEE Trans. Reliability, vol. 37, pp.65–69, Apr. 1988.

[2] K. K. Aggarwal, Y. C. Chopra, and J. S. Bajwa, “Capacity considerationin reliability analysis of communication systems,”IEEE Trans. Relia-bility, vol. 31, pp. 177–181, June 1982.

[3] L. R. Ford and D. R. Fulkerson,Flows in Network: Princeton UniversityPress, 1962.

[4] K. Kyandoghere, “A note on: Reliability modeling & performance ofvariable link-capacity networks,”IEEE Trans. Reliability, vol. 47, pp.44–48, Mar. 1998.

[5] S. H. Lee, “Reliability evaluation of a flow network,”IEEE Trans. Re-liability , vol. 29, pp. 24–26, Apr. 1980.

[6] K. B. Misra and P. Prasad, “Comments on: Reliability evaluation of aflow network,” IEEE Trans. Reliability, vol. 31, pp. 174–176, June 1982.

[7] L. Y. Qiu and C. H. Zhong, “A new algorithm for reliability evaluation oftelecommunication networks with link-capacities,”Microelectron. Reli-ability, vol. 34, pp. 1943–1946, 1994.

[8] S. Rai and S. Soh, “A computer approach for reliability evaluation oftelecommunication networks with heterogeneous link-capacities,”IEEETrans. Reliability, vol. 40, pp. 441–451, Oct. 1991.

[9] R. Schanzer, “Comment on: Reliability modeling and performance ofvariable link-capacity networks,”IEEE Trans. Reliability, vol. 44, pp.620–621, 1995.

[10] P. K. Varshney, A. R. Joshi, and P. L. Chang, “Reliability modeling andperformance evaluation of variable link capacity networks,”IEEE Trans.Reliability, vol. 43, pp. 378–382, Sept. 1994.

Seung Min Leereceived the B.S. degree (1976) in computer science and sta-tistics from Seoul National University, and the Ph.D. degree (1987) from StateUniversity of New York at Stony Brook.

He is Full Professor at Hallym University. His research areas include networkreliability, quality control, and probability theory.

Dong Ho Park received the B.S. degree (1968) in allied mathematics fromSeoul National University, and the M.S. (1980) and Ph.D. degrees (1982) instatistics from Florida State University.

He is Full Professor at Hallym University. He was an Associate Professor atUniversity of Nebraska until 1995, and is President of the Korean ReliabilitySociety. He has published research on reliability theory, accelerated life testing,and nonparametric statistics in a variety of international journals.

Dr. Park is a Member of ASA, IMS, ASQ, and the IEEE.