IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 4, DECEMBER 2004 481

Sequential Capacity Determination of Subnetworksin Network Performance Analysis

Seung Min Lee, Member, IEEE, Chong Hyung Lee, and Dong Ho Park, Member, IEEE

Abstract—In the process of performance evaluation for a sto-chastic network whose links are subject to failure, subnetworks arerepeatedly generated to reflect various states of the network, andthe capacity of each subnetwork is to be determined upon genera-tion. The capacity of a network is the maximum amount of flowwhich can be transmitted through the network. Although thereare existing algorithms for network capacity computation, it wouldcreate a great number of repetitions to compute the capacity ofeach subnetwork anew upon generation in the process. This is truebecause subnetworks are generated by combining certain links tothe current one, and hence each current subnetwork is embeddedin those new subnetworks. Recently, a number of methods havebeen proposed in the context of searching a method which effi-ciently computes the capacity of subnetworks by utilizing the giveninformation of minimal paths, and preferably without many rep-etitions in sequential computations. But, most of the methods stillhave drawbacks of either failing to give correct results in certainsituations, or computing the capacity of each subnetwork anewwhenever the subnetwork is generated.

In this paper, we propose a method based on the concepts ofsigned minimal path, and unilateral link, as defined in the text.Our method not only computes the capacity of each subnetworkcorrectly, but also eliminates the repetitive steps in sequential com-putations, and thereby efficiently reduces the number of subnet-works to consider for capacity computation as well. Numerical ex-amples are presented to illustrate the method. The drawbacks ofother methods are also discussed with counter examples.

Index Terms—Network capacity, signed minimal path, subnet-work, unilateral link.

ACRONYMS1

mp minimal pathsmp signed minimal pathsul saturated unilateral link

NOTATION

set of all links of the networka linka signed link,1 if , and 1 otherwisea mpa smp

Manuscript received May 8, 2001; revised April 2, 2002; June 26, 2003; andSeptember 23, 2003. This work was supported by a Research Grant from HallymUniversity, Korea. Associate Editor: W.-T. K. Chien.

S. M. Lee and D. H. Park are with the Department of Statistics, Hallym Uni-versity, Chunchon 200-702, Korea.

C. H. Lee is with the Department of Computer Engineering, Konyang Uni-versity, Nonsan 320-711, Korea.

Digital Object Identifier 10.1109/TR.2004.837306

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

set of unilateral links of the networkCapacity vectorflow vectora subnetworkset of all smp of subnetwork

I. INTRODUCTION

ANETWORK is modeled as a graph , which con-sists of a set of nodes, and a set of links. Each link

of the network may have different capacity, and the network isrequired to transmit a specified amount of flow from the sourcenode to the terminal node. In this case, performance of the net-work is not necessarily characterized by connectedness only, butby the amount of maximum capacity flow that can be trans-mitted through the network. The network reliability, which isone of the performance measures for a network, is the proba-bility of transmitting the required amount of flow successfullyfrom the source node to the terminal node. The performanceindex of the network is essentially the same as the -expectedvalue of the maximum capacity flow of the network. A diffi-culty in evaluating network performance stems from the com-plexity of capacity determination for numbers of subnetworksgenerated in the process. In the evaluation process, a sequence ofsubnetworks are generated in succession one by one by addingcertain links to the current one, and then its corresponding ca-pacity is computed for each subnetwork generated. See [1], [2],and [8]–[13] for references. We may prefer a method which ef-ficiently computes the capacity of each subnetwork by utilizingthe given information of mp, and without many repetitions insequential computations. Recently, a number of methods havebeen proposed in this context. But most of the methods still havedrawbacks of either failing to give correct results in certain sit-uations, or computing the capacity of each subnetwork anewwhenever the subnetwork is generated.

Reference [2] generates subnetworks by combining the rowsof a path matrix, and determines the capacity of a subnetworkby taking into account the common links to those rows beingcombined. But the method [2] leads to incorrect results whenthe subnetwork contains more mp other than those involved incombination. Reference [1] points out that the method [2] lacksgenerality, and modifies it by utilizing a vector, each elementof which is the remaing capacity of the corresponding link. Butthe method [1] may produce different results on different orderof mp being applied, and hence fails in general to give correctresults. Reference [12] discusses the drawbacks of the methodsin [1] and [13] with some counter examples, but does not pro-vide any procedure of capacity determination explicitly. Refer-ence [11] also presents the counter examples to show that the

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482 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 4, DECEMBER 2004

methods in [1] and [13] fail, and correctly computes the capacityof each subnetwork. But the method [11] needs additional infor-mation on minimal cuts of the network, and has some drawbacksin sequential computations which may cause incorrect results,as discussed in [9]. Recently, [7] proposes a method based onmaximal-residual-capacity to complement the methods in [1]and [13], but also yields incorrect results in certain situations,as shown by a counter example given in Section IV-B.

In this paper, we propose a method based on the new conceptsof smp, and unilateral link defined in the text. At each iteration,we select a smp which contains the fewest number of links, ad-just the current flow vector by utilizing the concept of smp, andthen exclude unnecessary smp by applying the concept of uni-lateral links. The next selection is made only from the set of re-maining smp, and thereby, the number of iterations needed forthe process would be reduced. This effect would be magnifiedespecially in sequential computations because we remove un-necessary smp, not only from the set of smp for the current sub-network, but also from the set of remaining smp to be possiblyadded later on to the current subnetwork. In Section II, we pro-vide the concepts, and definitions for smp, and unilateral link.Based on the concepts, we present an algorithm for capacity de-termination of any given subnetwork, and also discuss the casewhen the original network contains some directed links. Sec-tion III gives the detailed descriptions on the methodology andalgorithm for sequential computations utilizing the methods de-veloped in the previous section. Some numerical examples arepresented as well. In Section IV, we discuss the computationalcomplexity of our method, and also discuss the drawbacks ofother methods.

II. CAPACITY DETERMINATION

Assumptions:

1) The nodes are perfect, and each has no capacity limit.2) All the links are undirected, and each link flow is

bounded by the link capacity.3) No flow can be transmitted through a failed link.4) The mp of the network, considering connectivity only,

are known.

A. Signed Minimal Path and Unilateral Link

A mp is a minimal set of links connecting the source node, and the terminal node . Given a link , let and be

two incident nodes connected by . We distinguish two possibledirections of , and on by ‘ ’, and ‘ ’. Similarly,we distinguish two edges , and by , and . Forexample, suppose that we use ‘ ’ as for link , and hence,use as . Then, the flow of amount 2 on link is denotedby if it is moving , and by otherwise.Similarly, let be a mp which contains link . In traversing from

to on , link in uniquely appears as one of two edges,or . Then, the link in is converted to if it

appears as , and to otherwise.Definition 1: The set of signed links obtained from is

said to be the smp corresponding to .Definition 2: A link is said to be unilateral, if the signs of

are the same in all smp containing the link.

Fig. 1. Bridge network.

Converting each mp to its corresponding smp one by one inany order, we obtain the set of all smp for the network; and thereis a one-to-one correspondence between the given set of mp, andthe set of smp. We note that this conversion is simply regardingeach mp as an edge sequence from to , rather than just a set oflinks. For convenience, we always convert the first encounteredin the process to . Suppose that appears for the first timewhen we convert , and that appears as in .Then, we convert to in the smp corresponding to , andthroughout the process, link , which appears asin a mp, would be converted to in its corresponding smp. Wenote that the sign of a unilateral link would be always ‘ ’, andhence, if appears in the process, link is not a unilateral link.The ‘ ’ sign may be omitted for both flow and link.

Example 1: Consider the bridge network shown in Fig. 1.The mp of the network are given as: (1,4), (2,5), (1,3,5),(2,3,4). Then, the corresponding smp are: (1,4), (2,5), (1,3,5),(2, ,4). We note that all links, except 3, are unilateral, i.e.,

. Given , a flow of amount1 moving along smp (2, ,4) would change the current flowvector to .

B. Capacity Determination With Signed Minimal Paths

Let be the set of all smp of the network, and letbe a link set. The subnetwork is meant to be the subnetworkinduced by , and the capacity of to be the capacity of sub-network . Given a smp , we denote ifimplies . Here, indicates that is a smp of thesubnetwork . The computation of capacity of is carried outwith , where . Given a flow vector

, and a smp , we compute by

where

where is the maximum amount of additional flow that can besent from to by , and is the remaining capacity of link

in direction . The net flow is augmented by ; and then, foreach , we adjust by

Note that each , , is always nonnegative, and hence, sois . The process stops when there is no more smp such that

, and the current value of net flow is the capacity of thenetwork. To reduce the number of iterations, we may as wellchoose the shortest smp at each iteration. A smp with the fewestnumber of links will be referred to as the shortest smp in thesequel. Treating each smp as a possible augmenting path, theabove process resembles the existing shortest augmenting pathalgorithm [3]. See also [4]–[6] for references.

Here, we employ a flow vector which contains the infor-mation on both the amount of flow, and its moving direction on

LEE et al.: SEQUENTIAL CAPACITY DETERMINATION OF SUBNETWORKS 483

each undirected link; and then, utilize the concept of smp to de-termine the capacity of the network.

C. Unilatral Links in Capacity Determination

At each iteration in the above process, we augment the netflow along a smp by the amount of . On certain circum-stances, however, we can identify the smp, which could neveraugment the net flow without computing , and hence, excludethem from further consideration. Let be a unilateral link, i.e.,

. During the process, remains nonnegative, and neverdecreases. Once is saturated, i.e., , then remains sat-urated thereafter; and hence, any smp containing could neveraugment the net flow. Excluding such smp from further consid-eration, we reduce the number of iterations necessary for thecompletion of the process. This effect would be magnified es-pecially in sequential computations of the next section, becausewe exclude unnecessary smp not only from the set of smp forthe current subnetwork, but also from the set of remaining smpto be possibly added later on to the current subnetwork. A sulmeans a saturated unilateral link throughout this paper.

The whole process of capacity determination is describedbelow. At each iteration in the algorithm, we select the shortestsmp, and exclude smp containing sul from consideration. If thereare more than one smp of the same number of links, we may aswell select the one which saturates more of unilateral links.

Specific Notation for Algorithmvalue of net flowset of available smp

Algorithm: CAPACITY

1. Initialize: , , and.

Arrange all smp in in ascendingorder of its number of links, and tryone by one from the shortest.

2. Given smp , compute , and set.

For each , [adjust . If isa sul, then delete smp containingfrom .]If there is no more smp to try,

then STOP; else go to 2.

Example 2: Consider the bridge network in Fig. 1, which hasfour smp: (1,4), (2,5), (1,3,5), (2, ,4), and .Let the capacity vector be given as . Here,we show how our algorithm works by computing the capacityof subnetwork , i.e., the whole network. At each iteration, wehave the contents: . The process is sum-marized in Table I, and the capacity is computed as 20.

D. Directed Links

In graph theory terminology, two nodes and are said to beconnected if there is a path, not necessarily a directed path, be-tween them. Hence, mp of the network would be given the samewhether there are directed links or not in the network. Suppose

TABLE ICAPACITY DETERMINATION FOR BRIDGE NETWORK

that the network contains some directed links, and let be theset of directed links. In converting mp to smp, we assign ‘ ’ forthe given direction on . For example, let be directed as

in the network. If appears asin a mp, then we convert it to in the corresponding smp. Oth-erwise, convert to . Converting each mp to its correspondingsmp, we obtain and . Let a flow vector and a smp begiven. In computing for each , we need a slight modifi-cation if , because negative flow is not allowed on .We modify: if , and '.

If ' for , then remains the same as in the algo-rithm given in Section II-B. Furthermore, we need not modify

if is unilateral. We note that a directed link can beunilateral in the network with ‘ ’ sign, whereas an undirectedunilateral link is always with ‘ ’ sign. Let denote the setof unilateral links with ‘ ’ sign. For , remains zerothroughout the process, and hence, any smp containing couldnever augment the net flow. Such smp could be removed from

at initialization of the process. For , always ap-pears with ‘ ’ sign in the process, and hence, whether directedor not, it behaves exactly the same way in capacity determina-tion. Given a mixed network with a set of directed links, wesummarize the above arguments as follows.

Converting each mp to its corresponding smp, we obtain ,, and as well. Set , i.e., is the

set of directed links which are not unilateral. Setcontains and set . Given a

link set , we determine the capacity of with the algorithmCAPACITY computing as

where if and

otherwise

Note that needs to be modified only when , , appearsin the process. All other steps in CAPACITY remain thesame.

Example 3: For all of the bridge networks (a), (b), and(c) in Fig. 2, mp are given the same as: (1,4), (2,5), (1,3,5),(2,3,4). For network (a) with , we obtain

, and. Because , the process of capacity determina-

tion for any subnetwork of network (a) is exactly the same asthat of its undirected version in Fig. 1. For network (b) with

, we obtain ,, and . We proceed with

, and . Link 1 neverappears in the process. For network (c) with , we ob-tain , ,and . We proceed with , , and . Wemodify only when (2, ,4) is selected.

484 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 4, DECEMBER 2004

Fig. 2. Bridge networks with directed links.

Fig. 3.

III. SEQUENTIAL COMPUTATIONS

In this section, we present the detailed description of themethodology and algorithm for sequential computations, whichutilizes the methods described in the previous section. Givenand , we suppose that a new subnetwork is formed bycombining those links in to the current subnetwork . Wemay compute the capacity of the new subnetwork by applyingthe algorithm CAPACITY . Such a process, however,should make a number of repetitions if CAPACITY hasalready been processed, because . We note thateach current subnetwork is embedded in the new subnetworksgenerated by combining certain links to the current one. Toavoid such repetitions, we re-construct the process to determinethe capacity of , preparing for sequential computations.Setting and , suppose that we initiallycompute the capacity of . At each iteration, we choose theshortest smp in , and remove smp containing sul from . Wealso remove smp containing sul in , because those smp wouldnot augment the net flow even though they are added later onto . On completion of the process for , we have the contentsfor , where is the capacity of ,

is the resulting flow vector, and is the resulting setin which all smp containing sul have been removed. For ,we define .Given the contents of , we proceed to compute the capacityof by adding, to , only those smp in which arenewly introduced, where .Setting and , we restartthe process of augmenting the net flow with smp in . Therestarting process is presented as the algorithm COMPOSITE( , ) below.

A. Algorithm

Specific Notation for AlgorithmSet of smp newly introducedSet of remaining smp

Algorithm: COMPOSITE ( , )Given contents for .

1. / No increase in capacity /If , then [no increase pos-

sible on in capacity. RETURN.]

Set .If , then [ has the same

contents as . RETURN.]/ Initialization /Set , , ,

.Arrange all smp in , in ascending

order of its number of links, and tryone by one from the shortest.

2. / Augment net flow /Given smp , compute and set

.For each , [adjust . If is

a sul, then delete smp containingfrom and ].If there is no more smp to try,

then go to 3; else go to 2.3. / Prepare contents for /

Set , , and.

RETURN.

B. Use of Algorithm

In the process of evaluating network performance, subnet-works are often generated as unions of mp [1], [2], [9]–[13].Let be the mp of the network, and be theindex set of positive integers. We denote by the subnetworkinduced by those links in . Consider the bridge networkin Fig. 1. The mp are: , , ,

; and the corresponding smp are: ,, , . To exemplify the

use of the algorithm, we sequentially compute the capacitiesof subnetworks , , , and . To compute thecapacity of , we apply the algorithm COMPOSITE ( , )with the contents:

. We get, and hence, we proceed with , and

. The process is summarizedin Table II. We get the contents for

. The capacityof , , is 10. To compute the capacity of , we applythe algorithm COMPOSITE ( , ). We get ,and proceed with and . The process issummarized in Table III. The contents for are producedas:

; and the capacity of is computed as 20. At thisstage, since , there is no increase possible on in

LEE et al.: SEQUENTIAL CAPACITY DETERMINATION OF SUBNETWORKS 485

TABLE IIPROCESS FOR L WITH L = ;, AND L = P

TABLE IIIPROCESS FOR L WITH L = L AND L = P

TABLE IVPROCESS FOR L WITH L = ;, AND L = P

TABLE VPROCESS FOR L WITH L = L AND L = P

capacity, and we can conclude that the capacities of andare the same as that of , i.e., 20.

To illustrate various possibilities of sequential computationswhich might appear in the process of evaluating networkperformance, we may also consider the case of computingthe capacity of , and then that of . To compute thecapacity of , we apply the algorithm COMPOSITE ( , )with the contents:

. We get, and hence, proceed with ;

and . The process is sum-marized in Table IV, and the capacity of is 1. Note thatthere is no sul identified. To compute the capacity of ,we apply the algorithm COMPOSITE ( , ) with contents:

. We get ,and proceed with and

. Note that is itself the whole network. The processis summarized in Table V, and the capacity of is computedas 20.

IV. DISCUSSION

A. Computational Time Complexity

Notationnumber of nodes in the network

number of sul identified in the processnumber of smp in the networknumber of smp in

The algorithm COMPOSITE ( , ) utilizes , and , whichare obtained from the set of mp of the network; and becausethe number of links in each mp is at most , and canbe provided in time. The performance of COMPOSITE( , ) depends largely on the number of smp in , and on howmany and which links are identified as sul in the process, whichin turn relies not only on the structure of the network, but alsoon the link capacity.

We obtain in time, because each smp in canbe tested for being a subset of in constant time usingset theoretic operation. At initialization, arrangement of smp inascending order of its number of links requirestime. The number of iterations is at most , and the time com-plexity at each iteration is:

• computing and adjusting in time, and• removing smp containing sul in time.

Observe that , and that the time complexity ofthe procedure is in the worst case. We note that thegreater is, the faster and are likely to decrease;and consequently, the number of iterations would be reducedfor the current process as decreases, and for the forthcomingsequential processes as decreases.

B. Other Methods of Capacity Determination

Let be the mp of the network, and bethe index set of positive integers. Given a subnetwork

, we denote . References[1], [13] essentially compute the capacity of as follows.

Start with , and . At each iteration, find a mpin such that , where . Next, set

, and for each , adjust . If thereremains no such in , then the value of is the capacity of

.We point out that there are two main drawbacks in the above

procedure.

d1. The method considers the mp in only. However, ,as a subnetwork, may contain more mp other than those in

.d2. is nonincreasing in the procedure, and hence, once

, then thereafter for any containing .It means that all links are treated as if they are unilateral,which may not be true in general.

The drawback d1 is discussed in [11], [12]. Considering allmp contained in , i.e., , the methods still need toarrange mp in a certain order to derive the correct result. Refer-ence [7] suggests an order based on maximal-residual-capacity,which selects the longest mp first when each link has the samecapacity. But, such an arrangement fails to obtain the correct re-sult for the network shown in Fig. 4 because of the drawbackdescribed in d2. We also note that applying the shortest mp firstdoes not work for the same network either.

Reference [11] correctly computes the capacity of , butneeds additional information on minimal cuts of the network.To reduce the number of subnetworks to consider for capacity

486 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 4, DECEMBER 2004

Fig. 4. 10-link bridge network with ccc = (1; 1; � � � ; 1).

determination in sequential computations, [11] introduces theconcepts of key_cut and cross_link. But, the key_cut checksonly unions of two mp, and it may not even exist. Further, thecross_link checks a link common to mp in only, and hence,it does not work in general due to d1 as discussed in [9].

On the other hand, our method not only computes the capacityof correctly, but also eliminates the repetitive steps in se-quential computations. Furthermore, our method computes thecapacity of new subnetworks only when necessary by checking

and beforehand, and thereby, efficiently reduces thenumber of subnetworks to consider for capacity determination.

ACKNOWLEDGMENT

The authors would like to thank Associate Editor, Dr. K.Chien, and the referees for their valuable comments on earlierversions of this paper.

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Seung Min Lee received the B.S. degree (1976) in Computer Science and Sta-tistics from Seoul National University, and the Ph.D. degree (1987) in Statisticsfrom State University of New York at Stony Brook. He is a full professor atHallym University. His research interests include probability models, reliabilitytheory and its applications.

Chong Hyung Lee received his B.S. (1993), M.S. (1995), and Ph.D. (2001) inStatistics from Hallym University and was a post-doctoral researcher (2001) atSeoul National University. He is an assistant professor at Konyang University.His research areas include software reliability, system maintenance policy, andnetwork reliability.

Dong Ho Park is a full professor at Hallym University. He received his B.S.(1968) in Applied Mathematics from Seoul National University, and M.S.(1980) and Ph.D. (1982) in Statistics from Florida State University. He was anassociate professor at the University of Nebraska-Lincoln until 1995. He haspublished on reliability theory, accelerated life testing, life distributions andits applications, software reliability, and system maintenance policy in a widevariety of journals. He is a member of ASA, IMS, ASQ, ISI and IEEE.