On the multidensity gateway location problem for a multilevel high speed internetwork

On the multidensity gateway location problem for a multilevelhigh speed internetwork

Debashis Sahaa, Amitava Mukherjeeb

aComputer Science and Engineering Department, Jadavpur University, Calcutta 700 032, IndiabPrice Waterhouse Associates India Pvt Ltd., Salt Lake, Calcutta 700 091, India

Received 10 March 1995; accepted 19 April 1996

Abstract

This paper addresses the problem of locating the concentrator gateways of different densities in a high speed interconnection of networksthat are connected in a hierarchical fashion. The organization of gateways can be viewed as an uprooted tree, where gateways would beconnected to each other in a hierarchy. Gateways at different levels of the tree will have different densities, and the higher is the level, thegreater is the density. Such a multidensity gateway facilitates the sharing of a high speed, high capacity intergateway link among multiplelocal networks. The objective of this work is to minimize the cost of setting up plus the cost of operating multidensity gateways in aninternetwork, subject to some capacity constraint. A mathematical programming model of the problem is developed, and a subgradientheuristic is used to develop a solution procedure for the model. The algorithm is efficient, and produces near optimal solutions always.Extensive simulation studies were conducted to test the performance of the heuristic. Internetworks, consisting upto thousands of localnetworks and hundreds of potential gateways locations, were considered to verify the algorithm.q 1997 Elsevier Science B.V.

Keywords:Computer network; Multidensity gateways; Internetwork; Hierarchical structure; Clustering; Subgradient heuristic

1. Introduction

Internetworking has emerged in the past few years as animportant alternative to Wide Area Networks (WANs). Ingeneral, networks are connected via gateways [1] which canbe logically viewed as adapters for internetwork communi-cations. In a situation where multiple (say hundred orthousand) networks are to be connected to form an inter-network, a common solution is to identify a gateway foreach of these networks in the first step, and then to inter-connect these gateways using a suitable topology to form asupernet [2] of gateways in the second step.

1.1. Motivation

The most common topology for supernets is the treestructure. This kind of hierarchical design has been foundto be cost-effective for a single network [3] and has alreadybeen employed in practice for internetwork design too [2].However, for the multilevel design of internetworks, verylittle optimization study has been made thus far regardingthe allocation of gateways. But, with the advent of high-speed and high-capacity intergateway links, there is anincreasing possibility of concentrating traffics from two or

more networks into a single gateway before forwarding thecollected traffic to a remote gateway (and vice versa). Sucha gateway, equipped with a facility for traffic concentration,is termed as aconcentrator gatewayin this paper. Again,concentration in a gateway can be of various densities (ordegrees), when internetworking is done in a hierarchicalfashion. Thus, the motivation behind the proposed designphilosophy can be simply stated as follows.In an inter-network design, usually, one network is assigned per gate-way, but a high speed gateway can be made to handle trafficto/from more than one network, provided its capacitypermits it to do so. This, in turn, reduces the number ofgateways to be opened for an internetwork considerably,making the complete design more cost-effective. This ideaof optimally assigning multiple networks to a singlegateway in a hierarchical structure leads us to conceive ofmultidensity concentrator gatewaysin the present work.Since concentration will be an integral part of every multi-density gateway, the term concentration will not be usedexplicitly any more in the rest of the paper. Still, it will beimplied that a concentration facility is always built-in with amultidensity gateway.

In remainder of the paper, we describe a model for inter-connecting local networks throughmultidensity gateways,

Computer Communications 20 (1997) 576±585

0140-3664/97/$17.00q 1997 Elsevier Science B.V. All rights reservedPII S0140-3664(97)00016-9

where each gateway serves more than one adjacentnetworks.Multidensity gatewaysare used to facilitate thesharing of a high capacity intergateway line among multiplenetworks. Due to the strong economies of scale shown bytele-communication tariffs, this approach of assigningmultiple networks to a single high performance gatewaygenerally results in solutions far superior to the approachof assigning one network per gateway (where gateways areconnected in an arbitrary mesh topology). This is analogousto the case where remote terminal sites (clients) are con-nected to a central site (server) using concentrators ratherthan using point-to-point links from each terminal to thecentral site [4]. This also follows from the concept of theexisting telephone network. A telephone network usuallycontains an extremely large number of subscribers locatedat various locations in a large city. These subscribers arenaturally partitioned into a number of groups. A telephoneexchange (similar to a gateway) is allocated to each of thesegroups such that a subscriber belonging to one exchangemay communicate with a subscriber to another exchangevia trunk lines (similar to intergateway high speed lines)connecting the exchanges. The justification of this schemeis that the traffic among closely situated subscribers aremuch more than that between distantly located subscribers[5]. The number of trunk lines between a pair of exchangesdepends on the volume of the traffic. That is, a gateway(telephone exchange) is assigned to multiple networks (sub-scribers), and high-speed intergateway links are used forinternetwork communications.

1.2. Objective

It should be emphasized that the problem of inter-networking is usually an evolutionary one in which theinternetwork is continually expanded by adding new net-works and gateways. Usually, each gateway has a capacitylimit on the traffic that it can handle. The objective ofgateway location problems is to minimize the total cost bychoosing an appropriate supernet topology (i.e., number andlocation of gateways and their interconnection structure).The selection of link capacities is considered because highcapacity fiber links may not always be used due to the highcost of laying cables. Although the problem is supposed tobe faced by a network the present model can also be viewedas an integral part of a decision support system for networkmanagers too.

The problem can be expressed as follows.Given the geo-graphical distribution of local networks and the capacitiesand potential locations of gateways, the problem is: (1) todetermine the number and locations of gateways by cluster-ing the given networks based on their locality of geographi-cal distribution, and(2) to assign networks to gateways at aminimum cost, subject to the capacity constraint of indi-vidual gateways.This minimum cost refers to the totaldesign cost which is sometimes referred to as the costfunction of the problem in this paper.

1.3. Previous works

Recent works on internetworking [1,2] have addressedthe issue from a different viewpoint. Our methodologydeviates from the earlier approaches right from the formula-tion of the problem as an optimization problem. However,similar design problems in the context of a single network,consisting of remote terminal sites connected to a centralsite using concentrators, have been addressed in [4–10].Almost all of these studies describe heuristic procedures,and some of them have considered hierarchical structures.An important feature of most of these designs [5–10] is theidea of forming clusters of terminals and allocating oneconcentrator for each cluster. A similar design philosophyhas already been successfully applied to the topologicaldesign of a single hierarchical network in [3]. This successmotivates us to think of multidensity gateways in anenvironment of multilevel (hierarchical) design of inter-networks. So far, internetworking literature [2,11,12] lacksin such design approaches where multidensity gateways areconsidered for the hierarchical design of an internetwork.

1.4. Proposed work

In this paper, we address the problem of – (1) clusteringthe local networks into several groups, and (2) determiningthe gateway locations optimally for each cluster – in multi-ple levels, where the gateway connection to a portion of theinternetwork (or, partial internetwork) could occur at anylevel of the access structure (Fig. 1). This kind of tree basedaccess structure permits economies not only in design butalso in network management [4]. A gateway of density ncaters to all partial internetworks upto leveln. Partial inter-networks are referred to by their positional level in the treestructure (for example, 1st level internetwork, 2nd level

Fig. 1. Hierarchical gateway location structure.

577D. Saha, A. Mukherjee/Computer Communications 20 (1997) 576–585

internetwork and so on) for the ease in understanding. Thus,a gateway of density 3 serves to one 1st level internetwork,as well as one 2nd level internetwork (which may containmore than one 1st level internetworks). Therefore, in effect,a gateway of densityn (n . 1) serves for several 1st levelinternetworks, and a gateway of density n usually serves tomore 1st level internetworks than that a gateway of density(n ¹ 1) serves to, wheren . 1.

The following input data are assumed to be available forour problem:

• number and distribution of networks to be connected;• number and location of potential gateway sites;• traffic originating at each network;• set of allowable capacities and associated costs for the

gateways;• costs for linking network nodes to the potential gateway

sites;• costs of connecting gateway sites to each other.

A mathematical programming [13–18] formulation forthis problem is presented in this paper. The formulation issimilar to the well-known budget-constrained topologicaldesign problem, involving a set of terminal location(nodes) which are to be connected to a single central facilitydirectly or indirectly. This type of problem has been shownto be analogous to the classical design problem of capaci-tated minimal spanning tree by Gavish [9]. Papadimitrion[19] has already proved that the general problem of capaci-tated minimal spanning tree is NP-complete [15]. Therefore,due to the NP-completeness property of the presentproblem, it is unlikely that a polynomially boundedalgorithm will be developed for the internetwork designproblem proposed in this paper.

Since the design analog of this problem belongs to theNP-complete class of problems [9,19], it is desirable todevelop an effective heuristic procedure, that terminateswithin a reasonable amount of computing time, in order todeal with problems of significant sizes. This paper presents aheuristic methodology that comprises Lagrangean relaxa-tion [13,17] along with an efficient subgradient based pro-cedure [14,18] in which the tight bounds are obtained on theoptimal solution of the problem. The solution quality of theproposed heuristic is estimated by calculating the gapbetween the primal solution value and the Lagrangeanlower bound [17]. This gap is a conservative measure forthe performance of the solution procedure, since it is neverless than the gap between the heuristic and the optimalsolution values.

1.5. Organization of the paper

In Section 2, an overview of the proposed methodology ispresented in short, and the clustering algorithm is described.In Section 3, the model of the optimization problem isformulated. A Lagrangean relaxation of the problemis then described, and an efficient solution procedure is

presented in Section 4. Next, a heuristic procedure tosolve the original problem is outlined, and results of com-putational experiments are reported in Section 5. Finally,some concluding remarks are given in the concluding sec-tion. A list of variables used throughout this paper is foundin Appendix B.

2. Overview of the algorithm

We present a brief sketch of the complete designmethodology here. It consists of the following eight stepsof which steps 1 through 5 are executed repeatedly until wereach the root of the gateway access tree. The steps aregiven below:

Step 0. Initialization: k ¼ 1.Step 1. Find out the centre of gravity [3] for each network.Step 2. Run the clustering algorithm (to be discussed later on) on the set

of networks, based on the concept of centre of gravity, to formkth level internetworks.

Step 3. Find out the centre of gravity of eachkth level internetwork. Infact, the clustering algorithm is so designed that It also finds thecentre of gravity of the cluster formed simultaneously.

Step 4. Apply the optimization technique to select the optimum numberof gateways to be opened, considering one gateway perinternetwork. Neglect other gateway locations.

Step 5. Connect all the networks in akth level internetwork to thegateway opened for that internetwork. Call these gateways asgateways of densityk.

Step 6. If there is only one internetwork, then stop, else continue.Step 7. Consider eachkth level internetwork as a single network and the

gateways of densityk as potential gateway locations for the(kþ 1)th level internetwork.

Step 8. Go to step 1.

2.1. Clustering algorithm

A major contribution of this paper is the presentation of anovel clustering algorithm that constitutes the Step 2 of thecomplete design procedure described above. In order todescribe the clustering algorithm formally, we further definethe following notations.

1. A node set Ci is described by the set of nodes{ ni1,ni2,nim}, where ni1,ni2,…,nim [ N such thatCi ∩Cj ¼ f for all Ci,Cj.

2. Massof a node setCi is defined by its cardinalitylCil.3. Centre of gravityof each node setCi is denoted by (xi,yi).

We assume that a single node of masslCil is located at(xi,yi).

4. Distancebetween a pair of node setsCi andCj is denotedby d(i, j) where

d(i, j) ¼ (xi ¹ xj)2 þ (yi ¹ yj)2� �1=2:

5. SetH is defined as the superset ofall such node setsCi.6. A node set is acomplex nodewhen its mass is greater

than 1; else the node set represents a simple node.

578 D. Saha, A. Mukherjee/Computer Communications 20 (1997) 576–585

Now, we describe the algorithm. We compute thedistanced(i, j) between each pair of given nodes (Ci,Cj)and choose the one for whichd(i, j) is minimum. We deletethis pair of node sets fromH and form a new node set whichis the union off the nodes included in the deleted pair. Thisnew node set, thus formed, is added toH. This process iscontinued untillHl ¼ 1. If we consider that a cluster isrepresented by a node set, then at any point during theexecution of the algorithm the setH represents a set ofclusters. A formal version of the algorithm is as follows:

H ← c ;ni [ N{ ni}While lHl Þ 1

do

find dðr ; sÞ ¼ min;i;jdði; jÞ : Ci ;Cj [H, i Þ j;

C̄p ← Cr ∪ Cs

xp←(xlCr l þ xslCs)/ðlCr lþ lCslÞyp←(yr lCr lþ yslCsl)/(lCr l þ lCsl)H ← (H¹ Cr ¹ Cs) ∪ Cp

od.

Details of the algorithm can be found in [18].

2.2. Illustration of clustering algorithm

Here, an example will obviously clarify the algorithmmore vividly. So, out of the several test cases consideredin [18], an example of moderate size is presented here. Thedesign concerns 24 local networks which are to be con-nected by multidensity gateways. According to Step 1 (ofthe above procedure), the center of gravity of each networkmust be determined first by an algorithm already discussedin [3]. From there starts the actual clustering algorithm. Figs2 and 3 are used to describe the algorithm. In Fig. 2, thedistribution of the centres of gravity (represented as nodes)for the set of 24 randomly generated networks is shown. Thetotal number of centers of gravity is also equal to 24, and the

nodes are marked as 1 through 24 for the sake of presenta-tion (Fig. 2). It can be assumed that the nodei (1 # i # 24)represents the centre of gravity of networki. After runningthe clustering algorithm once, the 1st level clusters areidentified as shown below:

Node set Cluster{9, 11, 13, 7, 10, 8, 12} I{2, 5, 6, 1, 3, 4} II{20, 23, 22, 19, 24, 21} III{14, 17, 18, 15, 16} IV

They are marked as 1st level clusters in Fig. 3. Next, foreach 1st level cluster, the centre of gravity is calculated,and, based on the clustering of these newly generatedcentres of gravity, the 2nd level clusters are formed. Thereare only two 2nd level clusters. One contains the 1st levelclusters I and II and the other contains III and IV. They arealso indicated in Fig. 3. Finally, the centres of gravity of 2ndlevel clusters are found out and they are grouped into asingle 3rd level cluster that contains all the networks.

The clustering algorithm, in effect, covers Steps 1through 3 (in every iteration) of the above complete designprocedure. However, every iteration of the clusteringalgorithm is complemented by an associated optimizationproblem that deals with an appropriate gateway selectionstrategy. This optimization problem is discussed at lengthin the next section.

3. Optimization problem

As discussed earlier, every iteration of the clusteringalgorithm is associated with a level in the hierarchy of theinternetwork to be designed. To be specific, iterationk willgeneratekth level clusters (or internetworks), for each ofwhich a gateway of densityk has to be identified from a setof potential gateways. This identification must be done in anoptimal manner at every iteration. However, opening of newgateway sites must be completed only at the very first itera-tion (i.e.,k ¼ 1), because, once the gateway sites are openedat the initial stage (i.e., first level), from the second iteration

Fig. 2. Distribution of centres of gravity (represented as nodes) for 24randomly generated networks.

Fig. 3. 1st and 2nd level clusters for the networks of Fig. 2.


(or level) onwards, no new gateway sites will be opened; onlythe density of the identified gateways will increase by one.

3.1. Nature of the problem

The optimization problem must be solved at every levelimmediately after the clusters for that level are formed. Forinstance, with respect to Fig. 3 which contains four 1st levelclusters, if there are six (say) potential locations for openinggateway sites, an optimization problem arises regardingwhich three (out of six) gateway sites are to be selectedfor a cost-effective design. For the sake of further illustra-tion, let us assume that gateways at sites 1, 2, 4 and 5 areopened at this initial stage. Then, for the two 2nd levelclusters, there will be four (not six) potential gateways,namely gateways at sites 1, 2, 4 and 5, out of which onlytwo are to be selected optimally. Again, let us assume thatan optimal design chooses gateways at sites 2 and 5 for the2nd level. Then these gateways at sites 2 and 5 will be ofdensity 2 currently, whereas gateways at sites 1 and 4remain fixed at density 1 only. Similarly, for the single3rd level cluster, there will be two potential gateways atsites 2 and 5, out of which only one is to be selected opti-mally. If the gateway at site 5 is selected by the optimizationalgorithm, then this gateway, at site 5 will be of density 3,whereas the gateway at site 2 will remain of density 2. Inthis way, the optimization algorithm will continue until theclustering algorithm stops.

The interconnection among the gateways will be made ina hierarchical fashion such that every gateway of densityn(n . 1) will be a root of a subtree comprising one or moregateways of density (n ¹ 1). The local networks comprisingthe final internetwork will be the leaves of this tree structure,and they are connected directly to one of the gateways ofdensity 1. This interconnection structure is clearly shown inFig. 1 and will be more clear when another concreteexample will be taken up in the section containing compu-tational results. Before that, a mathematical model for thisoptimization problem need to be formulated and solved first.

3.2. Mathematical model

The model is formulated with the following aims: (1) todetermine for every potential gateway location whether toopen a gateway or not, (2) to determine which internetworkis to be connected to which of the open gateways, and (3) todecide for each open gateway, if it connects to the nexthigher level gateway either directly or indirectly via anothergateway. With reference to the above notations listed at thebeginning, the problem (P) is formulated as follows:

(P) Minimize Zp ¼

"∑j[G

∑i[S

pij uij þ∑j[G

vjwj

þ∑j[G

∑k[G

(rj þ hjk)xjk

#a

ð1Þ

subject to

0 #∑j[G

uij # 1, ;i [ S, (2)

∑j[G

fjk ¹∑k[G

fkj ¹∑i[S

siuij ¼ 0, ;j [ G, (3)

fjk # (cjrjxjk), ;j,k [ G, (4)

uij # wj , ;i [ S, ;j [ G, (5)

wj ¼∑k[G

xjk, ;j [ G, (6)

0 # wj , uij , rj # 1, ;i [ S, ;j [ G, (7)

0 # xjk # 1; fjk $ 0, ;j,k [ G, (8)

a . 1: (9)

The first term of the objective function represents the cost ofassigning gateways of densityk to kth level internetworkswherek $ 1. The second term captures the cost of locatinggateways for each levelk, k $ 1. The third term captures thecost of selecting the capacities of gateways of densityk andthe cost of assigning them to gateways of density (k þ 1),wherek $ 1. The fixed cost of establishing the gateway sitesis assumed to be independent of the gateway sizes. The termr j captures costs for acquisition of gateway capacities plusfor operation which are assumed to be directly related to thesize of the gateways. The termh jk captures the cost of con-necting a gateway of densityj to another gateway of densityk, wherej $ 1 andk ¼ (j þ 1).

Constraint Eq. (2) ensures that everykth level inter-network has only one gateway,;k $ 1. Constraint Eq. (3)captures the flow conservation property at the gateway sites.Constraint Eq. (4) captures the limits on the attainable gate-way capacities. Constraint Eq. (5) ensures that a networkcannot be connected to an unopened gateway. ConstraintEq. (6) ensures that an open gateway site is connected eitherto a gateway of a same level internetwork or to a gateway ofa higher level internetwork. Constraint eqns (7) and (8)determine the non-negativity of the variables. ConstraintEq. (9) makes the cost function nonlinear as well as convex.A heuristic solution procedure for the above formulation ispresented in the next section.

The cost parametera is mainly used to characterize thecomputational complexity of the present design algorithm.Although the polynomial form of the computational costwith a exceeding unity represents a loss of scale, it is pre-cisely this which makes the cost function nonlinear andconcave. On the contrary, for the economy of scale, ifa isassumed to be less than unity, the cost function becomesconvex for which only a near optimal solution is feasible[17]. However, our aim is to formulate the problem as anonlinear optimization problem for which an optimal solu-tion exists and then solve the problem by Lagrangean


relaxation and subgradient optimization. This is whya isintroduced and its range is assumed to be greater than unity.

4. Heuristic solution procedure

The procedure is divided into two subprocedures givenbelow.

4.1. Lagrangean relaxation of the problem

For the model described in the previous section as a non-linear combinatorial optimization problem, the Lagrangeanrelaxation is used first on the way to develop a heuristicsolution procedure. The Lagrangean relaxation scheme hasearlier been successfully applied for the same purpose tosimilar combinatorial design problems in References[3,13,14,16,17]. The Lagrangean relaxation (L) of thepresent problem (P) is formed by multiplying the constraintEqs. (3), and (5) with vectors of Lagrangean multipliersm ij

andl ij, respectively, and then by adding them to the objec-tive function. The complete expression forL is writtenbelow:

L ¼ min∑j[G

∑i[S

pij uij þ∑j[G

vjwj þ∑j[G

∑k[G

(rj þ hjk)xjk

" #a

þ mij

∑j[G

fjk ¹∑k[G

fkj ¹∑i[S

siuij

!

þ lij uij ¹ wj

ÿ �,

subject to the constraints in Eqs. (2), (4), (6)–(8).

The set of feasible solutions for the Lagrangean relaxa-tion L of the problem P (Eq. (1)) is a super set of the set offeasible solutions for the problem P [13]. For the givenvectors ~l { ¼ l ij} and ~m { ¼ m ij}, if the problem has afeasible solution~u { ¼ uij} then the following relationshipholds:

L(~u, ~l, ~m) # Zp(~u):

Thus, L(~u, ~l, ~m) is the lower bound ofZp(~u) for each foreach(~l, ~m). The best possible bound for a such procedureis given by the vector(~lp, ~mp) satisfying the following conditions:

L(~up, ~l, ~m) # L(~up, ~lp, ~mp) # L(~u, ~lp, ~mp):

The point (~up, ~lp, ~mp) is called the optimal point of theLagrangean because, for a unique point~up, the followingrelation holds:

L(~up, ~l, ~m) # L(~up, ~lp, ~mp):

Hence, a solution of Lagrangean relaxation can be obtainedby differentiatingL in a manner shown in Appendix A. Wesimply present the solution here:

uij ¼ ( ¹lij þ mij si ¹ aAa ¹ 1pij )=(a(a ¹ 1)Aa ¹ 2p2ij ),

wherea . 2

4.2. Subgradient heuristic

Let us suppose that (m *,l*) be an optimal solution of theLagrangean relaxationL. A subgradient optimizationalgorithm is used to derive lower bounds on the optimalprimal objective value usingL. In the subgradient optimiza-tion procedure, gradient method is adapted by replacing thegradients with subgradients [14,16,17]. If an initial multi-plier vector (m0,l0) is given, a sequence of multipliers isgenerated using the following expressions:

mnþ 1j ¼ mn

j þ tn∑j[G

f njk ¹

∑k[G

f nkj ¹

∑i[S

siunij

!

, ;j [ G,

and

lnþ 1j ¼ ln

j þ tn(unij ¹ wn

j ), ;i [ S, ;j [ G,

whereuijn, wj

n, andf jkn are obtained from an optimal solution

to Lagrangean relaxationL, and tn, a positive scalar, is astepsize which is given as follows [13]:

tn ¼

dn(Zp ¹ L)∑j[G

f njk ¹

∑k[G

f nkj ¹

∑i[S

siunij

!2

þ∑i[S

∑j[G

unij ¹ wn

j

ÿ �2

( )1=2,

wheredn is a scalar satisfying 0# dn # 2. Initially, thisscalar is set equal to 2, and it is then halved when the lower

Fig. 4. Flow-chart of subgradient algorithm.


bound does not improve in a given number of consecutiveiterations. The subgradient algorithm is terminated, if eitherthe gap between the upper bound (the value of L) and thebest primal feasible solution value is within a given speci-fied limit, or after 300 iterations, whichever is earlier. Ahigh level flowchart for the subgradient algorithm is givenin Fig. 4.

5. Computational results

As described above, the complete solution procedure con-sists of two distinct algorithms, namely clustering algorithmand optimization heuristics. The clustering algorithm hasbeen repeatedly run to form the partial internetworks ateach levelk, k $ 1. In the optimization heuristics, a solutionprocedure for the primal problemZp has been developed byusing Lagrangean relaxation and subgradient optimizationtechnique. This procedure generates a feasible solution forthe primal problem after every iteration of the subgradientoptimization algorithm. The best solution is automaticallyretained when this optimization procedure is terminated.This solution procedure is used to determine the gateways(of various densities) to be opened for connecting to inter-networks. To form an overall internetwork, this procedure

has to be repeatedly invoked to determine the gateways ofdensityk for connecting with the internetworks at each levelk $ 1. Extensive computational analysis has beenperformed to test the performance of the algorithm onseveral design problems of varied sizes up to 10 000terminals and 500 potential gateway locations. The programhas been coded in C and run on a UNIX-based computer,HP-9000.

Here, we briefly describe a representative problem and itssolution. The data points representing local networks andgateway locations have been drawn from a uniform distri-bution over a rectangle with sides 50 and 100 (Fig. 5). Foreach network, its centre of gravity is determined from thecoordinates of the terminals contained in that network. Thiscompletes the Step 1 of the design methodology describedin Section 2. In the second step, 1st level internetworks areformed by applying the clustering algorithm describedearlier. During cluster formation, the centres of gravity ofthe 1st level internetworks are also calculated. The distancedij between internetworki and potential gatewayj is used todefine costp ij ¼ dij·w1, wherew1 is a positive constant. Thegateway capacity requirements of internetworks are takenfrom a uniform distribution between 5·e1 and 10·e1, wheree1 is the size of internetwork [4]. The available gatewaycapacities (cj) are assumed to be 200, 300, 400, 500, and

Fig. 5. An example of hierarchical gateway location problem.


600 units, and fixed installation costs (r j) of these capacitiesare 20·w2, 30·w2, 38·w2, 44·w2 and, 49·w2, respectively,wherew2 is a positive constant. The reliabilities of gatewayscorresponding to the above capacities are taken to be 0.75,0.80, 0.85, 0.88 and 0.92, respectively. The fixed setup cost(vj) of locating a gateway, at the sitej is defined asvj ¼ w2 þ

(w2/2)·r 1, wherer 1 is a random number drawn from a uni-form distribution between 0 and 1. The costs of connectinggateway i to gateway j are defined as 2·gij·w1, 3·gij·w1,3.5·gij·w1, 3.9·gij·w1 and 4.2·gij·w1 for the gateway capaci-ties of 200, 300, 400, 500 and 600 units respectively, wheregij is the distance between gatewaysi and j. The results ofthe experiment are given in Tables 1, 2 and 3. The rows andcolumns in all the tables indicate the internetwork numberand the potential gateway number, respectively. The tableentriesuij values. Those values ofuij are selected (i.e., boldin tables) which are less than one because of the constraintEq. (7).

To test the computational efficiency and the scalability ofthe algorithm, we have executed the program for varyingsizes of internetworks. The local networks are randomlygenerated in all cases. A good performance measure of

the algorithm is the number of levels (L) generated by theclustering algorithm becauseL determines how many timesthe outer loop of the algorithm (that includes both theclustering as well as optimization procedures) will beexecuted. Fig. 6 shows a plot ofL versusS, the index setof internetworks, which is taken as the input data size,G. theindex set of potential gateway locations, is assumed to be inthe order of 20 times less thanS, i.e.,lGl > lSl/20. The graph(Fig. 6) indicates a sublinear (logarithmic) relationshipbetween L and S, i.e., the algorithmic complexity isO(log S). This speaks for a nice scalability of the techniqueupto at least thousands of networks.

6. Conclusion

In a word, the problem of locating multidensity gatewaysin an internetwork and connecting them hierarchically has

Table 1Gateway selection table for level-1

i \ j 1 2 3 4 5 6 7 8 9 10

1 1.87 1.76 1.61 1.21 1.13 0.98 1.02 1.18 1.47 1.792 2.15 1.98 1.76 1.68 1.51 1.37 1.05 0.96 1.26 1.433 1.73 1.61 1.17 1.09 1.18 1.27 1.41 1.01 0.97 1.144 1.54 1.43 1.16 0.98 1.01 1.03 1.04 1.12 1.22 1.475 1.46 1.31 1.05 1.19 1.49 1.62 1.74 1.45 1.21 0.996 0.98 1.15 1.25 1.39 1.31 1.41 1.52 1.72 1.61 1.79

Note: i ¼ internetworks;j ¼ potential gateways; table entries indicateuij.


i \ j (1) (6) (4) (8) (9) (10)1 2 3 4 5 6

1 0.95 1.31 1.27 1.59 1.42 1.372 1.22 1.03 0.98 1.47 1.41 1.553 1.59 1.33 1.19 1.11 0.98 1.09

Note: bracketed entries indicate the original gateway number.

Fig. 6. Computational complexity of the algorithm.


i/j (1) (4) (9)1 2 3

1 1.37 0.99 1.05

Note: bracketed entries indicate the original gateway number.


been studied in this paper. A non-linear combinatorialoptimization model is developed to minimize the totalcost of setting up gateway sites and assigning networks tothem. The Lagrangean relaxation of the model is presentedwith a view to develop a subgradient heuristic. Compu-tational results obtained for a problem consisting of 10 gate-ways with upto 200 networks are reported. The solutionprocedure is found to be effective in other simulationexperiments too [18].

Keeping in view that internetworking is fast becoming anintegral part of the modern communication scenario, anattempt has been made in this paper to pose the inter-networking problem as a mathematical programmingproblem that could be solved by an efficient subgradientbased heuristic. Recent studies [11,12] clearly indicatethat the problem of modelling an internetwork is a keyresearch field today and the problem is far from being solvedcompletely till now. Under the circumstances, the introduc-tion of the concept of multidensity concentrator gateways ina multilevel design of high speed internetworks will cer-tainly add a new dimension to the internetworking researchfield. Porting the concept to other models of internetworkingmay be an interesting topic of future work.

Acknowledgements

The authors deeply acknowledge the helpful commentsmade by the referees towards the improvement of thequality and the presentation of the paper.

Appendix A

DifferentiatingL partially with respect touij and setting]L/]uij equal to zero, we obtain

]L]uij

¼a[pij uij þ vjwj þ (rj þ hjk)xjk]a ¹ 1pij ¹ mij si þ lij ¼ 0

or

aAa ¹ 1(1þ pij uij =A)a ¹ 1pij ¹mij si þ lij ¼ 0,

whereA¼ vjwj þ (r j þ h jk)xjk.Expanding the above expression binomially and ignoring

the higher order terms [18], we obtain

aAa ¹ 1(1þ (a ¹ 1)pij uij =A)pij ¹ mij si þ lij ¼ 0:

Simplifying the expression, we obtain:

uij ¼ ( ¹ lij þ mij si ¹ aAa ¹ 1pij )=(a(a ¹ 1)Aa ¹ 2p2ij ):

For instance, ifa ¼ 2, then

uij ¼ ( ¹ lij þ mij si ¹ 2Apij )=(2ap2ij ):

Appendix B

The following notations are used throughout the paper:

S index set of internetworks;G index set of potential gateway locations;p ij cost of connecting internetworki with gatewayj;vj fixed setup cost of locating a gateway at sitej;r j cost of installing a gateway at the sitej;h jk cost of connecting a gateway at locationj to a gateway at locationk;si amount of gateway capacity needed to support an internetworki;cj capacity of a gateway that can be located at sitej;f ij communication traffic (flow) from internetworki to internetworkj;a computational parameter, (a . 1);uij ¼ 1, if is the gateway site for internetworki;

¼ 0, otherwise;wj ¼ 1, if a gateway is located at sitej;

¼ 0, otherwise;xjk ¼ 1, if a gateway is located at sitej and sitej is connected to sitek

which is at a higher level in the hierarchy;¼ 0, otherwise;

r j reliability of a gateway that can be located at sitej;Zp primal objective function (cost function).

References

[1] C.A. Sunshine, Network interconnection and gateways, IEEE JournalSelect. Areas Communication 8 (1) (1990) 4–11.

[2] P.E. Green (Ed.), Computer Network Interconnection and ProtocolConversion, IEEE Reprint series, IEEE Press, 1988.

[3] D. Saha, A. Mukherjee, Design of hierarchical computer communica-tion networks under node/link failure constraints, Computer Com-munications 18 (1995) 871–875.

[4] S. Narasimhan, H. Pirkul, Hierarchical concentrator location problem,Computer Commununications 15 (3) (1992) 185–191.

[5] P. McGregor, D. Shen, Network design: An algorithm for accessfacility location problem, IEEE Trans. Communication 25 (1977).

[6] G.M. Schneider, M.N. Zastrow, An algorithm for the design of multi-level concentrator networks, Comput Networks 6 (1982) 1–11.

[7] V.K. Konangi, T. Aidja, C.R. Dhas, On the multilevl concentratorlocation problem for local access networks, in: Proc IEEE Globecom1984, pp. 912–915.

[8] M. Zitterbart, A.N. Tantawy, D.N. Serpanos, A high performance trans-parent bridge, IEEE/ACM Trans Networking 2 (4) (1994) 352–362.

[9] B. Gavish, Formulations and algorithms for the capacitated minimaldirected tree problem, Journal of the ACM 30 (1983) 118–132.

[10] H. Pirkul, S. Narasimhan, P. De, Locating concentrators for primaryand secondary coverage in a computer communications network,IEEE Trans. Communication 36 (4) (1988) 450–58.

[11] S. Sherkar, Fundamental design issues for the future Internet, IEEEJournal Select. Areas Communication 13 (7) (1995) 1176–1188.

[12] E.W. Zegura, K.L. Calvert, S. Bhattacharjee, How to model an inter-network, in: Proc of INFOCOM’96, California, USA, 1996.

[13] M.L. Fisher, An applications oriented guide to Lagrangean relaxation,Interfaces 15 (1985) 10–21.

[14] H. Held, P. Wolfe, H.P. Crowder, Validation of subgradient optimiza-tion, Mathematical Programming 5 (1974) 62–68.

[15] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide tothe Theory of NP-Completeness, Freeman, San Francisco, 1979.

[16] D. Saha, A. Mukherjee, S.K. Dutta, Capacity assignment in computercommunication networks with reliable links, Computer Communica-tions 17 (1994) 871–875.

[17] A. Mirzaian, Lagrangean relaxation for the star-star concentrator


location problem: Approximation algorithms and bounds, Networks(1985) 1–20.

[18] D. Saha, A. Mukherjee, Some studies on the internetworking problemfor connecting high speed local networks, Internal Report, JU, CSE,India, 1994.

[19] C.H. Papadimitrian, The complexity of the capacitated tree problem,Networks 8 (1978) 217–230.

Debashis Saha was born in India in 1965. Hereceived the B.E. degree in Electronics &Telecommunication Engineering from theJadavpur University, Calcutta, India in1986, the M.Tech. and Ph.D. degrees inElectronics and Electrical CommunicationEngineering from the Indian Institute ofTechnology, Kharagpur, India in 1988 and1996, respectively. His Ph.D. dissertationwas concerned with the formal protocolconversion techniques based on thecommunicating finite state machine model.

Since July 1990 he has been at the Jadavpur University, Calcutta,India, where he is an Assistant Professor in the Department ofComputer Science and Engineering. He teaches computer communica-tion networks, computer organization, data communication systems,distributed systems, microprocessors and programming languages.His research interests include topological design of computer net-works, local area networks, formal models of protocols, internetwork-ing, protocol conversion and validation, and parallel algorithms.

Amitava Mukherjee received a Ph.D. (Com-puter Science) degree from JadavpurUniversity, Calcutta, India. He was in theDepartment of Electronics and Tele-communication Engineering at JadavpurUniversity, Calcutta, India from 1982 to1995. Presently, he is a senior manager inPrice Waterhouse Associates, Calcutta,India. His research interests are in thearea of High Speed Networks, Combinator-ial Optimization and Distributed Systems.His interests also include mathematical

modelling and their applications in the fields of societal engineeringand international relations.


On the multidensity gateway location problem for a multilevel high speed internetwork

Documents

Transcript of On the multidensity gateway location problem for a multilevel high speed internetwork