hybrid fiber co-axial catv network design with variable capacity optical network units

Theory and Methodology

Hybrid ®ber co-axial CATV network design with variable capacityoptical network units

Rakesh Gupta *, Hasan Pirkul

School of Management, The University of Texas at Dallas, Box 830688 JO51 Richardson, Texas 75083-0688, USA

Received 1 December 1997; accepted 1 December 1998

Abstract

Recent changes in telecommunication regulations and changing market forces are making the market for broadband

network services to the home an extremely lucrative and competitive area. Of the many competing technologies for such

broadband services, major CATV companies are banking heavily on Hybrid Fiber Co-axial (HFC) networks as a

delivery mechanism. In this paper we provide a mathematical model for the design of HFC networks. We extend the

model developed by Pirkul and Gupta (1997) to allow the location of Optical Network Units (ONUs) of di�erent

capacities. Analysis of the model shows that it belongs to the NP-Complete class of problems. We provide a heuristic

solution procedure for the design task. Computational testing of our procedure demonstrates that it is applicable for the

design of relatively large networks and provides solutions that are of reasonably good quality. Ó 2000 Elsevier Science

B.V. All rights reserved.

Keywords: Telecommunications; Location; Modelling; Heuristics; Lagrangian relaxation

1. Introduction

The market for broadband network services tothe home is emerging as an important and lucra-tive source of revenues for telecommunicationcompanies [6]. This is due in part to recent changesin telecommunication regulations and changingmarket forces that allow competition in the localloop [15]. Of the many competing technologies for

such broadband services, major Community An-tenna Television (CATV) companies are bankingheavily on Hybrid Fiber Co-axial (HFC) networksas a delivery mechanism [6]. Increasingly, CATVproviders are deploying high bandwidth networkswith the intention of (a) providing more interactiveservices to subscribers (such as Video on Demand),(b) providing access to the internet and the WorldWide Web (WWW) and (c) integration with thePublic Switched Telephone Network (PSTN) toprovide telephony services.

Since the concept of ``®ber to the home'' isconsidered to be cost prohibitive and thereforenot economically viable in the near future [6], a

European Journal of Operational Research 123 (2000) 73±85www.elsevier.com/locate/orms

* Corresponding author. Tel.: +1 972 883 2458; fax: +1 972

883 2089.

E-mail addresses: [email protected] (R. Gupta),

[email protected] (H. Pirkul).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 7 0 - 3

network consisting of a combination of optical®ber and co-axial cable i.e., HFC is considered bymany to be a good compromise between (a)providing higher bandwidth to the consumer(over twisted pair copper wire) and (b) cost ofdeployment [5,20]. The increased competition al-lowed in the local loop by the Telecommunica-tions Act of 1996 and the existence of a co-axialphysical plant to over 60% of the homes in theUnited States places CATV providers in a goodposition to provide higher bandwidth applicationsas well as telephone service to subscribers [13].The need for e�cient procedures to design suchHFC architectures is therefore both relevant andimportant.

In this paper we provide a mathematical modelfor the design of such HFC networks. This modelis an extension to that developed by Pirkul andGupta [24] and allows the location of OpticalNetwork Units (ONUs) of di�erent capacities inthe design. Our model is substantially more real-istic than the previous e�ort since it allows for thedesign of networks that are more e�cient withrespect to serving the needs of a variety of popu-lation distributions. For instance, by locatinglower cost and capacity equipment in sparselypopulated neighborhoods and higher capacityequipment in neighborhoods with high populationdensities the overall cost of the network can bereduced and cost per subscriber decreased. Anal-ysis of this model shows that it belongs to the NP-Complete class of problems. Given that optimalsolutions for realistic sized networks would beextremely di�cult to attain, we develop a heuristicsolution procedure. Computational testing of ourprocedure demonstrates that it is applicable for thedesign of relatively large networks serving areaswith about 100 subscriber neighborhoods andprovides solutions that are of reasonably goodquality.

This paper is structured as follows. Section 2provides some background and literature review inthis context. Section 3 provides a mathematicalformulation for the problem we consider and inSection 4 we prove that it belongs to a set ofdi�cult problems. Sections 5 and 6 describe aLagrangian relaxation of the model and a heuristicprocedure for solving it. Section 7 presents and

discusses the results from computational experi-ments with the heuristic on a number of problems.Finally, Section 8 concludes the paper and indi-cates directions for future research.

2. Background

A CATV network can be considered to be madeup of three main parts: the trunk, the feeder, andthe customer drop [5]. The trunk is usually in-tended to cover large distances of tens of miles.The feeder portion of the cable supports taps forsubscribers. Its maximum length is only a fewmiles since energy is tapped o� to feed homes(subscribers) requiring relatively high power levels.The drop is the ¯exible cable which goes to thehome. It has a maximum length of approximately500 feet and is made up of lower quality co-axialcable than the feeder or trunk. Older generationCATV systems used co-axial cables in the trunkand feeder portions of the network and thereforeexperienced problems related to (a) interferencefrom spurious radiation, (b) distortions introducedby ampli®ers and (c) limited bandwidth.

The topology of a generic HFC CATV networkis as shown in Fig. 1. The Time-Warner CATVnetwork [17] is only one of a number of CATVnetworks that display similar architectures. Thenetwork uses ATM and SONET for its switchingand transmission technologies. Each headend isconnected via SONET links to other headends andto gateways to the PSTN and Video on Demandservers. The signal is transmitted from each hea-dend (or local distribution hub) in a star likefashion to ONUs via ®ber optic trunc cables atOC-1 rates. In larger metropolitan areas there maybe economical reasons to justify connecting eachheadend to ``®ber nodes'' [17] which are thenconnected to ONUs, thus introducing anotherlevel in the hierarchy of the network. The use of®ber in the trunk provides several advantages overthe older generation co-axial systems. Firstly,trunk ampli®ers and their corresponding distor-tion e�ects are eliminated. Secondly, higherbandwidth access is facilitated. Further, since ®beritself is not subject to interference, better controlover signal quality is achieved.

74 R. Gupta, H. Pirkul / European Journal of Operational Research 123 (2000) 73±85

Each ONU distributes the signals over co-axialcables providing multi-tap connections to sub-scriber neighborhoods. Each subscriber neighbor-hood consists of a cluster of approximately 25±50homes. The co-axial distributor cable requires acascade of distributor ampli®ers to compensate forenergy tapped o� at each neighborhood. High ca-pacity ONUs are relatively expensive and thereforemust serve hundreds of subscribers so that costscan be shared and the cost per subscriber is notvery high. Lower capacity ONUs are also availablesuch as those within the HFC 2000 package fromLucent Technologies [13]. ONUs of various ca-pacities available in the HFC 2000 package arecapable of serving 120±480 subscribers. ThusONUs of various capacities are capable of serving avariable number of subscribers via a small co-axialnetwork (i.e., the feeder network) [5].

As CATV companies begin providing new ser-vices (such as telephony and access to the internet)in addition to traditional television, there is agreater need today to provide reliable CATV net-work architectures than in the past. By restrictingthe number of neighborhoods connected to eachONU (and thereby the number of subscribers), thee�ect of any network downtime is mitigated tosome extent. Requiring fewer subscriber neigh-borhoods within each segment of the distributorcable from the ONU provides further advantagesin reducing the number of distributor ampli®ers.Older generation co-axial networks were oftenprey to signi®cant distortion e�ects due to theslight non-linearities present in each distributorampli®er and the accumulated e�ects of such non-linearities when they where cascaded over a largedistance. Thus limiting the number of subscriber

Fig. 1. Generic HFC CATV network topology.

R. Gupta, H. Pirkul / European Journal of Operational Research 123 (2000) 73±85 75

neighborhoods in each sub-tree ful®ls a two foldobjective in maintaining better service quality aswell as reliability.

Major questions to be answered by any designprocedure include the following: (a) the locationsof headend sites and the SONET links connectingeach headend to others, the PSTN as well asservers providing Video on Demand, (b) thenumber, location and capacities of ONU nodeschosen from a discrete set of available types andtheir ®ber links to the headend, (c) identi®cation ofthe subscriber neighborhoods to be connected toeach ONU and, (d) the topology of the co-axialcable links connecting each neighborhood to anONU. Given that the overall problem is NP-Hard[10], a disaggregation approach (as used in othercommunication network design problems [1])seems appropriate.

The design problem we consider in this researchinvolves choosing the number, locations and ca-pacities of Optical Network Units and connectingthem via ®ber links to the regional headend anddeciding the topology of co-axial links betweensubscriber neighborhoods and an ONU in aCATV network. It is assumed that locations ofregional headends have already been decided andwe are given as input the set of subscriber neigh-borhoods to be connected to a particular headend.

Considerable work has been done towards de-signing centralized computer networks. Some ofthe most recent research e�orts are those due to[8,12,22,23]. Most such models have investigateddesigns requiring the use of dedicated lines be-tween concentration sites and demand nodes.Also, the task of connecting subscriber neighbor-hoods to an ONU with a restriction on themaximum number of such neighborhoods in eachsub-tree is similar to that of local access networkdesign and has been addressed by a number ofauthors in the past. Some of the recent such workis that of Hall [16] and Gouveia and Martins [14]both of whom have utilized cutting plane basedtechniques to solve the problem. Malik and Yu[18] on the other hand present a branch and boundbased algorithm which is useful for solving smallinstances of such problems. Further, Gavish andAltinkemer [11] have developed heuristic proce-dures based on parallel savings techniques which

provides good results for such problems. Altink-emer and Gavish presented a heuristic withconstant error guarantees in Ref. [2].

This paper considers a capacitated tree topol-ogy connecting nodes to the ONU sites, whileimposing capacity constraints on the ONUequipment. This particular model has not beenaddressed in the literature to date. Our heuristicsolution procedure provides good quality feasiblesolutions and provides a lower bound againstwhich the feasible solutions can be compared.The analysis presented in this paper applies to anew generation of CATV networks beingimplemented today and is therefore timely andrelevant.

3. The CATV network design problem (CATV)

In this section we present a mathematical modelfor connecting subscriber neighborhoods to ONUsvia co-axial cables through the use of multipointlinks. The ONUs are then connected to theregional headend through the use of dedicated ®-ber optic links. The mathematical formulation ispresented below.

ParametersP Index set of potential ONU locations,

jP j � m.I Index set of subscriber neighborhood

locations. jI j � n.W I [ P .R Discrete set of capacity levels avail-

able for ONUs.C Headend site.Qm Maximum number of neighborhoods

served by ONU of capacity m 2 R.K Number of neighborhoods to be

connected on each sub-tree in thefeeder network.

Cij

cost of connecting site i to j;where i; j 2 I ;

0 8 i 2 P ; j � C;1 8 i; j 2 P :

8>><>>:Hkm Cost of installing ONU of capacity m

at site k and connecting to headend.


Problem CATV:

MinimizeXi2W

Xj2W [fCg

CijXij �Xj2P

Xm2R

HjmYjm; �1�

Xj2W [fCg

Xij � 1 8 i 2 W ; �2�

fijk 6Xjk 8 i 2 I ; 8 j 2 W ; 8k 2 W [ fCg; �3�Xi2I

fijk 6K Xjk 8 j 2 I ; 8k 2 P [ fCg; �4�

Xi2I

fijC 6Xm2R

Qm Yjm 8 j 2 P ; �5�

Xj2W [C

fikj ÿXj2W

fijk � 1 8 i; k 2 I ; k � i; �6�

Xj2W [C

fikj ÿXj2W

fijk � 0 8k 6� i; k 2 W [ fCg; i 2 I ;

�7�Xj2W [C

fikj ÿXj2W

fijk � ÿ1 8 i 2 I ; k � C; �8�

Xm2R

Yjm6 1 8 j 2 P ; �9�

Xij6Xm2R

Yjm 8 j 2 P ; 8i 2 I ; �10�

Xij 2 f0; 1g 8i 2 W ; j 2 W [ fCg; �11�

Yjm 2 f0; 1g 8 j 2 P ;m 2 R; �12�

fijk P 0 8 i 2 I ; j 2 W ; k 2 W [ fCg: �13�The objective function (1) minimizes the cost

of connecting customer neighborhoods to eachother via the feeder network or to a ONU siteand that of installing ONUs and connecting themvia the trunk ®ber link to the headend.Constraint (2) enforces exactly one outgoing arcfor each node. Constraint set (3) links ¯owvariables with arc connections. Constraints (4)and (5) incorporate capacity constraints. Con-servation of ¯ow is enforced by constraint sets(6)±(8). Constraints (9) ensure that at most oneONU of some capacity m is installed in any sitej. Constraint set (10) ensures that if a subscriberneighborhood is connected to an ONU site thenan ONU must be installed at that site.Constraints (11)±(13) impose integrality and non-negativity restrictions.

4. Computational complexity of problem CATV

Lemma. The decision analog of problem CATV isNP-Complete.

Proof. We prove the NP-Completeness of CATVby restriction. Consider an instance of the CATVproblem where the following conditions exist:· There are ``n'' customer neighborhoods where n

is an even number.· The maximum number of customer neighbor-

hoods allowed in each sub-tree (K) is equal to 2.· There is only one kind of ONU available (i.e.jRj � 1) and the cost of location and connectionto the headend are equal among all potentialONU sites.

· Each pair of neighborhoods (1, 2), �3; 4� � � ��i; i� 1� � � � �nÿ 1; n� satis®es the following con-ditions:1. Ci;i�1 �maxfCi;j;Ci�1;j8j 2 P ; j 6� i; j 6� i� 1g< minfCi;k;Ci�1;kg8k 2 I ; k 6� i; k 6� i� 1;

2. Ci;i�1 < minfCi;j;Ci�1;j8j 2 i; j 6� i; j 6� i� 1g;3. Ci;C � 1 8 i 2 I :Due to the above restrictions, we can see that it

will always be economical to connect each cus-tomer neighborhood pair �i; i� 1; where i is odd)

Decision variables

Xij

1 if link exists between location i;and j; i 2 W ; j 2 W ;

0 otherwise:

8<:Yjm

1 if ONU is located at site j and isof capacity \m" and connected to C

0 otherwise:

8<:fijk Flow originating at subscriber neighbor-

hood i, on link�j; k�; i 2 I ; j 2 W ; k 2 W [ fCg:


together. If we represent each such pair of cus-tomer neighborhoods as one single combinedcustomer neighborhood then the problem reducesto the Capacitated Concentrator Location prob-lem (CCL). Since the decision analog of CCL isNP-Complete [19], the decision analog of CATV isalso NP-Complete. �

Note that CATV is signi®cantly more di�cultthan CCL because it contains as a sub-problem,the Capacitated Spanning Tree Problem (CMST).Both CCL and CMST (for 2< K 6 n=2) are knownto be di�cult NP-Complete problems [21,19].Therefore, CATV is a di�cult problem to solveand heuristic procedures need to be resorted to forits solution. Since commercial general purposeinteger programming codes can only solve smallinstances of this problem, a heuristic which canexploit the underlying problem structure success-fully would be more e�ective. In the next fewsections we describe such a heuristic solutionprocedure to solve problem CATV.

5. A Lagrangian relaxation of problem CATV

We employ a Lagrangian relaxation procedurethat has been utilized successfully in the past inother complex problems, see Ref. [9] for a de-scription of the procedure. Since we know that asolution to the problem CATV will result in aspanning tree con®guration for links between thesubscriber locations and ONU links to the hea-dend, we add the following redundant constraintsto the problem CATV:

Cycle breaking constraints on X 0ijs: �14�Further, to obtain tighter lower bounds we add

the following (redundant) constraints to problemCATV:Xi2I

fijk 6 �K ÿ 1� Xjk 8 j; k 2 I ; �15�

fijC 6Xm2R

Yjm 8 i 2 I ; 8 j 2 P : �16�

Relaxing constraints (3)±(5), (10), (15) and (16),multiplying them with non-negative multipliersb; a;x; d; c and / respectively and adding them

into the objective function we get the followingrelaxed problem CATV-LR.

Problem CATV-LR (a; b; d;x; c;/):

MinimizeXi2W

Xj2W [C

CijXij �Xj2P

Xm2R

HjmYjm

�Xi2W

Xj2W

Xk2W [C

bijkffijk ÿ Xjkg

�Xj2I

Xk2P[fCg

ajk

Xi2I

fijk

(ÿ K Xjk

)

�Xj2P

xj

Xi2I

fijC

(ÿXm2R

Qjm Yjm

)

�Xi2I

Xj2P

dij Xij

(ÿXm2R

Yjm

)

�Xj2I

Xk2I

cjk

Xi2I

fijk

(ÿ �K ÿ 1�Xjk

)

�Xi2I

Xj2P

/ij fijC

(ÿXm2R

Yjm

):

subject to: Eqs. (2), (6)±(9), (11)±(14).

While the Lagrangian dual problem isProblem CATV-LRD

Maximizea;b;d;-;c;/

ZL�a; b; d;-; c;/�

subject to:

bijk P 0 8i 2 I ; j 2 W ; k 2 W [ fCg; �17�cjk P 0 8j; k 2 I ; �18�ajk P 0 8j 2 I ; k 2 P [ fCg; �19�xj P 0 8j 2 P ; �20�dij P 0 8i 2 I ; j 2 P ; �21�/ij P 0 8i 2 I ; j 2 P : �22�

Problem CATV-LR can be decomposed intothe three independent sub-problems below:

Sub-problem 1:

MinimizeXj2P

Xm2R

Hjm

(ÿ xjQm ÿ

Xi2I

dij

ÿXi2I

/ij

)Yjm

subject to: Eqs. (9) and (12).


Sub-problem 1 is only constrained by constraintset (9) and non-negativity and integrality constraints(12). This means that we can ®nd the optimal solu-tion by simply choosing to set Yjm � 1 at a site j 2 Pfor the capacity level m 2 R at which the term

Vjm � Hjm ÿ xjQm ÿXi2I

dij ÿXi2I

/ij

is minimum (and less than zero). For the remain-ing capacity levels m0 2 R we set Yjm0 � 0. Similarlyif there is no capacity level m0 at which Vjm0 < 0 weset Yjm0 � 0 for all m0 2 R. We carry out the sameprocedure for all potential ONU sites j 2 P andstore the resulting optimal solution in vector Y �.

Sub-problem 2:

MinimizeXi2W

Xj2W [C

CijXij ÿXi2I

Xj2W

Xk2W [C

bijkXjk

ÿXj2I

Xk2I

cjk�K ÿ 1�Xjk

ÿXj2I

Xk2P[C

ajkKXjk �Xi2I

Xj2P

dijXij

subject to: Eqs. (2), (11) and (14).

Note that constraint set (2) applies to all sitesincluding potential ONU locations. Since the lo-cation and equipment cost for an ONU and thecost of connecting it to C are included in Hjm, costsfor links XjC are set to zero (for all j 2 P ). Further,the inclusion of redundant (in the original formu-lation) cycle breaking constraints (14) on the Xij

variables forces the solution of sub-problem 2 to bea spanning tree. Thus, an optimal solution to sub-problem 2 will be a directed minimum spanningtree rooted at the headend, C. We use an e�cientsolution procedure suggested by Bock [4] for ®nd-ing such a spanning tree. Let the minimal spanningtree solution obtained be stored in vector X �.

Sub-problem 3:This sub-problem consists of jI j shortest path

sub-problems. That is: 8i 2 I

MinimizeXj2W

Xk2W [C

bijkfijk �Xj2I

Xk2I

cjkfijk

�Xj2I

Xk2P[C

ajkfijk �Xj2P

xjfijC

�Xj2P

/ijfijC

subject to: Eqs. (7)±(9) and (14).

It can be solved optimally by ®nding out theshortest path between the each subscriber neigh-borhood i and the headend C. We have utilizedDijkstra's [7] procedure to ®nd the shortestpath. All ¯ow variables along this path are setequal to 1.

Let (a�; b�; d�; x�; c�; /�) be an optimal solu-tion to the dual problem CATV-LRD. Findingoptimal multipliers a�; b�; d�; x�; c�; /� is noteasy, except in a few special cases. In practice,good (but not necessarily optimal) multiplierscan be obtained by using a subgradient optimiza-tion method or various multiplier adjustmentmethods.

We employ a subgradient optimization algo-rithm to derive lower bounds on the optimal pri-mal objective value using problem CATV-LR(a; b; d; x; c; /). Refer to Fisher [9] for an expla-nation of the subgradient procedure.

6. Solution heuristic

At every iteration of the Lagrangian heuristicwe utilize the following procedure to attempt to®nd a feasible solution. Decision variables Y � andX � obtained from sub-problems 1 and 2 are uti-lized in the following manner:1. Set Yjm � 1 i� Y �jm � 1.2. Tentatively set Xij � 1 i� X �ij � 1.The resulting solution will most likely be infeasi-ble. Infeasibility could result from one or more ofthe following:1. Some subscriber neighborhoods are connected

to sites j0 at which ONUs are not located. Thatis the set

fiji 2 I ; j0 2 P ;m 2 R;Xij0 � 1; Yj0m � 0g 6� fg:2. Links between subscriber neighborhoods may

exceed the maximum allowed. That is jSt�i�j >K ÿ 1, where i 2 I , St�i�� set of nodes in thesubtree rooted at node i.

3. Some ONUs may be violating capacity restric-tions. i.e., d jSt�j�j > Qm, where j 2 P , m 2 R,Yjm � 1.

We use the following heuristic to achieve a feasiblesolution.


The following terminology will be utilized todescribe our procedure:

In general, the feasible solution heuristic pro-ceeds in three phases. In the ®rst phase all sub-scriber neighborhoods connected to unopenedONUs are connected to the least cost (based on thedelta cost de®ned above) alternative open ONUsites or subscriber neighborhoods. At the end ofthe ®rst phase all subscriber neighborhoods areconnected to ONUs that are open. The secondphase involves transferring subscriber neighbor-hoods from ONUs exceeding capacity to alterna-tive subscriber neighborhoods or to open ONUsites once again based on the lowest delta cost (aslong as capacity constraints for ONUs are notviolated). At the end of the second phase therefore,the only infeasibilities that exist are due to linksbetween subscriber neighborhoods in the feedernetwork exceeding capacity K. The third phasetherefore involves ®rst ®nding the largest subtreeand the connecting a subscriber neighborhood inthis tree to an alternative site at the lowest deltacost. This procedure is continued until the solutionis feasible.

Procedure FEAS:Phase I:

1. O � fjjj 2 P ; Yjm � 1 for some m 2 Rg, U �P ÿ O; V � fj 2 U jXij � 1 for some i 2 I : g

2. For each j 2 V :While jSt�j�j > 0 do

Find index pair (i, k) such that Dik is min-imum8 i 2 St�j� and 8k 2 I [ O [ fCg

Set Xip�i� � 0; Xik � 1St�j� � St�j� ÿ St�i�Update set V

endwhile.

Phase II:1. T � f�j;m�jj 2 O;m 2 R; Yjm � 1; St�j� > Qmg2. 8�j;m� 2 T :

While �jSt�j�j > Qm� doFind index pair (i, k) such that Dik is min-imum8 i 2 St�j� and 8 k 2 I [ O [ fCg andk 62 St�j� andthe following conditions are satis®ed:

if k 2 O and Ykm � 1 then�jSt�k�j � jSt�i�j�6Qm

if k 2 I thenif k 2 St�q� for some q 2 O

then �jSt�q�j � jSt�i�j�6Qm.Set Xip�i� � 0;Xik � 1Set St�j� � St�j� ÿ St�i�Update set T

endwhile.

Phase III:1. While fiji 2 I ; jSt�i�j > Kg doFind index u 2 I such that jSt�u�j is minimumand jSt�u�j > Kand p(u)2 O [ CFind index pair (r, q) such that Drq is minimum8r 2 St�u�; q 2 I [ O [ fCg and q 62 St�u�Set Xrp�r� � 0; Xrq � 1Set St�u� � St�u� ÿ St�r�

endwhile.

Overall solution procedure:The following steps describe the overall solu-

tion procedure:1. Initialize Lagrangian multipliers (a; b; d; x;

c; /), lower bound ZL and best feasible solutionvalue. Set iteration count � 1.

2. Solve the Lagrangian problem (sub-problems 1,2 and 3).

3. The value of the relaxed problem ZL�a; b; d; x;c; /� is the summation of the values of sub-problems 1, 2 and 3.

4. If the value of the relaxed problem is higherthan the lower bound, then update the lowerbound and the best Lagrangian multipliers.

5. Execute the feasible solution procedure to solvethe primal problem P and update the bestfeasible solution value (ZP ) to problem P , ifapplicable.

p(i) is the end point of the directed link awayfrom node i, i.e., Xip�i� � 1.

Dik is a delta cost function de®ned as: theadditional cost of removing link �i; p�i��and adding link �i; k�. If addition of link�i; k� results in a cycle or capacity violationsthen Dik � 1.


If gap between best feasible solution and low-er bound <1% or iteration count > IterationLimit

then Stop.6. Iteration count� interation count +1. Update

Lagrangian multipliers. Go to 2.The above procedure was executed for a maximumof (Iteration Limit) 800 iterations.

7. Computational results

The heuristic was coded in C and computa-tional experiments were carried out on a Sun Ultra30 porting SunOS 5.6. Tables 2±7 report compu-tational results for a suite of 100 problems thatwere generated as follows.

Coordinates for sites were generated from auniform distribution over a square of size100� 100. Costs between subscriber neighbor-hoods and those between such neighborhoods andONU sites are set equal to the Euclidean distance

between the sites. Costs for ONUs were manipu-lated as shown in Table 1. The capacity of ONUequipment for capacity level m 2 R was calculatedto be Qm � dW 2m � Ke, while ONU cost was equalto Hjm� (Euclidean distance of potential ONUlocation ``j'' to headend site) �W 1m+Fixed Costm

as shown in Table 1.The results from the computational testing are

presented in Tables 2±6 and show that the heu-ristic performs relatively well on large networkswith 100 customer neighborhoods (which wouldimply between 2500 to 5000 subscribers) and 15potential ONU locations. Gaps range from 5.63%to 14.1% in relation to the lower bound. As seenfrom Fig. 2 and Table 7, the value of heuristicsolution increases as the bound on the maximumnumber of neighborhoods in each sub-treedecreases.

Table 1

Cost structure for ONUs at di�erent capacity levels

Capacity level (m) W1m W2m Fixed costm

Cost structure A1

1 1.2 4.0 10.0

2 1.4 4.5 10.0

3 1.6 5.5 10.0

Cost Structure A2

1 1.2 4.0 30.0

2 1.4 4.5 30.0

3 1.6 5.5 30.0

Cost Structure B1

1 1.5 4 10.0

2 2.1 5.55 10.0

3 3.8 7.5 10.0

Cost Structure B2

1 1.5 4 30.0

2 2.1 5.55 30.0

3 3.8 7.5 30.0

Cost Structure C

1 1.2 4.0 100

2 1.4 4.5 25

3 1.6 5.5 50

Cost�Hjm�W1m�Euclidean Distance of ``j'' to C�Fixed

Costm.

Capacity�dQ� m � W 2m � Link Capacitye.

Table 2

Computational results for CATV, number of subscriber

neighborhoods� 100, potential ONU� 15, cost structure A1

Link

capacity

(K)

Number

of ONUs

opened

Time

(min)

UB LB %Gap

4 5 42.05 1163.74 1042.04 11.68

4 4 41.20 1136.15 1046.67 8.55

4 6 41.38 1145.17 1038.9 10.23

4 5 41.47 1102.14 1001.03 10.10

Mean 5.00 41.42 ± ± 10.14

5 4 42.02 1073.51 957.55 12.11

5 4 42.19 1030.19 962.72 7.01

5 5 42.25 1066.21 953.83 11.78

5 4 42.27 1028.34 925.87 11.07

Mean 4.25 42.18 ± ± 10.49

6 3 41.13 986.62 897.1 9.98

6 4 41.43 985.24 906.85 8.64

6 4 41.49 999.44 899.59 11.10

6 2 42.12 954.08 874.82 9.06

Mean 3.25 41.44 ± ± 9.70

7 2 42.30 932.96 856.68 8.90

7 3 41.22 947.69 859.3 10.29

7 3 41.36 964.12 857.06 12.49

7 2 41.12 923.33 833.53 10.77

Mean 2.50 41.40 10.61

8 2 41.39 890.63 821.86 8.37

8 2 41.02 901.45 833.5 8.15

8 3 42.37 906.4 826.27 9.70

8 2 42.24 905.34 801.42 12.97

Mean 2.25 41.56 ± ± 9.80


Our heuristic allows a designer to evaluate thetradeo� between cost of the network and the degreeof reliability of the network. As the number ofneighborhoods within each subtree (K) increases, sodoes the number of subscribers e�ected by anynetwork downtime. Table 7 shows this change incosts for a single network instance when the linkcapacity K is changed from 2 to 9. These results areplotted in Fig. 2. Fig. 2 therefore, provides thenetwork designer with some guidelines as to theeconomic tradeo�'s resulting from reliability re-quirements and is similar to the ``e�cient frontier''suggested by Balakrishnan and Altinkemer [3].While Fig. 2 depicts results for cost structure A-1,similar results hold for all other cost structurestested. The tradeo� in cost is clear from the graphwhich shows that the cost increases as the size of

each subtree is reduced (i.e., as the number of af-fected subscribers is decreased), this makes intuitivesense since increases in link capacity (K) result incheaper (albeit less reliable) designs since the ma-jority of the costs are due to interconnection ofcustomer neighborhoods and ONU sites. Thisholds true in practice since trenching and outsideplant installation costs dominate in most real situ-ations [18]. Note that once link capacity is increasedbeyond a certain level, the resulting network is onlymarginally cheaper than the previous design.

From Tables 2±4 we can see that the number ofONUs installed by the heuristic, decreases as thelink capacity (K) is increased. In fact, for certainvalues of K and the ®xed cost of ONUs, it may nolonger be economical to install any ONUs. Thisresults in a capacitated spanning tree topology

Table 3


neighborhoods� 100, potential ONU sites� 15, cost structure

A2

Link

capacity

(K)

Number

of ONUs

opened

Time

(min)

UB LB %Gap

4 4 42.12 1252.55 1119.08 11.93

4 4 41.24 1218.85 1122.12 8.62

4 3 41.35 1280.41 1122.21 14.10

4 2 41.45 1217.34 1066.17 14.18

Mean 3.25 41.44 ± ± 12.21

5 3 42.10 1138.43 1016.76 11.97

5 3 41.20 1146.13 1029.76 11.30

5 2 42.05 1136.83 1022.51 11.18

5 2 40.55 1094.35 978.7 11.82

Mean 2.50 41.37 ± ± 11.57

6 2 41.37 1054.81 939.23 12.31

6 1 41.06 1074.84 958.86 12.10

6 2 41.15 1055.04 953.63 10.63

6 1 41.15 1010.51 906.8 11.44

Mean 1.50 41.18 ± ± 11.62

7 0 41.36 979.16 891.08 9.88

7 1 41.44 987.59 905.93 9.01

7 1 42.30 1013.25 901.83 12.35

7 1 42.06 972.24 863.12 12.64

Mean 0.75 41.59 ± ± 10.97

8 1 42.03 931.31 850.63 9.48

8 1 42.07 936.13 865.28 8.19

8 1 40.57 966.78 858.59 12.60

8 0 41.15 936.77 824.8 13.58

Mean 0.75 41.35 ± ± 10.96

Table 4



B1

Link

capacity

(K)

Number

of ONUs

opened

Time

(min)

UB LB %Gap

4 4 42.16 1229.26 1094.62 10.95

4 4 42.17 1205.86 1093.63 9.31

4 6 42.53 1193.27 1097.87 7.99

4 3 42.10 1163.00 1037.97 10.75

Mean 4.25 42.24 ± ± 9.75

5 3 42.38 1127.69 992.34 12.00

5 4 41.57 1078.82 1002.82 7.04

5 4 42.40 1114.46 998.89 10.37

5 2 41.45 1061.13 956.83 9.83

Mean 3.25 42.15 ± ± 9.81

6 3 42.14 1000.93 925.27 7.56

6 3 41.54 1015.15 934.37 7.96

6 4 42.20 1029.52 930.08 9.66

6 2 42.10 978.75 888.99 9.17

Mean 3 42.09 ± ± 8.59

7 2 41.10 948.86 873.05 7.99

7 2 40.31 970.88 885.18 8.83

7 2 40.54 971.54 880.22 9.40

7 2 41.18 941.61 849.88 9.74

Mean 2 40.58 ± ± 8.99

8 2 41.59 902.74 851.91 5.63

8 1 41.09 917.47 839.85 8.46

8 2 39.53 926.98 844.93 8.85

8 2 41.45 914.33 810.33 11.37

Mean 1.75 41.11 ± ± 8.58


(see Tables 3 and 5) with no ONUs located. Byevaluating various such alternatives the networkdesigner can choose a topology that best ®ts his orher design criteria.

Computational times are not a�ected signi®-cantly by changes in capacity or cost structure andaverage less than 45 minutes of CPU time on a SunUltra 30. While such times may seem large, it ishelpful to keep in mind the complexity of theproblem being solved.

8. Concluding remarks

In this paper we have formulated a mathemat-ical model for the design of Hybrid Fiber Co-axialnetworks utilized for CATV. Analysis of the

problem shows that it is NP-Complete. Finally, wehave provided a heuristic solution procedure basedon Lagrangian Relaxation and show that it cane�ectively solve realistic sized problems.

CATV companies are currently undergoing atransformation due to the converging a�ects oftechnological and regulatory changes. As yet onlya few CATV providers have applied to the FederalCommunication Commission (USA) for licensingto provide local telephone services. A large num-ber of CATV companies are currently modernizingand upgrading their networks to either HFC orsimilar architectures [5]. Thus there are a numberof opportunities and areas in this ®eld that areopen for research. The problem we have addressedis only one of a hierarchical set of tasks within anydesign framework. Other such tasks would include

Table 6



C

Link

capacity

(K)

Number

of ONUs

opened

Time

(min)

UB LB %Gap

4 5 43.28 1137.85 1060.15 7.33

4 4 44.12 1136.99 1052.93 7.39

4 6 45.15 1134.09 1047.87 7.60

4 6 37.50 1154.48 1059.97 8.19

Mean 5.25 42.41 ± ± 7.63

5 3 40.12 1048.34 935.86 10.76

5 3 41.13 1023.49 916.83 10.42

5 5 38.32 1024.41 933.35 8.89

5 4 44.20 1037.26 936.47 9.72

Mean 3.75 41.04 ± ± 9.95

6 4 43.25 975.17 885.40 9.21

6 3 45.35 1014.27 925.52 8.75

6 3 41.24 969.98 902.00 7.01

6 2 43.06 988.09 920.75 6.81

Mean 3.00 43.22 ± ± 7.94

7 3 45.17 892.26 836.11 6.29

7 3 41.29 938.44 848.15 9.62

7 3 39.10 901.26 822.06 8.79

7 3 43.31 916.93 838.63 8.54

Mean 3.00 42.22 ± ± 8.31

8 2 39.10 920.86 829.03 9.97

8 2 42.30 923.03 829.15 10.17

8 2 43.22 842.43 774.32 8.08

8 2 44.25 912.09 826.50 9.38

Mean 2.00 42.22 9.40

Table 5


neighborhoods � 100, potential ONU sites� 15, cost structure

B2

Link

capacity

(K)

Number

of ONUs

opened

Time

(min)

UB LB %Gap

4 2 40.08 1320.24 1148.91 12.98

4 3 40.24 1296.68 1166.69 10.02

4 3 40.38 1311.14 1175.08 10.38

4 3 41.13 1238.74 1096.81 11.46

Mean 2.75 40.36 ± ± 11.21

5 2 40.48 1171.90 1046.79 10.68

5 3 42.23 1160.43 1066.69 8.08

5 2 41.03 1176.21 1068.20 9.18

5 2 41.43 1104.36 997.36 9.71

Mean 2.25 41.29 ± ± 9.41

6 2 41.18 1016.43 923.93 9.10

6 1 41.55 1070.27 982.61 8.19

6 2 40.44 1076.11 983.15 8.64

6 2 42.32 1013.29 920.95 9.11

Mean 1.75 41.37 ± ± 8.76

7 1 41.13 983.08 897.88 8.67

7 1 40.37 1008.68 919.49 8.84

7 2 41.13 1029.92 920.55 10.62

7 1 41.15 963.96 870.28 9.72

Mean 1.25 41.05 ± ± 9.46

8 0 41.16 929.96 859.88 7.54

8 0 40.30 948.66 873.27 7.95

8 0 41.13 974.57 871.71 10.55

8 0 40.47 926.79 833.54 10.06

Mean 0 40.56 ± ± 9.02


the placement and interconnection of headendnodes in a metropolitan area, interconnection withother servers and services such as Video on De-mand, the PSTN and nodes to the internet back-bone. Further, with an existing older generationco-axial plant in a large number of areas, there is aneed to explore models that address making ad-ditions or upgrades in an economical manner.These and other areas are future directions that arebeing explored by the authors.

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hybrid fiber co-axial catv network design with variable capacity optical network units

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