1

Hybrid Wireless-Optical Broadband Access Network(WOBAN): Network Planning Using Lagrangean Relaxation

Suman Sarkar1, Hong-Hsu Yen2, Sudhir Dixit3, and Biswanath Mukherjee1

1Department of Computer Science, University of California, Davis, CA 95616, USA2Department of Information Management, Shih Hsin University, Taiwan

3Nokia Siemens Networks, Mountain View, CA 94043, USAEmail: {sarkar, mukherje}@cs.ucdavis.edu, hhyen@cc.shu.edu.tw, sudhir.dixit@nsn.com

Abstract— The concept of a hybrid wireless-opticalbroadband access network (referred to as WOBAN in thisstudy) is a very attractive one. This is because it may becostly in several situations to run fiber to every home (orequivalent end user premises) from the telecom CentralOffice (CO); also, providing wireless access from the COto every end user may not be possible because of limitedspectrum. Thus, running fiber as far as possible from theCO towards the end user and then having wireless accesstechnologies take over may be an excellent compromise.How far should fiber penetrate before wireless takesover is an interesting engineering design and optimizationproblem, which we address in this paper.

We propose and investigate the characteristics of ananalytical model for network planning, namely optimumplacements of Base Stations (BS) and Optical NetworkUnits (ONU) in a WOBAN (called the Primal Model orPM). We develop several constraints to be satisfied: BS andONU installation constraints, user assignment constraints,channel assignment constraints, capacity constraints, andsignal-quality and interference constraints. To solve thisPrimal Model (PM) with reasonable accuracy, we use“Lagrangean Relaxation” to obtain the corresponding“Lagrangean Dual” model. We solve this dual problemto obtain a lower bound of the primal problem. We alsodevelop an algorithm (called the Primal Algorithm) tosolve the PM to obtain an upper bound. Via simulation,we compare this PM to a placement heuristic (called theCellular Heuristic), and verify that the placement problemis quite sensitive to a set of chosen metrics.

Index Terms— broadband access, optical network, wirelessnetwork, network planning, primal model, Lagrangean relaxation,duality gap.

I. INTRODUCTION

The hybrid wireless-optical broadband access network(WOBAN) can be employed to capture the best of bothworlds — the reliability, robustness, and high capacity ofwireline optical communication, and the flexibility (“anytime-anywhere” approach) and cost savings of wireless network.A WOBAN consists of a wireless network at the front end,and it is supported by an optical network at the back end.

A short, summarized version of this work was presented at the WCNCconference in Hong Kong in March 2007.

Noting that the dominant optical access technology today is thepassive optical network (PON), different PON segments can besupported by a telecom Central Office (CO), with each PONsegment radiating away from the CO. The head end of eachPON segment is driven by an Optical Line Terminal (OLT),which is located at the CO. The tail end of each PON segmentwill contain a number of Optical Network Units (ONUs),which typically serve end users in a standard PON architecture.However, for the proposed WOBAN, the ONUs will also serveas “gateways” to the wireless portion of this network and cansupport multiple Base Stations (BS). The wireless portion ofWOBAN may employ standard technologies (such as WiFi andWiMAX).

In a typical WOBAN (see Fig. 1), end users with wirelessdevices at individual homes or business premises are scatteredover a geographic area. Each of them will communicate witha wireless BS. These BSs are placed in an optimized mannerand are connected to ONUs. There could be several ONUsthat are strategically placed to efficiently manage the networktraffic [1], [2]. Please see [3] for an overview of the WOBANarchitecture and a variety of research challenges associatedwith it.

Our prior work in [1], [2] reported on simple approaches forWOBAN deployment. In these approaches, minimizing averagedistance (from ONU to users) is the optimization metric, andother aspects of WOBAN deployment (such as ONU and BScapacities, user assignment, channel assignment, interference,etc.) were not considered. (These terms will become clearerlater in this paper.) Therefore, a sophisticated deploymentapproach should capture the design interplay between variousaspects of a WOBAN (as indicated above) to achieve anoptimal solution. Additionally, a “good” estimation model ofdeployment cost of a WOBAN should be very important to anetwork designer.

To tackle this challenging problem, in this paper, we proposeand investigate the characteristics of an analytical model (calledPrimal Model or PM) with the deployment cost as an opti-mization metric. We identify six sets of constraints, viz., userassignment constraints, BS installation constraints, ONU in-stallation constraints, capacity constraints, channel assignmentconstraints, and signal-quality and interference constraints. Foranalytical tractability of this PM, we relax a few constraints(“Lagrangean Relaxation”) to transform the problem to thecorresponding “Lagrangean Dual” problem. Then we solve thedual problem to obtain a lower bound on the primal problem

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Fig. 1. Wireless-optical broadband access network (WOBAN).

(i.e., PM without relaxation). We also develop an algorithm(called Primal Algorithm) to solve the PM and obtain itsupper bound. By measuring the “duality gap”, which is thedifference between the solution to the primal problem andthe solution to its Lagrangean dual problem, we verify theaccuracy of our formulation. Then we explore and investigatethe characteristic of a heuristic, called “Cellular Heuristic(CH)” for the placement of BSs and ONUs to compare withthe PM solutions.

The rest of the study is organized as follows. In Section II,we briefly review the related research efforts and currentWOBAN deployment scenarios. In Section III, we presentthe analytical formulation (Primal Model) and its solutionapproach by “Lagrangean Relaxation” and Primal Algorithm.Section IV contains performance studies of the PM and Sec-tion V summarizes this work.

II. MOTIVATION AND RESEARCH OPPORTUNITIES

WOBAN is a new approach for the broadband access net-work. Recently, WOBAN has been gaining increasing attentionand early versions of the wireless part of WOBAN are beingdeployed as a municipal access solution. WOBAN saves onnetwork deployment cost because fiber (or wiring) does notneed to reach the end user, and it extends the coverage ofemerging optical access solutions, e.g., PON-based access solu-tions. The growing customer demands for bandwidth-intensiveservices are also accelerating the research efforts needed todesign an efficient “last mile” access network in a cost-effectivemanner. Thus, the radio-on-fiber (ROF) technology has gainedmomentum, where radio signals can be effectively carried overan existing optical fiber infrastructure (saving “last mile” costs)

by means of the “Hybrid Fiber Radio” (HFR) enabling technol-ogy. Recent research works propose ROF-based technologiesin millimeter-waveband (mm-waveband) [4], and demonstrateintegrated broadband services in a ROF downstream link [5].Among other research efforts, the authors in [6] investigatea bandwidth-allocation algorithm for an interactive video-on-demand (VoD) system over such a hybrid network.

With the advances in wireless technologies, IEEE802.11a/b/g (WiFi) deployments, which are very common, cansupport up to 54 Mbps today; and the emerging IEEE 802.16(WiMAX) can support much higher data rates (∼ 100 Mbps)over a long distance. In WOBAN, BSs are associated withONUs with fiber; so each user can connect to the back-endPON infrastructure via wireless channels. Then WOBANcan save a major part of the deployment cost, which is thehigh cost of laying fiber in the “last mile” from CO to user.In addition, due to high ONU capacity (e.g., 1 Gbps to 10Gbps), one ONU can support multiple BSs; and a BS, in turn,can support multiple users via wireless channels. WOBANcan also leverage a wireless network’s flexibility; i.e., the“anytime-anywhere” approach.

In the WOBAN architecture, the deployment cost of theoptical part of this network is much higher than its wirelesscounterpart. Besides the deployment cost, the deployment timefor the optical part of WOBAN is longer than its wireless part.Therefore, to keep minimum fiber penetration, the wireless partof the network should provide coverage as far as possible. Inother words, one ONU needs to support more wireless BSs.However, this kind of design strategy must be careful to meetthe following criteria.

- How to cope with the increasing user traffic demands:

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As the user’s traffic demand grows, we need to deployadditional BSs to serve this demand. However, due tothe capacity constraint of an ONU, the number of BSsthat an ONU can support is limited. Therefore, unlesscarefully planned, the increasing traffic demand may notbe properly served.

- How to avoid co-channel interference among BSs:The frequency spectrum in a wireless network is a limitedand valuable resource. If two adjacent BSs (or BSs inclose proximity) use the same channel to serve theirusers, they will incur co-channel interference. Since theco-channel interference deteriorates the signal quality, itreduces the maximum number of users a BS and anONU can support. If the signal quality is below a certainthreshold (called Carrier-to-Interference threshold or CIthreshold), then users’ information (such as data packets)will be dropped. In order to deal with the co-channelinterference, channel assignment for users communicatingwith BSs should be carefully planned in a WOBANdesign.

Note that there is a tradeoff between WOBAN’s deploymentcost and its performance. Hence, how to minimize the deploy-ment cost without degrading the performance is a challengingtask. Given the user traffic demands and the signal qualityrequirements, several decisions should be made (details ofwhich are elaborated in Section III).

- BS location and its transmission radius:That is, where to deploy the BSs and how to determinetheir coverage area (or assign transmission radius) tosatisfy the user traffic demand and avoid interference atthe same time.

- User homing decision:That is, which BS should serve a user to satisfy user’sbandwidth requirement.

- Carrier-to-Interference ratio for channel assignment:A BS’s channel assignment should avoid co-channel inter-ference, which is captured by Carrier-to-Interference ratio(CI ratio). Unlike the conventional approach to assigndifferent non-overlapping channels to nearby BSs, whichmay result in poor channel utilization and throughput,we tackle this issue with a more sophisticated approachutilizing Carrier-to-Interference (CI) ratio. CI ratio is alsolinked with signal quality and bit-error rate (BER). Thehigher the CI ratio, the lower is the BER and vice versa(see Section IV for more information).

- ONU location and fiber link deployment:The deployment of the optical part of the network shouldminimize the deployment cost and meet the BS’s trafficdemands.

From the above discussions, we note that a WOBAN deploy-ment is more challenging than only an optical or a wirelessaccess network deployment. This is because of the designinterplay between two very diverse access technologies (opticaland wireless). Table I captures a sample of research activitieson network deployment. In Table I, Reference [1] proposesa WOBAN architecture and investigates a Greedy Algorithmfor suitable placement of ONUs in a WOBAN, and Reference[2] retrofits the WOBAN’s deployment in a combinatorial

optimizer such as Simulated Annealing. However, researchon traditional access network placements can also be a goodstarting point for a WOBAN design. Thus, the rest of Table Isummarizes the research on network setup, where the archi-tecture is mainly focused on the wireless network. We observethat the placement research can be broadly divided into twocategories: indoor and outdoor locations. For both categories,several techniques have been employed, e.g., iterative methods(viz., quasi-Newton in [7], linear regression and least squarein [13], etc.), pruning-searching techniques (viz., Hooke-Jeevesin [7], Nelder-Mead in [8], etc.), and combinatorial optimizers(viz., genetic algorithm in [11], tabu search in [15], etc.).Various metrics have been used for network optimization,ranging from distance (in [9]) to signal strength (in [10]). Somestudies also focus on the trial-and-error deployment of BSs sothat no void region (a region with little or no signal coverage)exists. A campus-wide access network setup is captured in [12].

Noting that a WOBAN is a high-capacity cost-effectivebroadband network, recently its early incarnations are beingdeployed as an access solution in many cities around the world(viz., Chaska MN by Tropos; San Francisco CA by Earthlinkand Google; Athens GA, and Bristol UK by Belair; CulverCity CA, and Gordes France by Firetide; etc.) [16], [17], [18],[19], [20], [21], [22], [23]. Thus, WOBAN deployment is animportant problem in today’s network scenario.

III. MATHEMATICAL FORMULATION FOR OPTIMALPLACEMENT OF BSS AND ONUS

This study focuses on the optimal placement of BSs andONUs in the front end, and the fiber layout from BSs to ONUsand from ONUs to OLT/CO in the back end of a WOBAN. Inour mathematical formulation of this optimization problem, weconsider the cost of ONUs and BSs, though our performancestudies (in Section IV) indicate that the cost of laying fiberin a WOBAN is more significant than the costs of variousdevices. Also note that more ONUs can lead to additional OLTinstallation cost, because an OLT can generally drive a fixednumber of ONUs. Therefore, additional OLTs will increasethe cost of a WOBAN solution considerably as OLTs areexpensive.

We propose and investigate a “Primal Model (PM)”formulation, which is a pre-deployment network-optimizationscheme, where the cost of WOBAN design is minimized (byplacing reduced number of BSs and ONUs, and planning anefficient fiber layout). We also examine the interference amongmultiple BSs, and explore several installation and assignmentconstraints that have to be satisfied for a better-quality accesssolution and increased coverage. Our proposed model (PM) forWOBAN placement and its solution approach by “LagrangeanRelaxation” are shown below.

Given Parameters:L: set of possible locations for BSs1,O: set of possible locations for ONUs,Θi (Bi): installation cost of a BS at location i,Λk (Uk): installation cost of an ONU at location k,

1Here ”locations” mean cross-points on a square grid.

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TABLE IRESEARCH ACTIVITIES ON NETWORK PLACEMENT.

Research Work Objective Setting Cost Model Optimization Contributions (in brief)

Sarkar et al. [1] Optimumplacements ofONUs in aWOBAN.

Outdoor Distance Greedy Algorithm Divide-and-conquer heuristic;Converges fast, low complexity;

Produces good solution.Sarkar et al. [2] Outdoor Distance Simulated Annealing Placement retrofitted to optimizer;

Improved solution quality over Greedy [1].

Sherali et al. [7]

Optimumplacements ofBase Stations forminimizing thetotal cost ofnetwork setup.

Outdoor Signal strength Hooke-Jeeves, Quasi-Newton, Hill-climbing Minisum, Minimax, and combination model;Captures both single and multiple Tx problems.

Wright [8] Indoor Signal propagation Nelder-Mead Direct Search Generic model with attenuation;Finds local optimum.

Molina et al. [9] Outdoor Distance Greedy, Genetic, Greedy+Genetic Optimized cellular coverage;Combinatorial approach.

Hurley [10] Outdoor Signal, Distance, Traffic Simulated Annealing Multiple costs approach;Cell handover considered.

Nagy et al. [11] Indoor Motley-Keenan path loss Genetic Algorithm Empirical model;High complexity.

Hills [12] Indoor Signal strength Trial-and-error Cylindrical design approach;Deployed in Carnegie Mellon.

Chen et al. [13] Indoor Bahl’s path loss Linear Regression, Least Square Both signal-strength and location awareness;Generic model with attenuation.

Kamenetskym et al. [14] Outdoor Signal strength Uniform, Pruning, Simulated Annealing Empirical model;Minisum and Minimax approaches.

Battiti et al. [15] Outdoor Signal strength Tabu, Hill-climbing, Simulated Annealing Localization + signal-coverage model;Shows trade-off between two metrics.

Φik (Zik): fiber installation cost from BS i to ONU k,T : set of users,Kr: traffic demand of user r,F : set of available wireless channels,A: upper bound on number of channels assigned to a BS,Ei: maximum capacity of BS at location i,ρ: fraction of users served by BSs (user coverage ratio),Dri: distance between user r and BS at location i,Ψ(Dri): supported bandwidth (in Mbps) to user r from BS

i with distance Dri,Γ: discrete set of possible transmission radius of BS,J ′: upper bound of decision variable Jk,R′: upper bound of decision variable Ri,I: threshold of Carrier-to-Interference (CI) ratio, andG: an arbitrarily large number.

Decision Variables:

Bi: 1, if a BS is installed at location i, and 0 otherwise,Uk: 1, if an ONU is installed at location k, and 0

otherwise,Zik: 1, if an ONU at location k is connected to a BS at

location i, and 0 otherwise,Xji: 1, if channel j is assigned to BS at location i, and 0

otherwise,Yri: 1, if user r is assigned to BS at location i, and 0

otherwise,Jk: capacity of ONU at location k,Ri: transmission radius of BS at location i, andIii′ : interference factor of BS at locations i′ on BS at

location i (Iii′ =(

Ri′Dii′

)4

where Dii′ is the distancebetween BSs at i and i′).

Objective Function of Primal Model:

CP M = Min

(∑k∈O

Λk (Uk) +∑i∈L

Θi (Bi) +∑i∈L

∑k∈O

Φik (Zik)

)(1)

The objective (CPM ) is to minimize the sum of thefollowing items: installation cost for all ONUs required, plusinstallation cost for all BSs required, plus cost of connectingBSs to an ONU by fiber. These three items are the major costsinvolved in configuring a WOBAN. As discussed before, foranalytical tractability, we relax a few challenging constraints(constraints are described later) in Section III-A to transformthe primal model (CPM ) to the corresponding Lagrangeandual problem (CLR).

Constraints:Below are the constraints that need to be satisfied in PrimalModel. The goal of the first set of constraints is to enforceuser assignment constraints. Constraint 1 captures the binarydecision variable Yri. Each user is at most associated with onlyone BS (Constraint 2), and at least ρ (ρ ≤ 1) fraction of totalnumber of users needs to be served by BSs (Constraint 3).

1. Yri = 0 or 1 ∀r ∈ T, i ∈ L,2.

∑i∈L

Yri = 1 ∀r ∈ T , and

3.∑

r∈T

∑i∈L

Yri ≥ ρ|T | ∀r ∈ T, i ∈ L.

The goal of the second set of constraints is to enforce BSinstallation constraints. Constraint 4 specifies that the decisionvariable Bi must be binary. A BS must be installed firstbefore a user can be assigned to it (Constraint 5). In addition,the distance between user r and BS i must be within thetransmission radius of BS i (Constraint 6). The non-negativityconstraints of decision variable Ri are captured by Constraints7 and 8. Constraint 9 is used to enforce that the transmissionradius of a BS should be a discrete set.

4. Bi = 0 or 1 ∀i ∈ L,5. Yri ≤ Bi ∀r ∈ T, i ∈ L,6. DriYri ≤ Ri ∀r ∈ T, i ∈ L,7. Ri ≤ R′Bi ∀i ∈ L,8. 0 ≤ Ri ∀i ∈ L, and9. Ri ∈ Γ ∀i ∈ L.

The goal of the third set of constraints is to enforce channelassignment constraints of BS. Constraint 10 specifies that thedecision variable Xji must be binary. In WiFi or WiMAXtechnology, the channel can be CDMA codes or TDMA time

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slots in wireless frequency bands. Therefore, Constraint 11indicates that the number of channels assigned to each BS islarge enough to serve its users. Constraint 12 indicates that BSmust be installed first before channel assignment. Constraint 13indicates that the number of channels assigned to a BS shouldnot exceed the upper bound of channels assigned to any BS.

10. Xji = 0 or 1 ∀j ∈ F, i ∈ L,11.

∑r∈T

Yri ≤∑

j∈FXji ∀i ∈ L,

12. Xji ≤ Bi ∀j ∈ F, i ∈ L, and13.

∑j∈F

Xji ≤ A ∀i ∈ L.

The fourth set of constraints captures the ONU installationconstraints. Constraints 14 and 15 specify that the decisionvariables Zik and Uk, respectively, must be binary. Constraint16 indicates that an ONU must be installed first before anyBS can be connected to it. Each BS should be connectedto only one ONU, which can be captured by an equality∑

k∈O Zik = Bi, where ∀i ∈ L. This equality can be brokeninto two inequalities such as Constraints 17 and 18.

14. Zik = 0 or 1 ∀i ∈ L, k ∈ O,15. Uk = 0 or 1 ∀k ∈ O,16. Zik ≤ Uk ∀i ∈ L, k ∈ O,17.

∑k∈O

Zik ≤ Bi ∀i ∈ L, and

18. Bi ≤∑

k∈OZik ∀i ∈ L.

The fifth set of constraints enforces capacity constraints ofBSs and ONUs. Constraint 19 enforces that the maximumbandwidth of user r from BS i satisfies the user’s trafficdemand. Constraint 20 indicates that a BS should be ableto manage its users’ aggregate traffic demands. Constraint 21enforces that the capacity of each ONU is large enough to serveall traffic introduced by its associated BSs. The non-negativityconstraints of decision variable Jk are captured by Constraints22 and 23.

19. Kr ≤ Ψ (Dri) Yri ∀r ∈ T, i ∈ L,20.

∑r∈T

Ψ (Dri) Yri ≤ Ei ∀i ∈ L,

21.∑

i∈LEiZik ≤ Jk ∀k ∈ O,

22. Jk ≤ J′Uk ∀k ∈ O, and23. 0 ≤ Jk ∀k ∈ O.

The goal of the sixth set of constraints is to enforce signalquality constraints for each user. Since co-channel interferencewill significantly impact the signal quality, we need to take thisinto account when we decide on the channel assignment foreach BS. In Constraint 24, the left-hand side is the total co-channel interference introduced by other BSs using the samechannel j to BS at i. In Constraint 24, when Xji = 0, the right-hand side will be equal to G (a very large number). This makesConstraint 24 to be always satisfied. However, when Xji = 1,the right-hand side will be equal to 1

I . Hence, we can guaranteethe signal quality to be at least the threshold of acceptable CIratio. Constraints 25 is the non-negativity constraint of decisionvariable Iii′ .

24.∑

i′∈L,i′ 6=iIii′Xji′ ≤ G +

(1I −G

)Xji ∀i ∈ L, j ∈ F , and

25. 0 ≤ Iii′ ∀(i, i′) ∈ L, i 6= i′.

A. Lagrangean Relaxation and Lower Bound of Primal Model

This Primal Model (PM) formulation is very challengingsince we need to carefully plan the locations of BSs and ONUs,the transmission radius of BSs, the channel assignment of BSs,the assignment of users to BS, and the assignment of BSs toONU in order to satisfy the traffic requirement and signal-quality requirement of users at minimum cost.

We apply the Lagrangean Relaxation (LR) method to relaxsome of the constraints of our formulation. For analyticaltractability, we relax those constraints that make the PM hard.After the relaxation, we get the Lagrangean dual problem.The solution to this dual problem is the lower bound to theprimal problem. On the other hand, by developing the PrimalAlgorithm, we can obtain feasible solutions which give theupper bound to the primal problem. The Primal Algorithm(see Section III-B) is designed such that it can obtain the upperbound of the primal problem quickly. By measuring the dualitygap, i.e., the gap between the upper bound and the lower boundof our model, we can determine how optimal the solutions are.

Next we relax the ten Constraints 5, 6, 11, 12, 17, 18, 20, 21,22, and 24. The intuition behind relaxing these ten constraintsis that they make the Primal Model “hard” to solve, and afterrelaxation, the Lagrangean dual model will be analyticallytractable. The more we relax these constraints to make thePrimal Model simpler, the larger will be the duality gap. Thenthe obtained solution to the primal problem will be too farfrom its optimal solution. On the other hand, if we relax toofew constraints, we may not be able to solve the Lagrangeandual problem optimally. Then the solution obtained to thedual problem will not produce the true lower bound of theprimal problem. Hence, how to relax the minimum number ofconstraints to get the correct and tight duality gap between thelower bound and the upper bound solutions is very importantin Lagrangean relaxation scheme. Next, we will show howrelaxing the above-mentioned ten constraints will provide anoptimal solution to the our Lagrangean dual problem and atighter duality gap.

So, the Lagrangean dual problem (CLR) becomes:

CLR (µ) = Min∑k∈O

Λk (Uk) +∑i∈L

Θi (Bi) +∑i∈L

∑k∈O

Φik (Zik) ...(2)

+∑i∈L

∑r∈T

µ1ir (Yri − Bi) +

∑i∈L

∑r∈T

µ2ir (DriYri − Ri) ...

+∑i∈L

µ3i

(∑r∈T

Yri −∑j∈F

Xji

)+∑i∈L

∑j∈F

µ4ij (Xji − Bi) ...

+∑i∈L

µ5i

(∑k∈O

Zik − Bi

)+∑i∈L

µ6i

(Bi −

∑k∈O

Zik

)...

+∑i∈L

µ7i

(∑r∈T

Ψ (Dri) Yri − Ei

)+

∑k∈O

µ8k

(∑i∈L

EiZik − Jk

)...

+

∑k∈O

µ9k

(Jk − J

′Uk

)...

+∑i∈L

∑j∈F

µ10ij

( ∑i′∈L,i′ 6=i

Iii′Xji′ −G−(

1

I−G

)Xji

),

subject to Constraints 1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16,19, 23, and 25. Note that the µn’s (µn ≥ 0,∀n ∈ [1, 10]) areknown as “Lagrangean multipliers”.

We can decompose CLR into five independent subproblems(viz., CS1, CS2, CS3, CS4, and CS5) and the terms with givenparameters Ei and G. Subproblems are the smaller blocksof problems for a large problem, such as CLR, and eachsubproblem contains one or multiple decision variable(s). So,

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Eqn. (2) becomes:

CLR (µ) = CS1 + CS2 + CS3 + CS4 + CS5 −∑i∈L

µ7i Ei −

∑i∈L

∑j∈F

µ10ij G. (3)

The five subproblems are as follows.- Subproblem 1: for Bi and Ri

CS1 = Min∑i∈L

Θi (Bi)−∑i∈L

∑r∈T

µ1irBi −

∑i∈L

∑j∈F

µ4ijBi... (4)

−∑i∈L

(µ

5i − µ

6i

)Bi −

∑i∈L

∑r∈T

µ2irRi,

subject to Constraints 4, 7, 8, and 9.We can further decompose Subproblem 1 into |L| indepen-

dent subsubproblems. So, for BS at location i ∈ L, Subproblem1 becomes:

Min Θi (Bi)−

(∑r∈T

µ1ir +

∑j∈F

µ4ij + µ

5i − µ

6i

)Bi −

∑r∈T

µ2irRi, (5)

subject to Constraints Bi = 0 or 1, Ri ≤ R′Bi, 0 ≤ Ri, andRi ∈ Γ.

We can observe that, if we assign Bi = 0, then Ri =0 and the objective value of the subproblem is zero aswell. On the other hand, if we assign Bi = 1, we canassign Ri = R′ and this will lead to the smallest objectivevalue (since the coefficient of Ri is negative for a non-negative µ2

ir). Then the objective value is equal to Θi (Bi) −(∑r∈T µ1

ir +∑

j∈F µ4ij + µ5

i − µ6i

)−∑

r∈T µ2irR

′. If thisobjective value is smaller than zero, then the optimal solution isto assign Bi = 1 and Ri = R′; otherwise, the optimal solutionis to first assign Bi = 0 and then Ri = 0. The time complexityof solving this subproblem is on the order of (|T | + |F |) foreach BS i.

- Subproblem 2: for Uk and Zik

CS2 = Min∑k∈O

Λk

(Uk − µ

9kJ

′Uk

)... (6)

+∑i∈L

∑k∈O

(Φik (Zik) +

(µ

5i − µ

6i + µ

8kEi

)Zik

),

subject to Constraints 14, 15, and 16.Similarly, we can further decompose Subproblem 2 into |O|

independent subsubproblems. So, for ONU at location k ∈ O,Subproblem 2 becomes:

Min Λk (Uk)− µ9kJ

′Uk +

∑i∈L

(Φik (Zik) +

(µ

5i − µ

6i + µ

8kEi

)Zik

), (7)

subject to Constraints Zik = 0 or 1 ∀i ∈ L, Uk = 0 or 1, andZik ≤ Uk ∀i ∈ L.

When Uk = 0, all the corresponding decision variables Zik

will be zero. Then, the objective value of the subproblem willbe zero as well. If Uk = 1, then some of the Zik’s will beone. In this case, we will only select those Zik’s that havenegative coefficient, i.e.,

(µ5

i − µ6i + µ8

kEi

)< 0. Consider, for

i ∈ L′, L′ ⊆ L, ω =∑

i∈L′

(µ5

i − µ6i + µ8

kEi

)Zik < 0. Now,

if Λk (Uk)−µ9kJ ′Uk +

∑i∈L′ Φik (Zik)+ω < 0, then we will

assign Uk = 1, Zik = 1 ∀i ∈ L′, and Zik = 0 ∀i /∈ L′, whereL′ ⊆ L. Otherwise, we will get optimal solution at Uk = 0 and

Zik = 0 ∀i ∈ L. The time complexity of solving subproblem2 is on the order of (|L|) for each ONU k.

- Subproblem 3: for Jk

CS3 = Min

∑k∈O

(µ

9k − µ

8k

)Jk, (8)

subject to Constraint 23.Similarly, we can further decompose Subproblem 3 into |O|

independent subsubproblems. So, for ONU at location k ∈ O,Subproblem 3 becomes:

Min(µ

9k − µ

8k

)Jk, (9)

subject to Constraint 0 ≤ Jk.For each ONU k, if the coefficient of Jk is negative, i.e.,(

µ9k − µ8

k

)< 0, then we will assign Jk to its maximum

value, i.e., Jk = J ′ to minimize the objective value of thesubproblem, else assign Jk = 0. The time complexity ofsolving subproblem 3 is on the order of (1) for each ONUk.

- Subproblem 4: for Yri

CS4 = Min∑i∈L

∑r∈T

(µ

1ir + µ

2irDri + µ

3i + µ

7i Ψ (Dri)

)Yri, (10)

subject to Constraints 1, 2, 3, and 19.For each user r ∈ T , since the possible number of BS

locations is finite and fixed, we can exhaustively examine eachBS. Then, we identify the set of BSs that can satisfy the maxi-mum bandwidth requirement, and among those BSs, we choosethe one with the smallest

(µ1

ir + µ2irDri + µ3

i + µ7i Ψ(Dri)

)value. Then, we will select ρ|T | number of users to be servedby those BSs. The time complexity of solving this subproblemis on the order of (|T ||L|).

- Subproblem 5: for Iii′ and Xji

CS5 = Min∑i∈L

∑j∈F

(µ

4ij − µ

3i −(

1

I−G

)µ

10ij

)Xji... (11)

+∑i∈L

∑j∈F

∑i′∈L,i′ 6=i

µ10ij Iii′Xji′ ,

subject to Constraints 10, 13, and 25.After performing variable changing (Xji′ as Xji), we can

decompose Subproblem 5 into |L| independent subsubprob-lems. So, for BS at location i ∈ L, Subproblem 5 becomes:

Min∑j∈F

(µ

4ij − µ

3i −(

1

I−G

)µ

10ij +

∑i∈L,i6=i′

µ10i′jIi′i

)Xji, (12)

subject to Constraints Xji = 0 or 1 ∀j ∈ F ,∑

j∈F Xji ≤ A,and 0 ≤ Ii′i.

The transmission radius of each BS is a discrete and finiteset. So, for each transmission radius assignment, the interfer-ence factor Ii′i can be precomputed. Therefore, Xji is theonly decision variable in Subproblem 5. Since the number ofchannels assigned to the BS at location i can not exceed A,we can choose at most A channels. Being a finite set, we canexhaustively try all possible transmission radius assignmentsto determine the minimum cost one. The scheme for solvingSubproblem 5 optimally is as follows, with the time complexityon the order of (|L||Γ|).

7

I. Initialize set S = Null.II. Until all transmission radius assignments have been

taken into account, for a transmission radius Ri,calculate Ii′i, and do the following.

III. For channel j, calculate coefficient(µ4

ij − µ3i −

(1I −G

)µ10

ij +∑

i∈L,i 6=i′ µ10i′jIi′i

).

IV. For all possible channel assign-ments in j ∈ F , calculate∑

j∈F

(µ4

ij − µ3i −

(1I −G

)µ10

ij +∑

i∈L,i 6=i′ µ10i′jIi′i

).

Find which of these channel assignmentsproduces the smallest coefficient. Assumefor a particular channel assignment j ∈ ξ,the coefficient will be the minimum, smin =∑

j∈ξ

(µ4

ij − µ3i −

(1I −G

)µ10

ij +∑

i∈L,i 6=i′ µ10i′jIi′i

).

V. S = S ∪ {smin}.VI. Change the transmission radius Ri ←

(Ri + ∆(Ri)) ∈ Γ. Go back to Step II.VII. After getting all the smallest coefficient values for

possible Ri’s, find Min (S) and let the correspondingchannels Xji = 1 ∀j ∈ ξ.

Note that the Lagrangean Relaxation technique helps us tosuccessfully solve a non-convex formulation of Subproblem 5.

According to the weak Lagrangean duality theorem [24],[25] (which says “for any given set of nonnegative multipliers,the optimal objective function value of the Lagrangean dualproblem is a lower bound on the optimal objective functionvalue of the corresponding primal problem”), solving CLR (µ)will give the lower bound (LB) of CPM . Based on above ob-servations for each subproblem, we can solve the Lagrangeandual problem (CLR (µ)) optimally by using the subgradientmethod to get the tightest lower bound (LB) [26], [27] (seeSection III-C for details).

B. Primal Algorithm and Upper Bound of Primal ModelThough the solution to the Lagrangean dual problem

(CLR (µ)) is not an exact solution (since some of the con-straints of the PM have been relaxed), it can serve as a goodstarting point to get a feasible solution. The basic idea of thePrimal Algorithm is to install the BSs that can serve moreusers under the capacity constraints and interference constraintsuntil at least a fraction of the total number of users is covered(Constraint 3). Then, we will deploy the minimum number ofONUs to satisfy the traffic demands from BSs. Figure 2 showsthe schematic of the Primal Algorithm, which will give theupper bound (UB) of CPM .

We identify the sequence of channels to be assigned to usersin order not to violate the co-channel interference constraints inStep 2 of the algorithm. This is because users at close proximityshould be assigned non-interfering channels simultaneously toreduce cross-talk; the same channel can only be reassignedto users far apart from each other (channel reuse). The non-negative Lagrangean multiplier µ10

ij has a physical significanceof co-channel interference violation cost. Therefore, we candetermine the sequence of channel assignment for BS i bysorting µ7

ij in ascending order.In Step 4, we examine the capacity constraint of BS (i.e.,

Constraint 20). In Step 5, we examine the co-channel inter-ference constraint (i.e., Constraint 24). If these constrains are

satisfied, then we assign one channel to the user; otherwise, wecontinue to examine the other unvisited (unassociated) users.In Step 8, if Constraint 3 is not satisfied, we add a new BS tocover unvisited users. After Constraint 3 is satisfied, we canassign ONUs to cover all the traffic demands from installed BSs(Constraint 21). After the ONUs are identified, we construct aminimum-cost spanning tree (MST) to layout fibers from OLTto reach all ONUs and BSs.

Fig. 2. Primal Algorithm schematic (“T” means True, “F” means False).

C. Computing Upper Bound (UB) and Lower Bound (LB) ofPrimal Model

Next, we show how to compute the UB (solution producedby Primal Algorithm, see Section III-B) and the LB (solutionproduced by Lagrangean relaxation, see Section III-A) of ourPM formulation. We use the subgradient method as given be-low, where quiescence age is incremented if CLR (µ) does notimprove. When quiescence age becomes quiescence threshold,step size coefficient (or δ) becomes halved for the next itera-tion. The complexity of this method for each iteration is onthe order of (|L|(|L||Γ|+ |F |log|F |+ |T |)).

1. Begin2. Initialize Lagrangean multipliers, µn(0) = 0, ∀n ∈ [1, 10].3. UB = ∞ and LB = −∞.4. quiescence age = 0, δ0 = 2, and ε ∼ 0.5. Initialize quiescence threshold.6. For (m = 0; m ≤ max iteration; m = m + 1)

8

7. Solve sub-1, sub-2, sub-3, sub-4, and sub-5.8. Compute Cm

LR (µ) as in Eqn. (3).9. If Cm

LR (µ) > LB10. LB = Cm

LR (µ) and quiescence age = 0.11. δm+1 = δm.12. Else quiescence age = quiescence age + 1.13. End If14. If quiescence age = quiescence threshold15. δm+1 = δm

2and quiescence age = 0.

16. End If17. Run Primal Algorithm. Compute upper bound ub.18. If ub < UB, UB = ub19. End If.20. γm = g(Cm

LR (µ)). /*piecewise gradient computation*/21. tm =

δm∗(UB−LB)||γm||2

. /*step size update*/22. µn(m + 1) = µn(m) + tmγm. /*Lagrange multiplier

update*/23. If ||µn(m + 1)− µn(m)||1 ≤ ε /*stopping criteria*/24. Stop.25. End If26. End For27. /* Comments: ||.||1: Norm of 1 and ||.||2: Norm of 2. */28. End

IV. PERFORMANCE STUDY

We conducted several computational experiments to test thesolution quality and effectiveness of our approach. In thisstudy, we set max iteration and quiescence threshold as 1000and 30, respectively (using our experience after experiment-ing with various values for these parameters). We initializedstep size coefficient (or δ) as 2.

We placed 800 users randomly in an area of 5 × 5 square-miles. We assumed OLT to be located at (0, 0). We choseWiMAX as the front-end wireless solution for WOBAN. Thereare 50 available channels, and each channel operates at 20MHz. In non-line-of-sight (NLOS) WiMAX communication,when the channel operates at 20 MHz, we can get a maximumdata rate of 75 Mbps with a maximum transmission radius of 5miles [28]. WiMAX supports adaptive modulation schemes toadjust its data rates as needed inside a BS’s coverage area.More sophisticated modulation scheme (e.g., 64 QAM) areused in inner-most zone of the coverage area to provide bettersignal quality, which, in turn, leads to higher throughput andlower BER. On the other hand, moderate modulation schemes(e.g., QPSK, BPSK, etc.) are adopted in the outer zones ofa BS’s coverage area [29]. Table II shows typical values ofCarrier-to-Interference ratios (CI) in order to ensure a BER of10−6 for different WiMAX modulation schemes [30].

TABLE IIWIMAX MODULATIONS VS. CI.

Modulations QPSK 16QAM 32QAM 64QAMCI (dB) 16 20 23 27

Hence, WiMAX is expected to support much higher datarates in the inner-most zone of a BS’s coverage area, comparedto its outer zones. In other words, when the distance betweena BS and a user is larger, the WiMAX data rate will bereduced. From above observations, we set 2, 5, 10, 20, 30,40, 75 Mbps for transmission radius of 5, 4, 3, 2, 1.5, 1, 0.5miles, respectively [28], [29]. Traffic demand for each user ischosen between 1 Mbps and 75 Mbps. The maximum capacityfor each BS i and ONU k was set as 1 Gbps and 10 Gbps,

respectively (these are the futuristic inputs to our numericalstudy and research on achieving higher BS and ONU capacitiesis very active) [31].

Table III shows the expenses of various devices and fiberlayout, normalized to the cost of one ONU unit, which is takento be USD 100 at the time of this article.2

TABLE IIIDEVICE AND FIBER LAYOUT EXPENSES (IN NORMALIZED UNITS).

Device Cost (1 ONU unit)

ONU 1 [32]WiMAX BS 50 [33]Fiber 1000/mile [34], [35]OLT 50 [32]CPE 1

Note: The expenditure reported here is normalized to the cost of one ONU unit.Note: At the time of this article, one ONU unit cost is taken as 100 USD.

From Table III, we infer that Θi (Bi) for each BS i andΛk (Uk) for each ONU k are set to 5000 USD and 100 USD(US Dollar), respectively [32], [33]. Φik (Zik) for connectingBS i and ONU k is determined by the cost of laying fiber,which is chosen to be 100000 USD/mile [34], [35]. [Forthe sake of completeness of this study, we also show thenormalized cost of an OLT and a WiMAX Customer PremiseEquipment (CPE). Note that the CPE cost will be borne bycustomers, not the WOBAN designers.]

We assume that all ONUs connect to one OLT, locatedat (0, 0). We connect the OLT and ONUs/BSs through aminimum-cost spanning tree with OLT as the root.

A. Cellular Heuristic (CH)

Next, we develop a heuristic for the placement of BSs andONUs [called “Cellular Heuristic (CH)”] to compare with theoptimal primal solution. The Cellular Heuristic (CH) modelsthe BS’s “footprint” (or coverage area) as a hexagonal cell(similar to the concept of modeling a hexagonal cellulararchitecture) [36]. Given the number of available channelsand the Carrier-to-Interference ratio (CI), CH determines theminimum number of BSs and ONUs needed to satisfy the co-channel interference constraints.

In CH, based on the channel reuse criteria, a cell andits neighboring cells can not use the same channel due tointerference. The set of neighboring cells that do not use thesame channel is known as a cluster. For example, if a clustersize is 7, each cell and its six neighboring cells should usedifferent channels, and the same channel can only be reusedbeyond these cells. The channel reuse distance is

√3NR,

where N is the cluster size and R is the cell radius [36].Assuming that only surrounding BSs can interfere with eachother, Table IV estimates the minimum number of BSs neededto serve 800 users with 50 available channels that can beassigned in order to satisfy co-channel interference constraints.

In Table IV, cluster size (N ) is computed by N = u2 +uv + v2, where u and v are non-negative integers (we showpossible values of N up to 16). Assuming a BS can interfereonly with its six surrounding BSs (i.e., N = 7), CI can be

computed by CI = (√

3N)4

6 (recall that Iii′ =(

Ri′Dii′

)4

). When

2The normalized cost is less sensitive to ups and downs of the absolute cost.

9

TABLE IVESTIMATION OF CHANNEL INTERFERENCE AND NUMBER OF BSS BY CH.

N 1 3 4 7 9 12 13 16

Reuse Dist.√

3R 3R 2√

3R√

21R 3√

3R 6R√

39R 4√

3RCI 1.5 13.5 24 73.5 121.5 216 253.5 384CI (dB) 1.8 11.3 13.8 18.7 20.8 23.3 24 25.8Channels/BS 50 50/3 50/4 50/7 50/9 50/12 50/13 50/16BSs (Min) 16 48 64 112 144 192 208 256

all the BSs in the same cluster share the available channels,each BS can get at most 50/N number of channels. Therefore,we need at least

(800N

50

)number of BSs to serve all 800

users. This is the minimum number of BSs, computed on thebasis of co-channel interference, and other aspects such asusers’ traffic demands have not been taken into account. CHdeploys these BSs uniformly, determines their transmissionradius, and assigns channels to them so that CI constraintis satisfied. Then, it verifies if additional BSs are needed toserve all the users and to manage traffic demands from them.After deploying additional BSs, CH assigns channels to themwithout violating CI constraints for existing BSs. Finally, CHdetermines the number of ONUs needed to support BSs anddeploys ONUs uniformly. The complexity of CH is on the orderof (|L|(|L||Γ|+ |F |+ |T |)). Below, we summarize the detailsof CH.

1. Begin2. Construct a channel interference table (see Table IV).3. Derive minimum # of BSs, based on CI threshold (see Table IV).4. Deploy these BSs uniformly in the area.5. Assign channels to BSs to serve users.6. Determine transmission radius (R) of each deployed BS.7. If (All users are within BSs’ “footprint”)

AND (Users’ traffic demands are satisfied)8. Go to Step 13.9. Else10. Deploy additional BSs.11. Assign channels to them without violating CI constraints.12. End If13. Deploy ONUs uniformly to satisfy traffic demands from BSs.14. Construct a minimum-cost spanning tree (MST) from OLT to

ONUs and BSs.15. Calculate design cost, based on # of BSs, # of ONUs, and fiber

layout.16. End

In Sections IV-B and IV-C, we set the user coverage ratio,ρ = 1, and observe how a WOBAN’s deployment costvaries due to CI threshold (I) and available wireless channels(F ), respectively. In Section IV-D, we study the impact ofuser coverage ratio ρ on deployment cost of WOBAN. InSection IV-E, we examine how the WOBAN’s deployment costvaries in a non-homogeneous demography, where a majority ofusers is clustered in a small area, and the remaining users arefar away. By observing the duality gap (see Figs. 3, 4, 5, and6) of Primal Model (PM), we can infer a WOBAN’s optimumdeployment cost; this is because the optimum cost is upper andlower bounded by UB and LB, respectively. Also, we comparethe cost returned by the PM to a placement heuristic capturedby CH.

B. PM vs. CH: Impact of Carrier-to-Interference (CI) Thresh-old, I

Intuitively, the higher the distance between a BS and auser, the lower will be the signal quality and higher will be

the noise/interference. So, to serve all users with satisfactorychannel quality, we need sufficient numbers of BSs and ONUs.In other words, we need to deploy enough BSs (and ONUs) toaccommodate all the users if CI threshold is higher. In Table V,we show the number of BSs and ONUs needed for differentCI thresholds. Note that, when CI threshold is set to be 18dB or higher, Cellular Heuristic (CH) could not find feasiblesolutions.

TABLE VNUMBER OF BSS AND ONUS.

CI theshold (dB) 0 3 6 9 12 15 18 20

Primal Model (PM).BSs 21 26 32 39 48 58 70 86ONUs 3 3 4 4 5 6 7 9

Cellular Heuristic (CH).BSs 23 30 36 49 71 99 NA NAONUs 3 3 4 5 8 10 NA NA

NA stands for Not Applicable.

In Fig. 3, we examine the solution quality of our formulationwith respect to different CI thresholds (i.e., I in Section III).UB is the upper bound solution through the Primal Algorithmand LB is the lower bound solution with CLR (µ). Thereare three important observations. First, the cost increase athigher CI threshold is not very significant (though we needsignificantly more number of BSs and ONUs at higher CIthreshold as in Table V). This is because the fiber layout cost(100000 USD/mile) dominates the equipment costs of BSs andONUs. Also, ONUs and BSs will be diversely placed as usersare randomly distributed. Therefore, for smaller number of BSsand ONUs, the total fiber length in the minimum-cost spanningtree is comparable to the larger number of deployed BSs andONUs. This is the reason why the deployment costs do notshoot up while deploying large numbers of ONUs and BSs.

Fig. 3. Impact of channel interference on normalized deployment cost (withρ = 1 and |F | = 50 channels). If I ≥ 18 dB, no feasible solution exists forCH.

Second, the UB increases with higher CI threshold, but theLB does not increase significantly. This is because the co-channel interference constraint (Constraint 24) was relaxedin formulating the dual problem. Hence, the LB is not verysensitive to CI threshold.

Third, observe that the cost estimation by PM always out-performs the Cellular Heuristic (CH), especially at higher CI

10

threshold. This is because, in the first phase, CH estimates theminimum number of BSs required, distributes them homoge-neously in the area, and tries to cover as many users as possibleby tuning the transmission radius. But apart from these BSs, wemay need additional BSs to take care of other constraints suchas traffic demand. At higher co-channel interference constraint,the transmission radius of the additional BSs need to be smallin order to satisfy existing CI constraints. This results in thedeployment of a larger number of additional BSs. Note thatCH does not produce a feasible solution for CI threshold of18 dB or higher. This is due to the fact that a stringent CIthreshold makes the deployment of additional BSs (which areneeded to cover all the users and their traffic demands) difficultwithout violating the CI constraints of existing BSs, leading toan infeasible solution.

Note that WOBAN deployment cost is normalized to oneONU cost, which is taken to be USD 100 at the time of thisstudy.

C. PM vs. CH: Impact of Wireless Channel Pool, F

In some of the WiMAX standards (such as IEEE 802.16-2004), the frequency spectrum is not free. Therefore, we needto utilize channels efficiently to satisfy the traffic demands.In Fig. 4, we study the impact of total number of availablechannels on the solution quality where the CI threshold is setat 12 dB.

Fig. 4. Impact of available channel pool on normalized deployment cost (withρ = 1 and I = 12 dB). If |F | < 35 channels, no feasible solution exists forCH.

We observe that, when the channel resource is scarce, weneed to deploy more BSs and ONUs to cover all the users. Thisleads to higher deployment cost compared to when the channelresource is rich. For example, when there are 20 channels,WOBAN’s deployment cost is upper bounded by 75000 ONUunit cost; but when there are 50 channels, deployment costis reduced to around 45000 ONU unit cost for UB. Therefore,the solution quality is better when channel resource is rich thanwhen it is scarce.

Also note that the LB is less sensitive to channel resourcescompared to the UB. This is because we relax channel assign-ment constraints (Constraints 11 and 12), which indicates thatthe number of channels assigned to each BS is large enoughto serve its users.

The solution quality of the Primal Model is always superiorto that of CH for all types of channel resources (rich or scarce).Furthermore, when channel resource is scarce (i.e., less than35 channels), CH can not find a feasible solution. This is dueto the fact that the number of limited channels will incur severeco-channel interference, leading to an infeasible solution.

D. PM vs. CH: Impact of User Coverage Ratio, ρ

Another interesting property is the impact of user coverageratio. With low user coverage ratio, it is intuitive that fewerBSs and ONUs would be needed to satisfy the traffic demand.In Fig. 5, we observe that the deployment cost is decreasingwith respect to smaller user coverage ratio ρ. The threshold ofCI is set at 12 dB, and the number of available channels ischosen as 50.

Fig. 5. Impact of user coverage ratio on normalized deployment cost (withI = 12 dB and |F | = 50 channels).

Also note that the solution quality is better for smaller ρ. Forexample, when ρ = 0.5, the duality gap is 26.17% comparedto 35.35% at ρ = 1.0. Again, the solution quality of the PMoutperforms CH for all coverage ratios.

E. PM vs. CH: Impact of Non-Homogeneous Demography

In a practical situation, the user population density may notbe evenly distributed. More often than not, it is expected to besignificantly non-homogeneous and clustered, where a majorityof users resides in a small area. Hence, we study what fractionof the WOBAN deployment cost is needed to serve the extremeusers (who are far away and/or isolated).

For this study, we partition the test network (area of 5 × 5square-miles) into 100 equal grids, where each grid is of 0.5×0.5 square-miles area. We select 20 grids randomly (called hot-spots) and distribute 80% of the users in these hot-spots. Theremaining 20% of the users are scattered over other parts ofthe network. Therefore, 80% of the users reside in only 20%of the area. Figure 6 shows the WOBAN deployment cost withthis uneven user coverage ratio (ρ).

There are three important observations. First, the deploymentcost is almost linear to the user coverage ratio till we servethe nearest 80% of the users (ρ ≤ 0.8). After that, the costgrows superlinearly. This is expected because the farthest 20%

11

Fig. 6. Impact of non-homogeneous user coverage ratio on normalizeddeployment cost (with I = 12 dB and |F | = 50 channels). If ρ > 0.8,no feasible solution exists for CH.

of the users are scattered over a larger area, leading to higherdeployment cost (due to more expense for longer fiber layout).

Second, as expected, the PM outperforms CH, especiallyfor higher user coverage ratio. CH can not produce a feasiblesolution after 80% user-coverage ratio, because users’ popu-lation in hot-spots is too dense to be served by the minimumnumber of BSs (as calculated from Table IV). Therefore, weneed additional BSs, but it is hard to deploy them in a smallarea without violating the CI constraints of existing BSs. Thus,some users in hot-spots do not get served, leading to aninfeasible solution.

Third, the vast majority (80%) of users can be served at acost of approximately 35000 ONU units (cost at mid-point ofthe duality gap at ρ = 0.8). The cost for serving all 100% ofusers, however, is approximately 75000 ONU units (mid-pointof duality gap at ρ = 1). Thus, we observe that more than 50%of the deployment cost is used to serve the 20% of users whoare far away (or outliers). The additional cost is mainly due toexpensive fiber layout.

V. SUMMARY

Hybrid wireless-optical broadband access technology(WOBAN) is evolving fast as a cost-effective and high-capacity alternative solution to traditional PON-based accessnetworks. The early incarnations of the wireless part of aWOBAN are being deployed in many communities. Butcareful deployment needs efficient network planning of boththe wireless front end and the optical back end of WOBAN.

In this study, we proposed and investigated the characteris-tics of an analytical model (called Primal Model) for optimumplacements of Base Stations (BS) and Optical Network Units(ONU) so that the WOBAN deployment cost is minimized.We developed several constraints that need to be satisfied foroptimality: BS and ONU installation constraints, their capacityconstraints, user assignment constraints, channel assignmentconstraints, and channel interference constraints. For analyticaltractability of the primal problem, we used the “LagrangeanRelaxation” technique to relax some of the harder constraints,and obtained the corresponding Lagrangean dual problem. Wesolved this dual problem to obtain the lower bound of the PM.

We also developed a Primal Algorithm and found an upperbound of the PM. We verified the solution quality with respectto a set of chosen metrics such as user coverage ratio, numberof channels, and channel interference threshold. Specifically,we measured the “duality gap” between the upper and lowerbounds (UB and LB, respectively) of the PM, and comparedthe primal solutions to a Cellular Heuristic (CH). We found thatthe PM outperformed CH in all these metrics, and CH couldnot find a feasible solution in several challenging scenarios.

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Prentice-Hall, 2002.

Suman Sarkar (S’04) received the M.S. degree incomputer science from the University of California,Davis, in 2005, where he is currently working towardthe Ph.D. degree with the Computer Science Depart-ment. He is currently a Research Assistant with theNetworks Research Laboratory, University of Cali-fornia, Davis. His research interests include hybridwireless-optical broadband access networks, wirelessmesh networks, and broadband optical-access net-works.

Hong-Hsu Yen received his B.S. degree in IndustrialEngineering from National Tsing Hua University in1990, and M.S. degree in Electrical Engineering fromNational Taiwan University in 1995 and Ph.D. degreein Information Management from National TaiwanUniversity in 2001. He joined the faculty of theDepartment of Information Management at Shin HsinUniversity in 2001. His research interests include net-work planning, network optimization, performanceevaluation and QoS routing.

Sudhir Dixit (S’75-A’80-M’80-SM’95) received theB.E. degree from Maulana Azad National Instituteof Technology, Bhopal, India, the M.E. degree fromBirla Institute of Technology and Science, Pilani,India, the Ph.D. degree from the University of Strath-clyde, Glasgow, U.K., and the M.B.A. degree fromthe Florida Institute of Technology, Melbourne. Hewas with NYNEX Science and Technology (currentlyVerizon), GTE (currently Verizon), Codex Motorola,Wang, Harris, and Standard Telecommunication Lab-oratories (currently Nortel Europe Laboratories) in

the various management and research positions. From 1996 to October 2003,he was a Senior Research Manager, focusing on IP/ATM, wireless networks,content networks, and optical networks, and from November 2003 to March2007, he was a Research Fellow with Nokia Research Center. He is currentlyworking with Nokia Siemens Networks, Mountain View, CA. He has publishedor presented over 150 papers, published three books, and is the holder of 16patents. Dr. Dixit has been a Guest Editor over a dozen times in variousIEEE and other publications. He serves on the Editorial Board of the IEEECommunications Magazine, Wireless Personal Communications: An Inter-national Journal (WIRE) (Amsterdam, The Netherlands: Kluwer), WirelessCommunications and Mobile Computing Journal (Hoboken, NJ: Wiley), andis a Coeditor of The Cambridge Wireless Essentials Series (Cambridge, U.K.:Cambridge University Press). He also serves as an Advisory Committee Chairof Ubiquitous Computing and Monitoring System (UCoMS) for Discovery ofEnergy Resources which is a project funded by the U.S. Department of Energyand Board of Regents, State of Louisiana. He is a Fellow of the Institutionof Engineering and Technology, U.K., and the Institution of Electronics andTelecommunication Engineers, India. He represents Nokia on the SteeringBoard of the Wireless World Research Forum. He is also the Chair of theSpecial Interest Group on Self-Organization of Wireless World Systems.

Biswanath Mukherjee (S’82-M’87-SM’05-F’07)received the B.Tech. degree (with honors) from theIndian Institute of Technology, Kharagpur, India, in1980 and the Ph.D. degree from the University ofWashington, Seattle, in 1987. He held a GTE Teach-ing Fellowship and a General Electric FoundationFellowship at the University of Washington. In July1987, he was with the University of California, Davis(UC Davis), where he has been a Professor of com-puter science since July 1995, was a Chairman withthe Department of Computer Science from Septem-

ber 1997 to June 2000, and is currently holding the Child Family EndowedChair Professorship. To date, he has graduated nearly 25 Ph.D. students, withalmost the same number of M.S. students. Currently, he supervises the researchof nearly 20 scholars, mainly Ph.D. students and including visiting researchscientists in his laboratory. He is a Member of the Board of Directors ofIPLocks, Inc.: a Silicon Valley startup company. He has consulted for andhas served on the Technical Advisory Board (TAB) of a number of startupcompanies in optical networking. His current TAB appointments includethe following: Teknovus, Intelligent Fiber Optic Systems, and LookAheadDecisions Inc. He authored the textbook Optical Communication Networks(McGraw-Hill, 1997), which is a book that received the Association of Amer-ican Publishers, Inc.’s 1997 Honorable Mention in Computer Science. He is theAuthor of the textbook Optical WDM Networks (Springer, January 2006). Hisresearch interests include lightwave networks, network security, and wirelessnetworks. Dr. Mukherjee was the recipient of the 2004 Distinguished GraduateMentoring Award at UC Davis. Two Ph.D. dissertations (by L. Sahasrabuddheand K. Zhu), which he supervised, were winners of the 2000 and 2004 UCDavis College of Engineering Distinguished Dissertation Awards. He was thecorecipient of paper awards presented at the 1991 and the 1994 NationalComputer Security Conferences. He serves or has served on the EditorialBoards of the IEEE/ACM TRANSACTIONS ON NETWORKING, IEEENetwork, ACM/Baltzer Wireless Information Networks (WINET), Journal ofHigh-Speed Networks, Photonic Network Communications, Optical NetworkMagazine, and Optical Switching and Networking. He served as Editor-at-Large for optical networking and communications for the IEEE Communi-cations Society, as the Technical Program Chair of the IEEE INFOCOM’96conference, and as Chairman of the IEEE Communication Society’s OpticalNetworking Technical Committee from 2003 to 2005.