Optimization Models and Algorithms for Joint Uplink/Downlink UMTS Radio Network Planning With...

1612 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011

Optimization Models and Algorithms for JointUplink/Downlink UMTS Radio NetworkPlanning With SIR-Based Power Control

Amin Abdel Khalek, Student Member, IEEE, Lina Al-Kanj,Zaher Dawy, Senior Member, IEEE, and George Turkiyyah

Abstract—Universal mobile telecommunication system (UMTS)networks should be deployed according to cost-effective strategiesthat optimize a cost objective and satisfy target quality-of-service(QoS) requirements. In this paper, we propose novel algorithmsfor joint uplink/downlink UMTS radio planning with the objectiveof minimizing total power consumption in the network. Specifi-cally, we define two components of the radio planning problem:1) continuous-based site placement and 2) integer-based site selec-tion. In the site-placement problem, our goal is to find the optimallocations of UMTS base stations (BSs) in a certain geographicarea with a given user distribution to minimize the total powerexpenditure such that a satisfactory level of downlink and uplinksignal-to-interference ratio (SIR) is maintained with bounded out-age constraints. We model the problem as a constrained optimiza-tion problem with SIR-based uplink and downlink power controlscheme. An algorithm is proposed and implemented using patternsearch techniques for derivative-free optimization with augmentedLagrange multiplier estimates to support general constraints. Inthe site-selection problem, we aim to select the minimum set ofBSs from a fixed set of candidate sites that satisfies quality andoutage constraints. We develop an efficient elimination algorithmby proposing a method for classifying BSs that are critical fornetwork coverage and QoS. Finally, the problem is reformulatedto take care of location constraints whereby the placement ofBSs in a subset of the deployment area is not permitted dueto, e.g., private property limitations or electromagnetic radiationconstraints. Experimental results and optimal tradeoff curves arepresented and analyzed for various scenarios.

Index Terms—Cellular network planning, electromagnetic (EM)radiation exposure, network deployment, network optimization.

I. INTRODUCTION

IN universal mobile telecommunication system (UMTS)networks, the base station (BS) coverage and capacity are

a function of the user distribution, the signal-to-interference

Manuscript received July 2, 2010; revised November 1, 2010 and January24, 2011; accepted February 23, 2011. Date of publication March 28, 2011;date of current version May 16, 2011. This work was supported by a researchgrant from the National Council for Scientific Research (CNRS), Lebanon. Thereview of this paper was coordinated by Prof. W. A. Krzymien.

A. Abdel Khalek was with the American University of Beirut, Beirut 11072020, Lebanon. He is now with the Department of Electrical and ComputerEngineering, The University of Texas at Austin, TX 78712-0240 USA (e-mail:[email protected]).

L. Al-Kanj and Z. Dawy are with the Department of Electrical and Com-puter Engineering, American University of Beirut, Beirut 1107 2020, Lebanon(e-mail: [email protected]; [email protected]).

G. Turkiyyah is with the Department of Computer Science, AmericanUniversity of Beirut, Beirut 1107 2020, Lebanon (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2132745

ratio (SIR) requirements, and the interference level, which isan important coverage-limiting factor. The transmit powers ofmobile users are power controlled, depending on their dis-tance from the BS to reduce interference, avoid the near–farproblem, and ensure coverage for users close to the celledge. Given the high cost of network infrastructure invest-ments and spectrum licenses, operators should make informeddecisions on network deployment to satisfy performance re-quirements in a cost-efficient way. This drives the need foroptimized UMTS-specific planning tools that take into ac-count wideband code-division multiple-access air interfacecharacteristics.

UMTS radio network planning involves configuring thenetwork resources and parameters in a way that guaranteessatisfactory performance for the end users according to the fol-lowing three main quality attributes: 1) coverage; 2) capacity;and 3) quality of service (QoS). Radio network planning isconventionally approached as an iterative process that requiressetting target coverage and capacity objectives. The initialnetwork plan is obtained from geographic data, demographicdata, and propagation models, e.g., [1]–[3], and is then opti-mized by iterative updates of various network variables. Severalmodeling techniques are feasible and can be solved by mathe-matical and heuristic optimization algorithms such as simulatedannealing, greedy algorithms, genetic algorithms (GAs), as wellas linear and nonlinear programming.

A. Related Work

In [4]–[6], the authors proposed discrete optimization algo-rithms using randomized greedy procedures and a tabu search(TS) algorithm to plan the process of locating new BSs, consid-ering quality constraints for the uplink, which is argued to bemore stringent than the downlink for symmetric traffic. In [7],the previous work is extended to the downlink for asymmetrictraffic by applying SIR-based power control. Models spanningboth downlink and uplink with power control are also presentedin [8] and [9].

In [10] and [11], mixed integer linear programming is usedfor planning cost-efficient radio networks under network qual-ity constraints. Models based on set covering are used toobtain lower bounds on the number of required BSs to serve agiven fixed area, and an automatic two-phase network planningapproach based on successively solving instances of the modelis presented. In [12] and [13], two graph-theory-based models

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ABDEL KHALEK et al.: MODEL AND ALGORITHM FOR JOINT UPLINK/DOWNLINK UMTS RADIO NETWORK PLANNING 1613

are proposed, i.e., the maximum independent set model and theminimum dominating set model to select a satisfactory subsetout of a user-provided set of BS locations while ensuring thatat least a given percentage of the considered area is servedby the selected BSs. A large set of candidate BS sites is firstdetermined. In [14], a net-revenue maximization model for theselection of BS sites and the calculation of service capacity ispresented. The integer programming model takes the candidateBS locations and the traffic demand model as input and uses apriority-branching scheme to achieve a target optimization gaptolerance.

The majority of the contributions on optimized networkplanning focus on locating BSs according to the best tradeoffbetween network infrastructure costs and service coverage,whereas the electromagnetic (EM) field exposure levels arerarely considered. However, raising concerns about seriousconsequences for human health due to exposure to EM fieldshave led to precautionary regulations enforced by public ad-ministrators [15], [16]. The most widely accepted standards arethose developed by the IEEE [17] and the International Com-mission on Non-Ionizing Radiation Protection [18]. Due to theincreasing concerns about EM pollution in cellular networks, itshould be considered an important metric for network planningand optimization. The inclusion of EM radiation in the cellularnetwork planning problem has been addressed in [19] and [20],where two sequential metaheuristics were developed to limit thetotal EM field at selected test points and combined in a TS anda GA. The TS procedure is able to explore the solution spacedeeply enough, but it works on partial configurations, whereasthe GA procedure can manage the complete set of consideredparameters but is computationally expensive.

Current work in the literature focuses on selecting a minimalBS set from a larger candidate fixed BS set. The equallyimportant continuous version of the problem, which involvesfinding the optimal locations of these BSs in the network area,was not considered previously in the literature. By combiningthe two components of the problem, we can target more gen-eral application scenarios and further adapt and optimize thenetwork plan.

B. Contributions of the Paper

In this paper, the problem of joint uplink/downlink radionetwork planning is subdivided into two components. In thefirst component, which is referred to as the site-placementproblem throughout this paper, we are interested in finding theoptimal locations of a fixed number of UMTS BSs. In thesecond component, which is referred to as the site-selectionproblem, we are interested in finding the minimal cardinalityset from a set of BSs with fixed locations. It can be seenthat the site-placement problem is a continuous problem be-cause the problem variables are the physical BS locations,whereas the site-selection problem is a combinatorial problembecause the problem variables are the binary selection variablesfor each of the BSs. We solve the continuous problem offinding the optimal locations of BSs that minimize the totalpower expenditure in the network using robust pattern searchalgorithms for derivative-free optimization with nonsmooth

objectives. We define QoS targets and maintain bounded outagelevels on a networkwide and per-BS basis for both uplinkand downlink channels. Next, we formulate and propose analgorithm to solve the integer problem of selecting the small-est set of BSs from a fixed set of potential sites such thatthe cost function is minimized and the QoS requirements aresatisfied.

It is important to note that these two problems and theirsolution approaches are distinct; however, we propose that thetwo problems be combined into a hybrid integer/continuousalgorithm involving successive site selection/site placementuntil both components converge, i.e., until no relocation orelimination is feasible. This allows the operator to find the min-imal set of BSs needed for satisfactory coverage and the opti-mized deployment strategy in the network according to the userdistribution. An additional novel component of our work isthe development of a framework for radio network planningwith location constraints. Such constraints frequently arise inpractice due to private property limitations or EM radiation con-straints. We formulate the problem of radio network planningwith location constraints as an optimization problem. As part ofthe problem solution, we use Lagrangian multiplier estimatesand penalty parameters to construct and solve a sequence ofaugmented Lagrangian subproblems based on the augmentedLagrangian pattern search (ALPS) algorithm. Finally, we pro-vide optimal tradeoff curves under different user distributions,and we demonstrate the effectiveness of the proposed schemes,compared with conventional clustering.

C. Paper Organization

The rest of this paper is organized as follows: Section IIdescribes the system model. Section III presents the mathe-matical formulation for the site-selection and site-placementoptimization problems. Section IV presents the strategy and al-gorithms used to solve the optimization problems. In Section V,we report results for different user distributions and voice/datatraffic combinations, and we provide optimal tradeoff curvesfor the network configuration. In Section VI, we extend theoptimization algorithms to scenarios with location constraintsand/or EM radiation restrictions. Finally, Section VII providesconcluding remarks.

II. SYSTEM MODEL

The user distribution model is assumed to be snapshot based.A snapshot represents a set of users (or test points) using thephysical channel at a given instant of time. For a given distri-bution of currently active users or connections, we aim to finda network plan that guarantees minimal power consumption inthe network and satisfies coverage and QoS requirements in acost-effective manner.

In UMTS, users rely on channelization and scrambling codesto differentiate their own signal and combat the effect of multi-path and multiuser interference. To guarantee the required QoSlevel, a target minimum SIR value should be maintained for all


active connections. The downlink and uplink SIR expressionsfor user k can be expressed as follows:

SIRdk = SF

P dreceived,k

σ2 + λdIdin,k + Iout,k

(1)

SIRuk = SF

Pureceived,k

σ2 + λuIuin,k + Iu

out,k

(2)

where P dreceived,k is the downlink received power at mobile

station (MS) k; Pureceived,k is the uplink received power at the

BS from MS k; Idin,k and Id

out,k represent the downlink intracelland intercell interference affecting MS k, respectively; Iu

in,k andIuout,k represent the uplink intracell and intercell interference

affecting MS k, respectively; SF is the spreading factor; σ2

is the thermal noise power; and λ is the orthogonality factor(0.4 ≤ λd ≤ 0.9 in the downlink and λu = 1 in the uplink[21]). Since we need to satisfy the SIR constraint for all users,we more explicitly express the received power and interferencecomponents in (1) and (2) for user k as follows:(

SF+SIRdkλd

SIRdk

)gb(k),kP d

b(k),k

−

⎛⎝λdgb(k),k

∑j∈Cb(k)

Pb(k),j

⎞⎠−

N∑i=1,i �=b(k)

(gi,k

∑l∈Ci

P di,l

)=σ2

(3)(SF+SIRu

kλu

SIRuk

)gb(k),kPu

k

−

⎛⎝λu

∑j∈Cb(k)

gb(k),jPuj

⎞⎠−

N∑i=1,i �=b(k)

(∑l∈Ci

gi,lPdl

)=σ2

(4)

P dreceived,k = gb(k),kP d

b(k),k

Pureceived,k = gb(k),kPu

k (5)

Idin,k = gb(k),k

(P d

b(k) − P db(k),k

)Iuin,k =

∑j∈Cb(k),j �=k

gb(k),jPuj (6)

Idout,k =

N∑i=1,i �=b(k)

gi,kP di

Iuout,k =

N∑i=1,i �=b(k)

∑j∈Ci

gi,jPuj (7)

where N is the number of BSs, P di is the total transmit power

of BS i, P di,k is the power allocated by BS i to MS k (subject

to k being covered by BS i), gi,k is an estimate of the pathloss between MS k and BS i calculated according to the Cost231-Hata model [1], and b(k) is defined as the BS servingMS k and calculated by constructing the Voronoi tessellations

associated with the BS locations. Consequently, P db(k),k is the

transmit power allocated to MS k by its serving BS, Pb(k) isthe total transmit power of the BS serving user k, and gb(k),k isan estimate of the path loss between MS k and its serving BS.The notation j ∈ Ci means all MSs {j} covered by cell i. Con-sequently, j ∈ Cb(k) means all MSs covered by the same BS ask. The path loss coefficients gi,k can be written strictly in termsof the BS and MS locations in addition to some constants asgi,k = (1/keq)(di,k)−μ and di,k =

√(xi − uk)2 + (yi − vk)2,

where (xi, yi) are the BS locations and (uk, vk) are the fixedMS locations, with keq and μ chosen according to the Cost-231Hata model. In UMTS radio network planning, shadowing andfading are compensated for via link budget margins [1], [21].

Based on the preceding derivation, (1) can be rewritten interms of the powers P d

i,k allocated to MSs, which are thedownlink state variables, as shown in (3). Similarly, (2) can berewritten in terms of the MS transmit powers Pu

k , which arethe uplink state variables, as shown in (4), where SIRd

k andSIRu

k are the target SIRs to achieve the required QoS for thedownlink and the uplink, respectively. Equations (3) and (4) canbe expressed in matrix format as follows:

[Gd]U×U × [Pd]U×1 = [σ2]U×1 (8)

[Gu]U×U × [Pu]U×1 = [σ2]U×1 (9)

where Gd and Gu are square matrices with size U × U , Pd is acolumn vector with size U × 1 corresponding to the downlinkuser powers P d

b(k),k, Pu is a column vector with size U × 1corresponding to the uplink user powers Pu

k , σ2 is the thermalnoise power column vector with size U × 1, and U is the totalnumber of active users in the network. The kth row of Gd andGu corresponds to user k, with each row representing one ofthe U equations, and the columns can be separated into blockscorresponding to the BSs according to the number of users ineach BS. The first term of (3) and (4) appears in the diago-nal of Gd and Gu, respectively; the second term representsintracell interference and appears in the block corresponding tothe BS serving user k; and the third term represents intercellinterference and appears in the blocks corresponding to all BSsexcept the serving BS. Solving the power assignment problemwith SIR-based power control reduces to solving this set ofequations, which adjusts the transmit powers to meet the targetSIRs [1], [2].

III. OPTIMIZATION PROBLEM FORMULATION

In this section, we present a formulation for the joint up-link/downlink radio planning problems. The site-placementproblem is formulated as a continuous optimization problem,and the site-selection problem is formulated as an integeroptimization problem; the objective, variables, and constraintsfor each of the problems are defined.

A. Joint Uplink/Downlink Site-Placement Problem

The input to the site-placement problem is given as follows:1) the area of interest, 2) the fixed set of MS locations modelingthe typical distribution of active users in the area, 3) the fixed


number of BSs, and 4) the initial locations of the BSs. Thesite-placement algorithm optimizes the initial locations of theBSs to minimize the target objective while satisfying the QoSrequirements. The output is the set of optimal locations of BSs.The objective of joint uplink/downlink site placement is definedas a weighted linear combination of two objectives.

1) Minimize the downlink power expenditure: The totaldownlink power expenditure is the sum of powers allo-cated to each user by its serving BS, which is expressedas

∑Uk=1 P d

b(k),k. In addition to the inherent benefits ofminimizing power consumption, we will demonstrate thatthis reduces the variance of BS powers, thus providinga solution that balances the power load according to thepredicted user distribution.

2) Minimize the uplink outage: The uplink channel is lim-ited by the power capabilities of the MSs, which is byfar less than that of the BSs. Thus, it is important toensure that uplink transmissions can still reach the BSsat a reasonable power subject to their handset limita-tions while satisfying the minimum SIR threshold foracceptable performance. We formulate this componentas

∑Uk=1(P

uk − Pu

max)+, where (·)+ = max(·, 0), Pu

k isthe uplink power associated with MS k, and Pu

max is themaximum allowed MS power.

Effectively, each MS requiring power in excess of Pumax will

be considered in the second objective. If all MSs do not exceedthe threshold, this will be a vector of zeros, which is essentiallydisregarded from the objective. We do not directly consider thenumber of MSs that are in outage in the objective function tomaintain coherence in the units (watts) and avoid nonsmoothstep transitions in the function. However, this approach has thesame potential in minimizing the number of MSs in outagesince the algorithm will attempt to find the BS configura-tion that satisfies (Pu

k − Pumax)

+ = 0 ∀k, if such configurationexists.

In the site-placement problem, N is fixed, i.e., we are notinterested in reducing the initial number of BSs but in op-timizing their locations. Thus, the decision variables are thelocations of the BSs (xi, yi), i = 1, . . . , N , that minimize thecost function. Since the objective function is expressed in termsof the downlink powers assigned by the BSs to their MSs, weconsider the powers assigned by the BSs (Pi,k), i = 1, . . . , N ,k = 1, . . . ,Mi, as state variables. Note that

∑Ni=1 Mi = U ,

where Mi is the number of MSs served by BS i. Thus, thetotal number of variables in the problem is U + 2N , whereU � N . The power assignment variables relate to the decisionvariables through the SIR-based power control mechanism, asshown in (8) and (9). Since the matrices Gd and Gu consistof the path loss coefficients gi,k, it is worth mentioning that itcan be written strictly in terms of the BS and MS locations inaddition to some constants according to gi,k = (1/keq)(di,k)−μ

and di,k =√

(xi − uk)2 + (yi − vk)2. Thus, the power assign-ments for a given setting of the location variables can be foundby solving the U set of equations, which is an operation thatcosts O(U3).

Two important observations concerning Gd and Gu areworth pointing out: First, they are strongly diagonally domi-

nant. For example, the average value of the diagonal terms is atleast 103 times the average value of the nondiagonal terms witha spreading factor of 128 since the spreading gain appears onlyin the diagonal terms and the nondiagonal terms correspond tointerference components. This fact can be exploited by usingfast solvers that are suitable for diagonally dominant matrices.Second, if the SIR requirement for the kth user cannot besatisfied due to high interference at its location, the solution ofthe corresponding equation will yield a negative power, whichcan be used for spotting outages in the network.

The outage conditions are defined networkwide and for eachBS. The networkwide outage conditions ensure that the totalnumber of users in outage is less than ηnetworkU and that theBS outage conditions ensure that the number of users in outagefor every BS i is less than ηBSMi, where ηnetwork and ηBS aredesign parameters that satisfy 0 ≤ ηnetwork ≤ ηBS � 1.

The problem of joint uplink/downlink site placement is for-mulated as an optimization problem as in

minx,y

U∑k=1

P db(k),k + α

U∑k=1

(Puk − Pu

max)+ (10)

s.t.

[Gd]U×U × [Pd]U×1 = [σ2]U×1 (11)

[Gu]U×U × [Pu]U×1 = [σ2]U×1 (12)

U∑k=1

⎛⎝−

P dk,b(k)∣∣∣P dk,b(k)

∣∣∣⎞⎠

+

< ηnetwork · U (13)

U∑k=1

(− Pu

k

|Puk |

)+

< ηnetwork · U (14)

∑k∈Ci

⎛⎝−

P dk,b(k)∣∣∣P dk,b(k)

∣∣∣⎞⎠

+

< ηBS · Mi (15)

∑k∈Ci

(− Pu

k

|Puk |

)+

< ηBS · Mi (16)

gi,k =1

keqd−μ

i,k di,k =√

(xi − uk)2 + (yi − vk)2 (17)

b(k) = arg mini

di,k (18)

∑j∈Ci

Pi,j ≤ P dmax (19)

xmin ≤ xi ≤ xmax, ymin ≤ yi ≤ ymax (20)

i = 1, . . . , N ; k = 1, . . . , U (21)

where (10) represents the weighted objective and α is a con-stant that determines the relative weight of each of the twocomponents, (11) represents the matrix-form downlink SIRconstraint for all MSs, (12) represents the matrix-form uplinkSIR constraint for all MSs, (13) is the downlink network outagecondition, (14) is the uplink network outage condition, (15) isthe downlink BS outage condition for every BS, (16) is the


uplink BS outage condition for every BS, (17) is the path lossaccording to the Cost 231-Hata model, (18) is the distance-based assignment of MSs to BSs obtained by constructing theVoronoi tessellation corresponding to the BS locations, (19)is the maximum BS power constraint, and (20) is the areaof operation constraint. Note that MS k may be in outageeither because it requires a transmission power Pu

k higher thanPu

max to reach the closest BS or because the transmit powerPu

k , although lower than Pumax, causes high interference at

another MS, so that the SIR constraints cannot be satisfied.Equation (10) eliminates outage due to the former case, whereas(13)–(16) eliminate outage due to the latter case.

B. Joint Uplink/Downlink Site-Selection Problem

The input to the site-selection problem is given as follows:1) the area of interest, 2) the fixed set of MS locations modelingthe typical distribution of active users in the area, and 3) thecandidate set of BSs with fixed locations. The site-selection al-gorithm selects the minimal cardinality set of BSs that satisfiesthe coverage and SIR requirements. The output is the subsetof candidates corresponding to the selected BSs. The maindifference from the site-placement problem is that the decisionvariables are not the BS locations; instead, these locations arefixed, and the decision variables are Boolean ci, where ci = 1if BS i is to be used in the optimal network configuration andci = 0 otherwise. The objective of the problem is to minimizethe number of selected BSs, i.e.,

∑N0i=1 ci, where N0 is the size

of the candidate set of BSs. The problem can thus be formulatedas a nested optimization problem, as shown in

minc

N0∑i=1

ci (22)

s.t.

minU∑

k=1

cb(k) · P dk,b(k) + α

U∑k=1

cb(k) · (Puk − Pu

max)+ (23)

subject to

(11)−(20)

i = 1, . . . , N0; k = 1, . . . , U. (24)

The inner problem finds the set of BSs that minimizes the costfunction, and the outer problem finds the minimal cardinalityset of such BSs. We will later show that using the minimizationof the total power expenditure as the criterion for BS removalresults in better solutions than a removal purely based on thefeasibility of BS elimination (see Section V-B). Note that theconstraints (11)–(20) are only applied to the set of active BSssatisfying ci = 1.

IV. OPTIMIZATION STRATEGIES AND ALGORITHMS

Solving the initial site-placement optimization problem in-volves nonlinear equality constraints [see (11), (12), (17), and(18)], nonlinear inequality constraints [see (13)–(16) and (19)],and linear bound constraints [see (20)]. Given the current BS

Fig. 1. Pattern of change of objective function while moving a BS along acoordinate direction.

locations x and y, we use (18) to obtain the assignments ofMSs to their closest BS and (17) to obtain the path loss betweeneach MS and its serving BS. Finally, solving the SIR equations(11) and (12), we obtain the power allocated to each MS onthe uplink and downlink to satisfy QoS requirements. Thus,the four sets of equality constraints are implicitly includedin objective function evaluations, because they determine thepower allocation scheme throughout the network. The boundconstraint representing the area of operation is a simple linearconstraint that is taken care of by the algorithm by choosingappropriate steps that do not violate that constraint. To solve thesite-placement problem, we propose a pattern search algorithmbased on mesh adaptive direct search (MADS). Additionally,to ensure satisfying the BS power limit and the outage con-straints, we will describe how to extend the algorithm to includeany general nonlinear inequality constraints using the ALPSmethod. After presenting the algorithm for the site-placementproblem, we will present the solution for the nested optimiza-tion problem of site selection based on successive eliminationof BSs.

A. Algorithm for Site Placement With Uplink and DownlinkQoS Guarantees

The MADS class of derivative-free algorithms is effective forpractical nonlinear optimization problems with nonsmooth ob-jective functions, where the computation of derivatives is eithernot possible or not sufficiently representative of the variabilityof the function around a point due to the roughness of theobjective function. MADS has a well-developed convergencetheory based on the Clarke calculus and Rockafeller’s notion ofa hypertangent cone [22], [23].

To illustrate why the MADS algorithm is suitable for radioplanning problems, we present a sample pattern of the changeof the objective function value with respect to a change in oneof the variables (equivalent to moving one of the BSs along acoordinate direction). Fig. 1 shows that the objective functionhas many discontinuities and nonsmooth changes, which areexplained by the changes in user assignments due to the shiftin the Voronoi diagram. As new users are handed over to a


new BS, they become on the boundary of that BS, requiringhigher power to achieve the target SIR, thus causing the instantrise in power allocations and, correspondingly, in the objectivefunction value. Note that the change in one MS–BS assignmentdoes not affect only that user due to the changes in intercelland intracell interference effects experienced by other users,which are translated in the change of the structure and contentof path loss matrices Gd and Gu and the solutions to the set ofequations in (8) and (9). Although the total power expenditureincreases if the BS is moved any distance larger than 20 m,a gradient computed using the finite-difference method or theadjoint method is misleading because it will imply that theobjective function is decreasing, which is only valid 20 m awayfrom the BS. These kinds of peculiarities make it impracticalto follow a line search method in solving the problem, whichis why we advocate the use of derivative-free optimizationalgorithms, which are suitable for the nonsmooth objectives ofthe radio network planning problem.

The MADS algorithm operates by performing polling andsearching on a set of mesh points around the current locationof the decision variables. The mesh points are constructed bybuilding a pattern that is a basis set of vectors that the algorithmuses to determine which points to search at each iteration.The most common basis set is the 2n basis set, where n is thenumber of decision variables. In our problem, the number ofdecision variables is n = 2N ; thus, the basis set will contain4N vectors, each of size 2N . The vectors are defined as follows:v1 = [ 1 0 · · · 0 ], v2 = [ 0 1 · · · 0 ], . . ., v2N =[ 0 0 · · · 1 ], v2N+1 = [−1 0 · · · 0 ], v2N+2 =[ 0 −1 · · · 0 ], . . ., and v4N = [ 0 0 · · · −1 ].

Algorithm 1: Proposed solution for the joint uplink/downlink site-placement problem.Given N , U , {xi = xinitial,i}2N

i=1, {uk = ufixed,k}2Uk=1, Δ =

Δ0, Δth, α, μ, keq, σ2, λd, λu, SF , {SIRdk}U

k=1, and{SIRu

k}Uk=1.

Construct the 2n basis set of vectors v that the algorithm usesto determine which points to search at each iteration, wheren is the number of decision variables (2n = 4N).v1 = [ 1 0 · · · 0 ], v2 = [ 0 1 · · · 0 ], . . .,v2N = [ 0 0 · · · 1 ]v2N+1 = [−1 0 · · · 0 ], v2N+2 =[ 0 −1 · · · 0 ], . . ., v4N = [ 0 0 · · · −1 ].while Δ > Δth {While the mesh size is higher than theconvergence threshold} do

for m = 1 : 4N {For all possible patterns of movement}doxtemp = x + Δm · vp

Step 1. Construct the Voronoi tessellation correspondingto the BS locations {xm}, and find the distance betweeneach (BS, MS) pair.Step 2. Construct the path loss matrices Gd and Gu

based on the estimated path loss for each (BS, MS) pairin the uplink and downlink.Step 3. Solve the set of equations: [Gd] × [Pd] = [σ2]and [Gu] × [Pu] = [σ2] to find the downlink power thatshould be allocated to each MS and the uplink MS powerto achieve SIR-based power control for all users.

Step 4. Calculate the objective function at this mesh point

xm : fm =U∑

k=1

P dk,b(k) + α

U∑k=1

(Puk − Pu

max)+

end forif minm fm < f {Is there a mesh point with smaller

objective function} thenx = x + Δ · vm {Successful Poll: Advance to the pointthat minimizes the objective and expand mesh size}Δ = Δ × (Expansion Factor).

elseΔ = Δ × (Contraction Factor) {Unsuccessful Poll:reduce mesh size}.

end ifend while

At each iteration, the pattern search polls the points in thecurrent mesh for a point that improves the objective function bycomputing the objective function at the mesh points to check ifthere is one whose function value is less than the function valueat the current point. The mesh points are found by multiplyingeach pattern vector vp by a scalar Δm to generate a set ofdirection vectors and adding the direction vector to the currentpoint found at the previous step. The number of mesh points4N is simply due to the four coordinate directions of pollingfor each BS. These points can be expressed as xp = x + Δmvp,p = 1, . . . , 4N , where x is a vector of size 2N correspondingto the current locations of the N BSs, Δm is the mesh sizeat iteration m, and xp is the polled mesh point. Initially, themesh size is specified based on the scale of the problem and isexpanded and contracted during execution of the optimizationalgorithm. In this way, the algorithm finds a sequence of pointsx0, x1, x2, . . . that approach an optimal point. The convergencecriteria are satisfied when the mesh size is smaller than the meshtolerance such that minimizing the objective function furtherwould require moving any BS a distance smaller than thismesh tolerance threshold. This polling technique is effectivelyequivalent to moving the BSs in their neighborhood accordingto the mesh size at a given iteration. It is important to notethat, at each polled point, the objective function is calculated byreconstructing the Voronoi tessellation, building the path lossmatrices Gd and Gu, solving the set of equations correspond-ing to SIR-based power control, and computing the objectivefunction from Pd and Pu. A complete algorithmic descriptionis presented in Algorithm 1.

As previously mentioned, the algorithm previously describeddoes not account for the nonlinear inequality constraints inthe problem. To include the maximum BS power and theoutage constraints, we use the ALPS algorithm, which is arobust extension of pattern search algorithms for general con-straints. The algorithm operates by formulating a subproblemobtained by combining the objective function and nonlinearconstraint functions using Lagrange multiplier estimates andpenalty parameters. A sequence of such optimization problemsare approximately minimized using a pattern search algorithm,and the convergence to the optimal solution is guaranteed (seeSection VI-B for more details).


TABLE ISIMULATION PARAMETERS

B. Algorithm for Site Selection With Uplink and DownlinkQoS Guarantees

Algorithm 2: Proposed solution for the joint uplink/downlink site-selection problem.Given N = N0, U , F = ∅, I = ∅, {xi = xinitial,i}N0

i=1,{yi =yinitial,i}N0

i=1, α, μ, keq, σ2, λd, λu, {uk =ufixed,k}Uk=1,

{vk = vfixed,k}Uk=1, SF , {SIRd

k}Uk=1, and {SIRu

k}Uk=1.

while true dofor i = 1 : N {Try eliminating BS i}

xtemp = {x1, . . . , xi−1, xi+1, . . . , xN}ytemp = {y1, . . . , yi−1, yi+1, . . . , yN}Step 1. Construct the Voronoi tessellation corre-

sponding to the BS locations {xtemp, ytemp}, and find thedistance between each (BS, MS) pair.

Step 2. Construct the path loss matrices Gd and Gu

based on the estimated path loss for each (BS, MS) pair.Step 3. Solve the set of equations [Gd] × [Pd] =

[σ2] and [Gu] × [Pu] = [σ2] to find uplink and downlinkpower allocation achieving SIR-based power control for allusers.

Step 4.if Pd and Pu do not satisfy (13)-(16) {If eliminating

the BS causes significant DL or UL coverage loss} theni ∈ I {Place BS i in the infeasible set}

elsei ∈ F {Place BS i in the feasible set}Calculate the objective function at

(xtemp, ytemp): fi =∑U

k=1 P dk,b(k) + α

∑Uk=1(P

uk − Pu

max)+

end ifend forif F �= ∅ {If some BS can be eliminated} then

e = arg mini fi

x = {x1, . . . , xe−1, xe+1, . . . , xN}y = {y1, . . . , ye−1, ye+1, . . . , yN}N = N − 1

elsebreak {Converged: No BS can be eliminated without

jeopardizing coverage}end if

end while

The site-selection problem is a nested optimization problemwith the outer problem minimizing the number of BSs andthe inner problem selecting the set of BSs that minimize the

weighted cost function. The approach for solving the problemis based on successive elimination of BSs one at a time. Givena set of fixed BSs S and a user distribution model, at eachelimination step, we can divide the set of BSs S into two distinctsubsets F and I, where F is the feasible set (i.e., if BS i ∈ F ,it can be safely eliminated from the set), I is the infeasibleset (i.e., if BS i ∈ I, it cannot be eliminated from the setwithout jeopardizing coverage to a significant fraction of usersas defined by ηnetwork and ηBS), and F ∪ I = S. Formally,BS i ∈ F if, for the set {BS1, . . . , BSi−1, BSi+1, . . . , BSN},∃ a solution for the 2U equations in (8) and (9) such that(13)–(16) are satisfied. Out of the feasible set, we eliminate theBS that minimizes the weighted objective after elimination inaccordance with our initial cost function. This whole operationis repeated until, at some point, the elimination testing phaseyields an empty feasible set. It is worth noting that testingfeasibility requires solving the set of 2U equations in (8) and (9)for each BS. A complete algorithmic description is presented inAlgorithm 2.

To provide a general optimization framework for UMTSradio network planning, we combine the two optimizationproblems of site selection and site placement to find the min-imal set of BSs to cover the area and their optimal locationssubject to the user distribution. Thus, the two algorithms canbe successively executed as subproblems until the convergencecriteria for both are satisfied.

V. RESULTS AND INTERPRETATION

This section presents results and analysis for an area measur-ing 10 km × 10 km with U = 1000 active users having the sametarget per-bit SIR Eb/I0 (DL) = 7 dB and Eb/I0 (UL) = 5 dB.Initially, we assume that all MSs are operating a voice service.In Section V-D, we present results for network scenarios withtwo service classes (voice and data). For radio propagation, weconsider the COST-231 Hata model for metropolitan areas [1],and we compensate for shadowing, fading, and antenna lossesby adding a 16-dB margin to the link budget. This resultingtotal loss between BS i to MS k can be approximated to withinless than 0.1 dB in all regions of interest by gi,k = (1/keq)d

−μi,k ,

where keq = 2.75 × 1015, and μ = 3.52. The simulation para-meters are summarized in Table I.

Each BS is assumed to be equipped with an omnidirectionalantenna that is placed at the cell center. We point out that thealgorithms developed in this paper make no assumptions about


Fig. 2. BS distribution obtained for different user distributions with combined site selection and site placement. (Left) Uniform user distribution. (Middle)Gaussian user distribution. (Right) Four Gaussian hot spots. (Star) BS. (Dot) MS. (Boundary line) Voronoi region.

the type or directionality of the antennas. In fact, they can beeasily applied to a sectorized network by accounting for thedirectional antenna gains and constructing the Voronoi regionson a sector-by-sector basis. In this section, we assume that bothBSs and MSs are equipped with omnidirectional antennas.

A. Optimal BS Configurations for Different User Distributions

We present sample results of optimal BS deployments forthree different distributions by consecutively executing Algo-rithm 1 and Algorithm 2 to combine site selection and siteplacement, and find the minimal set of BSs and their optimallocations to cover the given area. For each of the three dis-tributions, we initially start with N0 = 100 randomly locatedBSs. N0 is chosen to be large enough to provide an initialconfiguration that satisfies outage and SIR requirements.

Fig. 2(a) shows the optimal BS distribution for a uniformuser distribution with 1000 active users distributed uniformlyover the entire area. Results show that 64% of the BSs wereeliminated, reaching 36 BSs with an average BS power of17.1 W and a standard deviation of BS powers of 1.26 W.Fig. 2(b) shows the optimal BS distribution for a Gaussian userdistribution modeling a hot spot with maximum user density atthe center and gradually decreasing toward the area boundary.Results show that 61% of the BSs were eliminated, reaching39 BSs with an average BS power of 13.1 W and a standarddeviation of BS powers of 4.34 W. Finally, Fig. 2(c) solves theproblem for four Gaussian user distributions modeling severalhot spots with 250 active users each. After execution, 55% ofthe BSs were eliminated, reaching 45 BSs with an average BSpower of 11.6 W and a standard deviation of 3.39 W.

B. Analysis of Site-Selection and Site-Placement Algorithms

Fig. 3 shows the average BS power during execution ofthe site-selection algorithm for different inner objectives witha Gaussian user distribution and N0 = 100 BSs. The outerobjective is to minimize the number of BSs and with the innerobjective to either minimize the sum of BS powers or eliminatethe first feasible BS. In the first approach, we eliminate theone that, among other BSs, minimizes the total power expen-diture, as described in Section III-B. This essentially providesa solution that minimizes the average BS power. In the sec-

Fig. 3. Improvement in the radio network plan due to the utilization of thenested objective in the site-selection algorithm.

ond approach, while executing the outer problem, we greedilyeliminate the first possible BS that does not cause significantcoverage loss, as defined in (13)–(16). Obviously, the averageBS power will increase under both approaches since BSs aregetting eliminated; however, what the algorithm optimizes isthe rate of this increase. The most noticeable difference is that46 BSs were eliminated with the first inner objective, whereasonly 20 BSs were eliminated with the greedy approach. Thisdemonstrates the impact of the elimination criteria on the rate ofincrease of the average BS power and, thus, on the effectivenessof the selection algorithm.

Another important observation is that minimizing the totalpower expenditure implicitly reduces the variance of BS powersby converging to a solution that nearly equally distributesthe power load among BSs. Fig. 4(a) shows the average BSpower during the execution of the site-placement algorithmwith a Gaussian user distribution for two different objectivefunctions: 1) minimize sum of BS powers and 2) minimizevariance of BS powers. The two objectives converge almost tothe same average BS power, suggesting that opting for equalBS powers does not greedily increase these powers to achieveequality; instead, the network configuration converges to a low-power solution due to the SIR-based power control mechanismutilized in the problem. Fig. 4(b) shows a similar scenario forthe site-selection algorithm with N0 = 100 initial BSs.


Fig. 4. Comparison of different cost functions according to the achieved average BS power. (Left) Site-placement algorithm. (Right) Site-selection algorithm.

TABLE IIEFFECT OF N0 ON THE SOLUTION QUALITY

We present a case study to analyze the effect of the initialnumber of BSs N0 on the solution. We consider a uniformuser distribution with U = 200 active users. Initially, the BSsare randomly located in the network. We successively executethe site-placement and site-selection algorithms as subproblemsuntil the convergence criteria for both are satisfied. This corre-sponds to the case where eliminating any BS causes the QoSconstraints to be violated, and moving any BS a distance largerthan the mesh size threshold will only increase the objective.The resulting network plan size, average transmit power perBS, global iterations, and total computation time are shown inTable II for N0 = 100, 60, 40, and 25. Global iterations repre-sent the number of times the site-selection and site-placementalgorithms are sequentially executed until the aforementionedconvergence criteria are satisfied.

The network plan size is the main metric for judging thequality of the solution. It can be observed from the resultsthat larger N0 generates a finer solution since there are moredegrees of freedom in selecting the candidate sites; thus, morecritical site locations can be picked during the optimization.Consequently, the solution is only slightly dependent on theinitial locations of BSs. On the other hand, a smaller N0 ismore likely to provide a locally optimal solution. Obviously,a solution with lower number of BSs would require higheraverage transmit power per BS that satisfies the BS maximumtransmit power constraint. Additionally, a larger N0 requiresmore global iterations and more computation time.

C. Radio Network Planning Tradeoffs

Combining site selection and site placement allows the op-erator to find the minimal set of BSs to cover the network areaand their optimal locations. Since eliminating BSs increases theaverage power per BS, we can think of the problem differently

by trading off the number of BSs and the average BS power. Thesolid lines of Fig. 5 show the set of pareto-optimal points thatprovide this tradeoff between the minimum average BS powerachieved and the number of deployed BSs. The Pareto sets aregenerated for a uniform user distribution in Fig. 5(a) and fora Gaussian user distribution in Fig. 5(b). Generating these setsis performed by running the site-selection algorithm to selecta target number of BSs Nmin from a set of 100 initial BSs.The site placement algorithm is then executed for these Nmin

BSs to find their optimal locations that minimize the total powerexpenditure.

In an attempt to validate our algorithm and demonstrate itseffectiveness, we compare these results with off-the-shelf hier-archical clustering algorithms. Hierarchical clustering initiallytreats each data point as a single cluster and then successivelymerges clusters according to the linkage criteria. In this prob-lem, the data points are the MSs, and each cluster correspondsto a single BS serving a set of MSs. In complete linkagehierarchical clustering, which is also called furthest neighbor,the two clusters whose merger has the smallest maximum pair-wise distance are merged. In single linkage clustering, which isalso called nearest neighbor, the two clusters with the smallestminimum pairwise distance are merged. Finally, in averagelinkage clustering, the two clusters with the smallest averageof pairwise distances (maximum cohesion) are merged.

For the considered problem, single linkage clustering is nota suitable approach due to its tendency to form long chainsthat cannot model BS coverage. Thus, we consider the othertwo linkage criteria. For a given number of clusters (i.e., BSs)Nmin, each of the two criteria is applied to construct the setsof MS clusters, and the BS location for each cluster is definedas the centroid of the set of cluster data points. The powerallocated to each user is determined by solving the equations[Gd] × [Pd] = [σ2] and [Gu] × [Pu] = [σ2], and the averageBS power is computed. The results for uniform and Gaussianuser distributions are shown in Fig. 5. For a uniform userdistribution in Fig. 5(a), average linkage and complete linkageclustering provide a solution close to the optimal set generatedfrom the site-placement and site-selection algorithms. Intu-itively, since each MS is equidistant from its neighbors, theclusters will be almost equal in size, and their centroids willapproximate a uniform distribution. However, for a Gaussian


Fig. 5. (Solid line) Optimal tradeoff set between the average BS power and the number of deployed BSs. (Dotted line) Comparison with average linkage andcomplete linkage clustering algorithms. (Left) Uniform user distribution. (Right) Gaussian user distribution.

Fig. 6. (Left) Percentage of users in outage versus number of BSs with and without the second component of (10) for Pmaxu = 1 W and a voice-only service.

(Right) Percentage of users in outage versus the percentage of data users for two network case studies. (a) Nmin = 36 BSs. (b) Nmin = 50 BSs.

user distribution in Fig. 5(b), hierarchical clustering experi-ences bad performance. Complete linkage and average linkageclustering can only generate feasible network configurations ofsizes at least 72 and 78 BSs, respectively, in comparison to39 BSs for our proposed algorithm. Additionally, the feasibleconfigurations incur significantly higher power consumption.This is explained by the fact that these linkage criteria do nottake into account the discrepancy in the SIR levels and powerallocations when the user density is not constant. Generally,these results demonstrate the effectiveness of the site-selectionand site-placement algorithms, particularly for nonuniform userdistributions.

While decreasing the number of BSs is desirable, it makes theuplink power requirement higher, thus increasing the chance ofcrossing the Pmax

u threshold. We demonstrate the improvementin the uplink outage due to the combined objective minimiza-

tion in Fig. 6(a). The outage for each network size is computedby solving the joint planning problem for α = 0 (single ob-jective) and α = 0.5 (weighted objective) with Pmax

u = 1 W.Results demonstrate that accounting for uplink outage in theobjective minimizes the number of users crossing the Pmax

u

threshold.

D. Concurrent Voice and Data Services

We consider generalizing the model to accommodate forconcurrent voice and data services. A given percentage ofdata users in the network is selected, and these data users areindependently and randomly picked from the set of all users.

To obtain insight into the operation of the algorithm undermultiple service rates, we assume that all data users require64 kb/s in the downlink and 32 kb/s in the uplink and that


all voice users require 12.2 kb/s in both directions. The re-quirement for higher data rate services naturally increases thenetwork load and makes the QoS requirements more stringent.In Fig. 6(b), we quantify this increase by considering thepercentage of voice/data users in outage versus the percentageof data users in the network for two scenarios. The first scenariocorresponds to the optimal plan for a uniform user distributionwith 36 BSs. Since this plan is optimal in terms of power con-sumption for a voice-only deployment, we expect that addingdata users will adversely impact the performance of the net-work. When the network is dominated by voice users, the datausers experience 19% outage, and the voice users experience3% outage. The rate of increase of outage as the percentageof data users increase is relatively fast for both traffic classes.Since this is an extreme case, we notice that we can operatethe network with a larger number of BSs to provide morerobustness to the network under concurrent voice/data traffic.Thus, in the second scenario, we relax Nmin to 50 BSs, and werun the site-placement algorithm to optimize the site locationssubject to the uniform voice/data user distribution. It can beseen that outage significantly drops, and QoS requirementsbecome less stringent to satisfy.

Finally, for the downlink, data traffic produces an increase inthe BS power so that all data users can also satisfy their QoSrequirements. Thus, the limiting factor becomes the total BSpower. Considering P d

max = 30 W in the simulations, we findthat a maximum of 15.5% of data users can be supported in thefirst scenario, whereas a maximum of 20% can be supportedin the second scenario. This analysis captures the interplaybetween the traffic classes, the QoS requirements, and the sizeof the network plan.

VI. RADIO NETWORK PLANNING

WITH LOCATION CONSTRAINTS

In realistic radio network planning scenarios, operators oftendo not have the liberty of placing BSs in the entire area ofthe network for two main reasons. First, the presence of roughterrains, private property such as universities, or city zoningrestrictions in the network region limits the scope of placing BSsites. Second, the presence of radiation-sensitive zones, such ashospitals or schools, places limitations on EM radiation in suchareas of the network. These limitations are referred to in thispaper as location constraints. The EM exposure limitations aredefined by specifying maximum permissible exposure levels fordifferent frequency ranges [17], [18]. In general, for frequencieslower than 300 MHz, exposure limits are specified in terms ofelectric field strength (in volts per meter) and magnetic fieldstrength (in amperes per meter), whereas for frequencies higherthan 300 MHz, exposure limits are specified in terms of powerdensity S (in milliwatts per square centimeter). Since UMTSoperates at 1800 MHz, we will define our constraints in terms ofpower density. The power density quadratically decays with thedistance from the BS in a free space environment. In general,measured data show that average power densities are generallyin the range of 0.001–0.01 μW/cm2. Health recommendationssuggest that the median exposure in urban areas be limited

to 0.005 μW/cm2 and that 95% of the urban population beexposed to less than 0.1 μW/cm2 [17], [18].

A. Problem Formulation

In this section, we extend the formulated radio networkplanning problem in Section III-A to include constraints on thefollowing: 1) locations of deployed BSs in a target BS-free areaand 2) peak power density in a target radiation-sensitive zone.We model the BS-free region as a rectangular area, which canbe formulated as follows:

xi �∈ [xL,min, xL,max], yi �∈ [yL,min, yL,max], i = 1, . . . , N(25)

where [xL,min, xL,max] and [yL,min, yL,max] represent the con-tinuous interval of points in the BS-free region.

To include the power density constraint in the formulation,we discretize the target area into a sufficiently representative setof sample points. Then, we calculate an estimate of the powerreceived at each of the points from each BS using the propaga-tion model. The total power received at each point is calculatedby summing the power received from all BSs. Based on thereceived power, we calculate the power density at all points,and the sample point with the peak power density is considered.When this constraint is satisfied, implicitly, the entire area willsatisfy the constraint. The constraint is formulated as follows:

maxsj

(N∑

i=1

((∑k∈Ci

P dk,i

)· 1keq

(di,sj)−μ

)· 4πf2

c2G

)≤ Sth

j = 1, . . . , p (26)

where sj is the sample point j in the target area with jbetween 1 and p, and p is the number of sample points takenin the target area, f is the carrier frequency (assumed to be1800 MHz for UMTS), c is the speed of light, G is the gainof the transmitting antenna of the BS in the direction of theradiation, Sth is the recommended power density threshold, andkeq and μ are parameters that depend on the path loss model.

The site-placement problem with location constraints canthen be formulated as follows:

minx,y

U∑k=1

P dk,b(k) + α

U∑k=1

(Puk − Pu

max)+ (27)

subject to (25), (26), (11)−(20).

Similarly, the site-selection problem with location con-straints can also be formulated as an extension to the initialsite-selection problem as follows:

minc

N0∑i=1

ci (28)

s.t.

minU∑

k=1

cb(k) · P dk,b(k) + α

U∑k=1

cb(k) (Puk − Pu

max)+ (29)

subject to (25), (26), (11)−(20)

i = 1, . . . , N0; k = 1, . . . , U. (30)


Fig. 7. Uniform user distribution with location constraints in an area measuring 4 km × 4 km. (Left) Total area. (Right) Radiation-limited area. The color scaledenotes the base-10 logarithm of the power density, e.g., −9 corresponds to S = 10−9 W/m2.

In both problems, (25) and (26) represent the location con-straints. Note that the constraints (11)–(20) from the initialproblems are also included in the formulation. The next sectiondescribes how to extend the algorithms developed in Section IVto solve the radio network planning problem with locationconstraints.

B. Algorithm for the Radio Network Planning Problem WithLocation Constraints

The location constraints as defined in (25) and (26) arenonlinear inequality constraints. To satisfy such constraints,we modify the algorithms presented in Section IV. A robustextension of pattern search algorithms for general constraints isthe globally convergent ALPS, which solves general problems.

Initially, the problem is modified to convert the inequalityconstraints into equality constraints by introducing nonnegativeslack variables. Next, we attempt to find Lagrange multiplierestimates for the equality constraints updated at each iteration,such that the estimates do not involve information about deriv-atives of the objective or constraints to be consistent with thederivative-free nature of pattern search algorithms.

The algorithm begins with an initial value for the penaltyparameter and the Lagrange multiplier estimates. The mathe-matical significance of these parameters is explained in [24]. Asubproblem is formulated by combining the objective functionand the nonlinear constraint function using the Lagrangianmultiplier estimates and the penalty parameters. A sequenceof such optimization problems are approximately minimizedusing a pattern search algorithm such that the linear constraintsand bounds are satisfied. When the subproblem is minimizedto a required accuracy and satisfies feasibility conditions, theLagrangian multiplier estimates are updated. Otherwise, thepenalty parameter is increased. These steps are repeated untilthe stopping criteria are met. A frequently used Lagrangianupdate is the first-order Hestenes–Powell multiplier update forthe augmented Lagrangian, which assumes no knowledge ofderivative information [24].

Convergence analysis for the ALPS algorithm can be foundin [24] and [25]. It is shown that, despite the absence of anyexplicit estimation of any derivatives, the pattern search aug-mented Lagrangian approach exhibits first-order global conver-gence properties. Although the subproblems are approximatelysolved and the stopping criterion of the subproblem is based onthe magnitude of a measure of first-order stationarity, the algo-rithm converges to Karush–Kuhn–Tucker points of the originalproblem. This important result establishes that proceeding bysuccessive inexact minimization of the augmented Lagrangianvia pattern search methods ensures convergence [24], [25].

Since our nonlinear constraints are written in terms of thepowers allocated to users, we need to solve the set of equations[Gd] × [Pd] = [σ2] and [Gu] × [Pu] = [σ2] for a given x (BSlocations) to find the uplink and downlink powers, computethe value of the constraints, and solve the inner augmentedLagrangian subproblem. Algorithm 1 and Algorithm 2 can beextended to ALPS by constructing the subproblem at each iter-ation, minimizing this subproblem over the mesh points insteadof minimizing the objective fm(x), updating the Lagrange mul-tipliers at each successful iteration using the Hestenes–Powellmultiplier update, and adjusting the penalty factor based on thesuccess or failure of the polls.

C. Analysis of Network Topologies With Location Constraints

To demonstrate the performance of the proposed algorithmto solve the network planning problem with location con-straints, we run the modified site-selection and site-placementalgorithms presented in Section VI-B consecutively to selectthe smallest set of BSs that cover the network area such thatthe location constraints and the QoS constraints are satisfied.To ensure limited EM radiation within the area of interest,we set the target peak power density in the area to Sth =0.005 μW/cm2. To compute the power density, we assume thatG = 1 and that f = 1800 MHz. All other parameters are thesame as those listed in Table I. Fig. 7 shows a particular casewith a uniform user distribution, where the radiation-limited


Fig. 8. Pareto-optimal set that provides the tradeoff between average BS power and the dimensions of the radiation-limited area. (Left) Uniform user distribution.(Right) Gaussian user distribution.

area is a square with dimensions of 4 km × 4 km defined asfollows: 3000 m ≤ x ≤ 7000 m, and 3000 m ≤ y ≤ 7000 m.The figure shows that the peak power density in the area is S =10−8.5 W/m2 = 3.16 × 10−9 W/m2 = 0.003 μW/m2, which isbelow the predefined threshold Sth. Additionally, it is clear thatall BSs satisfy as well the location constraints.

Finally, we generate pareto-optimal sets to show the tradeoffbetween the dimensions of the radiation-limited area and theachieved average BS power in the whole network. The largerthe area, the more power is required to achieve coverage to usersin the area and satisfy their SIR requirements, which eventuallyincreases the average BS power. Because the modified site-selection and site-placement algorithms are both executed, thecase with zero area dimensions is equivalent to the point ofoperation on Fig. 5(a) and (b) with the lowest number of BSsand highest average BS power. Fig. 8 shows the achievableaverage BS powers for sample uniform and Gaussian userdistributions. For the uniform case, the square is characterizedby its center at [5000, 5000]. For the Gaussian case, thesquare is characterized by the following: 1) yL,max touchesthe upper boundary. 2) The centroids of xL,min and xL,max lieat x = 5000.

We can see that, with a Gaussian user distribution, the slopeof increase is faster, which is why we cannot reach the 4-kmdimension as in the uniform case; otherwise, the increase inBS powers would prohibit satisfying the SIR requirements. Thealgorithm is executed with and without a maximum BS powerconstraint. The maximum BS power constraint corresponds to(19), with P d

max = 30 W. Such constraint will further limit thearea dimensions, because, even though the average BS power islower than the constraint, some BSs (specifically those coveringthe radiation-limited area) will require a very high power toachieve the required SIR for most users. The area dimensionsin the unconstrained power case are limited by interference,whereby further increasing BS power for some user makes theSIR requirement for other users nonachievable. On the otherhand, the area dimensions in the constrained power case are

limited by the physical maximum BS power limit, which turnsout to be a more stringent limitation. Obviously, these twosets represent specific case studies for the area locations thatwe considered. In fact, the slope of the curve greatly dependson the location of the square, particularly in the Gaussiandistribution case.

Another important observation is that placing a maximumBS power constraint decreases the average BS power for somefixed area dimensions. This is explained by the fact that whenBSs are not allowed to exceed a certain peak transmit power, thesite-selection algorithm will be forced to select a larger numberof BSs to cover the entire network, specifically to cover usersinside the constrained area. Looking back at Fig. 5, we recallthat the average BS power decreases as the number of BSsincreases for a given user distribution, which explains the gapbetween the two curves in each of Fig. 8(a) and (b).

VII. CONCLUSION

We have presented optimization-based formulations for theproblems of joint uplink/downlink site placement and siteselection in cellular networks. The formulations use an SIR-based power control mechanism with outage conditions to pro-vide quality guarantees. We have proposed algorithms to solvethe continuous component of the problem using derivative-free optimization techniques with general constraints. We havealso developed an optimization algorithm to solve the integercomponent of the problem based on a nested approach withan outer problem that minimizes the number of BSs and aninner problem that minimizes a cost function of the networkdeployment. Case studies have been presented and analyzed foruniform and nonuniform user distributions, and Pareto-optimalsets have been generated to trade off network configurationparameters. Finally, the formulation and solution have beenextended to provide a framework for solving general radionetwork planning problems with location constraints.


REFERENCES

[1] B. Matha, Radio Propagation in Cellular Networks. Boston, MA:Artech House, Nov. 1999.

[2] J. Laiho, A. Wacker, and T. Novosad, Radio Network Planning andOptimisation for UMTS. Hoboken, NJ: Wiley, 2006.

[3] A. Mishra, Fundamentals of Cellular Network Planning andOptimisation: 2G/2.5G/3G . . . Evolution to 4G. Hoboken, NJ:Wiley-Interscience, 2004.

[4] E. Amaldi, A. Capone, and F. Malucelli, “Improved models and algo-rithms for UMTS radio planning,” in Proc. IEEE Veh. Technol. Conf.,Oct. 2001, pp. 920–924.

[5] E. Amaldi, A. Capone, and F. Malucelli, “Optimizing base station sit-ing in UMTS networks,” in Proc. IEEE Veh. Technol. Conf., Oct. 2001,pp. 2828–2832.

[6] E. Amaldi, A. Capone, F. Malucelli, and F. Signori, “Optimization mod-els and algorithms for downlink UMTS radio planning,” in Proc. IEEEWCNC, Mar. 2003, pp. 827–831.

[7] E. Amaldi, A. Capone, and Malucelli, “Planning UMTS base stationlocation: Optimization models with power control and algorithms,” IEEETrans. Wireless Commun., vol. 2, no. 5, pp. 939–952, Sep. 2003.

[8] A. Eisenblätter, A. Fügenschuh, T. Koch, A. M. C. A. Koster, A. Martin,T. Pfender, O. Wegel, and R. Wessäly, “Modelling feasible network con-figurations for UMTS,” Konrad-Zuse-Zentrum für InformationstechnikBerlin (ZIB), Berlin, Germany, Tech. Rep. ZR-02-16, Mar. 2002.

[9] E. Amaldi, A. Capone, and F. Malucelli, “Radio planning and coverageoptimization of 3G cellular networks,” J. Wireless Netw., vol. 14, no. 4,pp. 435–447, Aug. 2008.

[10] A. Eisenblatter, A. Fugenschuh, H. Geerdes, D. Junglas, T. Koch, andA. Martin, “Integer programming methods for UMTS radio network plan-ning,” in Proc. WiOpt, Cambridge, U.K., Mar. 2004.

[11] A. Eisenblatter and H. Geerd, “Wireless network design: solution-orientedmodeling and mathematical optimization,” Wireless Commun., vol. 13,no. 6, pp. 8–14, Dec. 2006.

[12] P. Calegari, F. Guidec, P. Kuonen, B. Chamaret, S. Ubeda, S. Josselin,D. Wagner, and M. Pizarosso, “Radio network planning with combinator-ial optimization algorithms,” in Proc. ACTS Mobile Telecommun. Summit,Sep. 1996, pp. 707–713.

[13] P. Calegari, F. Guidec, P. Kuonen, and D. Wagner, “Genetic approach toradio network optimization for mobile systems,” in Proc. IEEE 47th Veh.Technol. Conf., May 1997, pp. 755–759.

[14] J. K. J. Kalvanes and E. Olinick, “Base station location and serviceassignments in WCDMA networks,” INFORMS J. Comput., vol. 18, no. 3,pp. 366–376, Summer 2006.

[15] J. Osepchuk and R. Petersen, “Safety standards for exposure to RFelectromagnetic fields,” IEEE Microw. Mag., vol. 2, no. 2, pp. 57–69,Jun. 2001.

[16] L. Catarinucci, P. Palazzari, and L. Tarricone, “Human exposure to thenear field of radiobase antennas—A full-wave solution using parallelFDTD,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 935–940,Mar. 2003.

[17] IEEE Standard for Safety Levels With Respect to Human Exposure toRadio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEEStd. C95. 1, 1999.

[18] “Guidelines for limiting exposure to time-varying electric, magnetic,and electromagnetic fields,” Health Phys., vol. 74, no. 4, pp. 494–522,Apr. 1998.

[19] B. Di Chiara, M. Ñonato, M. Strappini, L. Tarricone, and M. Zappatore,“Hybrid metaheuristic methods in parallel environments for 3G networkplanning,” in Proc. EMC Eur. Workshop, Rome, Italy, Sep. 2005.

[20] T. Crainic, B. Di Chiara, M. Nonato, and L. Tarricone, “Tackling elec-trosmog in completely configured 3G networks by parallel cooperativemeta-heuristics,” Wireless Commun., vol. 13, no. 6, pp. 34–41, Dec. 2006.

[21] H. Holma and A. Toskala, WCDMA for UMTS. Chichester, U.K.: Wiley,2000.

[22] E. Dolan, R. Lewis, and V. Torczon, “On the local convergence of patternsearch,” SIAM J. Optim., vol. 14, no. 2, pp. 567–583, Feb. 2003.

[23] R. Lewis, V. Torczon, and M. Trosset, “Why pattern search works,”Optima, vol. 59, pp. 1–7, Dec. 1998.

[24] R. Lewis and V. Torczon, “A globally convergent augmented Lagrangianpattern search algorithm for optimization with general constraints andsimple bounds,” SIAM J. Optim., vol. 12, no. 4, pp. 1075–1089,Jul. 2002.

[25] A. Conn, N. Gould, and P. Toint, “Globally convergent augmentedLagrangian algorithm for optimization with general constraints andsimple bounds,” SIAM J. Numer. Anal., vol. 28, no. 2, pp. 545–572,Apr. 1991.

Amin Abdel Khalek (S’05) received the B.E. degreein computer and communication engineering (withhighest distinction) from Notre Dame University,Beirut, Lebanon, in 2008 and the M.E. degree inelectrical and computer engineering from the Amer-ican University of Beirut, Beirut, Lebanon, in 2010.He is currently working toward the Ph.D. degree inelectrical and computer engineering with the Depart-ment of Electrical and Computer Engineering, TheUniversity of Texas, Austin.

His research interests include wireless communi-cations, multimedia signal processing, and optimization.

Lina Al-Kanj received the B.E. degree in electricaland communications engineering from the LebaneseUniversity, Beirut, Lebanon, in 2005 and the M.E.degree in computer and communications engineer-ing, in 2007, from the American University of Beirut,Beirut, Lebanon, where she is currently workingtoward the Ph.D. degree with the Department ofElectrical and Computer Engineering.

Her research interests include cooperative com-munications and radio network planning.

Zaher Dawy (SM’09) received the B.E. degreein computer and communications engineering fromthe American University of Beirut (AUB), Beirut,Lebanon, in 1998 and the and the M.Sc. andDr.-Ing. degrees in communications engineeringfrom Munich University of Technology (TUM),München, Germany, in 2000 and 2004, respectively.

In September 2004, he joined the Department ofElectrical and Computer Engineering, AUB, wherehe is currently an Associate Professor. His researchinterests are computational biology, information the-

ory, and wireless communications, focusing on genomic coding theory, genenetwork modeling, distributed and cooperative communications, cellular tech-nologies, radio network planning and optimization, and multimedia transmis-sion over communication networks.

Dr. Dawy is the Chair of the IEEE Communications Society LebanonChapter and a member of the Lebanese Order of Engineers. He was therecipient of the AUB 2008 Teaching Excellence Award, the Best GraduateAward from TUM in 2000, the Youth and Knowledge Siemens Scholarshipfor Distinguished Students in 1999, and the Distinguished Graduate Medal ofExcellence from the Harriri Foundation in 1998.

George Turkiyyah received the B.E. degree fromAmerican University of Beirut (AUB), Beirut,Lebanon, and the M.S. and Ph.D. degrees fromCarnegie Mellon University, Pittsburgh, PA.

He is currently a Professor and Chair of the De-partment of Computer Science, AUB. Prior to join-ing AUB, he was an Assistant and then AssociateProfessor with the University of Washington, Seattle.He is the author of more than 50 refereed publica-tions. He is the holder of three patents on geometricrepresentation technologies. He has supervised six

Ph.D. students, cofounded a software startup, and published a number of widelyused software systems. His research interests are high-performance computing,geometric modeling, physically based simulation, numerical optimization, andlarge-scale web-enabled data repositories.

Dr. Turkiyyah is a member of Association for Computing Machinery(ACM)and the Society for Industrial and Applied Mathematics (SIAM). He hasbeen a recipient of a number of awards, including the 2003 Transporta-tion Research Board K. B. Woods Award for Best Paper in Design andConstruction and Best Presentation/Poster Awards in the 2007 ACM Solid andPhysical Modeling and the 2006 Medicine Meets Virtual Reality Conferences.

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