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Optimization Models and Algorithms for Joint

Uplink/Downlink UMTS Radio Network Planning

with SIR-based Power ControlAmin Abdel Khalek, Lina Al-Kanj, Zaher Dawy, Senior Member, IEEE, and George Turkiyyah

Abstract UMTS networks should be deployed accordingto cost-effective strategies that optimize a cost objective andsatisfy target quality of service (QoS) requirements. In thispaper, we propose novel algorithms for joint uplink/downlinkUMTS radio planning with the objective of minimizing totalpower consumption in the network. Specifically, we define twocomponents of the radio planning problem: (1) Continuous-based site placement, and (2) Integer-based site selection. Inthe site placement problem, our goal is to find the optimal

locations of UMTS base stations in a certain geographic areawith 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 boundedoutage constraints. We model the problem as a constrainedoptimization problem with SIR-based uplink and downlink powercontrol scheme. An algorithm is proposed and implemented usingpattern search techniques for derivative-free optimization withaugmented Lagrange multiplier estimates to support generalconstraints. In the site selection problem, we aim to select theminimum set of base stations from a fixed set of candidatesites that satisfies quality and outage constraints. We developan efficient elimination algorithm by proposing a method forclassifying base stations that are critical for network coverageand quality of service. Finally, the problem is reformulated to

take care of location constraints whereby the placement of basestations 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, network optimiza-tion, network deployment, electromagnetic radiation exposure

I. INTRODUCTION

In UMTS networks, the base station (BS) coverage and

capacity are a function of the user distribution, the signal-

to-interference ratio (SIR) requirements, and the interference

level which is an important coverage-limiting factor. The trans-mit powers of mobile users are power controlled depending

Copyright (c) 2011 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

This work was supported by a research grant from the National Councilfor Scientific Research (CNRS), Lebanon.

A. Abdel Khalek was with the Department of Electrical and ComputerEngineering, American University of Beirut, Beirut, Lebanon. He is now withthe Department of Electrical and Computer Engineering, The University ofTexas at Austin, TX, USA (email: [email protected]). L. Al-Kanj andZ. Dawy are with the Department of Electrical and Computer Engineering,American University of Beirut, Beirut, Lebanon (email: [email protected],[email protected]). G. Turkiyyah is with the Department of Computer Science,American University of Beirut, Beirut, Lebanon (email: [email protected]).

on their distance from the BS to reduce interference, avoid

the near-far problem and ensure coverage for users close to

the cell edge. Given the high cost of network infrastructure

investments and spectrum licenses, operators should make

informed decisions on network deployment to satisfy perfor-

mance requirements in a cost efficient way. This drives the

need for optimized UMTS-specific planning tools that take

into account the WCDMA air interface characteristics.

UMTS radio network planning involves configuring the

network resources and parameters in a way that guarantees

satisfactory performance for the end-users according to the

following three main quality attributes: coverage, capacity

and quality of service. Radio network planning is conven-

tionally approached as an iterative process which requires

setting target coverage and capacity objectives. The initial

network plan is obtained from geographic data, demographic

data, and propagation models, e.g., [1], [2], [3], and is then

optimized by iterative updates of various network variables.

Several modeling techniques are feasible and can be solved

by mathematical and heuristic optimization algorithms such

as simulated annealing, greedy algorithms, genetic algorithms,linear and non-linear programming, etc.

A. Related Work

In [4], [5], [6], the authors propose discrete optimization

algorithms using randomized greedy procedures and a tabu

search algorithm to plan the process of locating new BSs

considering quality constraints for the uplink which is argued

to be more stringent than the downlink for symmetric traffic. In

[7], the previous work is extended to the downlink for asym-

metric traffic by applying SIR-based power control. Models

spanning both downlink and uplink with power control are

also presented in [8], [9].

In [10], [11], mixed integer linear programming (MILP) is

used for planning cost-efficient radio networks under network

quality constraints. Models based on set covering are used to

obtain lower bounds on the number of required BSs to serve a

given fixed area and an automatic two phase network planning

approach based on successively solving instances of the model

is presented. In [12] and [13], two graph theory based models

are proposed: Maximum independent set (MIS) model and

minimum dominating set (MDS) model to select a satisfactory

subset out of a user-provided set of BS locations, while

ensuring that at least a given percentage of the considered area

is served by the selected BSs. A large set of candidate BS sites

is first determined. In [14], a net-revenue maximization model


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for the selection of BS sites and the calculation of service

capacity is presented. The integer programming model takes

the candidate BS locations and the traffic demand model as

input, and uses a priority branching scheme to achieve a target

optimization gap tolerance.

The majority of the contributions on optimized network

planning focus on locating BSs according to the best trade-

off between network infrastructure costs and service coverage,whereas the electromagnetic (EM) field exposure levels are

rarely considered. However, raising concerns about serious

consequences for human health due to exposure to EM fields

have led to precautionary regulations enforced by public

administrators [15], [16]. The most widely accepted standards

are those developed by the IEEE [17] and the ICNIRP [18].

Due to the increasing concerns about EM pollution in cellular

networks, it should be considered an important metric for

network planning and optimization. The inclusion of EM

radiation in the cellular network planning problem has been

addressed in [19], [20] where two sequential meta-heuristics

were developed to limit the total EM field at selected test

points and combined in a tabu search (TS) and a genetic algo-

rithm (GA). The TS procedure is able to explore the solution

space deeply enough, but it works on partial configurations,

while the GA procedure can manage the complete set of

considered parameters, but is computationally expensive.

Current work in the literature focuses on selecting a minimal

BS set from a larger candidate fixed BS set. The equally-

important continuous version of the problem which involves

finding the optimal locations of these BSs in the network area

was not considered previously in the literature. By combining

the two components of the problem, we can target more

general application scenarios and further adapt and optimize

the network plan.

B. Contributions of the Paper

In this work, the problem of joint uplink/downlink radio

network planning is subdivided into two components. In the

first component, referred to as the site placement problem

throughout the paper, we are interested in finding the optimal

locations of a fixed number of UMTS base stations. In the

second component, referred to as the site selection problem,

we are interested in finding the minimal cardinality set from

a set of base stations with fixed locations. It can be seen that

the site placement problem is a continuous problem because

the problem variables are the physical BS locations, while

the site selection problem is a combinatorial problem because

the problem variables are the binary selection variables for

each of the BSs. We solve the continuous problem of finding

the optimal locations of base stations that minimize total

power expenditure in the network using robust pattern search

algorithms for derivative-free optimization with non-smooth

objectives. We define QoS targets and maintain bounded

outage levels on a network-wide and per BS basis for both

uplink and downlink channels. Next, we formulate and propose

an algorithm to solve the integer problem of selecting the

smallest set of BSs from a fixed set of potential sites such

that the cost function is minimized and the QoS requirements

are satisfied.

It is important to note that these two problems and their

solution approaches are distinct, however, we propose that the

two problems be combined into a hybrid integer/continuous

algorithm involving successive site selection/site placement

until both components converge, i.e., until no relocation or

elimination is feasible. This allows the operator to find the

minimal set of BSs needed for satisfactory coverage and the

optimized deployment strategy in the network according to theuser distribution. An additional novel component of our work

is the development of a framework for radio network planning

with location constraints. Such constraints arise frequently in

practice due to private property limitations or electromagnetic

radiation constraints. We formulate the problem of radio

network planning with location constraints as an optimization

problem. As part of the problem solution, we use Lagrangian

multiplier estimates and penalty parameters to construct and

solve a sequence of augmented Lagrangian subproblems based

on the Augmented Lagrangian Pattern Search (ALPS) algo-

rithm. Finally, we provide optimal tradeoff curves under dif-

ferent user distributions and we demonstrate the effectiveness

of the proposed schemes compared to conventional clustering.

C. Paper Organization

The rest of the paper is organized as follows: Section II

describes the system model. Section III presents the math-

ematical formulation for the site selection and site place-

ment optimization problems. Section IV presents the strategy

and algorithms used to solve the optimization problems. In

Section V, we report results for different user distributions

and voice/data traffic combinations and we provide optimal

tradeoff curves for the network configuration. In Section

VI, we extend the optimization algorithms to scenarios with

location constraints and/or EM radiation restrictions. Finally,Section VII provides concluding 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 the physical channel at a given instant of time. For a

given distribution of currently active users or connections, we

aim to find a network plan that guarantees minimal power

consumption in the network, and satisfies coverage and quality

of service requirements in a cost-effective manner.

In UMTS, users rely on channelization and scrambling

codes in order to differentiate their own signal and combatthe effect of multipath and multiuser interference. To guarantee

the required QoS level, a target minimum SIR value should be

maintained for all active connections. The downlink and uplink

SIR expressions for user k can be expressed as follows:

SIRdk = SFPdreceived,k

2 + dIdin,k + Idout,k

(1)

SIRuk = SFPureceived,k

2 + uIuin,k + Iuout,k

(2)

where Pdreceived,k is the downlink received power at mobilestation (MS) k, Pureceived,k is the uplink received power at

the BS from MS k, Idin,k and Idout,k represent the downlink


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SF + SIRdk

d

SIRdk

gb(k),kP

db(k),k

dgb(k),k

jCb(k)

Pb(k),j

N

i=1,i=b(k)

gi,k

lCi

Pdi,l

= 2 (3)

SF + SIRuk

u

SIRuk

gb(k),kP

uk

u

jCb(k)

gb(k),jPuj

N

i=1,i=b(k)

lCi

gi,lPdl

= 2 (4)

intracell and intercell interference affecting MS k, respectively,Iuin,k and I

uout,k represent the uplink intracell and intercell

interference affecting MS k, respectively, SF is the spreadingfactor, 2 is the thermal noise power, and is the orthogonalityfactor (0.4 d 0.9 in the downlink and u = 1 inthe uplink [21]). Since we need to satisfy the SIR constraint

for all users, we express the received power and interference

components in (1) and (2) more explicitly for user k asfollows:

Pdreceived,k = gb(k),kPdb(k),k; P

ureceived,k = gb(k),kP

uk (5)

Idin,k = gb(k),k Pdb(k)Pdb(k),k ; Iuin,k = jCb(k),j=k

gb(k),jPuj (6)

Idout,k =

Ni=1,i=b(k)

gi,kPdi ; I

uout,k =

Ni=1,i=b(k)

jCi

gi,jPuj (7)

where N is the number of BSs, Pdi is the total transmit powerof BS i, Pdi,k is the power allocated by BS i to MS k (subjectto k being covered by BS i), gi,k is an estimate of the pathlossbetween MS k and BS i calculated according to Cost 231-Hatamodel [1], and b(k) is defined as the BS serving MS k andcalculated by constructing the Voronoi tessellations associated

with the BS locations. Consequently, Pdb(k),k is the transmitpower allocated to MS k by its serving BS, Pb(k) is the totaltransmit power of the BS serving user k and gb(k),k is anestimate of the pathloss between MS k and its serving BS.The notation j Ci means all MSs {j} covered by cell i.Consequently, j Cb(k) means all MSs covered by the sameBS as k. The pathloss coefficients gi,k can be written strictly interms of the BS and MS locations in addition to some constants

as gi,k =1keq

(di,k), 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 chosen according to Cost-231Hata Model. In UMTS radio network planning, shadowing and

fading are compensated for via link budget margins [1], [21].

Based on the derivation above, (1) can be rewritten in terms

of the powers Pdi,k allocated to MSs which are the downlinkstate variables as shown in (3). Similarly, (2) can be rewritten

in terms of the MS transmit powers Puk which are the uplinkstate variables as shown in (4) where SIRdk and SIR

uk are the

target SIRs to achieve the required QoS for the downlink andthe uplink respectively. Equations (3) and (4) can be expressed

in matrix format as follows:GdUU

PdU1

=

2U1

(8)

[Gu]UU [ Pu ]U1 =

2U1

(9)

where Gd and Gu are square matrices with size U U, Pd is acolumn vector with size U 1 corresponding to the downlink

user powers Pdb(k),k, Pu

is a column vector with size U 1

corresponding to the uplink user powers Puk , 2 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 in

each BS. The first term of (3) and (4) appears in the diagonal

ofGd and Gu, respectively, the second term represents intracell

interference and appears in the block corresponding to the BS

serving user k, the third term represents intercell interferenceand appears in the blocks corresponding to all BSs except the

serving BS. Solving the power assignment problem with SIR-

based power control reduces to solving this set of equationswhich adjusts the transmit powers in order to to meet the target

SIRs [1], [2].

III. OPTIMIZATION PROBLEM FORMULATION

In this section, we present a formulation for the joint

uplink/downlink radio planning problems. The site placement

problem is formulated as a continuous optimization problem

and the site selection problem is formulated as an integer opti-

mization problem, and the objective, variables, and constraints

for each of the problems are defined.

A. The Joint Uplink/Downlink Site Placement Problem

The input to the site placement problem is: (1) The area

of interest, (2) the fixed set of MS locations modeling the

typical distribution of active users in the area, (3) the fixed

number of BSs, and (4) the initial locations of the BSs. The

site placement algorithm optimizes the initial locations of the

BSs to minimize the target objective while satisfying the QoS

requirements. The output is the set of optimal locations of

BSs. The objective of joint uplink/downlink site placement is

defined as a weighted linear combination of two objectives:

1) Minimize the downlink power expenditure: The total

downlink power expenditure is the sum of powers al-

located to each user by its serving BS, expressed asUk=1 P

db(k),k. In addition to the inherent benefits of

minimizing power consumption, we will demonstrate

that this reduces the variance of BS powers, thus pro-

viding a solution that balances the power load according

to the predicted user distribution.

2) Minimize the uplink outage: The uplink channel is

limited by the power capabilities of the MSs, which is

by far less than that of the BSs. Thus, it is important

to ensure that uplink transmissions can still reach the

BSs at a reasonable power subject to their handset limi-

tations while satisfying the minimum SIR threshold for

acceptable performance. We formulate this component

asU

k=1 (Puk Pumax)+ where ()+ = max(, 0), Puk is


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the uplink power associated with MS k, and Pumax is themaximum allowed MS power.

Effectively, each MS requiring power in excess of Pumaxwill be considered in the second objective. If all MSs do not

exceed the threshold, this will be a vector of zeros, essentially

disregarded from the objective. We do not consider the number

of MSs that are in outage directly in the objective function to

maintain coherence in the units (Watts) and avoid non-smoothstep transitions in the function. However, this approach has the

same potential in minimizing the number of MSs in outage

since the algorithm will attempt to find the BS configuration

that satisfies (Puk Pumax)

+ = 0 k, if such configurationexists.

In the site placement problem, N is fixed, that is, weare not interested in reducing the initial number of BSs but

in optimizing their locations. Thus, the decision variables

are the locations of the BSs (xi, yi), i = 1, , N thatminimize the cost function. Since the objective function is

expressed in terms of the downlink powers assigned by the

BSs to their MSs, we consider the powers assigned by theBSs (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, the total number of variables in theproblem is U + 2N where U N. The power assignmentvariables relate to the decision variables through the SIR-based

power control mechanism as shown in (8) and (9). Since the

matrices Gd and Gu consist of the pathloss coefficients gi,k,it is worth mentioning that it can be written strictly in terms

of the BS and MS locations in addition to some constants

according to the equations gi,k =1keq

(di,k), di,k =

(xi uk)2 + (yi vk)2. Thus, the power assignments for a

given setting of the location variables can be found by solvingthe U set of equations, an operation that costs O(U3).

Two important observations concerning Gd and Gu are

worth pointing out: First, they are strongly diagonally-

dominant. For example, the average value of the diagonal

terms is at least 103 times the average value of the non-diagonal terms with a spreading factor of 128 since the

spreading gain appears only in the diagonal terms and the

non-diagonal terms correspond to interference components.

This fact can be exploited by using fast solvers suitable for

diagonally-dominant matrices. Second, if the SIR requirement

for the kth user cannot be satisfied due to high interferenceat its location, the solution of the corresponding equation will

yield a negative power, which can be used for spotting outages

in the network.

The outage conditions are defined network-wide and for

each BS. The network-wide outage conditions ensure that the

total number of users in outage is less than networkU and theBS outage conditions ensure that the number of users in outage

for every BS i is less than BSMi where network and BS aredesign parameters that satisfy 0 network BS 1.

The problem of joint uplink/downlink site placement is for-

mulated as an optimization problem as in (10)-(20) where (10)

represents the weighted objective and is a constant whichdetermines the relative weight of each of the two components,

(11) represents the matrix-form downlink SIR constraint for all

MSs, (12) represents the matrix-form uplink SIR constraint for

all MSs, (13) is the downlink network outage condition, (14)

is the uplink network outage condition, (15) is the downlink

BS outage condition for every BS, (16) is the uplink BS

outage condition for every BS, (17) is the pathloss according to

Cost 231-Hata model, (18) is the distance-based assignment of

MSs to BSs obtained by constructing the Voronoi tessellation

corresponding to the BS locations, (19) is the maximum BSpower constraint, and (20) is the area of operation constraint.

Note that MS k may be in outage either because it requires atransmission power Puk higher than P

umax to reach the closest

BS, or because the transmit power Puk , although lower thanPumax, causes high interference at another MS so that the SIRconstraints cannot be satisfied. Equation (10) eliminates outage

due to the former case while (13)-(16) eliminate outage due

to the latter case.

minx,y

Uk=1

Pdb(k),k +

Uk=1

(Puk Pumax)

+ (10)

s.t.GdUU

PdU1

=

2U1

(11)

[Gu]UU [ Pu ]U1 =

2U1

(12)

Uk=1

Pdk,b(k)

|Pdk,b(k)|

+< network U (13)

Uk=1

Puk|Puk |

+< network U (14)

kCi

Pdk,b(k)

|Pdk,b(k)|

+< BS Mi (15)

kCi

P

uk

|Puk |

+< BS Mi (16)

gi,k =1

keqdi,k ; di,k=

(xi uk)2 + (yi vk)2(17)

b(k) = argmini di,k (18)jCi

Pi,j Pdmax (19)

xmin xi xmax, ymin yi ymax (20)

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

B. The Joint Uplink/Downlink Site Selection Problem

The input to the site selection problem is: (1) The area of

interest, (2) the fixed set of MS locations modeling the typical

distribution of active users in the area, (3) the candidate set of

BSs with fixed locations. The site selection algorithm selects

the minimal cardinality set of BSs that satisfies the coverage

and SIR requirements. The output is the subset of candidates

corresponding to the selected BSs. The main difference from

the site placement problem is that the decision variables are

not the BS locations, instead, these locations are fixed, and

the decision variables are booleans ci where ci = 1 if BS i isto be used in the optimal network configuration, and ci = 0otherwise. The objective of the problem is to minimize the

number of selected BSs, that is,N

0i=1 ci where N0 is the


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size of the candidate set of BSs. The problem can thus be

formulated as a nested optimization problem as shown in (22)-

(24). The inner problem finds the set of BSs that minimizes

the cost function and the outer problem finds the minimal

cardinality set of such BSs. We will later show that using the

minimization of the total power expenditure as the criterion for

BS removal results in better solutions than a removal based

purely on the feasibility of BS elimination (See Section V-B).Note that the constraints (11)-(20) are only applied to the set

of active BSs satisfying ci = 1.

minc

N0i=1

ci (22)

s.t.

minU

k=1

cb(k) Pdk,b(k) +

Uk=1

cb(k) (Puk P

umax)

+(23)

subject to

(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)

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

IV. OPTIMIZATION STRATEGIES AND ALGORITHMS

Solving the initial site placement optimization problem

involves nonlinear equality constraints: (11), (12), (17), and

(18), nonlinear inequality constraints: (13)-(16) and (19), and

linear bound constraints: (20). Given the current BS locations

x and y, we use (18) to obtain the assignments of MSs to

their closest BS, and (17) to obtain the pathloss between each

MS and its serving BS. Finally, solving the SIR equations

(11) and (12), we obtain the power allocated to each MS on

the uplink and downlink to satisfy QoS requirements. Thus,

the four sets of equality constraints are implicitly includedin objective function evaluations because they determine the

power allocation scheme throughout the network. The bound

constraint representing the area of operation is a simple linear

constraint that is taken care of by the algorithm by choosing

appropriate steps that do not violate that constraint. To solve

the site placement problem, we propose a pattern search algo-

rithm based on Mesh Adaptive Direct Search (MADS). Addi-

tionally, to ensure satisfying the BS power limit and the outage

constraints, we will describe how to extend the algorithm to

include any general nonlinear inequality constraints using the

Augmented Lagrangian Pattern Search (ALPS) method. After

presenting the algorithm for the site placement problem, we

will present the solution for the nested optimization problem

of site selection based on successive elimination of BSs.

A. Algorithm for Site Placement with Uplink and Downlink

QoS Guarantees

The MADS class of derivative-free algorithms is effective

for practical nonlinear optimization problems with nonsmooth

objective functions where the computation of derivatives is

either not possible, or it is not sufficiently representative

of the variability of the function around a point due to

the roughness of the objective function. MADS has a well

developed convergence theory based on the Clarke calculus

and Rockafellers notion of a hypertangent cone [22],[23].

To illustrate why the MADS algorithm is suitable for radio

planning problems, we present a sample pattern of the change

of the objective function value with respect to a change in

one of the variables (equivalent to moving one of the BSs

along a coordinate direction). Figure 1 shows that the objective

function has a lot of discontinuities and non-smooth changes

which are explained by the changes in user assignments due

to the shift in the Voronoi diagram. As new users are handed-over to a new BS, they become on the boundary of that BS,

requiring higher power to achieve the target SIR, thus causing

the instant rise in power allocations, and correspondingly,

in the objective function value. Note that the change in

one MS-BS assignment does not affect only that user due

to the changes in intercell and intracell interference effects

experienced by other users, translated in the change of the

structure and content of the pathloss matrices Gd and Gu and

the solutions to the set of equations in (8) and (9). Although

the total power expenditure increases if the BS is moved any

distance larger than 20 meters, a gradient computed using finite

difference method or adjoint method is misleading because

it will imply that the objective function is decreasing which

is only valid 20 meters away from the BS. These kinds of

peculiarities make it impractical to follow a line search method

in solving the problem, which is why we advocate the use of

derivative-free optimization algorithms suitable for the non-

smooth objectives of the radio network planning problem.

0 100 200 300 400 500180

181

182

183

184

185

186

187

188

Movement Distance (m)

TotalPowerExpen

ditureinthenetwork(W)

Fig. 1. Pattern of change of objective function while moving a base stationalong a coordinate direction.

The MADS algorithm operates by performing polling and

searching on a set of mesh points around the current location

of the decision variables. The mesh points are constructed by

building a pattern which is a basis set of vectors that the

algorithm uses to determine which points to search at each

iteration. The most common basis set is the 2n basis set wheren is the number of decision variables. In our problem, thenumber of decision variables is n = 2N, thus the basis set willcontain 4N vectors each of size 2N. The vectors are definedas follows: 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].

At each iteration, the pattern search polls the points in the

current mesh for a point that improves the objective function

by computing the objective function at the mesh points to

check if there is one whose function value is less than the

function value at the current point. The mesh points are found


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Algorithm 1 Proposed solution for the joint uplink/downlink

site placement problem.

Given N ; U ; {xi = xinitial,i}2Ni=1 ; {uk = ufixed,k}

2Uk=1 ; =

0 ; th ; ; ; keq ; 2 ; d ; u ; SF ; {SIRdk}

Uk=1 ; {SIR

uk}

Uk=1.

Construct the 2n basis set of vectors v that the algorithm uses to determinewhich points to search at each iteration where n is the number of decisionvariables (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 > th{While the mesh size is higher than the convergencethreshold} do

for m = 1 : 4N{For all possible patterns of movement} doxtemp = x + m vpStep 1. Construct the Voronoi tessellation corresponding to the basestation locations {xm} and find the distance between each (BS ,MS) pairStep 2. Construct the pathloss matrices Gd and Gu based on theestimated pathloss for each (BS , MS) pair in the uplink and downlinkStep 3. Solve the set of equations:

Gd

Pd

=

2

and

[Gu] [ Pu ] =

2

to find the downlink power that should beallocated to each MS and the uplink MS power to achieve SIR-basedpower control for all users.Step 4. Calculate the objective function at this mesh point

xm: fm =U

k=1

Pdk,b(k) + U

k=1

(Puk Pumax)

+

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

x = x+vm{Successful Poll: Advance to the point that minimizesthe objective and expand mesh size} = (Expansion Factor).

else = (Contraction Factor){Unsuccessful Poll: reduce meshsize}.

end ifend while.

by multiplying each pattern vector vp by a scalar m to gen-erate a set of direction vectors and adding the direction vector

to the current point found at the previous step. The number of

mesh points 4N is simply due to the four coordinate directionsof polling for each BS. These points can be expressed as

xp = x + mvp, p = 1, , 4N where x is a vector ofsize 2N corresponding to the current locations of the N BSs,m is the mesh size at iteration m, and xp is the polled meshpoint. Initially, the mesh size is specified based on the scale of

the problem, and is expanded and contracted during execution

of the optimization algorithm. In this way, the algorithm finds

a sequence of points, x0, x1, x2, that approach an optimalpoint. The convergence criteria are satisfied when the mesh

size is smaller than the mesh tolerance such that minimizing

the objective function further would require moving any BS

a distance smaller than this mesh tolerance threshold. This

polling technique is effectively equivalent to moving the BSs

in their neighborhood according to the mesh size at a given

iteration. It is important to note that at each polled point, the

objective function is calculated by reconstructing the Voronoi

tessellation, building the pathloss matrices Gd and Gu, solving

the set of equations corresponding to SIR-based power control,

and computing the objective function from Pd and Pu. A

complete algorithmic description is presented in Algorithm 1.

As mentioned previously, the algorithm described above

does not account for the nonlinear inequality constraints in the

problem. To include the maximum BS power and the outage

constraints, we use the ALPS algorithm which is a robust

extension of pattern search algorithms for general constraints.

Algorithm 2 Proposed solution for the joint uplink/downlink

site selection problem.

Given N = N0 ; U ; F = ; I = ; {xi = xinitial,i}N0i=1 ; {yi =

yinitial,i}N0i=1 ; ; ; keq ;

2 ; d ; u ; {uk =ufixed,k}

Uk=1 ; {vk = vfixed,k}

Uk=1 ; SF ; {SIR

dk}

Uk=1 ; {SIR

uk}

Uk=1.

while true dofor i = 1 : N{Try eliminating base station i} do

xtemp = {x1, , xi1, xi+1, , xN}ytemp = {y1, , yi1, yi+1, , yN}Step 1. Construct the Voronoi tessellation corresponding to the basestation locations {xtemp, ytemp} and find the distance between each(BS , MS) pairStep 2. Construct the pathloss matrices Gd and Gu based on theestimated pathloss for each (BS , MS) pairStep 3. Solve the set of equations:

Gd

Pd

=

2

and

[Gu] [ Pu ] =

2

to find uplink and downlink power allocationachieving SIR-based power control for all users.Step 4.if Pd and Pu do not satisfy (13)-(16) {If eliminating the base stationcauses significant DL or UL coverage loss} then

i I {Place BS i in the infeasible set}else

i F {Place BS i in the feasible set}Calculate the objective function at (xtemp,ytemp): fi =U

k=1 Pdk,b(k) +

Uk=1 (P

uk

Pumax)+

end ifend forif F = {If some base station can be eliminated} then

e = argminifix = {x1, , xe1, xe+1, , xN}y = {y1, , ye1, ye+1, , yN}N = N 1

elsebreak{Converged: No BS can be eliminated without jeopardizingcoverage}

end ifend while

The algorithm operates by formulating a subproblem obtained

by combining the objective function and nonlinear constraint

functions using Lagrange multiplier estimates and penalty

parameters. A sequence of such optimization problems are

approximately minimized using a pattern search algorithm and

the convergence to the optimal solution is guaranteed (See

Section VI-B for more details).

B. Algorithm for Site Selection with Uplink and Downlink QoS

Guarantees

The site selection problem is a nested optimization problem

with the outer problem minimizing the number of BSs and

the inner problem selecting the set of BSs that minimize the

weighted cost function. The approach for solving the problem

is based on successive elimination of BSs one at a time. Given

a set of fixed BSs S and a user distribution model, at eachelimination step, we can divide the set of BSs S into twodistinct subsets: F and I where F is the feasible set (i.e.,if BS i F, it can be safely eliminated from the set)and I is the infeasible set (i.e., if BS i I, it cannotbe eliminated from the set without jeopardizing coverage to

a significant fraction of users as defined by network andBS) and F I = S. Formally, BS i F if for the set{BS1, , BSi1, BSi+1, , BSN}, a solution for the2U equations in (8) and (9) such that (13)-(16) are satisfied.Out of the feasible set, we eliminate the BS that minimizes

the weighted objective after elimination in accordance with

our initial cost function. This whole operation is repeated until

at some point, the elimination testing phase yields an empty


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TABLE I. SIMULATION PARAMETERS

Parameter Value Description Parameter Value Description

U 1000 Number of active users xmax 10 Km Area dimensionskeq 2.75 10

15 Pathloss parameters ymax 10 Km 3.52 Eb/I0 (DL) 7 dB QoS parameters

Pdmax 30 W Maximum BS power Eb/I0 (UL) 5 dBPumax 1 W Maximum MS power network 0.05

SF 128 Service characteristics BS 0.05Rdvoice 12.2 Kbps

d 0.4 Orthogonality factors

Ruvoice 12.2 Kbps u 1

Rddata 64 Kbps 0.5 Multiobjective weightRudata 32 Kbps

2 2 1014 W Noise power

0 2 Km 4 Km 6 Km 8 Km 10 Km0

2 Km

4 Km

6 Km

8 Km

10 Km

0 2 Km 4 Km 6 Km 8 Km 10 Km0

2 Km

4 Km

6 Km

8 Km

10 Km

0 2 Km 4 Km 6 Km 8 Km 10 Km0

2 Km

4 Km

6 Km

8 Km

10 Km

Fig. 2. Base station 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. In the figures, stars denote base stations, dots denote mobile stations, and the boundarylines denote Voronoi regions.

feasible set. It is worth noting that testing feasibility requires

solving the set of 2U equations in (8) and (9) for each BS. Acomplete algorithmic description is presented in Algorithm 2.

In order to provide a general optimization framework for

UMTS radio network planning, we combine the two opti-

mization problems of site selection and site placement to

find the minimal set of BSs to cover the area and their

optimal locations subject to the user distribution. Thus, the

two algorithms can be successively executed as subproblems

until the convergence criteria for both are satisfied.

V. RESULTS AND INTERPRETATION

This section presents results and analysis for a 10 Km x10 Km area with U = 1000 active users having the sametarget per-bit signal-to-interference ratio Eb/I0 (DL) = 7dB and Eb/I0 (UL) = 5 dB. Initially, we assume all MSsare operating a voice service. In Section V-D, we present

results for network scenarios with two service classes (voice

and data). For radio propagation, we consider the COST-231

Hata model for metropolitan areas [1] and we compensate

for shadowing, fading, and antenna losses by adding a 16 dB

margin to the link budget. This resulting total loss between BS

i to MS k can be approximated to within less than 0.1 dB in allregions of interest by gi,k =

1keq

di,k where keq = 2.75 1015

and = 3.52. The simulation parameters are summarized inTable I.

Each BS is assumed to be equipped with an omni-directional

antenna that is placed at the cell center. We point out that the

algorithms developed in this paper make no assumptions about

the type or directionality of the antennas. In fact, they can be

easily applied to a sectorized network by accounting for the

directional antenna gains and constructing the Voronoi regions

on a sector by sector basis. In this section, we assume that both

BSs and MSs are equipped with omni-directional antennas.

A. Optimal Base Station Configurations for Different User

Distributions

We present sample results of optimal BS deployments for

three different distributions by executing Algorithm 1 and

Algorithm 2 consecutively to combine site selection and site

placement and find the minimal set of BSs and their optimal

locations to cover the given area. For each of the three

distributions, we start initially with N0 = 100 randomlylocated BSs. N0 is chosen to be large enough to provide aninitial configuration that satisfies outage and SIR requirements.

Figure 2(a) shows the optimal BS distribution for a uniform

user distribution with 1000 active users distributed uniformly

over the entire area. Results show that 64% of the BSs were

eliminated, reaching 36 base stations with an average BS

power of 17.1 W and standard deviation of BS powers of1.26 W. Figure 2(b) shows the optimal BS distribution for aGaussian user distribution modeling a hot spot with maximum

user density at the center and decreasing gradually towards

the area boundary. Results show that 61% of the BSs were

eliminated, reaching 39 BSs with an average BS power of

13.1 W and standard deviation of BS powers of 4.34 W.Finally, Figure 2(c) solves the problem for four Gaussian

user distributions modeling several hot spots with 250 active

users each. After execution, 55% of the BSs were eliminated,


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reaching 45 BSs with an average BS power of 11.6 W and astandard deviation of 3.39 W.

B. Analysis of Site Selection and Site Placement Algorithms

Figure 3 shows the average BS power during execution

of the site selection algorithm for different inner objectives

with a Gaussian user distribution and N0 = 100 BSs. The

outer objective is to minimize the number of BSs and withthe inner objective to either (1) minimize sum of BS powers

or (2) eliminate first feasible BS. In the first approach, we

eliminate the one that among other BSs minimizes total power

expenditure as described in Section III-B. This essentially

provides a solution that minimizes average BS power. In

the second approach, while executing the outer problem, we

greedily eliminate the first possible BS that does not cause

significant coverage loss as defined in (13)-(16). Obviously,

the average BS power will increase under both approaches

since BSs are getting eliminated, however, what the algorithm

optimizes is the rate of this increase. The most noticeable

difference is that 46 BSs were eliminated with the first inner

objective while only 20 BSs were eliminated with the greedy

approach. This demonstrates the impact of the elimination

criteria on the rate of increase of the average BS power, and

thus on the effectiveness of the selection algorithm.

0 5 10 15 20 25 30 35 40 452

3

4

5

6

7

8

Number of Eliminated Base Stations N0 N

Average

BSPower

1 N

U k

=1

Pd k

,b(k)

Inner objective: Min sum of BS powersInner objective: First feasible BS

26 more base stations eliminated

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

Another important observation is that minimizing the total

power expenditure implicitly reduces the variance of BS

powers by converging to a solution that distributes the power

load nearly equally among BSs. Figure 4(a) shows the average

BS power during the execution of the site placement algorithm

with a Gaussian user distribution for two different objective

functions: (1) Minimize sum of BS powers, (2) Minimize

variance of BS powers. The two objectives converge almost to

the same average BS power suggesting that opting for equal

BS powers does not greedily increase these powers to achieve

equality, instead, the network configuration converges to a low

power solution due to the SIR-based power control mechanism

utilized in the problem. Figure 4(b) shows a similar scenario

for the site selection algorithm with N0 = 100 initial BSs.We present a case study to analyze the effect of the initial

number of base stations N0 on the solution. We consider auniform user distribution with U = 200 active users. Initially,

the BSs are located randomly in the network. We execute

the site placement and site selection algorithms successively

as subproblems until the convergence criteria for both are

satisfied. This corresponds to the case where eliminating any

BS causes the quality of service constraints to be violated and

moving any BS a distance larger than the mesh size threshold

will only increase the objective. The resulting network plan

size, average transmit power per BS, global iterations, and total

computation time are shown in Table II for N0 = 100, 60, 40,and 25. Global iterations represents the number of times thesite selection and site placement algorithms are sequentially

executed until the aforementioned convergence criteria are

satisfied.

TABLE II. EFFECT OF N0 ON THE SOLUTION QUALITY

N0 Network Average BS Global Total comp.size (BSs) Tx power (W) iterations time (sec)

100 17 21.32 5 471.15

60 18 16.29 3 203.63

40 20 12.44 2 172.30

25 23 9.51 1 99.80

The network plan size is the main metric for judging the

quality of the solution. It can be observed from the results

that larger N0 generates a finer solution since there are moredegrees of freedom in selecting the candidate sites, thus, more

critical site locations can be picked during the optimization.

Consequently, the solution is only slightly dependent on the

initial 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 higher

average transmit power per BS that satisfies the BS maximum

transmit power constraint. Additionally, a larger N0 requires

more 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 area

and their optimal locations. Since eliminating BSs increases

the average power per BS, we can think of the problem

differently by trading-off the number of BSs and the average

BS power. The solid lines of Figure 5 show the set of pareto-

optimal points that provide this tradeoff between the minimum

average BS power achieved and the number of deployed BSs.

The pareto sets are generated for a uniform user distribution in

Figure 5(a) and for a Gaussian user distribution in Figure 5(b).

Generating these sets is performed by running the site selection

algorithm to select a target number of BSs Nmin from aset of 100 initial BSs. The site placement algorithm is then

executed for these Nmin BSs to find their optimal locationsthat minimize total power expenditure.

In an attempt to validate our algorithm and demonstrate its

effectiveness, we compare these results with off-the-shelf hi-

erarchical clustering algorithms. Hierarchical clustering treats

each data point initially as a single cluster, and then succes-

sively merges clusters according to the linkage criteria. In

this problem, the data points are the MSs and each cluster

corresponds to a single BS serving a set of MSs. In complete

linkage hierarchical clustering, also called furthest neighbor,


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0 5 10 15 20 25 30 352

3

4

5

6

7

8

9

10

Iteration

A

verageBSPower

1 N

U k

=1

Pd k

,b(k)

Objective: Min variance o f BS Powers

Objective: Min sum of BS Powers

0 5 10 15 20 25 30 35 40 452

3

4

5

6

7

8

9

10

Number of Eliminated Base Stations N0 N

A

verageBSPower

1 N

U k

=1

Pd k

,b(k)

Inner objective: Min variance of BS powersInner objective: Min sum of BS powers

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

30 40 50 60 70 80 90 100

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Target number of Base Stations Nmin

AverageBSp

ower

1 N

U k

=1

Pk

,b(k)

Site Selection and Site PlacementAverage Linkage ClusteringComplete Linkage Clustering

Operating point which minimizes number

of BSs for a uniform user distribution

30 40 50 60 70 80 90 100

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Target number of Base Stations Nmin

AverageBSp

ower

1 N

U k

=1

Pk

,b(k)

Site Selection and Site PlacementAverage Linkage ClusteringComplete Linkage Clustering

Operating point which minimizes number

of BSs for a Gaussian user distribution

Fig. 5. The optimal tradeoff set between the average BS power and the number of deployed base stations is shown by the solid lines. The dotted lines providecomparison with average linkage and complete linkage clustering algorithms. Left: uniform user distribution, Right: Gaussian user distribution.

the two clusters whose merger has the smallest maximum

pairwise distance are merged. In single linkage clustering, also

called nearest neighbor, the two clusters with the smallestminimum pairwise distance are merged. Finally, in average

linkage clustering, the two clusters with the smallest average

of pairwise distances (maximum cohesion) are merged.

For the considered problem, single linkage clustering is not

a suitable approach due to its tendency to form long chains

that cannot model BS coverage. Thus, we consider the other

two linkage criteria. For a given number of clusters (i.e.,

BSs) Nmin, each of the two criteria is applied to constructthe sets of MS clusters, and the BS location for each cluster

is defined as the centroid of the set of cluster data points.

The power allocated to each user is determined by solving the

equations Gd Pd = 2 and [Gu] [ Pu ] = 2 andthe average BS power is computed. The results for uniform

and Gaussian user distributions are shown in Figure 5. For

a uniform user distribution in Figure 5(a), average linkage

and complete linkage clustering provide a solution close to

the optimal set generated from the site placement and site

selection algorithms. Intuitively, since each MS is equidistant

from its neighbors, the clusters will be almost equal in size and

their centroids approximate a uniform distribution. However,

for a Gaussian user distribution in Figure 5(b), hierarchical

clustering experiences bad performance. Complete linkage and

average linkage clustering can only generate feasible network

configurations of sizes at least 72 and 78 BSs, respectively,

in comparison to 39 BSs for our proposed algorithm. Addi-

tionally, the feasible configurations incur significantly higher

power consumption. This is explained by the fact that these

linkage criteria do not take into account the discrepancy in theSIR levels and power allocations when the user density is not

constant. Generally, these results demonstrate the effectiveness

of the site selection and site placement algorithms, particularly

for non-uniform user distributions.

While decreasing the number of BSs is desirable, it makes

the uplink power requirement higher, thus increasing the

chance of crossing the Pmaxu threshold. We demonstrate theimprovement in the uplink outage due to the combined objec-

tive minimization in Figure 6(a). The outage for each network

size is computed by solving the joint planning problem for

= 0 (single objective) and = 0.5 (weighted objective)with Pmaxu = 1 W. Results demonstrate that accounting for

uplink outage in the objective minimizes the number of userscrossing the Pmaxu threshold.

D. Concurrent Voice and Data Services

We consider generalizing the model to accommodate for

concurrent voice and data services. A given percentage of

data users in the network is selected, and these data users are

independently and randomly picked from the set of all users.

To obtain insight into the operation of the algorithm under

multiple service rates, we assume all data users require 64Kbps in the downlink and 32 Kbps in the uplink and all voiceusers require 12.2 Kbps in both directions. The requirement for

higher data rate services naturally increases the network load


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10

30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Number of Base Stations N

PercentageofOutageUsers

With Uplink Outage Minimization (=0.5)Without Uplink Outage Minimization (=0)

5 10 20 30 40 50 60 70 80 90 950

10

20

30

40

50

60

Percentage of data users in the network

Percentageofo

utageforvoice/data

users

Voice users - 36 BS plan

Data users - 36 BS plan

Voice users - 50 BS plan

Data users - 50 BS plan

Data users

Voice users

Fig. 6. Left: Percentage of users in outage vs. number of base stations with and without the second component of (10) for Pmaxu = 1 W and a voice-onlyservice, Right: Percentage of users in outage versus the percentage of data users for two network case studies: (a) Nmin = 36 base stations, (b) Nmin = 50base stations.

and makes the QoS requirements more stringent. In Figure

6(b), we quantify this increase by considering the percentage

of voice/data users in outage versus the percentage of data

users in the network for two scenarios. The first scenario

corresponds to the optimal plan for a uniform user distribution

with 36 BSs. Since this plan is optimal in terms of power

consumption for a voice-only deployment, we expect that

adding data users will adversely impact the performance of

the network. When the network is dominated by voice users,

the data users experience 19% outage and the voice users

experience 3% outage. The rate of increase of outage as the

percentage of data users increase is relatively fast for both

traffic classes. Since this is an extreme case, we notice that

we can operate the network with a larger number of BSs

to provide more robustness to the network under concurrentvoice/data traffic. Thus, in the second scenario, we relax

Nmin to 50 BSs and we run the site placement algorithm tooptimize the site locations subject to the uniform voice/data

user distribution. It can be seen that outage drops significantly

and QoS requirements become less stringent to satisfy.

Finally, for the downlink, data traffic produces an increase

in the BS power so that all data users can also satisfy their QoS

requirements. Thus, the limiting factor becomes the total BS

power. Considering Pdmax = 30W in the simulations, we findthat a maximum of 15.5% of data users can be supported in the

first scenario whereas a maximum of 20% can be supported

in the second scenario. This analysis captures the interplay

between the traffic classes, the quality of service requirements,

and the size of the network plan.

V I. RADIO NETWORK PLANNING WITH LOCATION

CONSTRAINTS

In realistic radio network planning scenarios, operators often

do not have the liberty of placing BSs in the entire area of

the network for two main reasons. First, the presence of rough

terrains, private property such as universities, or city zoning

restrictions in the network region limits the scope of placing

BS sites. Second, the presence of radiation-sensitive zones

such as hospitals or schools places limitations on EM radiation

in such areas of the network. These limitations are referred

to in this paper as location constraints. The EM exposure

limitations are defined by specifying maximum permissible

exposure (MPE) levels for different frequency ranges [17],

[18]. In general, for frequencies lower than 300 MHz, exposure

limits are specified in terms of electric field strength (V/m) and

magnetic field strength (A/m), while for frequencies higher

than 300 MHz, exposure limits are specified in terms of power

density S (mW/cm2). Since UMTS operates at 1800 MHz,we will define our constraints in terms of power density. The

power density decays quadratically with the distance from the

BS in a free space environment. In general, measured data

shows that average power densities are generally in the 0.001 -

0.01 W/cm2 range. Health recommendations suggest that the

median exposure in urban areas be limited to 0.005 W/cm2

and that 95% of the urban population be exposed to less than

0.1 W/cm2 [17], [18].

A. Problem Formulation

In this section, we extend the formulated radio network

planning problem in Section III-A to include constraints on:

(1) Locations of deployed BSs in a target BS-free area, and

(2) peak power density in a target radiation-sensitive zone. We

model the BS-free region as a rectangular area which can be

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

set of sample points. Then, we calculate an estimate of the

power received at each of the points from each BS using the

propagation model. The total power received at each point is

calculated by summing the power received from all BSs. Based

on the received 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, all the


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11

area will satisfy the constraint. The constraint is formulated

as follows:

maxsj

Ni=1

kCi

Pdk,i

1

keq

di,sj

4f2

c2G

Sth

j = 1, , p (26)

where sj is the sample point j in the target area with j between1 and p and p is the number of sample points taken in thetarget area, f is the carrier frequency (assumed 1800 MHz forUMTS), c is the speed of light, G is the gain of the transmittingantenna of the BS in the direction of the radiation, Sth isthe recommended power density threshold, and keq and areparameters depending on the pathloss model.

The site placement problem with location constraints can

then be formulated as follows:

minx,y

U

k=1Pdk,b(k) +

U

k=1(Puk P

umax)

+ (27)

subject to (25), (26),

(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)

Similarly, the site selection problem with location con-

straints can also be formulated as an extension to the initial

site selection problem as follows:

minc

N0i=1

ci (28)

s.t.

min

U

k=1

cb(k) Pdk,b(k) +

U

k=1

cb(k)(Puk P

umax)

+ (29)

subject to (25), (26),

(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)

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

In both problems, (25) and (26) represent the location

constraints. Note that the constraints (11)-(20) from the initial

problems are also included in the formulation. The next section

describes how to extend the algorithms developed in Section

IV to solve the radio network planning problem with location

constraints.

B. Algorithm for the Radio Network Planning Problem withLocation Constraints

The location constraints as defined in (25) and (26) are

nonlinear inequality constraints. To satisfy such constraints,

we modify the algorithms presented in Section IV. A robust

extension of pattern search algorithms for general constraints

is the globally convergent ALPS which solves general prob-

lems.

Initially, the problem is modified to convert the inequality

constraints into equality constraints by introducing nonneg-

ative slack variables. Next, we attempt to find Lagrange

multiplier estimates for the equality constraints updated at each

iteration, such that the estimates do not involve information

about derivatives of the objective or constraints to be consistent

with the derivative-free nature of pattern search algorithms.

The algorithm begins with an initial value for the penalty

parameter and the Lagrange multiplier estimates. The mathe-

matical significance of these parameters is explained in [24]. A

subproblem is formulated by combining the objective function

and the nonlinear constraint function using the Lagrangian

multiplier estimates and the penalty parameters. A sequenceof such optimization problems are approximately minimized

using a pattern search algorithm such that the linear constraints

and bounds are satisfied. When the subproblem is minimized

to a required accuracy and satisfies feasibility conditions, the

Lagrangian multiplier estimates are updated. Otherwise, the

penalty parameter is increased. These steps are repeated until

the stopping criteria are met. A frequently used Lagrangian

update is the first order Hestenes-Powell multiplier update for

the augmented Lagrangian which assumes no knowledge of

derivative information [24].

Convergence analysis for the ALPS algorithm can be found

in [25], [24]. It is shown that despite the absence of any

explicit estimation of any derivatives, the pattern search

augmented Lagrangian approach exhibits first-order global

convergence properties. Although the subproblems are solved

approximately, and the stopping criterion of the subproblem is

based on the magnitude of a measure of first-order stationarity,

the algorithm converges to Karush-Kuhn-Tucker points of the

original problem. This important result establishes that pro-

ceeding by successive, inexact minimization of the augmented

Lagrangian via pattern search methods ensures convergence

[25], [24].

Since our nonlinear constraints are written in terms of

the powers allocated to users, we need to solve the set of

equations

Gd

Pd

=

2

and [Gu

] [ Pu

] =

2

for a given x (BS locations) to find the uplink and downlinkpowers, compute the value of the constraints, and solve the

inner augmented Lagrangian subproblem. Algorithm 1 and

Algorithm 2 can be extended to ALPS by constructing the

subproblem at each iteration, minimizing this subproblem over

the mesh points instead of minimizing the objective fm(x),updating the Lagrange multipliers at each successful iteration

using the Hestenes-Powell multiplier update, and adjusting the

penalty factor based on the success or failure of the polls.

C. Analysis of Network Topologies with Location Constraints

In order to demonstrate the performance of the proposed

algorithm to solve the network planning problem with loca-

tion constraints, we run the modified site selection and site

placement algorithms presented in Section VI-B consecutively

to select the smallest set of BSs that cover the network area

such that the 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 W/cm2. For computing the power density, we

assume G = 1 and f =1800 MHz. All other parameters are thesame as those listed in Table I. Figure 7 shows a particular case

with a uniform user distribution where the radiation-limited

area is a square with 4 Km x 4 Km dimensions defined as


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12

0 2 Km 4 Km 6 Km 8 Km 10 Km0

2 Km

4 Km

6 Km

8 Km

10 Km

Power Density (log10(W/m2))

11.5

11

10.5

10

9.5

9

8.5

8

7.5

7

3 Km 4 Km 5 Km 6 Km 7 Km3 Km

4 Km

5 Km

6 Km

7 Km

Power Density (log10(W/m2))

11

10.5

10

9.5

9

Fig. 7. A uniform user distribution with location constraints in a 4 Km x 4 Km area; Left: total area, Right: radiation-limited area. The color scale denotesthe base-10 logarithm of the power density, e.g., -9 corresponds to S = 109 W/m2.

0 500 1000 1500 2000 2500 3000 3500 400017

18

19

20

21

22

23

24

25

26

Radiation-limited area dimensions (m)

AverageBSpower

1 N

U k

=1

Pd k

,b(k)

Without Maximum Power Constraint

With Maximum Power Constraint

0 500 1000 1500 2000 2500 3000

14

16

18

20

22

24

26

Radiation-limited area dimensions (m)

AverageBSpower

1 N

k=1

Pd k,b(k)

Without Maximum Power Constraint

With Maximum Power Constraint

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.

follows: 3000 m x 7000 m, 3000 m y 7000 m.The figure shows that the peak power density in the area is

S = 108.5W/m2 = 3.16109W/m2 = 0.003 W/m2

which is below the predefined threshold Sth. Additionally, itis clear that all BSs satisfy as well the location constraints.

Finally, we generate pareto-optimal sets to show the tradeoff

between the dimensions of the radiation-limited area and the

achieved average BS power in the whole network. The larger

the area, the more power is required to achieve coverage to

users in the area and satisfy their SIR requirements, which

eventually increases the average BS power. Because the mod-

ified site selection and site placement algorithms are bothexecuted, the case with zero area dimensions is equivalent to

the point of operation on Figures 5(a) and 5(b) with the lowest

number of BSs and highest average BS power. Figure 8 shows

the achievable average BS powers for sample uniform and

Gaussian user distributions. For the uniform case, the square

is characterized by its center at [5000, 5000]. For the Gaussiancase, the square is characterized by: 1) yL,max touches theupper boundary, 2) The centroids of xL,min and xL,max lie atx = 5000.

We can see that with a Gaussian user distribution the slope

of increase is faster, which is why we cannot reach the 4 Km

dimension as in the uniform case; otherwise, the increase in BS

powers would prohibit satisfying the SIR requirements. The

algorithm is executed with and without a maximum BS power

constraint. The maximum BS power constraint corresponds

to (19) with Pdmax = 30 W. Such constraint will furtherlimit the area dimensions, because even though the average

BS power is lower than the constraint, some BSs (specifically

those covering the radiation-limited area) will require a very

high power to achieve the required SIR for most users. The

area dimensions in the unconstrained power case are limited

by interference whereby further increasing BS power for

some user makes the SIR requirement for other users non-

achievable. On the other hand, the area dimensions in theconstrained power case are limited by the physical maximum

BS power limit which turns out to be a more stringent

limitation. Obviously, these two sets represent specific case

studies for the area locations we considered. In fact, the slope

of the curve depends greatly on the location of the square,

especially in the Gaussian distribution case.

Another important observation is that placing a maximum

BS power constraint decreases the average BS power for some

fixed area dimensions. This is explained by the fact that when

BSs are not allowed to exceed a certain peak transmit power,

the site selection algorithm will be forced to select a larger

number of BSs to cover the entire network, specifically to


13/13

13

cover users inside the constrained area. Looking back at Figure

5, we recall that the average BS power decreases as the number

of BSs increases for a given user distribution which explains

the gap between the two curves in each of Figure 8(a) and

Figure 8(b).

VII. CONCLUSION

We presented optimization-based formulations for the prob-

lems of joint uplink/downlink site placement and site selection

in cellular networks. The formulations use an SIR-based power

control mechanism with outage conditions to provide quality

guarantees. We proposed algorithms to solve the continuous

component of the problem using derivative-free optimization

techniques with general constraints. We also developed an

optimization algorithm for solving the integer component

of the problem based on a nested approach with an outer

problem that minimizes the number of base stations and an

inner problem that minimizes a cost function of the network

deployment. Case studies were presented and analyzed for

uniform and non-uniform user distributions and pareto-optimalsets were generated to tradeoff network configuration param-

eters. Finally, the formulation and solution were extended

to provide a framework for solving general radio network

planning problems with location constraints.

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