Umts

Math Meth Oper Res (2006) 63: 1–29DOI 10.1007/s00186-005-0002-z

ORIGINAL ARTI CLE

Andreas Eisenblatter · Hans-Florian GeerdesThorsten Koch · Alexander MartinRoland Wessaly

UMTS radio network evaluationand optimization beyond snapshots

Received: December 2004 / Revised version: May 2005 / Published online: 16 December 2005© Springer-Verlag 2005

Abstract A new evaluation scheme for universal mobile telecommunications sys-tem (UMTS) radio networks is introduced. The approach takes the complex cou-pling of coverage and capacity through interference into account. Cell load esti-mates, otherwise obtained through Monte-Carlo simulation, can now be approx-imated without time-consuming iterative simulations on user snapshots. The twocornerstones are the generalization of interference coupling matrices from usersnapshots to average load and the emulation of load control by an analytical scal-ing scheme. Building on the new evaluation scheme, two novel radio networkoptimization algorithms are presented: an efficient local search procedure and amixed integer program that aims at designing the coupling matrix. Computationalexperiments for optimizing antenna tilts show that our new approaches outperformtraditional snapshot models.

Keywords UMTS radio interface · Network design · Network planning

1 Introduction

The universal mobile telecommunications system (UMTS) is a 3rd generation (3G)cellular system for mobile telecommunications. UMTS supports all services of the

A. Eisenblatter · H.-F. Geerdes · T. Koch · R. WessalyZuse Institute Berlin (ZIB),Takustrasse 7, D-14195 Berlin-Dahlem, GermanyE-mails: [email protected]; [email protected]

A. Eisenblatter (B) · R. Wessalyatesio GmbH, Sophie-Taeuber-Arp-Weg 27, D-12205, Berlin, GermanyE-mails: [email protected]; [email protected]

A. MartinDarmstadt University of Technology, Department of Mathematics, Schlossgartenstr. 7,D-64289 Darmstadt, GermanyE-mail: [email protected]

2 A. Eisenblatter et al.

worldwide predominant GSM and GPRS networks and is more powerful, moreflexible, and more radio spectrum efficient than its predecessors.

Universal mobile telecommunications system (UMTS) is a wideband code divi-sion multiple access (WCDMA) system. Radio transmissions are generally not sep-arated in the time or the frequency domain. Complex coding schemes are insteadused to distinguish different radio transmissions at a receiver. The capability todecode the desired carrier signal, however, requires that this is not “too deeply”buried among interfering radio signals. Technically speaking, the carrier signal tointerference ratio shall not drop below some threshold value. Interference, thus,needs to be carefully controlled during network planning and operation, becauseit is a limiting factor for network capacity.

1.1 Overview

This paper is organized as follows. The rest of this section introduces the task ofUMTS radio network optimization and addresses related work. Section 2 intro-duces the technical background in parallel with the notation used throughout thepaper. Section 3 introduces a method to perform fast approximate analysis of net-work capacity. This depends on systems of linear equations describing the interfer-ence coupling among cells. In section 4, we outline the skeleton of popular mixedinteger programming models for network optimization based on traffic snapshots(Eisenblatter et al. 2002, 2003a). We discuss the difficulties inherent to modelsof this type. A novel, alternative mathematical program, which overcomes somedrawbacks of the previous model, is presented in section 5. Section 6 containscomputational results for realistic planning scenarios. These results are obtainedby implementations based on the two mathematical models and from a local searchprocedure. We draw conclusions in section 7.

1.2 Network planning

Telecommunication operators are currently deploying UMTS across the world; byJanuary 2005, more than 60 3G networks based on WCDMA technology have beencommercially launched in 29 countries in Europe, Asia and the US. GSM alreadyoffers extensive coverage, high reliability, and acceptable prices. An incentive forthe user to change technology towards UMTS is the availability of more servicesand higher communication data-rates in large areas with competitive pricing. Dueto the change of the radio access technology from frequency and time multiplexingto code multiplexing even more complex planning tools than those for GSM areindispensable.

As in the case of frequency planning for GSM networks, it is to be expectedthat the quality of the planning results can be largely improved if the radio networkplanner is assisted by automatic optimization functions. First commercial productsin this domain are available, but it will certainly take a few more years to reach thematurity of GSM frequency planning. The paper contributes to this developmentwith respect to network evaluation and network optimization.

A central part in the initial deployment and the subsequent expansions of aUMTS radio network is to decide about the location and configuration of the base

UMTS radio network evaluation and optimization 3

stations including their antennas. Among others, the type of an antenna (and thusits radiation pattern), the mounting height, and its main radiation direction, theazimuth in the horizontal plane and the tilt in the vertical plane, have to be decided.This may look like a facility location problem at first sight, but it is not! The cov-erage area and the capacity of a base station may shrink significantly due to radiointerference.

1.3 Tilt optimization

All three optimization approaches proposed in this paper can be used to optimizeall aspects of antenna installation mentioned above. From the practical point ofview, however, tilt optimization is of primary interest. Operators often deploy aUMTS network in addition to 2G infrastructure. Since it is costly and increasinglydifficult to acquire new sites, UMTS antennas are often installed in places wherethere is already a GSM antenna. The two types of antennas are sometimes installedon the same mechanical structure, dual band antennas that operate for both GSMand UMTS are also in use. This basically fixes the antenna’s height and azimuth tothat of the GSM installation. In addition, operators typically have standard antennatypes. The one parameter that operators favor for optimization is tilt. We focus onoptimizing this parameter in our computational studies in section 6.

Antenna tilt comes in two flavors, mechanical and electrical tilt, which can bevaried independently. The mechanical tilt is the angle by which the antenna is tiltedout of the horizontal plane. For many UMTS antennas, the main radiation directionin the vertical plane can also be changed by electrical means. The resulting changeis called electrical tilt.

Tilt optimization alone has a considerable leverage on network coverage aswell as capacity, because it influences cell ranges and interference. If an antenna istilted down, the size of the coverage area and the covered traffic typically decrease.The amount of interference in neighboring cells is thereby often reduced.

1.4 Related work

This work is largely based on the authors’ participation in the EU-funded projectMomentum. The interdisciplinary project had mathematicians and engineers fromacademia as well as telecommunication operators in the consortium. The scope wasdeveloping models and algorithms for simulation and automatic planning of UMTSradio networks. The first version of the snapshot model presented in section 4 isformulated in Eisenblatter et al. (2002), refinements and detailed technical descrip-tions of the model parameters is given in Eisenblatter et al. (2003a); a summaryand first computational results are given in Eisenblatter et al. (2003b). Due to thepractical relevance of our topic we mention both works from the mathematical aswell as from the engineering side. In Whitaker and Hurley (2003) a survey chartingthe evolution of centralized planning for cellular systems is given.

There is a variety of mathematical work on UMTS radio network planning.Optimization models similar to our snapshot model are suggested in Amaldi et al.(2002, 2003a,b) together with computational results. The problems include siteselection and base station configuration. Heuristic methods such as tabu or greedy


search are used to solve instances. Integer programming methods for 3G networkplanning are presented in Mathar and Schmeink (2001). Some papers deal withsubproblems: pilot power optimization under coverage constraints using mathe-matical programming is performed in Siomina and Yuan (2004) and Varbrandtand Yaun (2003), power control and capacity issues with particular emphasis onnetwork planning are treated in Catrein et al. (2004) and Leibnitz (2003). To ourknowledge, however, there is no work focusing on the network tuning aspect undertechnical constraints; tilt optimization falls into this category.

There are also numerous publications by engineers on problems in networkplanning. The first and exemplary landmark monograph covering many technicalaspects of UMTS networks and some practice-driven optimization and tuning rulesis Laiho et al. (2001). Optimization of network quality aspects using meta-heuris-tics, but not involving site selection is treated in Gerdenitsch et al. (2002). Networkdesign with a special emphasis on optimizing cell geometry using evolutionaryalgorithms is presented in Jamaa et al. (2003) and Jedidi et al. (2004). A mixedinteger program for optimizing quality-of-service for users within a snapshot isdeveloped in Turke et al. (2003a).

Dimension reduction, as used in Turke et al. (2003a), had earlier been pro-posed for mono-service traffic in Mendo and Hernando (2001), and was extendedto a service mix in Catrein et al. (2004) and Catrein and Mathar (2003). Besidesa generalization to average user density maps, we contribute the concept of loadcontrol by scaling and an optimization model based on these premises. Earlierpapers addressing power control by considering systems of linear equations withpositive solutions are Grandhi et al. (1993, 1997) and Hanly (1995).

2 Radio network design for WCDMA

We give an introduction to the properties of UMTS technology insofar as they arerelevant to our paper. In mobile telecommunications, two transmission directionsare distinguished: the uplink (UL) and the downlink (DL). An UL transmissiontakes place from a mobile to a base station. A DL transmission takes place froma base station to a mobile. The type of radio network we consider with UMTS iscalled a cellular network: there is an infrastructure of base stations. One or moreantennas are installed at each base station. Each antenna serves users in a certainarea (typically the area where this antenna provides the strongest signal), this areais called a cell.

In a UMTS network, every antenna emits a pilot signal of fixed power, whichis used by the mobiles to measure which antenna is received best and for decidingwhich cell they try to register with. In this paper, we consider up- and downlinktransmissions on dedicated channels, which can be seen as point-to-point connec-tions between a mobile and a base station. There are some other channels, whichare shared among the mobiles in a cell. Apart from the pilot channel, we do notmodel them explicitly. In UMTS networks rolled out today, transmissions in uplinkand downlink do not interfere, because separate frequency bands are reserved foreach direction (frequency domain duplex, FDD).

Examples for mobile communication services are speech telephony or text mes-saging. Some of the additional services that will be offered on the basis of UMTSare video telephony, downlink data streaming or high-speed web access. This


Table 1 Notation

Symbol Domain Description

N Set of antenna installations (cells) in the networkI I ⊃ N Set of all possible installations for the networki i ∈ N , i ∈ I InstallationM Set of mobilesMi Mobiles served by antenna installation im m ∈ M MobileS Set of servicess s ∈ S ServiceA Planning areap p ∈ A Location in the planning areaAi Ai ⊂ A Cell area (best-server area) of installation iTs A → R+ Average spatial traffic distribution of service s

p↑m ∈ R+ Uplink transmit power from mobile m ∈ M↑

p↓im ∈ R+ Downlink transmit power from installation i to mobile m

p↓i ∈ R+ (Downlink) pilot transmit power from installation i

p↓i ∈ R+ (Downlink) common channels transmit power from installation i

p↓i ∈ R+ Total transmit power of installation i

p↑i ∈ R+ Total received power at installation i

�max↓i ∈ R+ Maximum total transmit power for installation i

γ↑mi [0, 1] Uplink attenuation factor between mobile m and installation i

γ↓im [0, 1] Downlink attenuation factor between installation i and mobile m

ηi, ηm ≥ 0 Noise at installation i/mobile m

α↑m, α

↓m [0, 1] Uplink/downlink activity factor of mobile m

ωm [0, 1] Orthogonality factor for mobile m

µ↑m, µ

↓m ≥ 0 Uplink/downlink CIR target for mobile m

µ↓m ≥ 0 Pilot Ec/I0 requirement for mobile m

diversification implies an increasing heterogeneity of the traffic in the network. Animportant difference between services is the requested data rate, especially as someof the new services require a substantially higher data rate than speech telephony.

2.1 Preliminaries and notation

For a complete account of the notation used in this paper please refer to Table 1.We consider a UMTS radio network with a set N of antennas (or cells) and a setM of mobile users in a traffic snapshot. We assume that each mobile is connectedto exactly one antenna, namely, the one which has the strongest pilot signal. Thepractical situation is more complicated, since a mobile can be linked to more thanone antenna at a time. We ignore this feature of UMTS called soft-handover forour presentation.

We denote a vector with components vj in bold, v. The notation diag (v) standsfor a diagonal matrix of matching dimension with the elements of v on the maindiagonal.

2.1.1 Antenna installations

The degrees of freedom when configuring an antenna (type, height, azimuth, tilt)and thereby determining the properties of the related cell have been mentioned


above. For the antenna of a certain cell, we call a fixing of the parameter values inthese dimensions an installation. The set of all considered installations is denotedby I. An actual network or network design is a selection of installations with theproperty that exactly one of the potential installations for each cell is chosen. Inparticular, N ⊂ I has to hold. We use the indices i and j for elements of both Nand I.

2.1.2 CIR inequality for a transmission

For a signal to be successfully decoded at the receiver with WCDMA technology,the ratio of the received strength of the desired signal to all interfering signals –including background noise exterior to the system – must reach a specific threshold.This ratio is called carrier-to-interference ratio (CIR), the threshold is called CIRtarget. For a transmission to take place, the following inequality must be satisfied:

Strength of desired signal

Noise +∑Strength of interfering signals

≥ CIR target (1)

Code multiplexing technology allows the right-hand side of this inequality to besmaller than one, i.e., the desired signal strength may be weaker than the interfer-ence. We next introduce the parameters that play a role in (1).

2.1.3 CIR targets and activity

The CIR target is service specific. In order to successfully decode a higher data ratea higher CIR target must be met. There is also a CIR target for the pilot channel.We denote the CIR targets for a user m ∈ M for uplink, downlink, and pilot byµ

↑m, µ

↓m, and µ

↓m, respectively.

The way that users access a service over time varies: in a speech conversation,each of the two users involved speaks roughly 50% of the time on average. No datais transmitted in silence periods (discontinuous transmissions). This is taken intoaccount in the form of activity factors. For each user m ∈ M we have two activityfactors, α

↑m for the uplink and α

↓m for the downlink. The activity factor can be used

to compute the average power that is sent in order to support a link. There areservices that cause traffic in only one direction, e. g., downlink data streaming. Thecorresponding activity factor for the other direction is assumed to be zero. Notethat there is always some control traffic, but this is negligible. This averaged valueis used for all interfering signals, since whether the interfering links are currentlyin an active period is not modeled at our level of detail. For the link in question,however, the CIR target needs to be met during active periods only. There is notransmission in inactive periods. The activity is thus only taken into account in thedenominator of (1), but not in the numerator.

2.1.4 Attenuation

The strength of a signal transmitted over a radio channel is attenuated. The receivedpower depends linearly on the output power at the transmitter. The attenuation onthe radio channel (excluding transmission and reception equipment) is called path


loss. Predicting the path loss in real-world settings is difficult and many methodsranging from rule-of-thumb formulas to sophisticated ray tracing methods using3D building data and vegetation are available. We refer the interested reader toBertoni (2000), Geng and Wiesbeck (1998) and Kurner (1999). Besides this losson the radio channel, further losses and gains due to the cabling, hardware, anduser equipment have to be considered. All this information is then summed up intotwo attenuation factors for each pair i ∈ N , m ∈ M: γ

↑im for the uplink and γ

↓mi

for the downlink.

2.1.5 Complete CIR constraints

We denote the uplink transmission power of a mobile m ∈ M by p↑m. The received

signal strength at installation i ∈ N is then γ↑mi p

↑m. Using ηi for the received back-

ground noise at installation i ∈ N , the uplink version of (1) for the transmissionfrom m to i with µ

↑m as CIR target reads

γ↑mip

↑m

ηi +∑n�=m γ

↑ni α

↑np

↑n

≥ µ↑m (2)

Writing

p↑i := ηi +

∑

m∈Mγ

↑mi α

↑m p↑

m (3)

for the average total received power at installation i ∈ N , this simplifies to

γ↑mi p

↑m

p↑i − γ

↑mi α

↑m p

↑m

≥ µ↑m (4)

In the downlink, the situation is more complicated. First of all, we have to takethe pilot and common channels into account, whose transmission power we denoteby p

↓i and p

↓i for installation i. Moreover, each cell selects orthogonal transmission

codes for the mobiles in its cell (we neglect the use of secondary scrambling codes).In theory, transmissions with orthogonal codes do not mutually interfere. Due topropagation phenomena, however, the signals partly lose this property. Signalsfrom other antennas do not have it at all. Hence, the interference from the same cellis reduced by an environment dependent orthogonality factor ωm ∈ [0, 1], withωm = 0 meaning perfect orthogonality and ωm = 1 no orthogonality. We definethe total average output power of installation i as

p↓i :=

∑

m∈Mi

α↓m p

↓im + p

↓i + p

↓i (5)

Writing ηm for the noise value and µ↓m for the downlink CIR target at mobile m,

we obtain the downlink version of (1) related to transmission from i to m:

γ↓im p

↓im

γ↓im ωim

(p

↓i − α

↓m p

↓im

)+∑

j �=i γ↓jm p

↓j + ηm

≥ µ↓m (6)


Note on downlink orthogonality. Strictly speaking, there is one control channel forwhich orthogonality to other channels in the cell does not apply, the synchroniza-tion channel (SYNC). We do not consider this detail for ease of presentation. Thisis acceptable from the engineering point of view, because its transmission poweris fairly low in comparison to the other common channels.

For the pilot, the situation is simpler. Technically speaking, the received chipenergy of the pilot signal – called the pilot channel’s Ec or CPICH Ec – relative tothe total power spectral density I0 has to lie above a threshold µ

↓m. When computing

the spectral density I0 as the denominator in the pilot version of (1), no benefit dueto orthogonality applies and even the pilot’s own contribution appears as interferer:

γ↓imp

↓i

ηm +∑i∈N γ

↓im p

↓i

≥ µ↓m (7)

The left-hand fraction is called the pilot channel’s Ec/I0 (CPICH Ec/I0).

2.1.6 Power control

The attenuation over a radio channel can actually not be pinned to a constant factorγ . It varies strongly over time due to a collection of dynamic phenomena groupedunder the terms shadowing and fading. In UMTS, fast fading is largely made upfor by power control. The receiver measures the signal strength in very short timeintervals and notifies the sender of any changes, which then adjusts the transmis-sion power in order to meet the CIR requirements (4) and (6). We do not take thismechanism into account, but assume perfect power control. Hence, we supposethat the transmitter instantaneously adjusts its power exactly to the level requiredto meet the CIR target.

2.2 Performance metrics for UMTS radio networks

The “quality” or “performance” of a UMTS radio network is measured by vari-ous scales (Holma and Toskala 2001; Laiho et al. 2001, 2002). There is no singleobjective for optimizing a radio network, but a variety of factors needs to be takeninto account.

2.2.1 Coverage

The user associates to the network based on the pilot channel. The quality of thepilot signal therefore determines the network’s coverage. The first condition forcoverage is that the pilot’s absolute signal strength Ec, the numerator in (7), issufficiently large to reliably detect it. The areas in which this is the case are said tohave (pilot) Ec coverage.

At locations with sufficient Ec coverage, the pilot signal can only be decodedif the pilot’s Ec/I0 is sufficient, that is, if (7) holds true; locations are said to have(pilot) Ec/I0 coverage when this is the case. Unlike Ec coverage, Ec/I0 coveragemay depend on the traffic load within the network. While Ec problems are likely tooccur at places that are too distant from any antenna, failure to reach the requiredEc/I0 ratio is often a sign of too much interference from other cells.


2.2.2 Pilot pollution

Mobiles continuously track the strength of the pilot signal not only of their owncell, but also for neighboring cells if these are received with a strength within acertain window (5 dB, say) below the own cell’s signal. This serves to allow a seam-less hand-over in case the user moves and crosses cell borders. Due to hardwareconstraints, however, the number of pilots that can be tracked simultaneously islimited. If too many pilot signals are “audible” in a certain location, the location issaid to suffer from pilot pollution.

2.2.3 Network load

Once that network coverage is given by sufficiently strong and clear pilot signals,users are capable of establishing a connection. An most important performancemeasure – and the one that is most involved to compute – is how many users thenetwork can carry. Services differ in their demand for radio resources. Whetherresources for all user requests are available is evaluated by network simulations.Monte-Carlo simulations are typically used for this analysis: User demand real-izations called traffic snapshots are drawn, and each snapshot is analyzed. Theanalysis of independent snapshots is ideally continued until network performanceindicators are statistically reliable Turke et al. (2003a).

The details of analyzing a snapshot differ. Each approach in essence has todecide which of the users are served by which cell(s), the power levels of eachactive link, and which users are out of coverage or unserved due to capacity orinterference reasons. The limits on user and base station equipment power, thevariations in CIR targets depending on service, equipment type, and speed, theeffects of various forms of soft hand-over, etc., have to be considered. As a result,the transmit and receive signal powers are determined for all cells. These powerlevels are in turn the basis for deriving Ec/I0 or service coverage maps.

The downlink load of an individual cell is measured as the fraction of the maxi-mum output power that the cell uses for transmission. The uplink load is measuredby the noise rise compared to an empty cell. The noise rise is the ratio of the totalreceived power at an antenna to the noise, which is always present. A noise rise oftwo (which means that the signals generated within the network reach the antennawith a strength equal to the noise) corresponds to 50% uplink load; the load is at100% if the noise rise reaches infinity.

In practice, the average cell load is limited to values significantly below 100%.This is to leave room for compensation of dynamic effects by the power controlmechanisms and to ensure the system’s stability. The maximum average load shouldtypically not exceed 50% in the uplink and 70% in the downlink. We denote thelimit on the total transmit power in the downlink by �max↓.

3 Computing network load

This section introduces systems of linear equations that describe the up- and down-link load per cell and the coupling among cells of a UMTS radio network. Thisdescription is idealized since no transmit power or noise rise limitations are taken


into account. For the sake of simplicity, we also assume that no mobile is in softhand-over. With the necessary precautions, solving these coupling systems never-theless allows to get an estimate of network load without Monte-Carlo simulations(see section 2.2). We use this method to quickly assess network load during localimprovement methods and at the core of the network optimization model proposedin section 5.

The coupling systems are introduced and extended to WCDMA in Catrein andMathar (2003), Catrein et al. (2004), Mendo and Hernando (2001) and Turke et al.(2003b) as a means to speed up a central operation in Monte-Carlo simulations.The original derivation for the equation systems is based on user traffic in the formof traffic snapshots. We follow this approach to introduce the systems. We thenextend it to derive the equation systems based on average traffic load distributionor to mixtures of average load and a traffic snapshot.

If the traffic demand is too high to be serviced by the network, the solutions tothe equation systems lose their meaning. The radio resource management (RRM)of the network applies call admission control, call congestion control, and packetscheduling to prevent such overloads. We introduce load scaling schemes for theup- and the downlink equation systems that mimic this network behavior.

3.1 Load coupling matrices

The central assumptions are that all users are served and that all CIR targets in theuplink (4) and downlink (6) are met at equality. We start from a network designwith installations i ∈ N and a traffic snapshot with mobiles m ∈ M, using thenotation from Table 1.

3.1.1 Uplink

Concerning the uplink at antenna installation i, recall that p↑i is the total amount of

received power including thermal and other noise. Under the above assumptionswe use elementary transformation of the equality version of (4) to derive two quan-tities for every mobile m served by installation i: First, the transmission power p

↑m

of mobile m given the total received power p↑i at the serving installation. Second,

the fraction of the total received power at the installation i originating in mobile m.

p↑m = 1

γ↑mi

µ↑m

1 + α↑mµ

↑m

p↑i (8)

α↑m γ

↑mi p

↑m

p↑i

= α↑m µ

↑m

1 + α↑m µ

↑m

(9)

We define the uplink user load l↑m of a mobile m as the right-hand side of (9).

l↑m := α↑m µ

↑m

1 + α↑m µ

↑m

(10)


We break down the contributions to the total received power p↑i at installation i

in dependence of all uplink connections (not just those served by i). Let Mi ⊆ Mdenote the set of users served by installation i. Then (3) reads as

p↑i =

∑

m∈Mi

γ↑mi α

↑m p↑

m +∑

j �=i

∑

m∈Mj

γ↑mj α↑

m p↑m + ηi (11)

Defining the installation uplink coupling factors C↑ii and C

↑ij (where i �= j ) as

C↑ii :=

∑

m∈Mi

l↑m and C↑ij :=

∑

m∈Mj

γ↑mi

γ↑mj

l↑m (12)

and substituting (8), the uplink transmission powers can be expressed as

p↑i = C

↑ii p

↑i +

∑

i �=j

C↑ij p

↑j + ηi (13)

Thus, the total power received at the installation is composed of three contribu-tions, those from the own cell, those from other cells, and noise. The quantity C

↑ii

measures the contribution from the own users, and C↑ij scales the contribution from

installation i. The matrix

C↑ := (C

↑ij

)1≤i,j≤|N | (14)

is called the uplink cell load coupling matrix.Collecting (13) for all installations and writing η↑ for the vector of noise val-

ues, we obtain the announced system of linear equations governing the uplink cellreception powers:

p↑ = C↑p↑ + η↑ (15)

Under the assumptions stated in the beginning of this section, the solution of (15)is the received powers at each installation. Necessary and sufficient conditions onC↑ for the existence of positive and bounded solutions to (15) are given in Catreinet al. (2004); Catrein and Mathar (2003).

3.1.2 Downlink

In the downlink case, we basically repeat what has just been done for the uplink.The starting point is the CIR constraint (6). Assuming that the constraint is tightfor all mobiles it can be rewritten as

1 + ωm α↓m µ

↓m

α↓m µ

↓m

α↓m p

↓im = ωm p

↓i +

∑

j �=i

γ↓jm

γ↓im

p↓j + ηm

γ↓im

(16)

We define the downlink user load of serving mobile m as:

l↓m := α↓m µ

↓m

1 + ωm α↓m µ

↓m

(17)


Similar to the uplink case, further notation is helpful to express the depen-dency of the power p

↓i on the downlink transmission power at all installations. We

introduce the downlink coupling factors

C↓ii :=

∑

m∈Mi

ωm l↓m and C↓ij :=

∑

m∈Mi

γ↓jm

γ↓im

l↓m (j �= i) (18)

for installations i and j as well as an installation’s traffic noise power

p(η)

i :=∑

m∈Mi

ηm

γ↓im

l↓m (19)

The transmit powers at the installations satisfy the expression

p↓i = C

↓ii p

↓i +

∑

j �=i

C↓ij p

↓j + p

(η)

i + p↓i + p

↓i (20)

The first term C↓ii captures the effects due to intra-cell interference. The sec-

ond term C↓ij quantifies the fraction of transmission power spent on overcoming

interference from other installations. The third quantity p(η)

i states how much trans-mission power is needed to overcome the noise at the receivers if no intra-systeminterference were present. We define the downlink load coupling matrix as

C↓ := (C

↓ij

)1≤i,j≤|N | (21)

Figure 1 shows a graphical representation of the downlink coupling matrix. Theconnection between pairs of antennas is shaded according to the absolute value of

Fig. 1 Effect of tilt adjusting on mutual interference, Berlin scenario. Connections between cellsindicate the absolute value of the corresponding elements of the coupling matrix (a) Originalnetwork (b) Optimized electrical tilts. Mutual interference coupling is reduced by adjusting tilts


the sum of the two corresponding off-diagonal elements, with darker shades indi-cating higher values. The difference between Figure 1a (fixed tilts) and b (adjustedelectrical tilts) illustrates decoupling effect achieved by tilting.

With the same qualifications as in the uplink case, equation (20) for all antennainstallations in the network form a linear equation system that describes the down-link transmit power in each cell:

p↓ = C↓p↓ + p(η) + p↓ + p↓ (22)

3.1.3 Generalized coupling matrices

We have to assess network load as part of network optimization in order to fathomwhether a tentative network design is capable of supporting the offered load. Inthis context, the above equation systems have two shortcomings:

– The results from the load calculation become meaningless once the network isin overload.

– The equation systems allow the determination of the network load for onesnapshot. In order to obtain statistically reliable results on the expected loaddistribution many snapshots have to be evaluated. This is computationally pro-hibitive if the load evaluation is to be used within a local search approach andhas to be executed during every step.

We address these two issues separately. An approach to target the first point isdeveloped in section 3.2. In the following, we derive the coupling matrices and thedownlink traffic noise power vector on the basis of average traffic load distribu-tions. This allows to obtain an estimate of the network load based on the analysisof only one equation system; in order to estimate the average transmit power, weform the above equation systems using the average coupling matrices.

The approach is appealing because the computational complexity is radicallydiminished. However, the speed-up comes at the expense of an estimation error.To our knowledge, there is no extensive research or upper bound for this error.It is reported (Meijerink et al. 2003) that for a mono-service situation the cellload calculation on the basis of the average load distribution leads to a systematicunderestimation of user blocking, but the corresponding error is estimated as lessthan 10%. In computational experiments based on the scenarios considered in thispaper, we have shown that the estimation error does not exceed 5% (Eisenblatteret al. 2005a). Our method combined with the scaling approach (see section 3.2)can attain a high accuracy for simplified load evaluation approaches (Eisenblatteret al. 2005b). A rigorous error analysis is, however, recommended to provide cellload estimates together with accuracy estimates.

Let A denote the total planning area and Ai ⊂ A the best-server area of cell i.Let S be the set of services under consideration, let Ts be the service-specific spatialtraffic distributions, and let Ts(p) denote the average traffic intensity of service sat location p (for some specific point in time, e. g., the busy hour). We assumethat the traffic intensity counts simultaneous calls at location p. The position pspecifies a two-dimensional coordinate at some reference height, e.g., 1.5 m, or athree-dimensional coordinate.

For ease of exposition, we assume that each service s has fixed CIR targetsµ

↑s , µ

↓s and activity factors α

↑s , α

↓s ; we also assume a location-specific noise ηp at


a mobile in position p. Instead of using γ↑mi, γ

↓im, ωm for the connection between

installation i and a mobile at location p, we simply write γ↑ip, γ

↓ip, ωp. With these

conventions the snapshot-based definitions of coupling matrices and downlinktraffic noise power can be easily generalized to traffic intensity distributions.

The basic load definitions (10) and (17) are replaced by

l↑p :=∑

s∈S

α↑s µ

↑s

1 + α↑s µ

↑s

Ts(p) , l↓p :=∑

s∈S

α↓s µ

↓s

1 + ωp α↓s µ

↓s

Ts(p) (23)

The uplink coupling matrix is given by

C↑ii :=

∫

p∈Ai

l↑p dp , C↑ij :=

∫

p∈Aj

γ↑ip

γ↑jp

l↑p dp (24)

The downlink coupling matrix and the traffic noise power are given by

C↓ii :=

∫

p∈Ai

ωp l↓p dp , C↓ij :=

∫

p∈Ai

γ↓jp

γ↓ip

l↓p dp , p(η)

i :=∫

p∈Ai

ηp

γ↑ip

l↓p dp (25)

The above definitions of coupling and traffic noise power can be generalizedfurther in two directions. First, the definitions make sense if some part of the trafficis taken according to average load distributions and another part from a snap-shot. This can be helpful for Monte-Carlo simulations: service usage well capturedby average behavior can be treated through average intensity maps, while highdata-rate users with bursty traffic can be analyzed at snapshot level. Second, thebest-server concept can be replaced by assignment probabilities, indicating theprobability with which some point is served from an installation.

3.2 Load control

In practice, a UMTS network controls its load. In the downlink, the maximumtransmit power a cell can emit is limited. In the uplink, the signal decoding at anantenna fails in case the received power is too high. With increasing load cells startto reject new users, downgrade the service level of the ones that are being served,and eventually drop users. Load control is, however, not considered in the aboveload calculation method. If too many users are present, the solutions to systems (15)and (22) may become prohibitively large or even negative. In this case, they canno longer be interpreted as received signal strength or transmit powers in a UMTSsystem.

Traditional Monte-Carlo based load analysis handles this by iteratively drop-ping users and reevaluating the load until powers stay in the feasible ranges. Thisapproach is computationally expensive, and it cannot be applied to the general-ized average coupling matrices as there is no notion of single users any longer.We propose to mimic the load reduction resulting from load control by scaling theload equation systems. For each cell, we multiply users’ load, as defined in (10)and (17), by some factor between 0 and 1. This scaling factor can be interpreted


as the percentage of load that cannot be served. Analytically, this results in scalingthe coupling matrices’ rows in (15) and their columns in (22).

As the downlink is expected to be the limiting direction in UMTS networks,we treat this case here. We give a recursion to determine sequences of load scalingfactors λ and downlink transmit power estimates p per cell. The sequences con-verge, and the resulting solution respects the maximum transmit power bounds perinstallation.

Proposition 1 (downlink load scaling) Provided that 0 < p↓i + p

↓i ≤ �

max↓i and

∑j C

↓ij > 0 for all i, then the initial settings

λ0i = 1

p0i = p

↓i + p

↓i

(26)

together with the update step

λt+1i = min

{λt

i,�

max↓i −p

↓i −p

↓i

C↓ii �

max↓i +p

(η)

i +∑j �=i C↓ij pt

j

}

pt+1i = 1

1−λt+1i C

↓ii

[p

↓i + p

↓i + λt+1

i

(p

(η)

i +∑j �=i C

↓ij p

tj

)] (27)

result in sequences with the properties:

1 = λ0i ≥ λ1

i ≥ λ2i ≥ · · · ≥ 0

p↓i + p

↓i = p0

i ≤ p1i ≤ p2

i ≤ · · · ≤ �max↓i

(28)

and

λti < 1 �⇒ pt

i = �max↓i (29)

The sequences of (λsi )s≥0 and (ps

i )s≥0 converge. Let λi and pi denote the limit-ing values, then the vector p is the solution to the system (22) with the load of eachcell i scaled down by λi , i.e.:

p = diag (λ) C↓p + diag (λ) p(η) + p↓ + p↓ (30)

As we are not aware of a reference for this result, we provide a sketch of aproof: Start by noting that the resulting system of linear equations is indeed thesystem (22) if all user loads l

↓m (or l

↓p) associated to installation i are scaled by

λi . Next, argue that the three properties hold at all t by induction on t . Finally,observe that both sequences (λs

i )s≥0 and (psi )s≥0 are component-wise monotone

and bounded, and thus converge. The update scheme for λ and p resembles theGauß-Seidel method for the iterative numerical solution of linear equation systems,cf. (Berman and Plemmons 1994, Chap. 7).

The complementarity property (29) governs the load control. The conditionimposes that cells have to serve all traffic upto the point where they would otherwiseexceed their limit. Determining load scaling factors that fulfill a similar comple-mentarity condition is more involved for the uplink. Columns instead of rows haveto be scaled. We use results from linear complementarity problem theory in orderto obtain scaling factors via solving a MIP. The details are a topic in a forthcomingpublication.


Fig. 2 Load assessment for a network design, Berlin scenario (a) Cell load p (b) Load scaling λ.Cells that reach 70% load are forced to drop users, corresponding to a load scaling value smallerthan 1

The interplay between load and scaling can be studied in Figure 2. The shadingof a cell’s area in Figure 2a corresponds to load after scaling, that is, the relatedcomponent of p. The counterpart to this, the scaling vector λ, is depicted in Fig-ure 2b. The value of �

max↓i corresponds to 70% of an antenna’s nominal maximum

output power (cf. section 2.2.3). In consequence, all cells that attain 70% load areassigned a scaling factor smaller than 1.

4 Mixed-integer programming model based on snapshots

In this section we sketch a snapshot-based MIP model for the UMTS planning prob-lem. Further details and a full account of the model can be found in Eisenblatteret al. (2002). The model resembles optimization models based on snapshots or testpoints that have been published. The problems described for this representativebasically apply to the other models as well.

Coverage and capacity requirements are modeled via users in different snap-shots. The snapshots are random samples. Only considering a large number ofthem allows conclusions with respect to the underlying distribution. For resultingnetworks to statistically perform well, many snapshots thus need to be consideredsimultaneously during optimization.

Model dimension, on the other hand, grows rapidly with the number of snap-shots. A decision is usually made for each user whether it is dropped (left withoutservice) or served, and if so, by which cell. For all served users, power levels haveto be determined for each link such that all CIR targets are met. Various objectivefunctions have been used with this type of model. Here we aim at maximizing thenumber of served users.

4.1 Outline of the Model

Binary variables zi, i ∈ I are used to indicate which installations are used for thenetwork. Exactly one potential installation is chosen for each cell. Binary variables


xmi for each pair of mobile m ∈ M and installation i ∈ I indicate whether m isserved by i. We require xmi ≤ zi to ensure that only selected installations servemobiles.

Bounded continuous variables are used for the uplink power p↑m of mobile m,

for the total received power p↑id at installation i in snapshot d, for the downlink

transmission power p↓im, and for the total downlink transmission power p

↓id of

installation i in snapshot d. The transmission powers of the pilot p↓i and the other

common channels p↓i are fixed.

If an installation serves a mobile, the required carrier-to-interference ratios forup- and downlink have to be fulfilled. To ensure this, we multiply the right-handside of the CIR inequalities in the uplink (4) and in the downlink (6) by xmi .The resulting quadratic inequalities are linearized using a big-M formulation. Weshow this exemplarily for the downlink. Defining the denominator of the modifieddownlink CIR inequality (6) as

φ(m, i) = ωim γ↓im (p

↓id − α↓

m p↓im) +

∑

j �=i

γ↓jm p

↓jd (31)

we write

γ↓im p

↓im ≥ µ↓

mφ(m, i) xmi + µ↓mηm xmi (32)

Setting big-M to

�im = ωim γ↓im �

max↓i +

∑

j �=i

γ↓jm �

max↓j + ηm (33)

a linearized version of (6) suitable for the MIP model is obtained:

γ↓im

µ↓m

p↓im − φ(m, i) − �im xmi ≥ ηm − �im (34)

A full optimization model consists mainly of constraints of this type, there is oneconstraint for each potential pair of antenna and mobile user.

Mixed integer rounding (MIR) cuts (Marchand and Wolsey 2001; Nemhauserand Wolsey 1990) can be added to the model in order to tighten the linear relaxationof the model. The following valid inequality is derived from (34):

xmi −(

1/µ↓m + ωimα↓

m

)γ↓im

ηm

p↓im ≤ 0 (35)

This is a relaxed version of a classical MIR cut, see Eisenblatter et al. (2003a,Section 4.6) for details.


4.2 Drawbacks of snapshot-based models

We have conducted extensive computational experiments (Eisenblatter et al. 2003b,2004a) with snapshot-based models using several different objectives. The goal wasto minimize network cost while maintaining acceptable quality. For the compu-tational results in this paper, we essentially fix the network cost and optimize thequality of the network. All our computational studies, including the ones presentedin section 6, reveal shortcomings of the snapshot-based model. In our experiments,snapshot-based models for problems of relevant size are not tractable as soon asmore than a few snapshots are considered. We are not aware of any report onsuccessful computational experiments on a similar model.

A serious issue is the very high dynamic range of the input data arising becauseattenuation values have to be used in linear scale. Relevant attenuation coefficientsγ vary between −60 and −160 db. They apply to output powers p

↑m of mobiles

in the range from −50 up to 21 dBm. This results in a dynamic range in receivedpowers of 171 dB; these are 17 orders of magnitude. Such dynamics are on the veryborder of precision for double precision IEEE floating point arithmetic (Goldberg1991). The problems with dynamics can only be circumvented by grossly simpli-fying interference calculation.

Another reason for poor solvability are excessive symmetries. The symmetriesare caused by relatively similar installations and by mobiles located close to eachother. Besides, the number of variables in the model grows with the number ofpotential installations times the number of mobiles. Even when applying elab-orate preprocessing to remove impossible combinations, this severely limits thescalability of the model.

To overcome the problems of poor numerics and scalability, we present a newoptimization model in the next section. This model is based on the compact eval-uation method presented in section 3 and on average load rather than snapshots.

5 Optimizing the coupling matrix

The performance properties of a particular network design on a given snapshotare to a large extent described by the coupling matrices C↑ and C↓ derived insection 3.1. The generalized average coupling matrices described in section 3.1.3provide an estimation of the network properties that is independent of single snap-shots. This leads to the idea of looking at radio network design as a problem ofdesigning a “good” average coupling matrix.

At the core of the model developed in this section resides a linear descriptionof the coupling matrix for the potential network designs. In addition, we imposecoverage constraints inspired by the performance metrics described in section 2.2.Our presentation will focus only on the downlink and on the average-load baseddescription with best-server areas as presented in section 3.1.3, an extension to theuplink direction is obvious.

Again, we describe any possible network design by an incidence vector z ∈{0, 1}|I|. Exactly one installation for each cell has to be chosen. We call the set ofnetwork designs that fulfill this condition F .


5.1 Determining the entries of the coupling matrix

Recall the entries of the average coupling matrix as given in (25). All entries in rowi of the coupling matrix are computed by integrating over the area Ai served bycell i. For determining the elements of the coupling matrix, it is therefore crucialto determine these best-server areas.

5.1.1 Server of single points

Consider a point p in the planning area. We introduce a decision variable c(p)

i thatexpresses whether p is served by installation i. This is the case if and only if (a)installation i is selected and (b) no installation with a stronger pilot signal at p isselected. In the example in Figure 3, p is served by i if and only if installation j(and not k) is selected for the right-hand cell. Assuming that the pilot powers of allantennas are set to the same value, this depends only on the attenuation values γ ↓.We denote by D

(p)

i the set of all installations that dominate i at p:

D(p)

i ={j ∈ I : γ

↓jp > γ

↓ip

}(36)

In the example, only k dominates i at p, so D(p)

i = {k}. We have the relation

c(p)

i = 1 ⇔ zi = 1 ∧ zj = 0 ∀ j ∈ D(p)

i (37)

This relation can be expressed by linear inequalities as follows:

c(p)

i ≥ zi −∑g∈D

(p)

izg

c(p)

i ≤ zi

c(p)

i ≤ 1 − zg ∀ g ∈ D(p)

i

(38)

Fig. 3 Partitioning of the planning area generated by installation i


5.1.2 Partitioning the planning area

The above construction can be carried out for all points in the planning area. Thearea served by installation i is then the set of all points p ∈ A with c

(p)

i = 1. Thislinear description of the area served by i may be very large. The description for asingle point p, however, depends only on the set D(p)

i of installations that dominatei at p: all points with the same set of installations dominating i lead to the sameinequalities (38). We aggregate these points into one set. This yields a coveringof the planning area A according to the installations that dominate i. For a givenset D ⊂ I, i /∈ D of installations, we define Ai(D) as the set of points in whichexactly the installations in D dominate i:

Ai(D) :={p ∈ A : D

(p)

i = D}

={p ∈ A : γ

↓jp > γ

↓ip for j ∈ D, γ

↓jp ≤ γ

↓ip for j /∈ D

}(39)

These sets do not necessarily form a partitioning of the area. There might be pointsreceiving the same pilot signal strength from different installations. This can beresolved by breaking ties arbitrarily (which is on all accounts valid if the set ofsuch points is small).A more refined version of this model should, however, includesoft hand-over and thereby deal with regions of similar signal strength from severalinstallations in a different way. Assuming that there are no ties we have

A =⊎

D⊂IAi(D) (40)

The example partitioning generated by installation i is shown in Figure 3. Eitherall points in Ai(D) are served by i or none, depending on whether any installationin D is selected. We define a binary variable c

(D)i ∈ {0, 1} stating whether this is

the case, and couple it to the decision variables z in analogy to (38). For a givennetwork design z ∈ F , we then know that

Ai(z) =⊎

D⊂I :c(D)i =1

Ai(D) (41)

We obtain a mathematical programming model that describes the best-serverarea of any network design. This is used to calculate both the main diagonal entriesof the downlink coupling matrix and the off-diagonal entries.

5.1.3 Calculating main diagonal entries and noise load

The main diagonal entries of C↓ are given as

C↓ii :=

∫

p∈Ai

ωp l↓p dp (42)


Using the partitioning (41) of Ai , this value is calculated for any network designz ∈ F by summing the contributions to C

↓ii on each set:

C↓ii (z) =

∑

D⊂I

(∫

p∈Ai(D)

ωp l↓p dp

)

c(D)i (43)

The noise load p(η)

i can be determined analogously:

p(η)

i (z) =∑

D⊂I

(∫

p∈Ai(D)

ηp

γ↑ip

l↓p dp

)

c(D)i (44)

5.1.4 Calculating off-diagonal entries

The same principle is used to determine the off-diagonal elements of C↓. Thereis one more difficulty, however. The main diagonal elements only depend on theservice area Ai of installation i, whereas the off-diagonal element C↓

ij are also influ-enced by the setting for installation j . We thus introduce another dependent binaryvariable c

(D)ij that specifies whether any contribution to C

↓ij is generated on Ai(D).

This is the case if and only if Ai(D) belongs to the service area of installation iand installation j is selected

c(D)ij = c

(D)i zj (45)

Three linear inequalities are used to replace this product of variables:

c(D)ij ≤ zj

c(D)ij ≤ c

(D)i (46)

c(D)ij ≥ c

(D)i + zj − 1

The entry C↓ij is determined in analogy to (43) for any network design z ∈ F :

C↓ij (z) =

∑

D⊂I

(∫

p∈Ai(D)

γ↓jp

γ↓ip

l↓p dp

)

c(D)ij (47)

Technically speaking, the matrix C↓(z) as defined here has dimension |I| × |I|.The network’s actual coupling matrix C↓ of dimension |N | × |N | is obtained bydeleting the all-zero rows and columns. The analogue applies to p(η)(z) and p(η).

5.1.5 Coverage

There is another condition for a point to be served by installation i: the pilot signalhas to be received with sufficient absolute strength (Ec coverage) and signal quality(Ec/I0 coverage) for the user to register with the network. We only treat the firstpoint here. Assuming a fixed pilot power level, this means that for a point p to beserved by installation i the attenuation γ

↓ip must not fall below a threshold value

γ . This condition can easily be added to the definition of Ai(D). For imposing


additional coverage constraints (e. g., 99% of the area A must be covered by thenetwork), we can use the area covered by the cell of installation i:

|Ai(z)| =∑

D⊂I|Ai(D)| c

(D)i (48)

The total area covered by the network is |A(z)| = ∑i∈I |Ai(z)|. The load covered

by individual cells can be read off the main diagonal of the matrix.

5.2 Objective and additional constraints

The model presented in the previous paragraphs can be used to design the down-link coupling matrix C↓ in a variety of settings. We present one way of usingthis approach for improving a network’s quality according to the criteria listed insection 2.2. We use an objective function that targets minimization of the averagedownlink transmit power while imposing Ec coverage constraints. A variation ofthis model and corresponding computational results are presented in Eisenblatteret al. (2005a).

We want to minimize the sum 1T p↓ of the components of the downlink powervector. This vector cannot simply be determined by designing the matrix, since itis the solution of the (scaled) downlink coupling equation (22). Solving (22) whiledesigning the matrix falls out of the scope of a linear optimization model. In thecase of network tuning, there is an original network design z0 on which we wantto improve. The associated power vector p↓(z0) and scaling vector λ(z0) can becomputed for this network design as described in Section 3.2. We propose to usethe following optimization model for network tuning:

minz∈F

1T(diag

(λ(z0)

)C↓(z) p↓(z0) + λ(z0)T p(η)(z)

)

s. t. |A(z)| ≥ |A(z0)| (49)

1T diag(C↓(z)

) ≥ 1T diag(C↓(z0)

)(50)

diag(λ(z0)

)Tdiag

(C↓(z)

) ≥ diag(λ(z0)

)Tdiag

(C↓(z0)

)(51)

The basic idea is to approximate the quantities that cannot be computed in alinear model by the corresponding values of the reference network. This is valid aslong as the optimization result is “similar” to the reference network. In the objec-tive function, we use the downlink power vector p↓(z0) in the right-hand side ofthe fix point form (22). Constraint (49) ensures that Ec coverage does not diminishcompared to the reference network. An analogue to this condition is stated in (50),where the covered area is weighted with user load. This makes sure that coveragein highly frequented areas is not traded for coverage in areas with little traffic.The last constraint (51) uses the input scaling vector as an approximation for theresulting network design’s load control vector and states that no further load shouldbe shifted to cells that are already in overload.

When producing good solutions in the sense of this objective function, differentissues play a role:


– Interference reduction is rewarded: cells minimize their influence on neighbors,– Noise load is reduced: cells minimize the attenuation to their own mobiles,– Load is shifted away from overloaded cells to emptier cells.

These to some extent contradictory points suggest that network designs improv-ing on the objective function have better performance properties. Computationalexperiments (cf. section 6) confirm this.

There are, however, some disadvantages to this approach. First, the optimiza-tion model depends largely on the input network. Different input networks lead todifferent optimization results. When using this model iteratively, no stabilizationof the result could yet be observed. Second, if the used approximation is not validon the entire set F , the objective function might be misleading: results of qualityinferior to the reference network design can be produced. This has been observedin computational experiments.

5.3 Model size

The full model as described above is of exponential size in the input length, sincethe contribution variables are indexed by all subsets D ⊆ I. The relevant sets ofdominating installations are, however, comparatively small in practice, and it is cru-cial to determine these sets in a preprocessing step. The area A is usually describedin the finite form of a pixel grid. We determine the dominator set D

(p)

i for eachelement p ∈ A and for all i ∈ I with γ

↓ip ≥ γ . At most |A| |I| such sets exist, lead-

ing to a maximum of |A| |I| variables c(D)i . This number is considerably reduced

in practice by Ec-coverage conditions. The constraint that one installation per cellmust be selected further reduces the number of relevant dominator sets. The numberof potential interference sources per pixel is at most |I|(|I|+1)/2. The number ofvariables c

(D)ij is thus O(|A| |I|3). At the expense of accuracy, those variables c

(D)i

and c(D)ij with small contributions

∫p∈Ai(D)

ωp l↓p dp and

∫p∈Ai(D)

(γ↓jp/γ

↓ip) l

↓p dp can

be deleted from the model.Model sizes for the planning scenarios from section 6 are listed in Table 2. In

all cases, cells at the border of the scenarios are fixed. The number |N | is smallerthan the total number of cells in the scenario as listed in Table 3. Three potentialinstallation configurations are allowed for all other cells, hence |I| = 3|N |. Thealternative installations for a cell differ only in tilt (mechanical and electrical).All dominator sets and hence all variables c

(D)i are included in the model. The

main diagonal elements and noise load are thus computed with the precision of theMIP solver used for solving the matrix design model. Variables c

(D)ij are deleted if

their potential contribution to the referring off-diagonal element is less than 10−4.

Table 2 Size of matrix design model

Scenario |N | |I| Binary vars Constraints Nonzeroes

The Hague 30 90 123,426 212,861 822,921Berlin 108 324 101,756 159,782 627,409Lisbon 110 330 108,929 182,133 724,842


Table 3 The scenarios

Name Sites Cells Avg. Total Areausers DL-Load km2

The Hague 12 36 673 20.38 16Berlin 65 193 2328 70.05 56Lisbon 60 164 3149 77.02 21

Despite this model reduction, the off-diagonal elements could be computed withan absolute error not larger than 10−3 in all tests.

In general, the model sizes are acceptable and tractable with standard MIPsolvers. The large size of the model for The Hague is surprising, because it hasthe smallest number of cells. This is due to radio signals propagating with littleattenuation across the scenario. Cell coupling is thus very high.

6 Computational results

This section reports on computational results from the models described in sec-tions 4 and 5. The results are analyzed with our evaluation method as outlined insection 3. We focus on adjusting the tilts of antennas in a radio network, startingfrom reference networks based on GSM settings. The potential impact of adjustingtilts is described in section 1.3, a graphical illustration can be found in Figure 1.

6.1 The scenarios

Three real-world scenarios developed within the Momentum Project (2003) areused as test cases. We give only a brief description of the scenarios, further detailscan be found in Eisenblatter et al. (2004b,c), Geerdes et al. (2003) and Rakocziet al. (2003).

6.1.1 General description

The scenarios cover downtown areas of The Hague, Berlin, and Lisbon; some keyinformation can be found in Table 3. The total (downlink) load per scenario asgiven in the table is

∫A

l↓p dp. The locations of base stations are called sites. There

are typically three cells and three antennas associated to each base station in thescenarios.

The smallest scenario is The Hague. There are 76 potential site locations inthe original setting; since it is a low traffic scenario, we select a subset of 12 sitesand 36 cells with a specialized set covering model and conducted the optimizationon the basis of this network. The Berlin scenario is the largest one featuring 65sites and 193 cells distributed over an area of 56 km2. Lisbon is the scenario withthe highest traffic density. With 60 sites and 164 cells, there is less infrastructureavailable than in Berlin, but the traffic load is higher. The load is concentrated onan area of only 21 km2, less than half of the Berlin’s scenario area.

The traffic mix of services is not listed in the table; it is similar for all thescenarios: about 55% speech telephony users, 17% data streaming users, and 7%video telephony users are present, the residual active users being distributed over


a variety of data services. While speech and video telephony users are relativelysymmetric in their resource requests with respect to up- and downlink, the notice-able presence of demanding downlink data streaming users makes the downlinkthe limiting direction.

6.1.2 Networks and reference network design

The radio networks to optimize were provided by the telecommunication operatorswithin the Momentum consortium. In the reference network designs, tilts are notadjusted specifically, but a global electrical tilt of 6◦ and mechanical tilt of 0◦ areused. For The Hague and Berlin these networks perform reasonably well in mostcells. Problems typically occur in fairly loaded cells. The Lisbon reference networkis suffering from overload in many cells. The optimization options are limited toelectrical tilts in Berlin and The Hague, in Lisbon also mechanical tilts can beadjusted.

6.2 Algorithms

We optimize the original networks with three different algorithms:

1. (Snapshot)A snapshot based MIP model as mentioned in section 4. Due to com-putational difficulties (see section 4.2), only one randomly generated snapshotis used.

2. (Matrix)A MIP model for designing the downlink coupling matrix as describedin section 5. The power and scaling vectors of the reference network are used asa starting point. The network for The Hague is treated in its entirety, while in thelarger scenarios tilt variations are allowed only in subareas that are problematicin the reference network design.

3. (Local search) A local search algorithm that repeatedly enumerates all possibleinstallations for each cell. For each variation, the performance metrics are com-puted with the methods presented in section 3. The variation that performs bestaccording to a weighting of the metrics is chosen to proceed. This is repeatedin passes over all cells, until no improvement is made within a pass.

The MIPs for the snapshot- and matrix-based models are solved using Cplex 9.0with a limit of about 2 days running time on standard PCs (3.2 GHz Intel P4 CPU,2 GB RAM). The local search terminates within minutes or a few hours.

6.3 Results

Table 4 lists relevant performance metrics as described in section 2.2 for the refer-ence networks as well as for the networks computed by the different algorithms.

6.3.1 Performance metrics

Lower values indicate better performance. The Ec and Ec/I0 columns give thepercentage of the total area with insufficient Ec and Ec/I0 coverage, respectively.


Table 4 Computational results

UL UL DL DL DL DLScenario Ec Ec/I0 Pilot Load Cpl. Ovl. Load Cpl. TPAlgorithm A% A% A% φ % φ % cells φ % φ % dBm

The HagueOriginal 0.00 10.39 22.76 23.78 50.23 9 45.34 61.88 326Snapshot 0.00 3.02 18.19 22.84 48.90 7 41.11 59.53 295Matrix 0.00 0.86 10.82 20.07 44.74 3 34.12 50.30 245Local Search 0.00 0.60 11.46 20.27 44.57 3 35.37 51.98 254

BerlinOriginal 0.68 2.95 8.78 12.35 40.43 12 24.46 30.99 942Snapshot 0.68 2.95 8.78 12.35 40.44 12 24.47 30.99 942Matrix 1.55 0.50 5.61 11.22 36.26 6 21.85 25.60 841Local Search 0.37 0.39 4.35 11.45 35.86 6 22.77 27.17 877

LisbonOriginal 1.55 48.59 4.33 16.24 34.70 32 33.48 38.49 1,096Snapshot 1.61 47.99 4.15 16.08 34.28 33 33.45 38.09 1,095Matrix 1.81 38.37 3.71 14.97 33.38 18 29.23 32.49 980Local Search 1.48 27.55 3.32 15.45 31.82 14 27.42 31.66 897

The column Pilot holds the percentage of the area with pilot pollution. A locationis here taken as suffering of pilot pollution if more than three of the strongest pilotsignals are within a 5 dB range. The Load columns show the averaged load inpercent of all cells in up- and downlink, respectively. The Coupling values (Cpl.),computed on the basis of the coupling matrices, give an indication of what per-centage of a cells’ transmission power is on average needed due to interference.For the downlink, we also list the number of overloaded cells (Ovl. cells), that is,cells with a scaling value λ < 1, and the total tranmission power (TP) of all cells.

6.3.2 Assessment of results

The results show that by adjusting tilts the load can be distributed better among cellsand more traffic can be handled. The Matrix and Local Search algorithm are ableto substantially improve on the performance of the reference networks in all cases.The number of overloaded cells is at least halved and the interference is reducedby about 18%. In the Lisbon scenario, the Local Search noticeably outperformsthe Matrix design algorithm. This is probably due to two effects. First, the Matrixmodel is only applied to those portions of the scenario that reveal particularly poorperformance. Not all cells are therefore subject to change, and heavy interferencesources may not be eliminated if the imposing cells are themselves not sufferingfrom quality problems. Second, the iterated application of the Matrix model doesnot work well for Lisbon for reasons described in section 5.2.

The Snapshot model basically fails to produce network improvements. Theresulting MIPs are hard to solve for reasons explained in section 4.2. The situationcan be improved by applying the model to parts of the original problem (as we dofor the Matrix method). This allows to increase the number of snapshots up to 5,say. According to our experience, five snapshots is still insufficient to achieve thedesired coverage. The Snapshot model might be employed within a framework,where it is applied iterative to subproblem and snapshots are “separated” treated as


model cuts. (If necessary, artificial snapshots could be added to enforce coverage.)Such an approach has not been investigated.

7 Conclusion

Estimating coverage and capacity are important steps in UMTS network design.In contrast to GSM, cell coverage and capacity in WCDMA networks can hardlybe considered independently since both are limited by interference from neigh-boring cells. Cell loads are typically determined using Monte-Carlo simulationsbased on user snapshots. The computational burden of such traditional simulationsis severely impairing powerful automatic network optimization.

In this paper, we introduce a new (fast) evaluation method that analyticallyapproximates cell load without simulations. Moreover, we propose a new optimi-zation model that overcomes the notorious problems of scalability and statisticalreliability of snapshot-based network optimization models. Our results are a sig-nificant step towards automatic planning of real-world UMTS networks.

The cell load analysis based on average user load in combination with the matrixscaling approach for load control allows to rapidly compute a good estimate of thecell load in UMTS networks. This is the key to our fast local search procedure.Moreover, the new network optimization formulation differs substantially fromprevious models and eliminates their main difficulties. The optimization results forthe adjustment of antenna tilts in realistic networks show that our approaches canbe efficiently implemented and outperform snapshot-based methods of networkdesign.

These developments give rise to a variety of questions. A careful assessment ofthe estimation errors of our average-based load calculation is pending. The scalingapproach has to be carried over to the uplink case. Our preliminary method relieson solving a MIP, but a more direct, iterative method is preferable. Based on ourpreliminary results, the matrix-based optimization model seems much better suitedthan the previous models based on snapshots (or demand points). The merits of themodel have to be analyzed in more detail. When the entire network is too big to beoptimized “in one shot,” the idea of iteratively solving small subproblems shouldbe pursued further. The problems with the iteration mentioned in section 6 shouldbe manageable, e. g., by using alternative objective functions or adding constraints.Finally, soft hand-over is still ignored in all known automatic planning methodsthat are not based on network simulation. An extension directly incorporating softhand-over seems to be possible for the matrix-based model.

Acknowledgements This research was partially supported by the DFG Research Center Mathe-on “Mathematics for key technologies: Modelling, simulation, and optimization of real-worldprocesses”.

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