COMP9334: Capacity Planning ofComputer Systems and Networks

Optimisation – Part 3

A/Prof. Chun Tung Chou

CSE, UNSW

Integer Programming - What have you seen?

A recurrent theme is to use integer programming to make binarydecisions

Examples of binary decisions

Week 10: Grid computing problemChoose a particular grid computing company or not

Week 11: Routing of flowsShould the flow be routed on a link or not?

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This week’s lecture

Integer programming for placement problem

Example: There are a number of potential places that I can putcertain devices, what are the best places to put them?

We will study a placement problem in wireless networks

Placement of wireless access points

For the revision problem, we will look at the controller placementproblem for software-defined networking

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Wireless Local Area Networks

Commonly known as Wireless LAN, WiFi etc.

Formal standards in IEEE 802.11, IEEE 802.11a/b/g/n/ac

Infrastructure mode: Wireless Access Points (APs) and Wirelessstations

AP

Station A Station B

AP

Station D Station CInternet

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Wireless Access Point CoverageDue to radio propagation loss and mandated limit on transmissionpower, wireless access points have only limited coverage

For example, a Cisco Aironet access point has a coverage of 304mwhen operating outdoor at 11 Mbps

Ideal coverage area is a circle. In the picture below, Station A cantalk to the access point but not Station B

AP

Station A Station B

Transmissionrange

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Coverage in practice

Note: High attenuation = Poor signal = Poor coverage

An area is covered only if the attenuation is smaller than a threshold

IEEE Wireless Communications • December 200612

sponding MILP model instance. The authorsoften use Zimpl [7] as a modeling tool. Alterna-tives are listed at the NEOS server [11].

The model instance is then loaded into anMILP solver such as SCIP [1]. Again, furthersolvers for MILPs and other problem types arelisted at the NEOS server [11]. The server offersthe service to upload model instances over theInternet and have them processed remotely.Benchmarks of various (free, academic, and com-mercial) mathematical optimization solvers areavailable [9] and reveal information on modelsize that are tractable and related running times.

EXAMPLES

We now give a few examples of how the tech-niques discussed above can be used in the con-text of realistic wireless network planning. Inparticular, we highlight to what extent the “wellbehaved” MILPs can contribute to solving theplanning problem and which other model com-ponents are needed. General presentations ofmodeling techniques and mathematical opti-mization algorithms, including search methodsused for wireless network designs in varioustechnologies, are given in [6, 8].

GSM COVERAGE PLANNINGThe typical planning process for GSM radio net-works separates coverage planning from capacityconsiderations. In the first planning step thebase station locations are decided. These deci-sions are instrumental to provide seamless net-work coverage (e.g., 98 percent outdoor coverageor 75 percent indoor coverage). For pure cover-age planning, the set covering model appliesdirectly. Here, we are in the fortunate situationthat a well-known problem is at the heart of theplanning problem. In the simplest version, loca-tions for base stations with a fixed sector/anten-

na configuration are to be selected. The task isto design a network that is as cost effective aspossible subject to the constraint that a givenfixed area A (in the form of pixels) needs to becovered. For each potential base station locationi, we are given a price ci for the deployment of asuitable base station, and we can derive the setof pixels Ci that could be covered with a refer-ence service by the base station from the propa-gation data. The classical set covering model isformally stated as

min Σicixis.t. Σi:p∈Ci xi ≥ 1 ∀ p ∈ A (Set Covering)

xi ∈ {0,1} ∀ i

The binary variables xi express whether location iis used.

The above type of model can be used with vir-tually any digital radio technology. An illustrationfor an indoor scenario is given in Fig. 3. Anindoor propagation prediction is shown in Fig.3a. The related coverage set Ci in the model isthe shaded area in Fig. 3b. Figure 3c shows auseful analysis by counting how many coveragesets overlap at each pixel. Further modeling con-sists in adapting this standard formulation to thereal-world situation under consideration. Alter-native base station configurations that differ inantenna types, azimuths, and tilts for the sectorsor even the sector number can be considered.The service area may only be covered to somethreshold percentage. Or each covered pixel maycarry an individual bonus, each base station apenalty, and the objective is to maximize benefit.These variants go beyond plain set covering mod-els, but typically retain enough of the structure tobe tackled by the methods described above.

Models of the above type are often well solv-able in practice. The first author has solved verylarge instances with superb quality guarantees incommercial consultancy projects. Thoseinstances capture the planning of thousands ofbase stations, each with tens of configurationoptions. For completing a network design, how-ever, a frequency assignment has to be found foreach transmitter.

MANAGED WIFI DEPLOYMENTWireless LAN (WLAN, WiFi) based on IEEE802.11 is a broadband radio access technologythat has become quite popular over the lastyears. In the infrastructure mode a WiFi networkcan be seen as a microcellular network. Accesspoints (APs) serve as base stations. ManagedWLANs accessible to automatic planning areused for hotspot areas and office buildings. Anexample for indoor propagation from an AP isgiven in Fig. 3a.

The key to understanding WiFi technology isthe radio access protocol. There is no distinctionbetween up- and downlink. All stations have tocontent for the wireless medium using a carriersense multiple access (CSMA) protocol with ran-dom backoff time. No communication can takeplace between two nodes if any station withinreception range communicates. To mitigate theeffects of contention, different frequencies canbe assigned to the APs. There are 13 channelsavailable at maximum, of which only three donot overlap. Interference from other stations

n Figure 3. Modeling coverage planning as a set covering problem: a)signalpropagation prediction for one potential AP location i; b)corresponding cover-age set Ci; c)AP location density.

110

40

15

0N

umbe

r of

pote

ntia

l ser

vers

Coveredarea

PotentialAPlocation

Att

enua

tion

[dB]

(a)

(b)

(c)

EISENBLAETTER LAYOUT 12/4/06 2:57 PM Page 12

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Covering a given area

Multiple access points may be required to cover a given area

Decisions to make

The number of access points required to cover the given areaThe position of the access points

IEEE Wireless Communications • December 200612

sponding MILP model instance. The authorsoften use Zimpl [7] as a modeling tool. Alterna-tives are listed at the NEOS server [11].

The model instance is then loaded into anMILP solver such as SCIP [1]. Again, furthersolvers for MILPs and other problem types arelisted at the NEOS server [11]. The server offersthe service to upload model instances over theInternet and have them processed remotely.Benchmarks of various (free, academic, and com-mercial) mathematical optimization solvers areavailable [9] and reveal information on modelsize that are tractable and related running times.

EXAMPLES

We now give a few examples of how the tech-niques discussed above can be used in the con-text of realistic wireless network planning. Inparticular, we highlight to what extent the “wellbehaved” MILPs can contribute to solving theplanning problem and which other model com-ponents are needed. General presentations ofmodeling techniques and mathematical opti-mization algorithms, including search methodsused for wireless network designs in varioustechnologies, are given in [6, 8].

GSM COVERAGE PLANNINGThe typical planning process for GSM radio net-works separates coverage planning from capacityconsiderations. In the first planning step thebase station locations are decided. These deci-sions are instrumental to provide seamless net-work coverage (e.g., 98 percent outdoor coverageor 75 percent indoor coverage). For pure cover-age planning, the set covering model appliesdirectly. Here, we are in the fortunate situationthat a well-known problem is at the heart of theplanning problem. In the simplest version, loca-tions for base stations with a fixed sector/anten-

na configuration are to be selected. The task isto design a network that is as cost effective aspossible subject to the constraint that a givenfixed area A (in the form of pixels) needs to becovered. For each potential base station locationi, we are given a price ci for the deployment of asuitable base station, and we can derive the setof pixels Ci that could be covered with a refer-ence service by the base station from the propa-gation data. The classical set covering model isformally stated as

min Σicixis.t. Σi:p∈Ci xi ≥ 1 ∀ p ∈ A (Set Covering)

xi ∈ {0,1} ∀ i

The binary variables xi express whether location iis used.

The above type of model can be used with vir-tually any digital radio technology. An illustrationfor an indoor scenario is given in Fig. 3. Anindoor propagation prediction is shown in Fig.3a. The related coverage set Ci in the model isthe shaded area in Fig. 3b. Figure 3c shows auseful analysis by counting how many coveragesets overlap at each pixel. Further modeling con-sists in adapting this standard formulation to thereal-world situation under consideration. Alter-native base station configurations that differ inantenna types, azimuths, and tilts for the sectorsor even the sector number can be considered.The service area may only be covered to somethreshold percentage. Or each covered pixel maycarry an individual bonus, each base station apenalty, and the objective is to maximize benefit.These variants go beyond plain set covering mod-els, but typically retain enough of the structure tobe tackled by the methods described above.

Models of the above type are often well solv-able in practice. The first author has solved verylarge instances with superb quality guarantees incommercial consultancy projects. Thoseinstances capture the planning of thousands ofbase stations, each with tens of configurationoptions. For completing a network design, how-ever, a frequency assignment has to be found foreach transmitter.

MANAGED WIFI DEPLOYMENTWireless LAN (WLAN, WiFi) based on IEEE802.11 is a broadband radio access technologythat has become quite popular over the lastyears. In the infrastructure mode a WiFi networkcan be seen as a microcellular network. Accesspoints (APs) serve as base stations. ManagedWLANs accessible to automatic planning areused for hotspot areas and office buildings. Anexample for indoor propagation from an AP isgiven in Fig. 3a.

The key to understanding WiFi technology isthe radio access protocol. There is no distinctionbetween up- and downlink. All stations have tocontent for the wireless medium using a carriersense multiple access (CSMA) protocol with ran-dom backoff time. No communication can takeplace between two nodes if any station withinreception range communicates. To mitigate theeffects of contention, different frequencies canbe assigned to the APs. There are 13 channelsavailable at maximum, of which only three donot overlap. Interference from other stations

n Figure 3. Modeling coverage planning as a set covering problem: a)signalpropagation prediction for one potential AP location i; b)corresponding cover-age set Ci; c)AP location density.

110

40

15

0

Num

ber o

fpo

tent

ial s

erve

rs

Coveredarea

PotentialAPlocation

Att

enua

tion

[dB]

(a)

(b)

(c)

EISENBLAETTER LAYOUT 12/4/06 2:57 PM Page 12

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The coverage problem - definition

Given

A number of potential access point locations L1, L2, ..., LpA number of stations s1, s2, s3, ..., snBinary constant δij where

δij =

{1 if station si is covered by an AP at location Lj0 otherwise

See the next page for a graphical explanation for δij

Find the minimum number of access points required so that

All stations are covered

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Explanation of δijExample: δ10,5 = 1, δ3,5 = 0

PotentialAP location #5

= location of station

Station 3 Station 10

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The coverage problem: Verbal formulationDecision variables:

xj =

{1 if an AP is to be installed at location Lj0 otherwise

Integer programming formulation:

min The number of access points (An expression in xj)

subject to

Each station is covered (One expression for each station, need xj and δi,j)

xj ∈ {0, 1}

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The coverage problem: FormulationDecision variables:

xj =

{1 if an AP is to be installed at location Lj0 otherwise

Integer programming formulation:

min

p∑j=1

xj

subject top∑j=1

δijxj ≥ 1 ∀i = 1, ..., n

xj ∈ {0, 1}

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Integer programming and optimisation:Summary

What you have learnt

How to formulate integer programming problemsHow to solve them using AMPLExamples of using integer programming for network design andanalysis

There are a lot more to learn but this will give you a starting point...

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Acknowledgments

Picture credits:

Andreas Eisenblatter and Hans-Florian Geerdes, ”WirelessNetwork Design: Solution-oriented modeling and mathematicaloptimization”, IEEE Wireless Communications, December 2006

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