Mathematical Challenges in Telecommunicati on a Tutorial

http://www.zib.de/[email protected]

Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (FZT 86) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Martin Grötschel

Mathematical Challenges in Telecommunication

a Tutorial

Martin Grötschel

IMA workshop on Network Management and Design

Minneapolis, MN, April 6, 2003

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Comment

This is an abbreviated version of my presentation.

The three films I showed are not included.

I also deleted many of the pictures and some of the tables: in some cases I do not have permission to publish

them since they contain some confidential material or information not for public use,

in other cases I am not sure of the copy right status (and do not have the time and energy to check it).

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The ZIB Telecom Group

Andreas Bley

Andreas Eisenblätter

Martin Grötschel

Thorsten Koch

Arie Koster

Roland Wessäly

Adrian Zymolka

ZIB Associates

Manfred Brandt

Sven Krumke

Frank Lutz

Diana Poensgen

Jörg Rambau

and more:

Clyde Monma (BellCore, ...)

Mechthild Opperud (ZIB, Telenor)

Dimitris Alevras (ZIB, IBM)

Christoph Helmberg (Chemnitz)

ZIB Telecom Team

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ZIB Partners from Industry

Bell Communications Research

Telenor (Norwegian Telecom)

E-Plus (acquired by KPN in 01/2002)

DFN-Verein

Bosch Telekom (bought by Marconi)

Siemens

Austria Telekom (controlled by Italia Telecom)

T-Systems Nova (T-Systems, Deutsche Telekom)

KPN

Telecel-Vodafone

Atesio (ZIB spin-off company)

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Contents

1. Telecommunication: The General Problem2. The Problem Hierarchy: Cell Phones and

Mathematics3. The Problem Hierarchy: Network Components

and Math4. Network Design: Tasks to be solved

Addressing Special Issues:5. Frequency Assignment6. Locating the Nodes of a Network 7. Balancing the Load of Signaling Transfer Points8. Integrated Topology, Capacity, and Routing

Optimization as well as Survivability Planning9. Planning IP Networks10.Optical Networks11.Summary and Future

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Contents






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What is the Telecom Problem?

Design excellent technical devices and a robust network that survivesall kinds of failures and organize the traffic such that high quality telecommunication betweenvery many individual units at many locations is feasibleat low cost!

Speech DataVideo Etc.

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What is the Telecom Problem?

Design excellent technical devices and a robust network that survivesall kinds of failures and organize the traffic such that high quality telecommunication betweenvery many individual units at many locations is feasibleat low cost!

This problem This problem is is

too general too general to be solved to be solved

in in one step.one step.

Approach in Practice:• Decompose whenever possible• Look at a hierarchy of problems• Address the individual problems one by one• Recompose to find a good global solution

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Contents


Mathematics3. The Problem Hierarchy: Network Components and

Math4. Network Design: Tasks to be solved


Optimization as well as Survivability Planning

9. Planning IP Networks10.Optical Networks11.Summary and Future

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Cell Phones and Mathematics

Designing mobile phones•Task partitioning•Chip design (VLSI)•Component design

•Computational logic•Combinatorial optimization•Differential algebraic equations

Producing Mobile Phones•Production facility layout•Control of CNC machines•Control of robots •Lot sizing•Scheduling•Logistics

•Operations research•Linear and integer programming•Combinatorial optimization•Ordinary differential equations

Marketing and Distributing Mobiles•Financial mathematics •Transportation optimization

cellphonepicture

Chip Design

CMOS layout for four-transistor static-memory cell

CMOS layout for two four-transistor static-memory cells.

Compacted CMOS layout for two four-transistor static-memory cells.

PlacementRouting

Compactification

Schematic for four-transistor static-memory cell

ZIB-Film 9:01 – 9:41

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Design and Production of ICs and PCBs

Integrated Circuit (IC)Printed Circuit Board (PCB)

Problems: Logic Design, Physical DesignCorrectness, Simulation, Placement of

Components, Routing, Drilling,...

Production and Mathematics: Examples

CNC Machine for 2D and 3D cutting and welding

(IXION ULM 804)Sequencing of Tasks

and Optimization of Moves

Mounting DevicesMinimizing Production Time

via TSP or IP

Printed Circuit Boards

Optimization of Manufacturing

SMD

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Drilling 2103 holes into a PCB

Significant Improvementsvia TSP

(Padberg & Rinaldi)

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Siemens Problem

before after

printed circuit board da1

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Siemens Problem

before after

printed circuit board da4

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Mobile Phone Production Line

Fujitsu Nasu plant

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Contents






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Network Components

Design, Production, Marketing, Distribution:Similar math problems as for mobile phones

•Fiber (and other) cables•Antennas and Transceivers •Base stations (BTSs)•Base Station Controllers (BSCs)•Mobile Switching Centers (MSCs)•and more...

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Component „Cables“

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Component „Antennas“

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Component „Base Station“

Nokia MetroSite

Nokia UltraSite

Component„Mobile Switching Center“:

Example of an MSC Plan

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Contents






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Network Design: Tasks to be solved Some Examples

Locating the sites for antennas (TRXs) and

base transceiver stations (BTSs)

Assignment of frequencies to antennas

Cryptography and error correcting encoding for wireless communication

Clustering BTSs

Locating base station controllers (BSCs)

Connecting BTSs to BSCs

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Network Design: Tasks to be solved Some Examples (continued)

Locating Mobile Switching Centers (MSCs)

Clustering BSCs and Connecting BSCs to MSCs

Designing the BSC network (BSS) and the MSC network (NSS or core network) Topology of the network Capacity of the links and components Routing of the demand Survivability in failure situations

Most of these problems turn out to be Combinatorial Optimization or

Mixed Integer Programming Problems

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Connecting Mobiles: What´s up?

BSC

MSC

BSC

BSC

BSC

BSC

BSC

BSC

MSC

MSCMSC

MSC

BTS

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Connecting Computers or other Devices

The mathematical problems are similar

but not identical

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Contents


Mathematics3. The Problem Hierarchy: Network Components and

Math4. Network Design: Tasks to be solved




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FAP

Film

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Antennas & Interference

xx

antenna

backbone network

xx

xxsite

xx

cell

co- & adjacentchannel

interferencecell

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Interference

Level of interference depends on distance between transmitters geographical position power of the signals direction in which signals are transmitted weather conditions assigned frequencies

co-channel interference adjacent-channel interference

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Separation

Parts of the spectrum forbidden

at some locations: government regulations, agreements with operators in

neighboring regions, requirements military forces, etc.

Site

Blocked channels

Frequencies assigned to the same

location (site) have to be separated

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Frequency Planning Problem

Find an assignment of frequencies to transmitters that satisfies all separation constraints all blocked channels requirements

and either avoids interference at all

or minimizes the (total/maximum) interference

level

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Minimum InterferenceFrequency Assignment Problem

Integer Linear Program:

min

. . 1

1 , ( )

1 ,

1 , 1

, , 0,1

co ad

v

co co ad advw vw vw vw

vw E vw E

vff F

dvf wg

co covf wf vw v w

ad advf wg vw

co advf vw vw

c z c z

s t x v V

x x vw E f g d vw

x x z vw E f F F

x x z vw E f g

x z z

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A Glance at some Instances

k 267 56,8 2 151,0 238 3 69B-0-E-20 1876 13,7 40 257,7 779 5 81

f 2786 4,5 3 135,0 453 12 69h 4240 5,9 11 249,0 561 10 130

E-Plus Project

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Region Berlin - Dresden

2877 carriers

50 channels

Interference

reduction: 83.6%

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Region Karlsruhe

2877 Carriers

75 channels

Interference

Reduction:83.9 %

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The UMTS Radio Interface

Completely new story

see talk by Andreas Eisenblätter on Monday

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Contents






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G-WiN Data

G-WiN = Gigabit-Wissenschafts-Netz of the DFN-Verein

Internet access of all German universities and research institutions

• Locations to be connected: 750• Data volume in summer 2000: 220 Terabytes/month • Expected data volume in 2004: 10.000 Terabytes/month

Clustering (to design a hierarchical network):• 10 nodes in Level 1a 261 nodes eligible for• 20 nodes in Level 1b Level 1• All other nodes in Level 2

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G-WiN Problem

• Select the 10 nodes of Level 1a.• Select the 20 nodes of Level 1b.

• Each Level 1a node has to be linked to two Level 1b nodes.• Link every Level 2 node to one Level 1 node.

• Design a Level 1a Network such that•Topology is survivable (2-node connected)•Edge capacities are sufficient (also in failure situations)•Shortest path routing (OSPF) leads to balanced capacity use (objective in network update)

• The whole network should be „stable for the future“.• The overall cost should be as low as possible.

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Potential node locations for the 3-Level Network of the G-WIN

Red nodes are potentiallevel 1 nodes

Cost: Connection between nodes Capacity of the nodes

Blue nodes are all remaining nodes

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Demand distribution

The demand scales with the height of each red line

AimSelect backbone nodes and

connect all non-backbone nodes to a backbone node

such that theoverall network cost is minimal

(access+backbone cost)

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G-WiN Location Problem: Data

set of locations

set of potential Level 1a locations (subset of )

set of possible configurations at

location in Level 1a

For , and :

connection costs from to

tr

p

p

ip

i

V

Z V

K

p

i V p Z k K

w i p

d

affic demand at location

capacity of location in configuration

costs at location in configuration

1 if location is connected to (else 0)

1 if configuration is used at loc

kp

kp

ip

kp

i

c p k

w p k

x i p

z k

ation (else 0)p

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G-WiN Location/Clustering Problem

min

1

1

p

k kip ip p p

p Z i V p Z k K

ipp

k ki ip p p

i k

kp

k

kp

p

w x w z

x

d x c z

z

z const

Each location i must be connected to a Level 1 node

Capacity at p must be large enough

Only one configuration at each Location 1 node

All variables are 0/1.

# of Level 1a nodes

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G-WiN Location Problem: Solution Statistics

The DFN problem leads to ~100.000 0/1-variables.Typical computational experience:

Optimal solution via CPLEX in a few seconds!

A very related problem at Telekom Austria has~300.000 0/1-variables plus some continuous variables

and capacity constraints.Computational experience (before problem specific fine tuning):

10% gap after 6 h of CPLEX computation,60% gap after „simplification“ (dropping certain capacities).

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Contents






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Re-Optimization of Signaling Transfer Points

Telecommunication companies maintain a signaling network (in adition to their communication transport network). This is used for management tasks such as:

Basic call setup or tear down

Wireless roaming

Mobile subscriber authentication

Call forwarding

Number display

SMS messages

Etc.

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Signaling Transfer Point (STP)

CCD

CCD

CCD

CCD CCD

CCD

CCD

CCD

CCD

Link-Sets STP

Cluster

Cluster

Cluster

CCLK

CCD=Common Channel Distributors, CCLK=Common Channel Link Controllers

CCD=routing unit, CCLK=interface card

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STP – Problem description

Target

Assign each link to a CCD/CCLK

Constraints

At most 50% of the links in a linkset can be assigned to a single cluster

Number of CCLKs in a cluster is restricted

Objective

Balance load of CCDs

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STP – Mathematical model

DataC set of CCDs j

L set of links i

Di demand of link i

P set of link-sets

Q set of clusters

Lp subset of links in link-set p

Cq subset of CCDs in cluster q

cq #CCLKs in cluster q

Variables

0,1 , , ijx i L j C

1ijx if and only iflink i is assigned to CCD j

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STP – Mathematical model

2

min

1

,

0,1

p

p q

q

ijj C

i iji L

i iji L

iji L j C

ij qi L j C

ij

L

y z

x i L

D x y j C

D x z j C

x p P q Q

x c q Q

x

Assign each link

Upper bound of CCD-load

Diversification

Lower bound of CCD-load

CCLK-bound

Integrality

Min load difference

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STP – current (former) solution

Minimum: 186 Maximum: 404 Load difference: 218

0

50

100

150

200

250

300

350

400

450

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18CCD

Tra

ffic

Lo

ad

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STP – „Optimal solution“

Minimum: 280 Maximum: 283 Load difference: 3

0

50

100

150

200

250

300

350

400

450

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18CCD

Tra

ffic

Loa

d

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STP – Practical difficulty

Problem: 311 rearrangements are necessary to migrate to the optimal

solution

Reformulation with new objective

Find a best solution with a restricted number of

changes

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STP – Reformulated Model

*, ( )

2

min

1

,

0,1

p

p q

q

ijj C

i iji L

i iji L

iji L j C

ij qi L j C

ij

iji L j C j j i

L

y z

x i L

D x y j C

D x z j C

x p P q Q

x c q Q

x

x B

Assign each link


Diversification


CCLK-bound

Integrality

Restricted number of changes!

Min load difference

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STP – Alternative Model

*, ( )

2

min

1

,

0,1

p

p q

q

iji L j C j j i

ijj C

i iji L

i iji L

iji L j C

ij qi L j C

ij

L

x

x i L

D x y j C

D x z j C

x p P q Q

x c q Q

x

y z D

Assign each link


Diversification


CCLK-bound

Integrality

Restricted load difference

Min # changes

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STP – New Solutions

0

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

CCD

Tra

ffic

Lo

ad White: D=50, (alternative) Minimum: 257 Maximum: 307 D=Load difference: 50 Number of changes: 12

Orange: B=8, (reformulated)

Minimum: 249 Maximum: 339 Load difference: 90 Number of changes=B: 8

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STP – Experimental results

Max changes 0 5 10 15 20

Load differences

218 129 71 33 14

1 hour application of CPLEX MIP-Solver for each case

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STP - Conclusions

It is possible to achieve 85%

of the optimal improvement with less than

5% of the changes necessary to obtain a load balance optimal solution !

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Contents






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Network Optimization Film

ZIB Film 1994, 9:40

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Network Optimization

Networks

Capacities Requirements

Cost

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What needs to be planned? Topology Capacities Routing Failure Handling (Survivability)

IP Routing Node Equipment Planning Optimizing Optical Links and Switches

DISCNET: A Network Planning Tool(Dimensioning Survivable Capacitated

NETworks)

Atesio ZIB Spin Off

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The Network Design Problem

200

65

258

134

30

42Düsseldorf

Frankfurt

Berlin

Hamburg

München

Communication Demands

Düsseldorf

Frankfurt

Berlin

Hamburg

München

Potential topology &

Capacities

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Capacities

PDH

2 Mbit/s

34 Mbit/s

140 Mbit/s

SDH

155 Mbit/s

622 Mbit/s

2,4 Gbit/s

... WDM (n x STM-N)

Two capacity models : Discrete Finite Capacities Divisible Capacities

(P)SDH=(poly)synchronous digital hierarchy

WDM=Wavelength Division MultiplexerSTM-N=Synchronous Transport Modul with N STM-1 Frames

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Survivability

H B

D

F

M

120

H B

D

F

M

3030

60Diversification„route node-disjoint“

Path restoration„reroute affected demands“(or p% of all affected demands)

6060

H B

D

F

M

H B

D

F

M

H B

D

F

M

60

60

Reservation„reroute all demands“(or p% of all demands)

120

H B

D

F

M

H B

D

F

M

H B

D

F

M

60

60

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Model: Data & Variables

SsOperating

states:

suvs PDuvSs ,,

0)( Pf s

uvPath variables:

Capacity variables: Ee}1,0{),( tex

Supply Graph: G=(V,E)

Z0

eCeT

ee CC 1

s

uvP PValid Paths:

Demand Graph: H=(V,D) Duvduv ,

Duvuv ,Duvluv ,

Duvuv ,

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Model: Capacities

Capacity variables : e E, t = 1, ..., Te {0,1}t

ex

Cost function :

1

mineT

t te e

e E t

k x

Capacity constraints : e E 0 11 0eTe e ex x x

0

eTt t

e e et

y c x

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Model: Routings

Path variables :

Capacity constraints :

Demand constraints :

, , ss uvs S uv D P

( ) 0suvf P

0

0

:

( )uv

e uvuv D P e P

y f P

e E

0

0 ( )uv

uv uvP

d f P

uv D

Path length restriction

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Model: Survivability (one example)

Path restoration „reroute affected demands“

H B

D

F

M

120

60

60

H B

D

F

M

for all sS, uvDs

for all sS, eEs

60

60

H B

D

F

M

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Mathematical Model

topology decisison

capacity decisions

normal operation routing

component failure routing

suvs PDuvSs ,,0)( Pf s

uv

)(:

0

0

PfyDuv PeP

uve

uv

Ee

0

)(0

uvPuvuv Pfd Duv

1

mineT

t te e

e E t

k x

eTtEe ,,1,

, 1, , ee E t T {0,1}tex

te

te xx 1

eT

t

te

tee xcy

0Ee

LP-based Methods

Feasibleintegersolutions

LP-based relaxation

Convex hull

Objectivefunction

Cuttingplanes

Feasibleintegersolutions

LP-based relaxation

Convex Convex hullhull

Objectivefunction

Cuttingplanes

Flow chart

Optimalsolution

InitializeLP-relaxation

Runheuristics

SolveLP-relaxation

Separation algorithms

AugmentLP-relaxation

Solve feasibilityproblem


Inequalities?

No

Yes

x variablesinteger?

Yes

No

Feasibleroutings?

Yes

No

Polyhedral combinatorics

Valid inequalities (facets)


Heuristics

Feasibility of a capacity vector

LP-based approach:

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Finding a Feasible Solution?

Heuristics Local search Simulated Annealing Genetic algorithms ...

Nodes Edges Demands Routing-Paths

15 46 78 > 150 x 10e6

36 107 79 > 500 x 10e9

36 123 123 > 2 x 10e12

Manipulation of– Routings– Topology– Capacities

Problem Sizes

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How much to save?

Real scenario• 163 nodes• 227 edges• 561 demands

34% potential savings!==

> hundred million dollars

PhD Thesis: http://www.zib.de/wessaely

[email protected]

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Contents






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Comment


The lecture ended after about 100 minutes. The last three topics above were not covered.

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Summary

Telecommunication Problems such as

• Frequency Assignment• Locating the Nodes of a Network Optimally• Balancing the Load of Signaling Transfer Points• Integrated Topology, Capacity, and Routing


• Planning IP Networks• Optical Network Design• and many others

can be succesfully attacked with optimization techniques.

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SummaryThe mathematical programming approach

• Helps understanding the problems arising• Makes much faster and more reliable planning

possible• Allows considering variations and scenario

analysis• Allows the comparision of different technologies• Yields feasible solutions• Produces much cheaper solutions than traditional

planning techniques• Helps evaluating the quality of a network.

There is still a lot to be done, e.g., for the really important problems, optimal solutions are way out of reach!

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The Mathematical Challenges

Finding the right ballance between flexibility and controlability of future networks

Controlling such a flexible network

Handling the huge complexity

Integrating new services easily

Guaranteeing quality

Finding appropriate Mathematical Models

Finding appropriate solution techniques (exact, approximate , interactive, quality guaranteed)

http://www.zib.de/[email protected]

Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (FZT 86) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Martin Grötschel

Mathematical Challenges in Telecommunication

The End

Martin GrötschelIMA workshop on

Network Management and Design

Minneapolis, MN, April 6, 2003

Documents

Mathematical Challenges in Telecommunicati on a Tutorial