hendrik schmidt france telecom nsm/rd/resa/net hendrik.schmidt@orange-ftgroup

research & development

Hendrik Schmidt France Telecom NSM/RD/RESA/[email protected]

SpasWin07, Limassol, Cyprus16 April 2007

Comparison of Network Trees in Deterministic and Random Settings using

Different Connection Rules

SpasWin07 - 16 April 2007 - H. Schmidt – p2 research & development France Telecom Group

Introduction and motivation

Geometric support: Models and their fitting

Comparison of network trees

Infrastructure and costs

Outlook and conclusion

1

2

3

Overview

4

5


1Introduction and motivation


Introduction

Real data

Study areas in Paris

A single study area


Place lower level devices (LLDs) in a serving zone Each LLD is connected to the corresponding higher level device (HLD) Length distribution LLD → HLD influences costs and technical possibilities

Serving zone (two levels of network devices), connection along infrastructure

Network devices in the plane, Euclidean distance connection

Distribution of distances LLD → HLD

Introduction

HLD LLD

HLD

LLD


Geometric considerations are essential: The access network … … runs along the infrastructure … contributes mainly to total network costs

Telecom providers are confronted with new challenges Network analysis of competing providers / in different countries New technologies / data

Need for simple and global modeling tools Fast comparison of scenarios Fast technical and cost evaluations Minimal number of parameters, maximal information about reality

One solution: Stochastic-geometric modeling Disregard too detailed information for the sake of clarity Study random objects and their distribution Take into account the spatial geometric structure of networks

IntroductionStochastic Subscriber Line Model


IntroductionSSLM: Main roads

Main roads

Cells: Subscribersare situated there


IntroductionSSLM: Main roads and side streets

Main roads

Two level hierarchyof side streets


IntroductionSSLM: Infrastructure, subscriber, serving zones

A serving zone

HLD


The SSLM consists of 3 parts

Random objects (infrastructure, equipment, topology) provide a statistically equivalent image of reality Are defined by few parameters Allow to study separately the three parts of the network

Geometric Support(infrastructure)

Network equipment (nodes, devices)

Topology of connections

RSRCPCS

IntroductionSSLM: Summary


2Geometric support: Models and their fitting


Stationary non-iterated Poisson tessellations Characterized by one parameter, called intensity (measured per unit

area) PLT (Poisson Line Tessellation): … mean total length of edges PVT (Poisson Voronoï Tessellation): … mean total number of cells PDT (Poisson Delaunay Tessellation): … mean total number of vertices

Non-iterated tessellations

PLT PVT PDT


Non-iterated tessellationsMean value relationships

Consider facet characteristics They can be expressed in

terms of the intensity

Mean values Model →per unit area ↓

PLT [L]-1

PDT [L]-2

PVT [L]-2

Mean number of vertices L-2 2/ 2

Mean number of edges [L]-2 2 2/ 3 3

Mean number of cells [L]-2 2/ 2

Mean total length of edges [L]-1 32 /(3 ) 2


The mean total length of edges is always

0600400204 ...

PLT/PLT PLT/PVT PLT/PDT0= 0.02 1= 0.04 0= 0.02 1= 0.0004 0= 0.02 1= 0.0001388

Nesting of tessellations


PLT / PVT with Bernoulli thinning PLT multi-type nesting

Nesting of tessellationsGeneralizations

Bernoulli thinning: Nesting in cell with probability p Multi-type nesting: Different nestings in different cells


Mean value relationships X0 / pX1

with and hence

Immediate application toPVT/(PLT, PVT, PDT), PDT/(PLT, PVT, PDT) and PLT/(PLT, PVT, PDT)

Nestings of tessellations


Raw data Preprocessed data

Model fitting


Estimation of characteristics Choice of a distance function Class of tessellation models Minimization of distance function

Realisation of the optimal tessellation: PLT 0 /PLT 1

Preprocessed data

Model fitting


Model fittingUnbiased Estimation


Solution of minimization problem analytically for non-iterated models numerical methods for nested models, e.g. Nelder-Mead algorithm

• fast• easy to implement• minimum depends on initial point → random variation

Example: Simulated PLT/PLT model ( )

Model fittingNumerical Minimisation

06.01.0 10


Monte Carlo test Null hypothesis H0 : The optimal model is PLT 0= 0.02384 / PLT 1= 0.013906 Decision: H0 is not rejected

Main roads Side streets

Model fittingExample

Fitting strategy: Exploit hierarchical data structure


3Comparison of network trees



Geometric support

Two levels of network devices:• Lower level devices (LLD)• Higher level devices (HLD)

Two connection rules:• Euclidean distance • Connection along geometric support


LLD and HLD in the planeConnection according to

Euclidean distance

LLD and HLD on the roads Connection along infrastructure


LLD and HLD on optimal geometric support

Connection along infrastructure

Note: Run time of simulations is very

long!



Comparison of network treesExample 1: Influence of fitting procedure LLD and HLD on optimal

geometric supportConnection along infrastructure

LLD and HLD on other geometric support



Comparison of network treesExample 1: Different models – different distributions

50 km

20 km

… geometric supports

Comparisons: Different … … intensities


LLD and HLD in the plane Connection according to

Euclidean distance

LLD and HLD on the roadsConnection along infrastructure




Note: Run time of simulations is

very long!



Comparison of network treesExample 2: Influence of fitting procedure



LLD and HLD on other geometric support



Comparison of network trees Example 2: Different models – different distributions

Comparisons: Different …

50 km

20 km

… geometric supports … intensities


Comparison of network treesExample 2: Non-iterated vs. iterated models

LLD and HLD on the roadsConnection along infrastructure

Optimal geometric support:Non-iterated model

Optimal geometric support:Iterated model


4Infrastructure and costs


An example of the SSLM Geometric support: Stationary PLT Network devices: 2 layer model of stationary Poisson point processes

• Lower level devices (LLD) • Higher level devices (HLD)

Topology of connection• Logical connection: LLD connected to closest HLC• Physical connection: Shortest path along the infrastructure

Questions What are the mean shortest path costs from LLD to HLD? Is a parametric description of the distribution possible?

Geometric support

(infrastructure) Network equipment (devices)

Topology

RSRCPCS

The model


Geometric support: Assume stationary PLT Xl with intensity (> 0)

Infrastructure and costsGeometric support …

Geometric support: PLT


Road system: Assume stationary PLT Xl with intensity

Higher level devices (HLD) Stationary point process (independent of Xl ) Poisson process on Xl (Cox process) with

linear intensity

Stationar planar point process XH with planar intensity

Infrastructure and costs… and network devices

1H

HLD


Road system: Assume stationary PLT Xl with intensity

Higher level devices (HLD) Stationary point process (independent of Xl ) Poisson process on Xl (Cox process) with

linear intensity

Stationar planar point process XH with planar intensity

Lower level devices (LLD) Stationary point process (indep. of Xl and XH) Poisson process on Xl (Cox process) with linear

intensity Stationar planar point process with planar intensity

Infrastructure and costs… and network devices

1H

X~ 2L

LLD


Random placement of HLD along the lines Each LLD is connected to the closest HLD Serving zones induce a Cox-Voronoi tessellation (CVT)

Infrastructure and costsLogical connection


Infrastructure and costsPhysical connection (1)


Infrastructure and costsPhysical connection (2)


Infrastructure and costsMean shortest path length (1)

Natural approach

Disadvantages



Alternative approach

Disadvantages Simulation not clear Not very efficient



Application of Neveu

Independent from

The typical serving zone (the typical cell of a CVT) has to be simulated



Estimation of

Note: The integrals can be calculated analytically


Infrastructure and costsMean shortest subscriber line length


Infrastructure and costsApplication

501

50

LH NA50c .

.* .

Mean length from LLD to HLD [km]

104501

550

LH VNA77390c ..

.* . Study zone: Area A [km2]

Geometric support: Within the study zone of length V [km], type PLT

Placement of network devices: LLD on the geometric support (number N0) HLD on the geometric support (number N1)

Mean total length from LLD to HLD in A [km]

104501

550

0 VNAN77390 ..

.*LH .L

Logical connection: LLD connected to closest HLD according to Voronoi principle Physical connection: Shortest path along the geometric support

Intensity of PLT (est.) [km-1]=V/A

Mean length from LLD to HLD [km] in case of spatial placement


5Outlook and conclusion


Analysis of shortest paths Formulas for other types of geometric support Not only mean values but (parametric) distributions of cost functions

Typology of infrastructure Within the cities Nationwide extension

Deterministic PDT in France (level préfectures and sous-préfectures)

Main roads: Optimal intensity of nested tessellation (within PDT)

Outlook


Analysis of shortest paths Formulas for other types of geometric support Not only mean values but (parametric) distributions of cost functions

Typology of infrastructure Within the cities Nationwide extension

Analysis of inhomogeneities Intensity maps

Intensity map of Paris (suppose underlying PLT)

Outlook


C. Gloaguen, H. Schmidt, R. Thiedmann, J.-P. Lanquetin and V. Schmidt (2007). Comparison of Network Trees in Deterministic and Random Settings using Different Connection Rules, Proceedings of "SpasWin07", 16 April 2006, Limassol, Cyprus

C. Gloaguen, F. Fleischer, H. Schmidt and V. Schmidt (2006). Fitting of stochastic telecommunication network models via distance measures and Monte-Carlo tests. Telecommunication Systems 31, pp.353-377, http://dx.doi.org/10.1007/s11235-006-6723-3

C. Gloaguen, F. Fleischer, H. Schmidt and V. Schmidt (2007). Analysis of shortest paths and subscriber line lengths in telecommunication access networks, Networks and Spatial Economics, to appear

H. Schmidt (2006). Asymptotic analysis of stationary random tessellations with applications to network modelling, Ph.D. Thesis, Ulm University, http://vts.uni-ulm.de/doc.asp?id=5702

http://www.geostoch.de

Bibliography


This presentation is based on collaborative work withC. Gloaguen, J.-P. Lanquetin – France Telecom R&D,

Paris&Belfort, FranceF. Fleischer, V. Schmidt, R. Thiedmann – Institute of Stochastics,

Ulm University, Germany

hendrik schmidt france telecom nsm/rd/resa/net hendrik.schmidt@orange-ftgroup

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