Comparison of Network Trees in Deterministic and Random ...

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

n Introduction and motivation

n Geometric support: Models and their fitting

n Comparison of network trees

n Infrastructure and costs

n Outlook and conclusion

1

2

3

Overview

4

5


Introduction and motivation


Introduction

Real data

Study areas inParis

A single study area


n Place lower level devices (LLDs) in a serving zone n Each LLD is connected to the corresponding higher level device (HLD) n 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

HLDHLDLLDLLD

HLDHLD

LLDLLD


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

n Telecom providers are confronted with new challengesn Network analysis of competing providers / in different countriesn New technologies / data

n Need for simple and global modeling toolsn Fast comparison of scenariosn Fast technical and cost evaluationsn Minimal number of parameters, maximal information about reality

n One solution: Stochastic-geometric modelingn Disregard too detailed information for the sake of clarityn Study random objects and their distributionn 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


n The SSLM consists of 3 parts

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

Geometric Support(infrastructure)

Network equipment (nodes, devices)

Topology of connections

RSRCPCS

IntroductionSSLM: Summary


Geometric support: Models and their fitting


n Stationary non-iterated Poisson tessellationsn Characterized by one parameter, called intensity γ (measured

per unit area)n PLT (Poisson Line Tessellation): γ … mean total length of edgesn PVT (Poisson Voronoï Tessellation): γ … mean total number of cellsn PDT (Poisson Delaunay Tessellation): γ … mean total number of vertices

Non-iterated tessellations

PLTPLT PVTPVTPDTPDT


Non-iterated tessellationsMean value relationships

n Consider facet characteristicsn They can be expressed in

terms of the intensity γ

Mean total length of edges

Mean number of cells

Mean number of edges

Mean number of vertices

Mean values Model ?per unit area ?

232 /(3 π)γλ4 [L]-1

γ2 γγ 2/ πλ3 [L]-23γ3γ2 γ 2/ πλ2 [L]-22 γγγ 2/ πλ1 [L]-2

PVT γ [L]-2

PDT γ [L]-2

PLT γ [L]-1

γγ


The mean total length of edges is always

0600400204 ... =+=λ

PLT/PLT PLT/PVT PLT/PDT

γ0= 0.02 γ1= 0.04 γ0= 0.02 γ1= 0.0004 γ0= 0.02 γ1= 0.0001388

Nesting of tessellations


PLT / PVT with Bernoulli thinning PLT multi-type nesting

Nesting of tessellationsGeneralizations

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


n Mean value relationships X0 / pX1

with and hence

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

Nestings of tessellations


Raw data Preprocessed data

Model fitting


n Estimation of characteristicsn Choice of a distance functionn Class of tessellation modelsn Minimization of distance function

Realisation of the optimal tessellation: PLT γ0 /PLT γ1

Preprocessed data

Model fitting


Model fittingUnbiased Estimation


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

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

n Example: Simulated PLT/PLT model ( )

Model fittingNumerical Minimisation

06.01.0 10 == γγ


n Monte Carlo testn Null hypothesis H0 : The optimal model is

PLT γ0= 0.02384 / PLT γ1= 0.013906n Decision: H0 is not rejected

n Main roads n Side streets

Model fittingExamplen Fitting strategy: Exploit hierarchical data structure


Comparison 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 roadsConnection 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 othergeometric 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 accordingto Euclidean distance





Note: Run time of simulations is

very long!



Comparison of network treesExample 2: Influence of fitting procedure



LLD and HLD on othergeometric support



Comparison of network treesExample 2: Different models – different distributions

Comparisons: Different …

50 km

20 km

… geometric supports … intensities


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


Optimal geometric support:Non-iterated model

Optimal geometric support:Iterated model


Infrastructure and costs


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

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

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

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

Geometricsupport

(infrastructure) Network equipment(devices)

Topology

RSRCPCS

The model


n Geometric support: Assume stationary PLT Xlwith intensity γ (> 0)

Infrastructure and costsGeometric support …

Geometric support: PLT


n Road system: Assume stationary PLT Xl withintensity γ

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

linear intensity λ1

n Stationar planar point process XH with planarintensity

Infrastructure and costs… and network devices

γλλ ⋅= 1H

HLDHLD


n Road system: Assume stationary PLT Xl withintensity γ

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

linear intensity λ1

n Stationar planar point process XH with planarintensity

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

intensity λ2

n Stationar planar point process with planarintensity

Infrastructure and costs… and network devices

γλλ ⋅= 1H

X~

γλλ ⋅= 2L

LLDLLD


n Random placement of HLD along the linesn Each LLD is connected to the closest HLDn 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)

n Natural approach

n Disadvantages



n Alternative approach

n Disadvantagesn Simulation not clearn Not very efficient



n Application of Neveu

n Independent from λ2

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



n Estimation of

n Note: The integrals can be calculated analytically


Infrastructure and costsMean shortest subscriber line length


Infrastructure and costsApplication

501

50

LH NA

50c .

.* .=

Mean length from LLD to HLD [km]

104501

550

LH VNA

77390c ..

.* .= 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 VNA

N77390 ..

.*LH .L =

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

ð

Intensity of PLT (est.) γ [km-1]=V/A

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


Outlook and conclusion


n Analysis of shortest pathsn Formulas for other types of geometric supportn Not only mean values but (parametric) distributions of cost functions

n Typology of infrastructuren Within the citiesn Nationwide extension

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

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

Outlook


n Analysis of shortest pathsn Formulas for other types of geometric supportn Not only mean values but (parametric) distributions of cost functions

n Typology of infrastructuren Within the citiesn Nationwide extension

n Analysis of inhomogeneitiesn Intensity maps

Intensity map of Paris (suppose underlying PLT)

Outlook


n 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

n 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

n 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

n 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

n 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

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