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Comparison of Network Trees in Deterministic and Random Settings using Different Connection Rules. Hendrik Schmidt France Telecom NSM/RD/RESA/NET [email protected] SpasWin07, Limassol, Cyprus 16 April 2007. Overview. 1. Introduction and motivation - PowerPoint PPT Presentation
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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
SpasWin07 - 16 April 2007 - H. Schmidt – p3 research & development France Telecom Group
1Introduction and motivation
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Introduction
Real data
Study areas in Paris
A single study area
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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
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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
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IntroductionSSLM: Main roads
Main roads
Cells: Subscribersare situated there
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IntroductionSSLM: Main roads and side streets
Main roads
Two level hierarchyof side streets
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IntroductionSSLM: Infrastructure, subscriber, serving zones
A serving zone
HLD
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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
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2Geometric support: Models and their fitting
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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
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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
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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
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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
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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
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Raw data Preprocessed data
Model fitting
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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
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Model fittingUnbiased Estimation
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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
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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
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3Comparison of network trees
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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
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LLD and HLD in the planeConnection according to
Euclidean distance
LLD and HLD on the roads Connection along infrastructure
Distribution of distances LLD → HLD
LLD and HLD on optimal geometric support
Connection along infrastructure
Note: Run time of simulations is very
long!
Comparison of network trees
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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
Connection along infrastructure
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Comparison of network treesExample 1: Different models – different distributions
50 km
20 km
… geometric supports
Comparisons: Different … … intensities
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LLD and HLD in the plane Connection according to
Euclidean distance
LLD and HLD on the roadsConnection along infrastructure
Distribution of distances LLD → HLD
LLD and HLD on optimal geometric support
Connection along infrastructure
Note: Run time of simulations is
very long!
Comparison of network trees
SpasWin07 - 16 April 2007 - H. Schmidt – p28 research & development France Telecom Group
Comparison of network treesExample 2: Influence of fitting procedure
LLD and HLD on optimal geometric support
Connection along infrastructure
LLD and HLD on other geometric support
Connection along infrastructure
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Comparison of network trees Example 2: Different models – different distributions
Comparisons: Different …
50 km
20 km
… geometric supports … intensities
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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
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4Infrastructure and costs
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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
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Geometric support: Assume stationary PLT Xl with intensity (> 0)
Infrastructure and costsGeometric support …
Geometric support: PLT
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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
SpasWin07 - 16 April 2007 - H. Schmidt – p35 research & development France Telecom Group
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
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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
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Infrastructure and costsPhysical connection (1)
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Infrastructure and costsPhysical connection (2)
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Infrastructure and costsMean shortest path length (1)
Natural approach
Disadvantages
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Infrastructure and costsMean shortest path length (2)
Alternative approach
Disadvantages Simulation not clear Not very efficient
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Infrastructure and costsMean shortest path length (3)
Application of Neveu
Independent from
The typical serving zone (the typical cell of a CVT) has to be simulated
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Infrastructure and costsMean shortest path length (4)
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Infrastructure and costsMean shortest path length (5)
Estimation of
Note: The integrals can be calculated analytically
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Infrastructure and costsMean shortest path length (6)
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Infrastructure and costsMean shortest path length (7)
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Infrastructure and costsMean shortest path length (8)
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Infrastructure and costsMean shortest subscriber line length
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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
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5Outlook and conclusion
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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
SpasWin07 - 16 April 2007 - H. Schmidt – p51 research & development France Telecom Group
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
SpasWin07 - 16 April 2007 - H. Schmidt – p52 research & development France Telecom Group
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
SpasWin07 - 16 April 2007 - H. Schmidt – p53 research & development France Telecom Group
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