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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
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
SpasWin07 - 16 April 2007 - H. Schmidt – p3 research & development France Telecom Group
Introduction and motivation
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Introduction
Real data
Study areas inParis
A single study area
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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
SpasWin07 - 16 April 2007 - H. Schmidt – p6 research & development France Telecom Group
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
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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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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
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Geometric support: Models and their fitting
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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
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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
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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
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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
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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
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Raw data Preprocessed data
Model fitting
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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
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Model fittingUnbiased Estimation
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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
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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
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Comparison 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 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
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Comparison of network treesExample 1: Influence of fitting procedure
LLD and HLD on optimal geometric support
Connection along infrastructure
LLD and HLD on othergeometric 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 accordingto 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
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Comparison of network treesExample 2: Influence of fitting procedure
LLD and HLD on optimal geometric support
Connection along infrastructure
LLD and HLD on othergeometric support
Connection along infrastructure
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Comparison of network treesExample 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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Infrastructure and costs
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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
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n Geometric support: Assume stationary PLT Xlwith intensity γ (> 0)
Infrastructure and costsGeometric support …
Geometric support: PLT
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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
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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
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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
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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)
n Natural approach
n Disadvantages
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Infrastructure and costsMean shortest path length (2)
n Alternative approach
n Disadvantagesn Simulation not clearn Not very efficient
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Infrastructure and costsMean shortest path length (3)
n Application of Neveu
n Independent from λ2
n 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)
n Estimation of
n 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 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
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Outlook and conclusion
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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
SpasWin07 - 16 April 2007 - H. Schmidt – p51 research & development France Telecom Group
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
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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
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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