research & development Hendrik Schmidt France Telecom NSM/RD/RESA/NET SpasWin07, Limassol, Cyprus 16 April 2007 Comparison of Network Trees in Deterministic and Random Settings using Different Connection Rules

research & development France Telecom Group SpasWin April H. Schmidt – p2 Introduction and motivation Geometric support: Models and their fitting Comparison of network trees Infrastructure and costs Outlook and conclusion Overview 4 5

research & development France Telecom Group SpasWin April H. Schmidt – p3 1 Introduction and motivation

research & development France Telecom Group SpasWin April H. Schmidt – p4 Introduction Real data Study areas in Paris A single study area

research & development France Telecom Group SpasWin April H. Schmidt – p5 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 HL D LLD HL D LLD

research & development France Telecom Group SpasWin April H. Schmidt – p6 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 Introduction Stochastic Subscriber Line Model

research & development France Telecom Group SpasWin April H. Schmidt – p7 Introduction SSLM: Main roads Main roads Cells: Subscribers are situated there

research & development France Telecom Group SpasWin April H. Schmidt – p8 Introduction SSLM: Main roads and side streets Main roads Two level hierarchy of side streets

research & development France Telecom Group SpasWin April H. Schmidt – p9 Introduction SSLM: Infrastructure, subscriber, serving zones A serving zone HLD

research & development France Telecom Group SpasWin April H. Schmidt – p10 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 R SR CP CS Introduction SSLM: Summary

research & development France Telecom Group SpasWin April H. Schmidt – p11 2 Geometric support: Models and their fitting

research & development France Telecom Group SpasWin April H. Schmidt – p12 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

research & development France Telecom Group SpasWin April H. Schmidt – p13 Non-iterated tessellations Mean 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 33 Mean number of cells  [L] -2  2 /  2  Mean total length of edges  [L] -1  32 /(3  ) 2

research & development France Telecom Group SpasWin April H. Schmidt – p14 The mean total length of edges is always PLT/PLT PLT/PVTPLT/PDT  0= 0.02  1= 0.04  0= 0.02  1=  0= 0.02  1= Nesting of tessellations

research & development France Telecom Group SpasWin April H. Schmidt – p15 PLT / PVT with Bernoulli thinning PLT multi-type nesting Nesting of tessellations Generalizations Bernoulli thinning: Nesting in cell with probability p Multi-type nesting: Different nestings in different cells

research & development France Telecom Group SpasWin April H. Schmidt – p16 Mean value relationships X0 / pX1 with and hence Immediate application to PVT/(PLT, PVT, PDT), PDT/(PLT, PVT, PDT) and PLT/(PLT, PVT, PDT) Nestings of tessellations

research & development France Telecom Group SpasWin April H. Schmidt – p17 Raw data Preprocessed data Model fitting

research & development France Telecom Group SpasWin April H. Schmidt – p18 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

research & development France Telecom Group SpasWin April H. Schmidt – p19 Model fitting Unbiased Estimation

research & development France Telecom Group SpasWin April H. Schmidt – p20 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 fitting Numerical Minimisation

research & development France Telecom Group SpasWin April H. Schmidt – p21 Monte Carlo test Null hypothesis H 0 : The optimal model is PLT  0 = / PLT  1 = Decision: H 0 is not rejected Main roads Side streets Model fitting Example Fitting strategy: Exploit hierarchical data structure

research & development France Telecom Group SpasWin April H. Schmidt – p22 3 Comparison of network trees

research & development France Telecom Group SpasWin April H. Schmidt – p23 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

research & development France Telecom Group SpasWin April H. Schmidt – p24 LLD and HLD in the plane Connection 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

research & development France Telecom Group SpasWin April H. Schmidt – p25 Comparison of network trees Example 1: Influence of fitting procedure LLD and HLD on optimal geometric support Connection along infrastructure LLD and HLD on other geometric support Connection along infrastructure

research & development France Telecom Group SpasWin April H. Schmidt – p26 Comparison of network trees Example 1: Different models – different distributions 50 km 20 km … geometric supports Comparisons: Different … … intensities

research & development France Telecom Group SpasWin April H. Schmidt – p27 LLD and HLD in the plane Connection 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

research & development France Telecom Group SpasWin April H. Schmidt – p28 Comparison of network trees Example 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

research & development France Telecom Group SpasWin April H. Schmidt – p29 Comparison of network trees Example 2: Different models – different distributions Comparisons: Different … 50 km 20 km … geometric supports … intensities

research & development France Telecom Group SpasWin April H. Schmidt – p30 Comparison of network trees Example 2: Non-iterated vs. iterated models LLD and HLD on the roads Connection along infrastructure Optimal geometric support: Non-iterated model Optimal geometric support: Iterated model

research & development France Telecom Group SpasWin April H. Schmidt – p31 4 Infrastructure and costs

research & development France Telecom Group SpasWin April H. Schmidt – p32 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 R SR CP CS The model

research & development France Telecom Group SpasWin April H. Schmidt – p33 Geometric support: Assume stationary PLT X l with intensity  (> 0) Infrastructure and costs Geometric support … Geometric support: PLT

research & development France Telecom Group SpasWin April H. Schmidt – p34 Road system: Assume stationary PLT X l with intensity  Higher level devices (HLD) Stationary point process (independent of X l ) Poisson process on X l (Cox process) with linear intensity  Stationar planar point process X H with planar intensity Infrastructure and costs … and network devices HL D

research & development France Telecom Group SpasWin April H. Schmidt – p35 Road system: Assume stationary PLT X l with intensity  Higher level devices (HLD) Stationary point process (independent of X l ) Poisson process on X l (Cox process) with linear intensity  Stationar planar point process X H with planar intensity Lower level devices (LLD) Stationary point process (indep. of X l and X H ) Poisson process on X l (Cox process) with linear intensity  Stationar planar point process with planar intensity Infrastructure and costs … and network devices LLD

research & development France Telecom Group SpasWin April H. Schmidt – p36 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 costs Logical connection

research & development France Telecom Group SpasWin April H. Schmidt – p37 Infrastructure and costs Physical connection (1)

research & development France Telecom Group SpasWin April H. Schmidt – p38 Infrastructure and costs Physical connection (2)

research & development France Telecom Group SpasWin April H. Schmidt – p39 Infrastructure and costs Mean shortest path length (1) Natural approach Disadvantages

research & development France Telecom Group SpasWin April H. Schmidt – p40 Infrastructure and costs Mean shortest path length (2) Alternative approach Disadvantages Simulation not clear Not very efficient

research & development France Telecom Group SpasWin April H. Schmidt – p41 Infrastructure and costs Mean shortest path length (3) Application of Neveu Independent from  The typical serving zone (the typical cell of a CVT) has to be simulated

research & development France Telecom Group SpasWin April H. Schmidt – p42 Infrastructure and costs Mean shortest path length (4)

research & development France Telecom Group SpasWin April H. Schmidt – p43 Infrastructure and costs Mean shortest path length (5) Estimation of Note: The integrals can be calculated analytically

research & development France Telecom Group SpasWin April H. Schmidt – p44 Infrastructure and costs Mean shortest path length (6)

research & development France Telecom Group SpasWin April H. Schmidt – p45 Infrastructure and costs Mean shortest path length (7)

research & development France Telecom Group SpasWin April H. Schmidt – p46 Infrastructure and costs Mean shortest path length (8)

research & development France Telecom Group SpasWin April H. Schmidt – p47 Infrastructure and costs Mean shortest subscriber line length

research & development France Telecom Group SpasWin April H. Schmidt – p48 Infrastructure and costs Application Mean length from LLD to HLD [km] in case of spatial placement

research & development France Telecom Group SpasWin April H. Schmidt – p49 5 Outlook and conclusion

research & development France Telecom Group SpasWin April H. Schmidt – p50 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

research & development France Telecom Group SpasWin April H. Schmidt – p51 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

research & development France Telecom Group SpasWin April H. Schmidt – p52 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 , 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, Bibliography

research & development France Telecom Group SpasWin April H. Schmidt – p53 This presentation is based on collaborative work with C. Gloaguen, J.-P. Lanquetin – France Telecom R&D, Paris&Belfort, France F. Fleischer, V. Schmidt, R. Thiedmann – Institute of Stochastics, Ulm University, Germany