Stochastic geometry and telecommunications network architecture - - PowerPoint PPT Presentation

12-98 - N°1

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Stochastic geometry and telecommunications network architecture

From research works of

• M. Klein, C. Gloaguen, I. Le Madec, R. Muret, FT / CNET / DAC
• F. Baccelli*, K. Tchoumatchenko*, S. Zuyev**, INRIA Sophia Antipolis
• M. Lebourges, FX Godron, FT / BD / DPS
• S. Capitaneu*** CNAM

(+ D. Kofman, JL Rougier (ENST), I. Molchanov (Glasgow), ...) *Currently Ecole Normale Supérieure ** Currently Glasgow University *** Back in Roumanie Marc Lebourges FT / BD / DPS

12-98 - N°2

Branche Développement - Direction du Plan et de la Stratégie

Stochastic geometry et and telecom network architecture

• Competition in telecommunications : need of simple, realistic and global telecommunication

network cost models, using aggregated data for :

– evaluation of entry in new markets and a new countries, – productivity benchmark, – adequation of tariffs structures with costs structures, – regulatory obligations : tariffs of interconnection services, cost of universal service, – specifications and evaluation of new network technologies.

• The objectives of stochastic geometry network models is to lead to explicit mathematical

relationships :

• network cost driver (infrastructure length, copper and fiber pair length, transmission and switching system

capacity, ... ) = fct(Surface of network area, number of subscribers, global traffic volumes, network architecture parameters) synthetic model of the production & cost functions of telecommunication services, explicit formulation of the optimal architecture of the network (architecture, dimensioning) formal description, evaluation and optimisation of network technology (IP multicast routing protocols ..)

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Contents

• Stochastic geometry models of :

– subscribers, nodes, access infrastructure (fix & mobiles), – transport infrastructure, – traffic streams and routing (fix, mobiles, multicast ...) – homogeneous or inhomogeneity models ?

• Exemple of applications

– Specification and optimisation of a multicast internet protocol – Superposition of technical, tarification, regulatory, competitive zones on the same territory – Optimisation of access nodes localisation as a function of the heterogeneous customer demography

• Open questions and conclusions

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Telecommunications networks

• Classical access : a copper pair/subscriber, through a hierarchy of cables of increasing capacity goes from the

subscriber premise to the nearest network node,. There, an electronic card transforms, when needed, the analog electrical signal into a bit stream

• Switching - routing : Access nodes concentrate numerical signals from a set of subscribers and send it to routers /

switches, which route numerical signals towards their destination(s). The concentration and the routing functions may be located in the same building, or may be distant. The number of switches / routers seen by the traffic, the network level needed to route the trafic, depends on the distance between origin and destination(s)

• Transmission :access nodes, switches and routers are machines linked by cables of optical fibers. The cable network

geometry between buildings is analog to an road infrastructure between towns. At the extremities of fiber are opto- electronic systems, which convert electronic numerical signals produced by switching machines into optical numerical signals travelling in the fibers.

• Mobiles :equivalent to fixed network, but with radio wireless access and databases managing the localisation of mobiles

and controlling the routings executed by switches .

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A Poisson process model of network nodes

• For networks operators, localisation of nodes & subscribers are random : operators know how many

nodes they need but have constraints on nodes localisation and cannot choose them exactly

• Total network cost = fct(number of nodes, pattern of nodes & customers) : network cost depend on

parameters of the geometric laws of nodes and edges of the network, not of a specific realisation

• Poisson process : reference stochastic point process in a plan. Any point process + random -> Limit -> Poisson process
• Voronoï tesselation : each point of the plan parented to the nearest point of the process => convex polygons zones

covering the area

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Hierarchical network model

• N level of hierarchy from subscribers to top level nodes :

– Simple / dual /triple parenting :

• Distance (nearest network node) =
• Distance (Nst nearest network node) =

– Polynomial cost fonction (lengths, capacity) : explicit expression of the optimal intensity of intermediary nodes : Optimisation of access networks, mobile handover databases , multicast IP routings

( )

0 5 , . . Surface typical zone

( ) ( ) ( )

Dis ce N nearest Node N N tan . . . . − ⋅ − + − 1 2 1 1 2 1

12-98 - N°7

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Network connectivity

• Node (Poisson process), Zones (Voronoï tesselation) + Delaunay triangulation for physical Connectivity :

reference model for phyical network infrastructure connecting nodes

• Delaunay triangulation : maximum physical connectivity between network nodes : shortest path on a Delaunay graph is a

good approximation of a straight line : 5% difference on average

• Network connectivity costs : Delaunay triangulation costly, minimal weight tree cheapest :
• Asymptitic length =
• Delaunay triangulation : k = 3,395
• Minimum weight tree : k = 0,656

k Number Network

Nodes Surface

. .

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Traffic - Circuit - Leased Lines routing

• Demand volume (euclidian distance origin - extremities) :

– Geometric analysis -> Parametric form + demand statistics -> Value of parameters

Demand law (euclidian distance)

• Routing 1 to 1 (classical traffic, leased lines)

– Technical zones (Local zone, transit zone, ...) = Voronoï tesselation (Number of zones) – Demand law(Euclidian distance) + geometric calculation (Voronoï tesselation / Technical zones)

Routing factors (euclidian distance)

• Network nodes and link capacity :

– Demand law (euclidian distance) * Routing factors (euclidian distance)

Load statistics on each network node and link category

=> Traffic - circuits- LL global production function

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Homogeneous ou inhomogeneous models ?

• Realism of homogeneous Poisson process vs inhomogeneous human demography ?

– First thought homogenous model : reference theoritical model note directly applicable but

• Homogeneous model may be more realistic than forecasted in well defined cases

– There are objects and scales where homogenity is a reasonable assumption – One of the effect of network organisation and hierarchy is to erase inhomogenity (concentration of demand) – Interesting results are often relative measurements, ratios, for which homogeneous models give approximately unbiaised values (eg intersection of zones) Good agreement between stochasticgeometry models and network statistics in all cases tested until now

• Inhomogeneous model

– Empirical approach : co-existence of several Poisson process of different intensities (Markov Modulated Poisson Process). Currently used – Space transformation : mathematical transformation Inhomogeneous -> Homogenous - resolution of the problem -> Conversion of the solution back in the inhomogeneous space – Stochastic optimisation : optimal heterogenous density of network node as a function of optimal heterogeneous intensity of customer demand.

12-98 - N°10

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Optimal Internet multicast routing

• Multicast routings in the INTERNET : n -> n’ (n & n’ from units to tens of thousands)

sources, destinations and spatial repartition in permanent evolution. How to model and optimise multicast routing protocol ? no classical model of the problem. Stochastic geometry network model : – stochastic routing protocol specification – parametric optimisation of routing protocol algorithm : Case 1 : hierarchical Center Based Tree (HCBT) Not all routers have multicast functions General optimisation result : 3 levels of hierarchy is enough whatever the size of the multicast tree

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Optimal Internet multicast routing (2)

Case 2 : non-hierarchical Center Based Tree (all routers have multicast functions) Under study

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Superposition of zones (1)

• Several independant type of zones co-exists on the same territory (e.g. France) :

– Technical zones of a network operator : Distribution Frame zone, Switch-router zone, transit zone, ... – Tariffs zones : local tariff zone, regional tariff zone, ... – Regulatory : Zones Locales de Tri (ZLT) in France, LATA in the USA... – Technical zones of several network operators

• How do these types of zones interact ?

– Tariff /Cost - Market / Regulatory - Cost / Regulatory analysis

• Model :

– Two types of zones -> Two Independent Poisson process -> Two independent Voronoï tesselations Zones interaction <=> Geometrical characteristics of the tesselation intersection of the two Voronoï tesselation Statistics concerning the interaction between the two zones.

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Superposition of zones (2)

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Stochastic optimisation : application

• Question

– Customer : inhomogeneous poisson process – Polynomial access cost function – Known total number of access nodes

– Optimal localisation access nodes ?

• General result of the stochastic optimisation theory

– The intensity of the access node process is a power function of the intensity of customer process

– The parameter of the optimal power function is an explicit function of the parameters of the cost function

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Open question and conclusions

• Statistical tests

– Network statistics OK – Statistical nature of actual point processes : in progress

• Integration of all theoritical results for network modelling
• Very short and interactive relations between :

– Pure mathematics – Applied mathematics – Industry applications

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Web references

• International seminar , Dagshtul, Mars 1998 :

– http://www.mathematik.uni-ulm.de/stochastik/dagstuhl

• INRIA-ENS :

– http:// www.inria.fr/mistral et www.ens.fr:mistral

– Pages de F. Baccelli et S. Zuyev

• RNRT (Réseau National de Recherches en Télécommunications) : «Georges» Project

– Responsable : catherine.gloaguen@francetelecom.fr