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 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°1
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 ..) Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°2
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 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°3
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 . Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°4
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 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°5
Hierarchical network model • N level of hierarchy from subscribers to top level nodes : – Simple / dual /triple parenting : ( ) 0 5 , . Surface typical zone . Distance (nearest network node) = • ( ) − + 2 . N 1 1 ( ) − ⋅ Dis tan ce N 1 . nearest Node . ( ) • Distance (N st nearest network node) = − 2 . N 1 – Polynomial cost fonction (lengths, capacity) : explicit expression of the optimal intensity of intermediary nodes : � Optimisation of access networks, � mobile handover databases , � multicast IP routings Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°6
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 : k . Number . Network Nodes Surface • Asymptitic length = • Delaunay triangulation : k = 3,395 • Minimum weight tree : k = 0,656 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°7
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 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°8
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. Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°9
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 Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°10
Optimal Internet multicast routing (2) Case 2 : non-hierarchical Center Based Tree (all routers have multicast functions) Under study Branche Développement - Direction du Plan et de la Stratégie 12-98 - N°11
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