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1 Analysis of Random Mobility Models with PDE's Michele Garetto Emilio Leonardi Politecnico di Torino Italy MobiHoc Firenze

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2 Introduction We revisit two widely used mobility models for ad-hoc networks: Random Way-Point (RWP) Random Direction (RD) Properties of these models have been recently investigated analytically Steady-state distribution of the nodes Perfect simulation [Vojnovic, Le Boudec ‘05]

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3 Motivation and contributions Open issues in the analysis of mobility models: 1) Analysis under non-stationary conditions 2) How to design a mobility model that achieves a desired steady-state distribution (e.g. an assigned node density distribution over the area) We address both issues above using a novel approach based on partial differential equations We introduce a non-uniform, non-stationary point of view in the analysis and design of mobility models

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4 Random waypoint (RWP) and Random Direction (RD) Pause Nodes travel on segments at constant speed The speed on each segment is chosen randomly from a generic distribution Random Way Point (RWP) : choose destination point Random Direction (RD) : choose travel duration Wrap-around Reflection

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5 Analysis of a mobility model using PDE Describe the state of a mobile node at time t Write how the state evolves over time Try to solve the equations analytically, under given boundary conditions and initial conditions at t = 0 At the steady-state In the transient regime

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6 = pdf of being in the move phase at position x, with speed v, at time t = pdf of being in the pause phase at location x, at time t Example: Random Direction model with exponential move/pause times Move time ~ exponential distribution ( ) Pause time ~ exponential distribution ( ) { position, phase (move or pause), speed } Note :

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7 Example: Random Direction in 1D Pause Move

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8 Wrap-around Random Direction: boundary conditions

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9 Reflection Random Direction: boundary conditions

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10 Random Direction model We have extended the equations of RD model to the case of general move and pause time distributions multi-dimensional domain We have proven that the solution of the equations, with assigned boundary and initial conditions, exists unique details in the paper…

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11 RD – Steady state analysis We obtain the uniform distribution (true in general for RD):

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12 Can we design a mobility model to achieve a desired node density distribution ? desired distributions:, The PDE formulation allows us to define a generalized RD model to achieve this goal: 1)scale the local speed of a node by the factor 2)Set the transition rate pause move to: Generalized RD model

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13 Generalized RD - example A metropolitan area divided into 3 rings R4R4 R3R3 R2R2 R1R1 Area 20 km x 20 km 8 million nodes Desired densities:

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14 Generalized RD - example

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15 Transient analysis of RD model Methodology of separation of variables Candidate solution : ( With wrap-around boundary conditions )

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16 Wrap-around conditions require that: Transient analysis of RD model For any, the equations are satisfied only for specific values of All are negative, except

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17 Transient analysis of RD model The initial conditions can be expanded using the standard Fourier series over the interval Each term of the expansion (except k = 0) decays exponentially over time with its own parameter As, all “propagation modes” k > 0 vanish, leaving only the steady-state uniform distribution ( k = 0 )

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18 Transient analysis of RD model Can be extended to : Rectangular domain (requires 2D Fourier expansion) Reflection boundary condition General move/pause time, through phase-type approximation details in the paper…

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19 Transient example – t = 0 RD Parameters : move ~ exp(1), pause ~ exp(1), V uniform [0,1]

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20 Transient example – t = 0.5

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21 Transient example – t = 1

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22 Transient example – t = 2

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23 Transient example – t = 4

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24 Transient example – t = 8

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25 Transient example – t = 16

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26 Application of the transient analysis Controlled simulations under non-stationary conditions (i.e. with time-varying node density) Capacity planning Network resilience and reliability Obtain a given dispersion rate of the nodes as a function of the parameters of the model e.g.: people leaving a crowded place (a conference room, a stadium, downtown area after work)

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27 Application of the transient analysis Stability of a wireless link Estimate of the initial location of the mobile node at time t = 0 Still in range of the access point at time t ?

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28 Conclusions The proposed PDE framework allows to: Define a generalized RD model to achieve a desired distribution of nodes in space (at the equilibrium) Analytically predict the evolution of node density over time (away from the equilibrium) The ability to obtain non-uniform and/or non-stationary behavior (in a predictable way) makes theoretical mobility models more attractive and close to applications

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29 The End Thanks for your attention questions & comments…

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