# 1 Analysis of Random Mobility Models with PDE's Michele Garetto Emilio Leonardi Politecnico di Torino Italy MobiHoc 2006 - Firenze.

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1 Analysis of Random Mobility Models with PDE's Michele Garetto Emilio Leonardi Politecnico di Torino Italy MobiHoc 2006 - Firenze

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]

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

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

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

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 :

7 Example: Random Direction in 1D Pause Move

8 Wrap-around Random Direction: boundary conditions

9 Reflection Random Direction: boundary conditions

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…

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

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

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: 

14 Generalized RD - example

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

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

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 )

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…

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

20 Transient example – t = 0.5

21 Transient example – t = 1

22 Transient example – t = 2

23 Transient example – t = 4

24 Transient example – t = 8

25 Transient example – t = 16

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)

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 ?

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