Analysis of Random Mobility Models with PDE's Michele GarettoEmilio Leonardi

Politecnico di TorinoItalyMobiHoc 2006 - Firenze

IntroductionWe 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 analyticallySteady-state distribution of the nodes Perfect simulation [Vojnovic, Le Boudec 05]

Motivation and contributions Open issues in the analysis of mobility models: Analysis under non-stationary conditionsHow 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

Random waypoint (RWP) and Random Direction (RD) Pause 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

Analysis of a mobility model using PDEDescribe 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

Example: Random Direction model with exponential move/pause times Move time ~ exponential distribution ()Pause time ~ exponential distribution () { position, phase (move or pause), speed }= 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 tNote:

Example: Random Direction in 1DPause Move

Random Direction: boundary conditionsWrap-around

Random Direction: boundary conditionsReflection

Random Direction modelWe 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

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

Generalized RD model 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:

scale the local speed of a node by the factor

Set the transition rate pause move to:

A metropolitan area divided into 3 ringsGeneralized RD - exampleR4R3R2R1 Area 20 km x 20 km 8 million nodes Desired densities:

Generalized RD - example

Transient analysis of RD modelMethodology of separation of variablesCandidate solution:( With wrap-around boundary conditions )

Transient analysis of RD model

Wrap-around conditions require that:

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 Transient analysis of RD modelAs , all propagation modes k > 0 vanish, leaving only the steady-state uniform distribution ( k = 0 )

Can be extended to : Rectangular domain (requires 2D Fourier expansion) Reflection boundary condition General move/pause time, through phase-type approximation

Transient analysis of RD modeldetails in the paper

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

Transient example t = 0.5

Transient example t = 1

Transient example t = 2

Transient example t = 4

Transient example t = 8

Transient example t = 16

Controlled simulations under non-stationary conditions (i.e. with time-varying node density)Capacity planningNetwork resilience and reliability

Obtain a given dispersion rate of the nodes as a function of the parameters of the modele.g.: people leaving a crowded place (a conference room, a stadium, downtown area after work)

Application of the transient analysis

Stability of a wireless link

Application of the transient analysisStill in range of the access point at time t ?

ConclusionsThe 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

The EndThanks for your attentionquestions & comments