1 Analysis of Random Mobility Models with PDE's Michele Garetto Emilio Leonardi Politecnico di Torino Italy MobiHoc 2006 - Firenze
Feb 03, 2016
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