Analysis of Random Mobility Models with PDE's
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Transcript of Analysis of Random Mobility Models with PDE's
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Analysis of Random Mobility Models with PDE's Michele GarettoEmilio Leonardi
Politecnico di TorinoItalyMobiHoc 2006 - Firenze
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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]
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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
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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
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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
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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:
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Example: Random Direction in 1DPause Move
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Random Direction: boundary conditionsWrap-around
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Random Direction: boundary conditionsReflection
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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
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RD Steady state analysis We obtain the uniform distribution (true in general for RD):
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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:
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A metropolitan area divided into 3 ringsGeneralized RD - exampleR4R3R2R1 Area 20 km x 20 km 8 million nodes Desired densities:
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Generalized RD - example
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Transient analysis of RD modelMethodology of separation of variablesCandidate solution:( With wrap-around boundary conditions )
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Transient analysis of RD model
Wrap-around conditions require that:
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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 )
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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
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Transient example t = 0RD Parameters : move ~ exp(1), pause ~ exp(1), V uniform [0,1]
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Transient example t = 0.5
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Transient example t = 1
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Transient example t = 2
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Transient example t = 4
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Transient example t = 8
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Transient example t = 16
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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
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Stability of a wireless link
Application of the transient analysisStill in range of the access point at time t ?
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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
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The EndThanks for your attentionquestions & comments