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					                                MobiHoc 2006 - Firenze


Analysis of Random Mobility Models
             with PDE's

          Michele Garetto
          Emilio Leonardi

        Politecnico di Torino
                Italy



                                                 1
                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]
                                                       2
         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
                                                               3
Random waypoint (RWP) and
  Random Direction (RD)
                  Nodes travel on segments
                  at constant speed

                  The speed on each
         Pause    segment is chosen
                  randomly from a generic
                  distribution

                 Random Way Point (RWP) :
                   choose destination point
     Pause
                  Random Direction (RD) :
                   choose travel duration
                       Wrap-around
                       Reflection           4
   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
                                                         5
       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 t

   Note:
                                                           6
Example: Random Direction in 1D
                            Move




                            Pause




                                    7
    Random Direction: boundary conditions

   Wrap-around




                                            8
    Random Direction: boundary conditions

   Reflection




                                            9
        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…
                                                    10
              RD – Steady state analysis




   We obtain the uniform distribution (true in general for RD):




                                                                   11
          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:

    1) scale the local speed of a node by the factor


    2) Set the transition rate pause move to:


                                                       12
     Generalized RD - example
   A metropolitan area divided into 3 rings

             R1
             R2             Area 20 km x 20 km
             R3             8 million nodes
             R4             Desired densities:


                           



                                                  13
     Generalized RD - example

1800
1600
1400
1200
1000
 800
 600
 400                                              10
 200                                      5
   0
   -10                                0
         -5                                   Y
                  0              -5
              X       5
                           -10
                          10
                                                       14
     Transient analysis of RD model

                                       ( With wrap-around
                                      boundary conditions )




   Methodology of separation of variables



   Candidate solution:


                                                       15
    Transient analysis of RD model



   Wrap-around conditions require that:




   For any         , the equations are satisfied
    only for specific values of

   All   are negative, except
                                                    16
    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 )
                                                   17
    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…
                                         18
        Transient example – t = 0
RD Parameters : move ~ exp(1), pause ~ exp(1), V uniform [0,1]
 0.18
                                                                                                0.18
 0.16
                                                                                                0.16
 0.14                                                                                           0.14
 0.12                                                                                           0.12
                                                                                                0.1
  0.1
                                                                                                0.08
 0.08                                                                                           0.06
 0.06                                                                                           0.04
                                                                                                0.02
 0.04                                                                                           0
 0.02

  -50
     -4
          -3
               -2
                    -1
                         0                                                                  5
                             1                                                      3   4
                                 2                                          1   2
                                     3                             -1   0
                                         4               -3   -2
                                             5 -5   -4                                           19
    Transient example – t = 0.5
0.18
                                                                            0.12
0.16
0.14                                                                        0.1
0.12                                                                        0.08
 0.1                                                                        0.06
0.08                                                                        0.04
0.06                                                                        0.02
0.04                                                                        0
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                        5
                   1                                            3   4
                       2                                1   2
                           3                   -1   0
                               4
                                 5 -5 -4 -3 -2


                                                                            20
    Transient example – t = 1
0.18
                                                                            0.09
0.16                                                                        0.08
0.14                                                                        0.07
0.12                                                                        0.06
                                                                            0.05
 0.1                                                                        0.04
0.08                                                                        0.03
0.06                                                                        0.02
                                                                            0.01
0.04                                                                        0
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                        5
                   1                                            3   4
                       2                                1   2
                           3                   -1   0
                               4
                                 5 -5 -4 -3 -2


                                                                            21
    Transient example – t = 2
0.18
                                                                            0.06
0.16
0.14                                                                        0.05
0.12                                                                        0.04
 0.1                                                                        0.03
0.08                                                                        0.02
0.06                                                                        0.01
0.04                                                                        0
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                        5
                   1                                            3   4
                       2                                1   2
                           3                   -1   0
                               4
                                 5 -5 -4 -3 -2


                                                                            22
     Transient example – t = 4
0.18
                                                                              0.03
0.16
0.14                                                                          0.025
0.12                                                                          0.02
 0.1                                                                          0.015
0.08                                                                          0.01
0.06                                                                          0.005
0.04                                                                          0
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                          5
                   1                                              3   4
                       2                                  1   2
                           3                     -1   0
                               4
                                   5 -5 -4 -3 -2


                                                                               23
     Transient example – t = 8
0.18
                                                                              0.022
0.16                                                                          0.02
0.14                                                                          0.018
                                                                              0.016
0.12                                                                          0.014
 0.1                                                                          0.012
                                                                              0.01
0.08                                                                          0.008
0.06                                                                          0.006
                                                                              0.004
0.04                                                                          0.002
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                          5
                   1                                              3   4
                       2                                  1   2
                           3                     -1   0
                               4
                                   5 -5 -4 -3 -2


                                                                               24
     Transient example – t = 16
0.18
                                                                              0.016
0.16                                                                          0.015
0.14                                                                          0.014
                                                                              0.013
0.12                                                                          0.012
                                                                              0.011
 0.1                                                                          0.01
0.08                                                                          0.009
                                                                              0.008
0.06                                                                          0.007
0.04                                                                          0.006
                                                                              0.005
0.02
   0
  -5
    -4
      -3
        -2
          -1
               0                                                          5
                   1                                              3   4
                       2                                  1   2
                           3                     -1   0
                               4
                                   5 -5 -4 -3 -2


                                                                               25
    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)


                                                             26
     Application of the transient analysis

   Stability of a wireless link
                                      Still in range of the
                                   access point at time t ?



                                              Estimate of the
                                             initial location of
                                           the mobile node at
                                                      time t = 0




                                                              27
                 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

                                                          28
        The End




Thanks for your attention



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