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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
   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

 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

   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
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