Analysis of Random Mobility Models with PDE's

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

2. 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]

3. Motivation and contributions • Open issues in the analysis of mobility models: • Analysis under non-stationary conditions • 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

4. Random waypoint (RWP) and Random Direction (RD) Nodes travel on segments at constant speed The speed on each segment is chosen randomly from a generic distribution Pause • Random Way Point (RWP) : • choose destination point Pause • Random Direction (RD) : • choose travel duration • Wrap-around • Reflection

5. 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

6. 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 pausephase at location x, at time t • Note:

7. Example: Random Direction in 1D Move Pause

8. Random Direction: boundary conditions • Wrap-around

9. Random Direction: boundary conditions • Reflection

10. 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…

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

12. 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:

13. Generalized RD - example • A metropolitan area divided into 3 rings R1 • Area 20 km x 20 km • 8 million nodes • Desired densities: R2 R3 R4

14. Generalized RD - example

15. Transient analysis of RD model ( With wrap-around boundary conditions ) • Methodology of separation of variables • Candidate solution:

16. For any , the equations are satisfied only for specific values of • All are negative, except Transient analysis of RD model • Wrap-around conditions require that:

17. 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 )

18. 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…

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

20. Transient example – t = 0.5

21. Transient example – t = 1

22. Transient example – t = 2

23. Transient example – t = 4

24. Transient example – t = 8

25. Transient example – t = 16

26. 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)

27. Estimate of the initial location of the mobile node at time t = 0 Application of the transient analysis • Stability of a wireless link Still in range of the access point at time t ?

28. 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

29. The End Thanks for your attention questions & comments…