1. 1 Perfect Simulation and Stationarity of a Class of Mobility Models Jean-Yves Le Boudec (EPFL) & Milan Vojnovic (Microsoft Research Cambridge) IEEE Infocom 05, Miami FL, March 2005.

2. 2 Examples

3. 3 RWP: random waypoint (Johnson and Maltz, 1996)

4. 4 RWP on general connected domain

5. 5 RWP on general connected domain (contd) called city-section (Camp et al, 2002)

6. 6 a restricted RWP ( Blažević et al, 2004)

7. 7 a restricted RWP (Jardosh et al, 2003) (contd)

8. 8 random walk with wrapping

9. 9 random walk with reflection

10. 10 What do we know about these models ? RWP considered harmful by Yoon et al (IEEE Infocom 2003) –speed decay: in ns-2 simulations, average speed decays with time –fix: redefine the speed distribution (at waypoints) Avoid transience : initialize mobility state, so that mobility is in steady-state throughout a simulation ( = perfect simulation) –Partial fix for RWP by Yoon et al (ACM Mobicom 2003): initialize the speed to a sample from its time-stationary distribution –Complete fix for RWP on a rectangle by Lin et al (IEEE Infocom 2004): initialize also node position to a sample drawn from the time- stationary distribution of position

11. 11 Problems that we study The speed decay is due to non existence of steady-state –Under what conditions there exists a steady-state ? –If exists, unique ? I am interested in steady-state of my mobility model –What are steady-state distributions of mobility states for my model ? I want to run perfect simulations of mobility –How do I initialize my simulation so that it is perfect, i.e. free of transients ?

12. 12 Why do we care about transients ? Or: why do we wish to run perfect simulations of mobility ? Simulations of mobility are commonly run with initial transient The simulation traces are then truncated and initial part thrown away in order to alleviate the transience effects How do we know where to truncate ? Initial transient may last as long as typical simulation duration ! next couple of slides …

13. 13 On transience longevity Example: revisit the restricted RWP instance: mobile always moves speed fixed to 1.25 m/s destination vertex drawn at random paths are shortest-length between vertices pairs default initialization: mobile placed at a random vertex (as in Jardosh et al) Q: How long it takes for this probability to converge to steady-state? Consider: Prob((Path at time t) = p)

14. 14 On transience longevity (contd) Transient phase lasts 1000s of seconds Typical simulation run is of the order 1000 seconds Prob((Path at time t) = path) Initial transient lasts as long as a typical simulation duration

15. 15 Outline Definition: The Random Trip Mobility Model –many existing mobility models in one (all on these slides), and new ones –easy-to-check conditions that guarantee existence of a unique time-stationary distribution –time-stationary distributions and their properties Perfect sampling algorithm –for the broad class of random trip mobility models –novelty: requires no knowledge of geometric normalization constants when they are difficult to compute Conclusion Pointer to randomtrip tool to use with ns-2

16. 16 The Random Trip Mobility Model (basic definitions) domain A Path P n : [0,1] A trip duration S n M n =P n (0) M n+1 =P n+1 (0) trip start TnTn trip end T n+1 Trip selection rule: at a trip transition instant T n, choose (P n,S n )

17. 17 Path and Trip duration (P n,S n ) Example (RWP on a convex domain*): Path: P n (u) = u M n + (1-u) M n+1, u [0,1] Trip duration: S n = (length of P n ) / V n V n = numeric speed drawn from a given distribution *convex domain := a domain such that line segment between any two points in the domain lies in the domain

18. 18 Path and Trip duration (P n,S n ) (contd) Example (Random Walk Models): Pick a movement direction Draw a trip duration S n Path specified by the direction and trip duration + additional rules Additional rules: wrapping reflection

19. 19 The Random Trip Mobility Model (further definitions) The trip selection rule is driven by phases I n Phases I n is a Markov chain –Example (RWP): I n = either pause or move Mobility state: (I(t),P(t),S(t),U(t)) U(t) = fraction of time elapsed on the trip at time t

20. 20 The Random Trip Mobility Model (assumptions) (H1)(P n,S n ) independent of all past, conditional on (M n,I n )

21. 21 The Random Trip Mobility Model (assumptions contd) (H2) Either is true: (H2a) M n+1 independent of past phases I n,I n-1, … and n, conditional on I n (renewal points) for a set of selected transitions of I n, M n+1 independent of all past, conditional on I n or (H2b) M n independent of I n and n (S n,I n+1 ) independent of all past, conditional on I n

22. 22 Random Trip Mobility Model (assumptions contd) (H3)Markov chain I n positive recurrent True, in particular: 1. state space finite & 2. all states communicate Remark: (H1)-(H3) true for all examples on these slides

23. 23 When does a time-stationary distribution of mobility state exist and is it unique ? Theorem: Under (H1)-(H3), a random trip mobility model has a time-stationary distribution, if and only if the mean trip duration sampled at trip transition instants, E 0 (S 0 ), is finite. Whenever it exists, a time-stationary distribution is unique. Proof: shows that (I n,P n,S n ) has a unique stationary distribution verifies conditions of Slivnyaks inverse construction

24. 24 When the conditions fails ? Example: RWP as was implemented in ns-2 At trip endpoints, numeric speed is independent of trip distance => Numeric speed is uniformly distributed on an interval (0,vmax] => Found and called harmful by Yoon et al (IEEE Infocom 2003) The theorem tells us that for this RWP, no steady-state exists This clarifies the issue for the first time

25. 25 What is time-stationary distribution of mobility state ? Theorem: Assume (I(t),P(t),S(t),U(t)) has a unique time-stationary distribution (provided by our previous theorem). The time-stationary distribution of (I(t),P(t),S(t),U(t)) is U(0) is independent of (I(0),P(0),S(0)) and uniform on [0,1] Proof: Palm inversion formula. Prob 0 (I 0 = i) E 0 (S 0 | I 0 = i)

26. 26 RWP time-stationary distributions Theorem: Under the time-stationary distribution: Conditionally on the phase I(t)=(l,l,r,move) 1.Numerical speed is independent of path and position; speed density = 2.dP(P(t)(0)=m 0,P(t)(1)=m 1 )=K ll d(m 0,m 1 ) 3.Given (P(t)(0) =m 0,P(t)(1) =m 1 ), position X(t) uniform on the segment [m 0,m 1 ] Conditionally on the phase I(t)=(l,l,r,pause), 1.Position and remaining pause time are independent 2.Position is uniform in A 3.Density of the remaining pause time = Remark: the independency property in item 1 is new (previously conjectured)

27. 27 Perfect Sampling Algorithm We want to initialize mobility state at time = 0 to a sample drawn from the time-stationary distribution To do this, we need to know to draw a sample from the time- stationary distribution –One technique: Rejection Sampling Previous work: Rejection sampling for RWP on a rectangle (Lin et al, Infocom 2004) –requires knowing geometric constants such as average distance between two random points on a rectangle Geometric constants are known in closed-form for some elementary geometrical objects If closed-form unknown, can be a priori estimated by Monte Carlo simulations –time complexity ?

28. 28 Perfect Sampling Algorithm (contd) Our algorithm: Perfect sampling without necessarily knowing geometric constants –If average distance between two trip endpoints is uknown, use a bound on this distance (diameter) In many cases, diameter is easy to compute Example (the restricted RWP):

29. 29 Illustration: Perfect samples of positions for some of our examples Restricted RWPs: RWP on a non convex domain:

30. 30 Perfect sampling for random walk models By definition, for RWP models, we know distributions of the mobility state at trip transition instants For random walk models we need first to find these distributions Theorems: For random walk with wrapping, if M 0 is uniformly distributed on the domain A, so is M n for any n>0. The same holds for random walk with reflection. Proof: By periodicity of the wrapping and reflection mappings.

31. 31 Perfect sampling for random walk models (contd) For RW with wrapping: Similar result obtained for RW with reflection

32. 32 Conclusion Proposed: the Random Trip Mobility Model –contains many existing and new mobility models in one Gave conditions for the Random Trip Mobility Model that guarantee existence and uniqueness of a time-stationary distribution Proposed a perfect sampling algorithm to sample mobility state from its time-stationary distribution (whenever exists) –The sampling algorithm is for a broad set of the random trip mobility models –The sampling algorithm does not require knowing normalization constants when they are difficult to compute – a bound on trip distance suffices –The sampling algorithm is implemented to use with ns-2, which enables to run perfect simulations of mobility

33. 33 Conclusion (contd) By-products: Demonstrated that transience for some mobility models may last as long as a typical simulation duration --- a compelling reason to run perfect simulations of mobility Proved that in steady-state of RWP models, node position and numerical speed are independent --- previously conjectured Showed new distribution invariance properties for random walk models with wrapping and reflection, which yield perfect sampling algorithm for these models

34. 34 ns-2 code Project Web Page: The Random Trip Mobility Model http://ic1wwww.epfl.ch/RandomTrip/ Links to: download randomtrip –ns-2 code of random trip, with perfect simulation (by S. PalChaudhuri, Rice University)