Perfect Simulation and Stationarity of a Class of Mobility Models

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. Examples

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

4. RWP on general connected domain

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

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

7. a restricted RWP (Jardosh et al, 2003) (cont’d)

8. random walk with wrapping

9. random walk with reflection

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. 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. 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. 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) Consider: Prob((Path at time t) = p) Q: How long it takes for this probability to converge to steady-state?

14. Initial transient lasts as long as a typical simulation duration On transience longevity (cont’d) • Transient phase lasts 1000’s of seconds • Typical simulation run is of the order 1000 seconds Prob((Path at time t) = path)

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. Tn+1 Tn Trip selection rule: at a trip transition instant Tn, choose (Pn,Sn) The Random Trip Mobility Model (basic definitions) Mn+1=Pn+1(0) trip end Path Pn : [0,1]  A trip duration Sn Mn=Pn(0) domain A trip start

17. Path and Trip duration (Pn,Sn) Example (RWP on a convex domain*): Path: Pn(u) = u Mn + (1-u) Mn+1, u[0,1] Trip duration: Sn = (length of Pn) / Vn Vn = 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. Path and Trip duration (Pn,Sn)(cont’d) Example (Random Walk Models): • Pick a movement direction • Draw a trip duration Sn • Path specified by the direction and trip duration+ additional rules Additional rules: • wrapping • reflection

19. The Random Trip Mobility Model (further definitions) • The trip selection rule is driven by phases In • Phases In is a Markov chain • Example (RWP): In = 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. The Random Trip Mobility Model(assumptions) (H1) (Pn,Sn) independent of all past, conditional on (Mn,In)

21. The Random Trip Mobility Model (assumptions cont’d) (H2) Either is true: (H2a) • Mn+1 independent of past phases In,In-1, … and n, conditional on In • (renewal points) for a set of selected transitions of In,Mn+1 independent of all past, conditional on In or (H2b) • Mn independent of In and n • (Sn,In+1) independent of all past, conditional on In

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

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, E0(S0), is finite. Whenever it exists, a time-stationary distribution is unique. Proof: • shows that (In,Pn,Sn) has a unique stationary distribution • verifies conditions of Slivnyak’s inverse construction

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. 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] Prob0(I0 = i) E0(S0 | I0 = i) Proof: Palm inversion formula.

26. RWP time-stationary distributions Theorem: Under the time-stationary distribution: Conditionally on the phase I(t)=(l,l’,r,move) • Numerical speed is independent of path and position;speed density = • dP(P(t)(0)=m0,P(t)(1)=m1)=Kll’ d(m0,m1) • Given (P(t)(0) =m0,P(t)(1) =m1), position X(t) uniform on the segment [m0,m1] Conditionally on the phase I(t)=(l,l’,r,pause), • Position and remaining pause time are independent • Position is uniform in A • Density of the remaining pause time = Remark: the independency property in item 1 is new (previously conjectured)

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. Perfect Sampling Algorithm (cont’d) • 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. Illustration: Perfect samples of positionsfor some of our examples Restricted RWPs: RWP on a non convex domain:

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 M0 is uniformly distributed on the domain A, so is Mn for any n>0. • The same holds for random walk with reflection. Proof: By periodicity of the wrapping and reflection mappings.

31. Perfect sampling for random walk models (cont’d) • For RW with wrapping: • Similar result obtained for RW with reflection

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. Conclusion (cont’d) 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. 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)