Properties of Random Direction Models

1. Properties of Random Direction Models Philippe Nain, Don Towsley, Benyuan Liu, Zhen Liu

2. Main mobility models • Random Waypoint • Random Direction

3. Random Waypoint • Pick location x at random • Go to x at constant speed v Stationary distribution of node location not uniform in area

4. Random Direction • Pick direction θat random • Move in direction θat constant speed v for time τ • Upon hitting boundary reflection or wrap around

5. Reflection in 2D

6. Wrap around in 2D

7. Question: Under what condition(s) stationary distribution of node location uniform over area?

8. Notation • Tj: beginning j-th movement • τj = Tj+1 - Tj: duration j-th mvt • sj: speed in j-th mvt • θ(t) : direction time t • θj = θ(Tj) : direction start j-th mvt • γj : relative direction • (Tj , sj , γj)j1: mvt pattern

9. γj = relative direction • 1D:γj {-1,+1}θj = θ (Tj-)γj Wrap around:θj = θj-1γj • 2D:γj[0,2π) θj = θ(Tj-)+ γj -2π(θ(Tj-) + γj)/2π Wrap around: θj = θj-1+ γj -2π (θ(Tj-) + γj)/2π

10. Result I (1D & 2D; Refl. & Wr)If location and direction uniformly distributed attime t=0 then these properties hold at any time t>0 under any movement pattern.

11. Proof (1D =[0,1) & Wrap around) • Mvt pattern (Tj , sj , γj)j1fixed • Assumption: P(X(0) < x, θ(0) = θ) = x/2 • Initial speed = s0 • 0≤t<T1 : X(t) = X(0) + θ(0)s0t- X(0) + θ(0)s0t P(X(t) < x, θ(t) = θ) = ½ ∫[0,1] 1u + θ(0)s0t- u + θ(0)s0t < xdu = x/2 (X(t),θ(t)) unif. distr. [0,1)x{-1,1}, 0≤t<T1.

12. Proof (cont’ - 1D & Wrap around) • For wrap aroundθ(T1)= θ(0)γ1 X(T1) = X(0) + θ(0)γ1s0T1- X(0) + θ(0)γ1s0T1 • Conditioning on initial location and direction yields (X(T1), θ(T1)) uniformly distributed in [0,1){-1,+1}. Proof for [0,1) & wrap around concluded by induction argument.

13. Proof (cont’ - 1D & Reflection) • Lemma: Take Tjr = Tjw, γjr = γjw, sjr = 2sjw If relations Xr(t) = 2Xw(t), 0 ≤ Xw(t) < ½ = 2(1-Xw(t)), ½ ≤ Xw(t) <1θr(t) = θw(t), 0 ≤ Xw(t) < ½ = -θw(t), ½ ≤ Xw(t) < 1 hold at t=0 then hold for all t>0. • Use lemma and result for wrap around to conclude proof for 1D and reflection.

14. Proof (cont’ - 2D Wrap around & reflection). • Area: rectangle, disk, … • Wrap around: direct argument like in 1D • Reflection: use relation between wrap around & reflection – See Infocom’05 paper.

15. Corollary • N mobiles unif. distr. on [0,1] (or [0,1]2) with equally likely orientation at t=0 • Mobiles move independently of each other Mobiles uniformly distributed with equally likely orientation for all t>0.

16. Remarks (1D models) • Additive relative direction ok θj = (θj-1 + γj ) mod 2 , γj {0,1} γj = 0 (resp. 1) if direction at time Tj not modified • θj = -1 with prob. Q = +1 with prob 1-q Uniform stationary distr. iffq=1/2

17. How can mobiles reach uniformstationary distributions for location and orientation starting from any initial state?

18. Mvt vector {yj= (τj,sj,γj,j)j}{j}j : environment (finite-state M.C.) • Assumptions{yj}j aperiodic, Harris recurrent M.C., with unique invariant probability measure q.

19. {yj}j,yjY, Markov chain • {yj}j -irreducible if there exists measure  on (Y) such that, whenever (A)>0, then Py(return time to A) > 0 for all y A • {yj}j Harris recurrent if it is -irreducible and Py(j1 1{yjA} = ) = 1 for all y A such that (A)>0.

20. Z(t) = (X(t), θ(t), Y(t)): Markov process Y(t) = (R(t),S(t),γ(t),(t) R(t) = remaining travel time at time t S(t) = speed at time t γ(t) = relative direction at time t(t) = state of environment at time t Result II (1D, 2D -- Limiting distribution)If expected travel times τ finite, then {Z(t)}thas unique invariant probability measure. In particular, stationary location and direction uniformly distributed.

21. Outline of proof (1D = [0,1]) • {zj}j has unique stationary distribution p A=[0,x){θ}[0,τ)  S {γ}  {m}q stat. distr. of mvt vector{yj}j p(A)=(x/2) q([0,τ)  S {γ}  {m}) • Palm formula Lim t P(Z(t)  A) = (1/E0[T2])  E0[∫[0,T2] 1(Z(u)  A) du] =(x/2)  ∫[0,τ)(1-q([0,u)S{γ}{m}) du

22. Outline of proof (cont’ -1D) S = set of speeds A = [0,x)  {θ} [0,)  S {γ} {m} Borel set • Lim t P(Z(t)  A) =(x/2)  ∫[0,)(1-q([0,u)S{γ}{m}) du • Lim t P(X(t)<x, θ(t) =θ) = γ,mLim t P(Z(t)  A) = x/2 for all 0≤ x <1, θ {-1,+1}.

23. Outline of proof (cont’ -2D = [0,1]2) • Same proof as for 1D except that set of directions is now [0,2) • Lim t P(X1(t)<x1, X2(t)<x2, θ(t)<θ) = x1x2 θ/2 for all 0≤ x1,x2 <1, θ [0,2).

24. Assumptions hold if (for instance): • Speeds and relative directions mutually independent renewal sequences, independent of travel times and environment {τj ,j}j • Travel times modulated by {j}j , jM:{τj (m)}j , mM, independent renewal sequences, independent of {sj , γj ,j}j, with density and finite expectation.

25. ns-2 module available from authors

26. Sorry, it’s finished 