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

Main mobility models Random Waypoint Random Direction

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

Random Direction Pick direction θ at random Move in direction θ at constant speed v for time τ Upon hitting boundary reflection or wrap around

Reflection in 2D

Wrap around in 2D

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

Notation T j : beginning j-th movement τ j = T j+1 - T j : duration j-th mvt s j : speed in j-th mvt θ(t) : direction time t θ j = θ( T j ) : direction start j-th mvt γ j : relative direction (T j, s j, γ j ) j  1 : mvt pattern

γ j = relative direction 1 D: γ j  {-1,+1} θ j = θ ( T j -) γ j Wrap around: θ j = θ j-1 γ j 2D: γ j  [0,2 π ) θ j = θ(T j -)+ γ j -2π  ( θ(T j -) + γ j )/2π  Wrap around: θ j = θ j-1 + γ j -2π  ( θ(T j -) + γ j )/2π 

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

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

Proof (cont’ - 1D & Wrap around) For wrap around θ(T 1 )= θ(0)γ 1 X(T 1 ) = X(0) + θ(0)γ 1 s 0 T 1 -  X(0) + θ(0)γ 1 s 0 T 1  Conditioning on initial location and direction yields (X(T 1 ), θ(T 1 )) uniformly distributed in [0,1)  {-1,+1}. Proof for [0,1) & wrap around concluded by induction argument.

Proof (cont’ - 1D & Reflection) Lemma: Take T j r = T j w, γ j r = γ j w, s j r = 2s j w If relations X r (t) = 2X w (t), 0 ≤ X w (t) < ½ = 2(1-X w (t)), ½ ≤ X w (t) <1 θ r (t) = θ w (t), 0 ≤ X w (t) < ½ = -θ w (t), ½ ≤ X w (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.

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.

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.

Remarks (1D models) Additive relative direction ok θ j = (θ j-1 + γ j ) mod 2, γ j  {0,1} γ j = 0 (resp. 1) if direction at time T j not modified θ j = -1 with prob. Q = +1 with prob 1-q Uniform stationary distr. iff q=1/2

How can mobiles reach uniform stationary distributions for location and orientation starting from any initial state?

Mvt vector {y j = (τ j,s j,γ j,  j ) j } {  j } j : environment (finite-state M.C.) Assumptions {y j } j aperiodic, Harris recurrent M.C., with unique invariant probability measure q.

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

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)} t has unique invariant probability measure. In particular, stationary location and direction uniformly distributed.

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

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,m Lim t  P(Z(t)  A) = x/2 for all 0≤ x <1, θ  {-1,+1}.

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(X 1 (t)<x 1, X 2 (t)<x 2, θ(t)< θ ) = x 1 x 2 θ/2  for all 0≤ x 1, x 2 <1, θ  [0,2  ).

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

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