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

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

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)j1 : mvt pattern

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)/2πWrap around:

θj = θj-1+ γj -2π (θ(Tj-) + γj)/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 (1D =[0,1) & Wrap around) Mvt pattern (Tj , sj , γj)j 1 fixed 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] 1u + θ(0)s0t - u + θ(0)s0t < xdu = 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(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.

Proof (cont’ - 1D & Reflection) Lemma: Take Tj

r = Tjw, γj

r = γjw, sj

r = 2sjw

If relations Xr(t) = 2Xw(t), 0 ≤ Xw(t) < ½ = 2(1-Xw(t)), ½ ≤ Xw(t) <1 θr(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.

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

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. iff q=1/2

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

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.

{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(j1 1{yjA} = ) = 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]) {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

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}.

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).

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 , jM:{τj (m)}j , mM, independent renewal sequences, independent of {sj , γj ,j}j, with density and finite expectation.

ns-2 module available from authors

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