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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.

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Examples

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

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RWP on general connected domain

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

6a restricted RWP (Blažević et al, 2004)

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

8random walk with wrapping

9random walk with reflection

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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

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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 ?

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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 …

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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)

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On transience longevity (cont’d)

• Transient phase lasts 1000’s of seconds

• Typical simulation run is of the order 1000 seconds

Pro

b((

Path

at

tim

e t

) =

path

)

Initial transient lasts as long as a typical simulation duration

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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

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The Random Trip Mobility Model (basic definitions)

domain A

Path Pn : [0,1] A trip duration Sn

Mn=Pn(0)

Mn+1=Pn+1(0)

trip start Tn

trip end Tn+1

Trip selection rule: at a trip transition instant Tn, choose (Pn,Sn)

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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

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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

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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

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The Random Trip Mobility Model(assumptions)

(H1) (Pn,Sn) independent of all past, conditional on (Mn,In)

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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

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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

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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 tripduration 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

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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

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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.

Prob0(I0 = i)

E0(S0 | I0 = i)

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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)=m0,P(t)(1)=m1)=Kll’ d(m0,m1)

3. 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), 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)

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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 ?

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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):

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Illustration: Perfect samples of positions

for some of our examples

Restricted RWPs:

RWP on a non convex domain:

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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.

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Perfect sampling for random walk models (cont’d)

• For RW with wrapping:

• Similar result obtained for RW with reflection

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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

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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

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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)