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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)