The Random Trip Mobility Model

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The Random Trip Mobility Model

• Milan Vojnovic (Microsoft Research)

with Jean-Yves Le Boudec (EPFL) part of
simulations by Santashil PalChaudhuri (Rice
University)
Computer Lab Seminar, University of Cambridge,
UK, Nov 2004
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Examples
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RWP random waypoint (Johnson and Maltz, 1996)
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RWP on general connected domain
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RWP on general connected domain (contd)called
city-section (Camp et al, 2002)
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Space graphs readily available from road-map
databases
Example Houston section, from US Bureaus TIGER
database(S. PalChaudhuri et al, 2004)
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a restricted RWP (Blaževic et al, 2004)
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a restricted RWP (Jardosh et al, 2003) (contd)
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random walk with wrapping
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random 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, is it 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 a 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)

Consider Prob((Path at time t) p)
Q How long it takes for this probability to
converge to steady-state?
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On transience longevity (contd)

• Transient phase lasts 1000s of seconds
• Typical simulation run is of the order 1000
  seconds

Prob((Path at time t) path)
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Does transience of mobility affect performance of
a protocol that I study ?

• Example DSR protocol with restricted RWP on the
  Houston section

• Numeric speed is random, uniform on 0.01 to 9.99
  m/s
• Pause time is random, uniform on 0 to 100
  seconds
• 50 mobiles
• Default initialization t0 is a trip transition
  instant, each mobile initially in move phase
• 20 data connections, each with packet sent rate
  1 pkt/spacket length 512 B

Performance measure packet delivery ratio (
of received packets) / ( of transmitted
packets), over a time interval
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transience of DSR
default initialization (non perfect mobility
simulation)
t (sec)
Packet delivery ratio
perfect mobility simulation
t (sec)
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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)
Mn1Pn1(0)
trip end
Path Pn 0,1 ? A
trip duration Sn
MnPn(0)
domain A
trip start
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Path and Trip duration (Pn,Sn)

• Example (RWP on a convex domain)
• Path Pn(u) u Mn (1-u) Mn1, u?0,1
• Trip duration Sn (length of Pn) / Vn
• Vn numeric speed drawn from a given
  distribution

convex domain a domain such that for any two
points in the domain, the line segment between
these two points lies in the domain
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Path and Trip duration (Pn,Sn)(contd)

• 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) is independent of all past,
  conditional on (Mn,In)

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The Random Trip Mobility Model (assumptions
contd)

• (H2) Either is true
• (H2a)
• Mn1 independent of past phases In,In-1, and n,
  conditional on In
• (renewal points) for a set of selected
  transitions of In, Mn1 independent of all
  past, conditional on In
• or
• (H2b)
• Mn independent of In and n
• (Sn,In1) independent of all past, conditional on
  In

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Random Trip Mobility Model (assumptions contd)

• (H3) Markov chain In is positive recurrent
• True, in particular, if the state space of In is
  finite, and all the states communicate.

Remark (H1)-(H3) true for all examples on these
slides
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When a time-stationary distribution of mobility
state exists and is unique ?

• Theorem Under (H1)-(H3), a random trip mobility
  model has
• a time-stationary distribution, if and only if
  the mean trip
• duration 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 Slivnyaks inverse
  construction

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When the conditions fail ?

• Example RWP as was implemented in ns-2
• At trip endpoints, numeric speed is independent
  of trip distancegt
• Numeric speed is uniformly distributed on an
  interval (0,vmax
• gt
• Found and called harmful by Yoon et al (IEEE
  Infocom 2003)
• The theorem tells us that for this RWP, no
  steady-state exists
• Renders many simulations results unreliable

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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

Prob0(I0 i)
E0(S0 I0 i)
Proof Palm inversion formula.
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What is Palm inversion formula ?

• A mean-value formula of Palm calculus ( a set
  of results for stationary point processes)
• Palm inversion formula relates time-stationary
  distribution and event-stationary distribution (
  as seen at instants of a point process)
• Holds in general for a stationary point process,
  not only for renewal processes as assumed in
  previous work

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Knowing Palm inversion formula, the rest is easy

• Time-stationary distribution of phase

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Knowing Palm inversion formula, the rest is easy
(contd)

• Intermediate step

• Time-stationary distribution of (phase, trip
  duration, and trip elapsed time), conditional on
  phase

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RWP time-stationary distributions

• Theorem Under the time-stationary distribution
• Conditionally on the phase I(t)(l,l,r,move)
• Numerical speed is independent of path and
  positionspeed density
• dP(P(t)(0)m0,P(t)(1)m1)Kll d(m0,m1)
• 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),
• Position and remaining pause time are
  independent
• Position is uniform in A
• Density of the remaining pause time

Remark the independency property in item 1
previously only conjectured
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Perfect sampling

• Goal draw a sample from the time-stationary
  distribution (provided it exists) of the mobility
  state (I(t),P(t),S(t),U(t))
• Recall

normalization constants
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Perfect sampling (contd)

• For i specifying move phase, and numerical speed
  and distance on a trip independent

for RWP-like models this is a geometric constant

• For RWP with domain rectangle, the geometric
  constant is average distance between two points
  on a rectangle (known in closed-form by Ghosh
  (1951))
• Such geometric const. are known for some
  elementary domains http//mathworld.wolfram.com/t
  opics/GeometricConstants.html
• They are in general difficult to compute, if not
  impossible

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Rejection sampling lemma

• For perfect sampling, we do not need to know
  geometric constants, when they are difficult to
  compute

• We want to sample a random vector (J,Y) on a
  space (J,Rd) with density
• Suppose we know a factorization
• where gi(.) is a probability density

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Rejection sampling lemma (contd)

• Twist the distribution of J as follows

• The sample is drawn from the given density of
  (J,Y)

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Perfect sampling for restricted RWP with one
sub-domain A1

• average distance between two random points on
  a domain A1
• bound on distance between any two points in
  A1

• The general case with an arbitrary number of
  sub-domains is in principle similar, but with
  description complexity

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What do I gain with this perfect sampling
algorithm ?

• When geometric constants are unknown, we may
  estimate them by Monte Carlo
• This may be time consuming
• The proposed perfect sampling algorithm needs
  onlya bound on any possible trip distance,
  under a given phase
• In many cases these bounds are easy to compute

Example (the restricted RWP)
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Illustration Perfect samples of positionsfor
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 A, so is Mn for any ngt0.
• 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 (contd)

• 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 (contd)

• 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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Pointers

• The Random Trip Mobility Model described in
• Perfect Simulation and Stationarity of a Class
  of Mobility Models, J.-Y. Le Boudec and V.
• IEEE Infocom 2005 (to appear)available as
  EPFL Technical Report IC/2004/59http//ic2.epfl
  .ch/publications/abstract.php?ID200459

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Additional pointers

• Web Page The Random Trip Mobility Model
• http//ic1wwww.epfl.ch/RandomTrip/
• On this web page
• Download randomtrip
• ns-2 code of random trip, with perfect
  simulation (by S. PalChaudhuri, Rice
  University)