The Random Trip Mobility Model - PowerPoint PPT Presentation

1 / 45
About This Presentation
Title:

The Random Trip Mobility Model

Description:

For random walk with wrapping, if M0 is uniformly distributed on A, so is Mn for ... The same holds for random walk with reflection. ... – PowerPoint PPT presentation

Number of Views:34
Avg rating:3.0/5.0
Slides: 46
Provided by: mil1150
Category:
Tags: mobility | model | random | trip | walk

less

Transcript and Presenter's Notes

Title: The Random Trip Mobility Model


1
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
2
Examples
3
RWP random waypoint (Johnson and Maltz, 1996)
4
RWP on general connected domain
5
RWP on general connected domain (contd)called
city-section (Camp et al, 2002)
6
Space graphs readily available from road-map
databases
Example Houston section, from US Bureaus TIGER
database(S. PalChaudhuri et al, 2004)
7
a restricted RWP (Blaževic et al, 2004)
8
a restricted RWP (Jardosh et al, 2003) (contd)
9
random walk with wrapping
10
random walk with reflection
11
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

12
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 ?

13
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

14
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?
15
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)
16
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
17
transience of DSR
default initialization (non perfect mobility
simulation)
t (sec)
Packet delivery ratio
perfect mobility simulation
t (sec)
18
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

19
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
20
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
21
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

22
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

23
The Random Trip Mobility Model(assumptions)
  • (H1) (Pn,Sn) is independent of all past,
    conditional on (Mn,In)

24
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

25
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
26
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

27
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

28
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.
29
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

30
Knowing Palm inversion formula, the rest is easy
  • Time-stationary distribution of phase

31
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

32
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
33
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
34
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

35
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

36
Rejection sampling lemma (contd)
  • Twist the distribution of J as follows
  • The sample is drawn from the given density of
    (J,Y)

37
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

38
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)
39
Illustration Perfect samples of positionsfor
some of our examples
  • Restricted RWPs

RWP on a non convex domain
40
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.
41
Perfect sampling for random walk models (contd)
  • For RW with wrapping
  • Similar result obtained for RW with reflection

42
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

43
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

44
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

45
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)
Write a Comment
User Comments (0)
About PowerShow.com