Random tripleb nov05

Random Trip
Stationarity, Perfect Simulation and Long
Range Dependence
Jean-Yves Le Boudec (EPFL)
joint work with
Milan Vojnovic (Microsoft Research Cambridge)

 This slide show
http://ica1www.epfl.ch/RandomTrip/slides/RandomTripLEBNov05.ppt
 Documentation about random trip model, including ns2 code for download
http://ica1www.epfl.ch/RandomTrip/
 This slide show is based on material from
[L-Vojnovic-Infocom05] J.-Y. Le Boudec and M. Vojnovic
Perfect Simulation and Stationarity of a Class of Mobility
Models
IEEE INFOCOM 2005
http://infoscience.epfl.ch/getfile.py?mode=best&recid=30089
[L-04] Tutorial on Palm calculus applied to mobility models
http://lcawww.epfl.ch/Publications/LeBoudec/LeBoudecV04.pdf
Resources

Abstract
The simulation of mobility models often cause problems due to long
transients or even lack of convergence to a stationary regime ("The
random waypoint model considered harmful"). To analyze this, we define
a formally sound framework, which we call the random trip model.
It is a generic mobility model for independent mobiles that contains as
special cases: the random waypoint on convex or non convex domains,
random walk, billiards, city section, space graph and other models. We
use Palm calculus to study the model and give a necessary and sufficient
condition for a stationary regime to exist. When this condition is satisfied,
we compute the stationary regime and give an algorithm to start a
simulation in steady state (perfect simulation). The algorithm does not
require the knowledge of geometric constants. For the special case of
random waypoint, we provide for the first time a proof and a sufficient
and necessary condition of the existence of a stationary regime. Further,
we extend its applicability to a broad class of non convex and multi-site
examples, and provide a ready-to-use algorithm for perfect simulation.
For the special case of random walks or billiards we show that, in the
stationary regime, the mobile location is uniformly distributed and is
independent of the speed vector, and that there is no speed decay. Our
framework provides a rich set of well understood models that can be
used to simulate mobile networks with independent node movements.
Our perfect sampling is implemented to use with ns-2, and it is freely
available to download from http://ica1www.epfl.ch/RandomTrip.
Abstract
The simulation of mobility models often cause problems due to long
transients or even lack of convergence to a stationary regime ("The
random waypoint model considered harmful"). To analyze this, we define
a formally sound framework, which we call the random trip model.
It is a generic mobility model for independent mobiles that contains as
special cases: the random waypoint on convex or non convex domains,
random walk, billiards, city section, space graph and other models. We
use Palm calculus to study the model and give a necessary and sufficient
condition for a stationary regime to exist. When this condition is satisfied,
we compute the stationary regime and give an algorithm to start a
simulation in steady state (perfect simulation). The algorithm does not
require the knowledge of geometric constants. For the special case of
random waypoint, we provide for the first time a proof and a sufficient
and necessary condition of the existence of a stationary regime. Further,
we extend its applicability to a broad class of non convex and multi-site
examples, and provide a ready-to-use algorithm for perfect simulation.
For the special case of random walks or billiards we show that, in the
stationary regime, the mobile location is uniformly distributed and is
independent of the speed vector, and that there is no speed decay. Our
framework provides a rich set of well understood models that can be
used to simulate mobile networks with independent node movements.
Our perfect sampling is implemented to use with ns-2, and it is freely
available to download from http://ica1www.epfl.ch/RandomTrip.

Contents
1. Issues with mobility models
2. The random Trip Model
3. Stability
4. Perfect Simulation
5. Long range dependent examples

Mobility models are used to evaluate system
designs
 Simplest example: random waypoint:
Mobile picks next waypoint Mn uniformly in area, independent of past
and present
Mobile picks next speed Vn uniformly in [vmin, vmax]
independent of past and present
Mobile moves towards Mn at constant speed Vn
Mn-1
Mn

Issues with this simple Model
 Distributions of speed, location, distances, etc change with
simulation time:
Distributions of speeds at times 0 s and 2000 s
Samples of location at times 0 s and 2000 s
Sample of instant speed for one
and average of 100 users

Why does it matter ?
 A (true) example: Compare impact
of mobility on a protocol:
Experimenter places nodes
uniformly for static case,
according to random waypoint for
mobile case
Finds that static is better
 Q. Find the bug !
 A. In the mobile case, the nodes
are more often towards the
center, distance between nodes is
shorter, performance is better
 The comparison is flawed. Should
use for static case the same
distribution of node location as
random waypoint. Is there such a
distribution to compare against ?
Random waypoint
Static

Issues with Mobility Models
 Is there a stable distribution of the simulation state ( = Stationary
regime) reached if we run the simulation long enough ?
 If so,
how long is long enough ?
If it is too long, is there a way to get to the stable distribution without
running long simulations (perfect simulation)

The Random Trip model
 Goals: define mobility models
1. That are feature rich, more realistic
2. For which we can solve the issues mentioned earlier
 Random Trip [L-Vojnovic-Infocom05] is one such model
mobile picks a path in a set of paths and a speed
at end of path, mobile picks a new path and speed
driven by a Markov chain
domain A
Path Pn : [0,1] → A trip duration Sn
Mn=Pn(0)
Mn+1=Pn+1(0)
trip start
trip end
Here Markov
chain is Pn
Here Markov
chain is Pn

Random Waypoint is a Random Trip Model
 Example (RWP):
Path:
Pn = (Mn, Mn+1)
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
 Other examples of random trip in next slides

RWP with pauses on
general connected
domain
Here Markov chain is
(Pn, In)
where
In = “pause” or In =“move”
(Pn, In)
where

Space graphs are readily available from road-map databases
Example: Houston section, from US Bureau’s TIGER database
(S. PalChaudhuri et al, 2004)

Restricted RWP (Blažević et al, 2004)
(Pn, In, Ln, Ln+1, Rn)
Where
Ln = current sub-domain
Ln+1 = next subdomain
Rn = number of trips in this
visit to the current domain
(Pn, In, Ln, Ln+1, Rn)
Where
Ln = current sub-domain
Ln+1 = next subdomain
Rn = number of trips in this
visit to the current domain

Random walk on torus with pauses

Assumptions for Random Trip Model
 Model is defined by a sequence of paths Pn and trip durations Sn,
and uses an auxiliary state information In
 Hypotheses
(Pn, In) is a Markov chain (possibly on a non enumerable state space)
Trip duration Sn is statistically determined by the state of the Markov
chain (Pn, In)
(Pn, In) is a Harris recurrent chain
i.e. is stable in some sense
 These are quite general assumptions
Trip duration may depend on chosen path

Solving the Issue
1. Is there a stationary regime ?
 Theorem [L-Vojnovic-Infocom 05]:
there is a stationary regime for random trip iff the expected trip time is
finite
If there is a stationary regime, the simulation state converges in
distribution to the stationary regime
 Application to random waypoint with speed chosen uniformly in
[vmin,vmax]
Yes if vmin >0, no if vmin=0
Solves a long-standing issue on random waypoint.

A Fair Comparison
 If there is a stationary regime, we
can compare different mobility
patterns provided that
1. They are in the stationary regime
2. They have the same stationary
distributions of locations
 Example: we revisit the
comparison by sampling the static
case from the stationary regime of
the random waypoint
Run the simulation long enough,
then stop the mobility pattern
Random waypoint
Static, from uniform
Static, same node location as RWP

The Issues remain with Random Trip Models
 Do not expect stationary distribution to be same as distrib at trip
endpoints
 Samples of node locations from stationary distribution
(At t=0 node location is uniformly distributed)

Random waypoint
on sphere
In some cases it is very simple
 Stationary distribution of location is uniform for
Random walk Billiards if speed vector is
completely symmetric
(goes up/down [right/left]
with equal proba)

Solving the Issue
2. How long is
long enough ?
 Stationary regime can be
obtained by running
simulation long enough
but…
 It can be very long
Initial transient longs at
least as large as typical
simulation runs

Palm Calculus gives Stationary Distribution
 There is an alternative to running the simulation long enough
 Perfect simulation is possible (stationary regime at time 0) thanks
to Palm calculus
 Relates time averages to event averages
Inversion Formula
by convention T0 · 0 < T1
Time average of observation X
Event average, i.e. sampled at end trips

Example : random waypoint
Inversion Formula Gives Relation between
Speed Distributions at Waypoint and at Arbitrary Point in Time

Distribution of Location was Previously Known
only Approximately
 Conventional approaches finds that closed form expression for
density is too difficult [Bettstetter04]
 Approximation of density in area [0; a] [0; a] [Bettstetter04]:

Stationary Distribution of Location Is also
Obtained By Inversion Formula
back

Stationary Distribution of Location
 Valid for any convex area

The stationary distribution of random waypoint
is obtained in closed form [L-04]
Contour plots of density of stationary distribution

But we do not need complex formulae
 The joint distribution of (Prev(t), Next(t), M(t)) is simpler
 True for any random trip model :
Stationary regime at arbitrary time has the simple generic,
representation:
For random waypoint we have Navidi and Camp’s formula

Perfect Simulation follows immediately
 Perfect simulation := sample stationary regime at time 0
 Perfect sampling uses generic representation and does not
require geometric constants
Uses representation seen before + rejection sampling
Example for random waypoint:

Example: Random Waypoint
No Speed Decay

Why Long Range Dependent Models ?
 Mobility models may exhibit some aspects of long range
dependence
See Augustin Chaintreau, Pan Hui, Jon Crowcroft, Christophe Diot,
Richard Gass, and James Scott. "Impact of Human Mobility on the
Design of Opportunistic Forwarding Algorithms".
 The random trip model supports LRD

Long Range Dependent Random Waypoint
 Consider the random waypoint without pause, like before, but
change the distribution of speed:

 The random trip model provides a rich set of
mobility models for single node mobility
 Using Palm calculus, the issues of stability and
perfect simulation are solved
 Random Trip is implemented in ns2 (by S.
PalChaudhuri) and is available at
web site given earlier
Conclusion

Random tripleb nov05

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