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Simulation de modèles de mobilité : paradoxes et étrangetés

Jean-Yves Le Boudec

EPFL

En collaboration avec

Milan Vojnović

Microsoft Research

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Résumé

Les ingénieurs qui développent des systèmes de communication mobile ont souvent recours à la simulation dans les phases de conception et de simulation. Bien que conceptuellement très simple, la simulation peut poser des problèmes parfois déroutants. Par exemple, des simulations de durées différentes donnent des résultats différents, et plus la simulation est longue, plus les résultats sont différents. Ces phénomènes peuvent être expliqués, et quelque fois entièrement évités, par la théorie des probabilités, et en particulier le calcul de Palm pour les processus ponctuels stationnaires – une théorie initialement développée dans le cadre des files d’attente.

[LV06] The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation, J.-Y. Le Boudec and Milan Vojnović, ACM/IEEE Trans. on Networking, Dec 06

[L04] Understanding the simulation of mobility models with Palm calculus, J.-Y. Le Boudec, Performance Evaluation, 2007

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Outline

Simulation Issues with mobility models

Palm calculus

Stability

Stationary distributions and Perfect Simulation

Random Waypoint (Johnson and Maltz`96)

Used at IETF to evaluate performance of ad-hoc routing protocols

Node picks next waypoint Xn+1 uniformly in area

Picks speed Vn uniformly in [vmin,vmax]

Moves to Xn+1 with speed Vn

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Xn

Xn+1

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Path Pn

Xn

Xn+1

Swiss Flag [LV05]

Non convex domain

Random Waypoint on Non Convex Area

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City-section, Camp et al [CBD02]

More Realistic Model

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Issue about speed

Distributions of node speed, position, distances, etc change with time

Node speed:

100 users average

1 user

Time (s)

Spe

ed (

m/s

)900 s

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Node Position

Distributions of node speed, position, distances, etc change with time

Distribution of node position:

Time = 0 sec Time = 2000 sec

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Why does it matter ?A. In the mobile case, the nodes are more often towards the center, distance between nodes is shorter, performance is betterThe 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

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 !

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Issues with Mobility Models

Is there a stable distribution of the simulation state (time-stationary distribution), 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) ?

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Outline

Simulation Issues with mobility models

Palm calculus

Stability

Stationary distributions and Perfect Simulation

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Palm Calculus

Relates time averages versus event averagesAn old topic in queueing theory

Now well understood by mathematicians under the name Palm Calculus

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Palm Calculus Framework

A stationary process (simulation) with state St.

Some quantity Xt measured at time t. Assume that

(St;Xt) is jointly stationary

I.e., St is in a stationary regime and Xt depends on the past, present and future state of the simulation in a way that is invariant by shift of time origin.

ExamplesSt = current position of mobile, speed, and next waypoint

Jointly stationary with St: Xt = current speed at time t; Xt = time to be run until next waypoint

Not jointly stationary with St: Xt = time at which last waypoint occurred

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Palm ExpectationConsider some selected transitions of the simulation, occurring at times Tn.

Example: Tn = time of nth trip end

Definition : the Palm Expectation is

Et(Xt) = E(Xt | a selected transition occurred at time t)

By stationarity:

Et(Xt) = E0(X0)

Example: Tn = time of nth trip end, Xt = instant speed at time t

Et(Xt) = E0(X0) = average speed observed at a waypoint

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Event versus Time Averages

E(Xt) = E(X0) expresses the time average viewpoint.

Et(Xt) = E0(X0) expresses the event average viewpoint.

Example: Tn = time of nth trip end, Xt = instant speed at time t

Et(Xt) = E0(X0) = average speed observed at trip end

E(Xt)=E(X0) = average speed observed at an arbitrary point in time

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Two Palm Calculus Formulas Intensity of selected transitions: := expected number of transitions per time unit Intensity Formula:

where by convention T0 · 0 < T1

Inversion Formula

The proofs are simple in discrete time – see [L04]

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A Classical Example

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Outline

Simulation Issues with mobility models

Palm calculus

Stability

Stationary distributions and Perfect Simulation

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Necessary Condition for Existence of a Stationary Regime

Apply the intensity formula to Tn = trip end times

Thus: if the random trip has a stationary regime it must be that the mean trip duration sampled at trip end times is finite

On bounded area, means: mean of inverse of speed is finite

Converse is true [LV06]

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A Random waypoint model that has no stationary regime !

Assume that at trip transitions, node speed is sampled uniformly on [vmin,vmax]

Take vmin = 0 and vmax > 0

Mean trip duration = (mean trip distance)

Mean trip duration is infinite !

Was often used in practice

Speed decay: “considered harmful” [YLN03]

max

0max

1v

v

dv

v

What happens when the model does not have a stationary regime ?

The simulation becomes old

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Outline

Simulation Issues with mobility models

Palm calculus

Stability

Stationary distributions and Perfect Simulation

Stationary Distribution of Speed

Closed Form Assume a stationary regime exists and simulation is run long enough

Apply inversion formula and obtain distribution of instantaneous speed V(t)

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Removing Transient MattersA. In the mobile case, the nodes are more often towards the center, distance between nodes is shorter, performance is betterThe 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

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 !

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Removing Transients May Take Long

If model is stable and initial state is drawn from distribution other than time-stationary distribution

The distribution of node state converges to the time-stationary distribution

Naïve: so, let’s simply truncate an initial simulation duration

The problem is that initial transience can last very long

Example [space graph]: node speed = 1.25 m/sbounding area = 1km x 1km

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Perfect simulation is highly desirable (2)

Distribution of path:

Time = 100s

Time = 50s

Time = 300s

Time = 500s

Time = 1000s

Time = 2000s

Solution: Perfect Simulation

Def: a simulation that starts with stationary distribution

Usually difficult except for specific models

Possible if we know the stationary distribution

Sample Prev and Next waypoints from their joint stationary distributionSample M uniformly on segment [Prev,Next]Sample speed V from stationary distribution

Stationary Distrib of Prev and Next

Stationary Distribution of Location

Stationary Distribution of Location

There is a closed form for stationary distribution of location but it is ugly and hard to sample from – not to be used in practice [LV04]

A Fair Comparison

We revisit the comparison by sampling the static case from the stationary regime of the random waypoint

Random waypoint

Static, from uniform

Static, same node location as RWP

No Speed Decay

Conclusions

Les simulations peuvent ne pas avoir de régime stationaire par vieillissement plutôt qu’explosion

Si régime stationaire existe, il faut éliminer les transitoires ou faire une simulation parfaite

Le calcul de Palm permet de faire une simulation parfaite pour ce type de modèles

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References

[ARMA02] Scale-free dynamics in the movement patterns of jackals, R. P. D. Atkinson, C. J. Rhodes, D. W. Macdonald, R. M. Anderson, OIKOS, Nordic Ecological Society, A Journal of Ecology, 2002

[CBD02] A survey of mobility models for ad hoc network research, T. Camp, J. Boleng, V. Davies, Wireless Communication & Mobile Computing, vol 2, no 5, 2002

[CHC+06] Impact of Human Mobility on the Design of Opportunistic Forwarding Algorithms, A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, J. Scott, IEEE Infocom 2006

[E01] Stochastic billiards on general tables, S. N. Evans, The Annals of Applied Probability, vol 11, no 2, 2001

[GL06] Analysis of random mobility models with PDE’s, M. Garetto, E. Leonardi, ACM Mobihoc 2006

[JBAS+02] Towards realistic mobility models for mobile ad hoc networks, A. Jardosh, E. M. Belding-Royer, K. C. Almeroth, S. Suri, ACM Mobicom 2003

[KS05] Anomalous diffusion spreads its wings, J. Klafter and I. M. Sokolov, Physics World, Aug 2005

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References (2)

[L04] Understanding the simulation of mobility models with Palm calculus,J.-Y. Le Boudec, accepted to Performance Evaluation, 2006

[LV05] Perfect simulation and stationarity of a class of mobility models, J.-Y. Le Boudec and M. Vojnovic, IEEE Infocom 2005

[LV06] The random trip model: stability, stationary regime, and perfect Simulation, J.-Y. Le Boudec and M. Vojnovic, MSR-TR-2006-26, Microsoft Research Technical Report, 2006

[M87] Routing in the Manhattan street network, N. F. Maxemchuk, IEEE Trans. on Comm., Vol COM-35, No 5, May 1987

[NT+05] Properties of random direction models, P. Nain, D. Towsley, B. Liu, and Z. Liu, IEEE Infocom 2005

[PLV05] Palm stationary distributions of random trip models, S. PalChaudhuri, J.-Y. Le Boudec, M. Vojnovic, 38th Annual Simulation Symposium, April 2005

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References (3)

[RMM01] An analysis of the optimum node density for ad hoc mobile networks, ICC 2001

[S64] Principles of random walk, F. Spitzer, 2nd Edt, Springer, 1976

[SMS06] Delay and capacity trade-offs in mobile ad hoc networks: a global perspective, G. Sharma, R. Mazumdar, N. Shroff, IEEE Infocom 2006

[SZK93] Strange kinetics (review article), M. F. Shlesinger, G. M. Zaslavsky, J. Klafter, Nature, May 1993

[YLN03] Random waypoint considered harmful, J. Yoon, M. Liu, B. Noble, IEEE Infocom 2003