Empirical tests Numerical evaluations Theoretical proofs for stationary settings Looking for a MAC...

Empirical tests Numerical evaluations Theoretical proofs

for stationary settings

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Relocation Analysis of

Stabilizing MAC

Algorithmsfor Large-Scale Mobile Ad Hoc

NetworksPierre Leone, Geneva (Switzerland)

Marina Papatriantafilou, Chalmers (Sweden)

Elad Michael Schiller, Chalmers (Sweden)

Challenges

• Collisions • When neighboring nodes simultaneously broadcast

Collision

Collision

Collision

CollisionCollision

Collision

Challenges

• Collisions

• Mobile nodes

• Random moves

Challenges

• Collisions

• Mobile nodes

• Random moves• Possibly adversarial

Challenges

• Collisions

• Mobile nodes

• Transient faults

Modeling the location of mobile nodes

• Arbitrary violations of the assumptions that model the locations

Short-lived malfunctions• hardware/software

Opportunities

• Synchronization

• Clock synchronization algorithms

and/or GPS

Opportunities

• Synchronization

• Wireless broadcast

A powerful primitive ensuring that:

• Nodes reach nearby nodes and receive the same messages

Our approach

• Stabilization • Assume that after the last transient fault the system state is random

• Steady state behaviors do not depend on that random state

• Natural self-stabilizing extensions

• arbitrary starting state

• guaranteed system recovery

• using periodic restarts

• Our negative results hold for self-stabilizing systems

Outline

3

Bounded Relocation

Rate

2

Negative Results

1

Relocation

Analysis

Relocation Analysis

• Random moves • Each mobile node randomly moves in the Euclidian plane

• Two mobile nodes can directly communicate if their distance is less than a threshold

In the short run

Gt , Gt+1 ∊ G

Many mobile nodes have similar neighborhoods in Gt and Gt+1

e.g., large communication radius

In the long run this similarity disappears

There are independent random relocations of the mobile nodes

e.g., Gt,Gt+x are independent when x → ∞

Relocation Analysis

• Random moves

• Evolving graphs

• Ferreira’04, Avin et al.’08

G=(G1, G2, …)

• In time t , graph Gt ∊ G models the communications and interferences

Relocation Analysis

• Between Gt, Gt+1 ∊ G, α|V| nodes

relocate to new neighborhoods

– α ∊in [0, 1] is the relocation rate

• Relocating nodes and their new neighborhoods are chosen randomly

• Random moves

• Evolving graphs

• Relocation rate

Relocation Analysis





• Random moves

• Evolving graphs

• Relocation rate

Our assumptions are different from:

• Random walks

do not consider short-term

(independent) random relocations

• Population protocols

do not consider long-term

neighborhood similarity

Relocation Analysis





• Random moves

• Evolving graphs

• Relocation rate

Relocation Analysis





• Random moves

• Evolving graphs

• Relocation rate

High rate=more collisions

Relocation Analysis





Useful in analyzing MAC algorithms

• Random relocation causes unexpected

interferences

• Expressiveness

• A single parameter defines the

rate of unexpected interferences

• Simpler proofs than Kinetic models

• Random moves

• Evolving graphs

• Relocation rate

Outline

3

Bounded Relocation

Rate

1

Relocation Analysis

2

Negative Results

Impossibility Result

Collision

Claim 1: There is no efficient and

deterministic MAC algorithm

Impossibility Result

Claim 1: For arbitrary relocation rate, there is no efficient and deterministic MAC algorithm

CollisionCollision

Focus on randomized MAC algorithms

Lower Bound

Claim 2: For arbitrary relocation rate, oblivious strategies are the best that you can hope for …

Oblivious strategies ignore

the broadcast history

•Consider random relocation of all

nodes after every algorithm step

•Learning the history is of no use

Focus on bounded relocation rate

Outline

1

Relocation Analysis

2

Negative Results

3

Bounded Relocatio

n Rate

Throughput Related Trade-off

• Oblivious • Ignores the history of broadcasts


• Oblivious

• Non-oblivious

P =

P =

P =

P =

•E.g., based on vertex-coloring– Luby '93


P =

P =

P =

P =

Can I use ?

Can I use ?

Can I use ?

Can I use ?

Great! I will keep

Great! I will keep

I will pick another one

CollisionI will pick

another one

• Oblivious

• Non-oblivious

•E.g., based on vertex-coloring– Luby '93– Color uniqueness:

•A node has a color different than its neighbors, i.e., no collisions


•E.g., based on vertex-coloring– Luby '93– Color uniqueness:

•A node has a color different than its neighbors, i.e., no collisions

P =

Next round

• Oblivious

• Non-oblivious


• Oblivious

• Non-oblivious

• Trade-off

Critical-threshold relocation

rate

• Above which oblivious is better

• Below which non-oblivious is better

Simplifying assumptions

1. No dependencies among neighbors

2. All relocations occur once in every round

3. Collision detection is easy

Broadcasts can inform about color choices

Later today: Remove these assumptions

How many nodes no longer have

unique colors after a round?

Collision

Collision Collision

Uniform distribution of colors

• Assume uniformity in the starting

configuration

• Show uniformity in every

configuration that follows

At most ~α|V| nodes are expected to no longer have unique colors after a round

From Stationary to Non-stationarySimplifying assumptions






How many nodes start having

unique colors after a round?

From Stationary to Non-stationary

Using the vertex-coloring algorithm

(1- β) ≅1/e• Let y be the number of nodes with

conflicting colors

• Let (1- β)y be the expected

number of nodes whose colors

become unique within a

broadcasting round

Conflicting Unique


• When the recovery is slower than the relocation (1 - β) < α • Convergence

Recovery

Relocation

Conflicting Unique


• When the recovery is slower than the relocation (1 - β) < α

• When the relocation is slower than the recovery (1 - β) > α

• Convergence

Recovery

Relocation

Throughpu

t

80%

60%

40%

20%

0.2 0.4 0.6 0.8

Oblivious strategy

non-oblivious strategy

0.2 0.4 0.6 0.8

24.0c

Too good to be true!We discover a critical-threshold relocation rate

Guaranteed throughput of non-oblivious strategies

Eventual Throughput Simplifying assumptions






We remove the simplifying assumptions and bound the recovery ratio

(1- β) ∊ [σ(1- α) ,1/e], where σ=(5+3/e)/32

Relocation Rate α

• CSMA/CA

Existing Approaches

• Such as Herman Tixeuil ‘04• Divided ratio time

– Overhead– TDMA time slots

• CSMA/CA in overhead part for

• Local leader election

• Vertex coloring

• When nodes relocate in every broadcasting round

• No guarantees for leader election

Overh

ea

dOverh

ea

d

RTS

Existing Approaches

• CSMA/CA

• RTS/CTS dialog

• Request to Send / Clear to Send• Facilitates short exposure time

– period during which a transmitted packet might be intercepted

– shorter than a time slot

Defer

CTS

Existing Approaches

• CSMA/CA

• RTS/CTS dialog




RTS

DeferDefer

ACK

Data

Existing Approaches

• CSMA/CA

• RTS/CTS dialog




DeferDefer

Our Approach

• Divided time slots

• Competition part – MaxRnd rounds

• DATA part

Competition rounds

DATA packetDATA packet

Competition rounds

DATA packet

Competition rounds

DATA packet DATA packet

Competition rounds

slot 1 slot 2 slot 3 slot 4

round 1

round 2

round 3

round 4

Max competition rounds, e.g., MaxRnd = 4

Our Approach


• Round based competition

Timeslot

Data partCompetition partRecovery is facilitated because of:

1. Simple winner and losers

2. “Unlucky winners” and “lucky losers”

Neighbors may choose the same slot

On the k competition round,

• competitors send RTS with

probability 2 (k-MaxRnd )

Our Approach







Timeslot

Data partCompetition part

RTC

CTS

DATA

Simple winner and losers

The simple winner chooses the slot as its “permanent” one

The simple losers are aware of the winner’s broadcast and continue to look for other broadcasting slots





Our Approach



Timeslot


CTS

“Unlucky winners” and “lucky losers”

RTC

RTCDATA

DATA

Collision

“Unlucky winners” are not aware of their coalitions and continue to compete for this slot on the next round

“Lucky losers” are aware of the winners’ collisions and stop competing for this slot on the next round





Our Approach



Timeslot


Simple winner and losers +

“Unlucky winners” + “Lucky losers” =

Recovery rate of (5+3/e)/8

Simplifying assumptions






Recovery rate of (5+3/e)/8

Recovery rate of (1- α) (5+3/e)/8

Recovery rate of (1- α) (5+3/e)/32

Confused?!

μ-M

AC

AI-LM

AC

B-

MA

C

BitM

AC B

MA

CC

-M

AC CM

AC

CS

MA

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PS A

RC

DM

AC E

2-

MA

C

EM

AC

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f-MA

C

FLA

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MA

C HM

AC LM

AC LPL

MF

P

MM

AC M

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C O-

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CT PC

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adio

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AC P

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Let us help you!

Consider your favorite decentralized vertex-coloring algorithm

The good news: the algorithm possibly can migrate from stationary settings to non-stationary ones

Conclusions

• Novel throughput-related trade-off– between oblivious and non-oblivious strategies

– depends on the relocation rate of mobile nodes

• Circumventing the difficulties of– collision detection

– modeling the locations of mobile nodes

• A study of a fault-tolerant and “stateful” algorithm – Extendable to consider self-stabilization

Algosensors’09: 5th International Workshop onAlgorithmic Aspects of Wireless Sensor Networks

July 10-11, 2009

Thank you for your attention

Contact info. [email protected]

TR-2008:23, Department of Computer Science and Engineering, Chalmers University of Technology

Empirical tests Numerical evaluations Theoretical proofs for stationary settings Looking for a MAC...

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