Inspiteofnumerousfaultmodes Definition(Self-stabilizingSystem) Definition(Self-stabilizingSystem) Maximizeusefullifetimeofsystem Self-stabilizationandSensorNetworks While(batteriessupplypower) Definition(ClassicalSystem, Definition(ClassicalSystem, a.k.a. a.k

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization and Sensor Networks

M Potop Butucaru, F Petit, S Tixeuil

UPMC Sorbonne Universités [email protected]

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMA

Motivation Algorithm stack Clustering

Density

Self-stabilizing Clustering Simulation Results Conclusion

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Sensor Networks

Iprocessor + sensors + radio

I2 AA batteries, on/off switch

I3 LEDs for debugging

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Sensor Networks

While (batteries supply power)

ICollect, aggregate and reduce data

Ilog into memory

In spite of numerous fault modes

IPermanent sensor failures, node failures

Irestarts, radio failures

Itransient faults, reconfigurations

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Distributed Systems

Definition (Classical System,

a.k.a.

Non stabilizing)

Starting from aparticularinitial configuration, the system immediatelyexhibits correct behavior.

Definition (Self-stabilizing System)

Starting fromanyinitial configuration, the system eventuallyreaches a configuration from with its behavior is correct.

ISelf-stabilization permits to recover from transient failures

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Distributed Systems

Definition (Classical System,

a.k.a.

Non stabilizing)

Starting from aparticularinitial configuration, the system immediatelyexhibits correct behavior.

Definition (Self-stabilizing System)

Starting fromanyinitial configuration, the system eventuallyreaches a configuration from with its behavior is correct.

ISelf-stabilization permits to recover from transient failures

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization

“Correct”

Time Configurations

Stabilization Time

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Complexity Criteria

Maximize useful lifetime of system

I“maximise useful”: correct quickly from illegitimate state

ISelf-stabilization, scalability

I“maximise lifetime”: use minimal energy to preserve batteries

Ilocal vs. global preserving

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

System Specifics

Ionly one radio frequency Ino collision detect Iaccess technique: CSMA/CA Iuse CRC to detect collision Ino directional send/receive Imsg. are small (30 bytes) Iradio range about 1 meter Inumber of neighbors < 10 Icould be large number of nodes (perhaps > 100000)

Iunique node IDs (probably) Icost a fewU(someday) Islow processor (4 MHz) Ilimited memory (4 KB RAM) Iitem nodes have real-time

clocks⌘drift between 1 msec and 100 msec per second

Iseveral power modes available

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

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f h g

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a b

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f h g

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a b

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

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f h g

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

Self-stabilizing model

IRead neighborhood state,

Icompute and update local state

Sensor Network model

IRead local state,

Icompute and broadcast to neighborhood

ICollisions may appear

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks

Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

IPros: reuse existing SS algorithms

ICons: potentially inefficient, overhead

I[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

IPros: potentially efficient

ICons: ignore previous SS work

I[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks

Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

IPros: reuse existing SS algorithms

ICons: potentially inefficient, overhead

I[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

IPros: potentially efficient

ICons: ignore previous SS work

I[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks

Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

IPros: reuse existing SS algorithms

ICons: potentially inefficient, overhead

I[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

IPros: potentially efficient

ICons: ignore previous SS work

I[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Basic Algorithm

IEach nodephas a variablevp IEach neighborqofphas a variablecqvp

Icqvpis the cached value ofvpatq

IWheneverpassignsvp,palso broadcasts the new value to the neighborhood

IWhenever a neighborqofpreceivesvp,qupdatescqvp

accordingly

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Definition (Cache coherence)

For all neighborspandq,cqvp=vp

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Periodic retransmit

IEach nodepperiodically broadcastsvpto its neighborhood

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Message Corruption

IEach neighborqofphas a Boolean variablebqvp IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

IIfqreceivesvpcorrectly,bqvpbecomes true

IG!Abecomes

for all neighborsqofp,bpvqandG! A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Periodic Retransmit Message Corruption Lemma (Self-stabilization)

If started from an arbitrary state, the self-stabilizing model is eventually simulated

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

IEach nodephas a clock variablevp IFor every neighborspandq,|vp-vq|61

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

IEach nodephas a clock variablevp IFor every neighborspandq,|vp-vq|61

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 0 8 1 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 0 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 1 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 2

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 3 3 8 3 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 4 4 8 4 4

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 5 5 8 5 5

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 6 6 8 6 6

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

7 7 6 8 7 7

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

8 7 6 8 8 8

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

8 7 7 8 9 8

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Self-stabilizing Unison

Ifor every neighborq,vq>vp!vp:=vp+1

non activatable activatable legitimate

8 8 7 8 9 9

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Specification

IEach nodephas a clock variablevp IFor every neighborspandq,|vp-vq|61

Self-stabilizing Unison

with Collisions

Ifor every neighborq,vq>vp!vp:=vp+1

Ifor every neighborq,cpvq>vp!vp:=vp+1

IOnly correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Specification

IEach nodephas a clock variablevp IFor every neighborspandq,|vp-vq|61

Self-stabilizing Unison

with Collisions

Ifor every neighborq,cpvq>vp!vp:=vp+1

IOnly correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Specification

IEach nodephas a clock variablevp IFor every neighborspandq,|vp-vq|61

Self-stabilizing Unison

with Collisions

Ifor every neighborq,cpvq>vp!vp:=vp+1

IOnly correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 2

1 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 3

2 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

3 4

3 3

4 3

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

4 4

3 4

4 4

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Cache coherence weakening

IFor every neighborspandq,cpvq6vq

Self-stabilizing Unison with collisions

IUnison and Weak cache coherence are preserved by program executions

IUnison and Weak cache coherence eventually hold

ISome extra work is expected to get bounded clock values

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks

Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

I[Herman 03]Cached Sensornet Transform

IOverhead is not upper bounded

Design self-stabilizing algorithms for the sensor networks model

I[Herman 03]Unison with collisions

IProof in the model is specific to the problem