1. Introduction to Self-Stabilization
Stéphane Devismes (CNRS, LRI)

2. Example of Self-Stabilizing System
Dijkstra’s Token Ring

3. Model Locally Shared Memory Guarded Action: Action:
Executed only if its guard is true (enabled)
The execution is asynchronous but each step is atomic

4. Topology: Rooted Oriented Ring

5. Algorithm (K = 7) 1 2 1 1 1 1 1

6. Transient Faults… (undefinitive and rare)2 1 2 1 3 1  1 2 3 1 1 
The system retreives by itself a correct behavior: Self-stabilization 3 2 1 1

7. Self-Stabilization Self-Stabilization [Dijkstra, 1974]:
Starting from any configuration, a self-stabilizing system reaches in a finite time a configuration c such that any suffix starting from c satisfies the intended specification.

8. Self-Stabilization Closure Illegitimate States Legitimate States
Convergence
System States

9. Advantages
Fault-Tolerance
Initialization
Dynamic Topology

10. Disavantages Initial inconsistencies (stabilization time) Overcost
No detection of stabilization

11. Around Self-Stabilization
Probabilistic Self-Stabilization
Robust Stabilization
Weak-Stabilization
Pseudo-Stabilization
Snap-Stabilization
Fault-Containment …

12. The system stabilizes even if some processes crash
Robust Stabilization
The system stabilizes even if some processes crash

13. Pseudo-Stabilization
Pseudo-Stabilization [Burns, Gouda, and Miller, 1993]:
Starting from any configuration, any execution of a pseudo-stabilizing system has a non-empty suffix that satisfies the intended specification.
Self ≠ Pseudo ?

14. Specification = {(i,i,i,…),(j,j,j,…)}
Self- vs Pseudo-
Specification = {(i,i,i,…),(j,j,j,…)} i r i j j

15. Robust Stabilizing Leader Election (SSS’07)
Carole Delporte-Gallet (LIAFA)
Stéphane Devismes (CNRS, LRI)
Hugues Fauconnier (LIAFA)
LIAFA

16. 14-16 November 2007, Paris, France
SSS’07 (+WRAS)
9th International Symposium on Stabilization, Safety, and Security of Distributed Systems
14-16 November 2007, Paris, France

17. Related Works on Robust Stabilization
Gopal and Perry, PODC’93
Beauquier and Kekkonen-Moneta, JSS’97
Anagnostou and Hadzilacos, WDAG’93
In partial synchronous model ?
Les deux premiers: systèmes synchrones
Le troisième: résultat d’impossibilité dans les réseaux asynchrones

18. Model Network: fully-connected n Processes (numbered from 1 to n):
timely
can crashed (an arbitrary number of processes may crash)
Variables: initially arbitrary assigned
Links:
Unidirectional
Initially not necessarily empty
No order on the message delivrance
Variable reliability and timeliness assumptions

19. Communication-Efficiency
[Larrea, Fernandez, and Arevalo, 2000]:
« An algorithm is communication-efficient if it eventually only uses n - 1 unidirectional links »

20. Can we implement Self-Stabilizing Leader Election in a full synchronous network?
Yes, it can be communication-efficiently implemented

21. Principle of the algorithm
A process p periodically sends ALIVE to every other if Leader = p
Any process q such that Leader <> q always chooses as leader the process from which it receives ALIVE the most recently
When a process p such that Leader = p receives ALIVE from q, then Leader := q if q < p
On Time out, a process p sets Leader to p

22. Can we implement Communication-Efficient Self-Stabilizing Leader Election in a system where at most one link is asynchronous? No

23. Impossibility of Communication-Efficiency in a system with at most one asynchronous link
Claim: Any process p such that Leader <> p must periodically receive messages within a bounded time otherwise it chooses another leader

24. Can we implement (non communication efficient) Self-Stabilizing Leader Election in a system where some links are asynchronous?
Yes

25. Self-Stabilizing Leader Election in a system with a timely routing overlay
For each pair of alive processor (p,q), there exists at least two paths of timely links:
From p to q
From q to p

26. Principle of the algorithm
Each process computes the set of alive processes and chooses as leader the smallest process of this set
To compute the set:
Each process p periodically sends ALIVE,p to every other process
Any ALIVE,p message is repeated n - 1 times (any other process periodically receives such a message)

27. Can we implement Self-Stabilizing Leader Election in a system without timely routing overlay ?No

28. Can we implement a Communication-Efficient Pseudo-Stabilizing Leader Election in a system where Communication-Efficient Self-Stabilizing Leader Election is not possible ?
Yes
In a system having a timely source and fair links (adaptation of an algorithm of [Aguilera et al, PODC’93])

29. Algorithm for systems with Source + fair links
A process p periodically sends ALIVE to every other if Leader = p
Each process stores in an Active set the IDs of each process from which it recently receives ALIVE
Each process chooses its leader among the processes in its Active set
Problem: we cannot use the IDs to choose a leader

30. Accusation Counter
p stores in Counter[p] how many times it was suspected to be crashed
When p suspects its leader:
it sends an ACCUSATION to LEADER
And chooses as new leader the process in its Active set with the smallest accusation counter (we use IDs to break ties)
p periodically sends ALIVE,Counter[p] to every other if Leader = p
Problem: assuming that LEADER=s, the source s can volontary stop sending ALIVE

31. Phase Counter
Each process maintains in Phase[p] the number of times it looses the leadership
p periodically sends ALIVE,Counter[p],Phase[p] to every other if Leader = p
p increments Counter[p] only when receiving ACCUSATION,ph with ph = Phase[p]

32. Can we implement a Communication-Efficient Pseudo-Stabilizing Leader Election in a system having only a timely source?
No, but a non communication efficient pseudo-stabilizing leader election can be done (techniques similar to those used in the algorithm of [Aguilera et al, PODC’93])

33. Result Summary ce-SS SS ce-PS PS Synchronous Yes Timely bi-source No
Timely routing ?
Timely source + fair links
Timely source
Totally asynchronous

34. Perspectives
Communication-efficient leader election in a system with timely routing
Extend these results to other topologies and models
Robust stabilizing decision problems ?
Exemple de modèle: only a finite number of processes crashed