Introduction to Self-Stabilization

1. Introduction to Self-Stabilization StéphaneDevismes(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 P0 P1 P5 P2 P4 P3

5. Algorithm (K = 7) 1 0 2 0 1 0 1 1 0 0 1 1 0

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

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

15. LIAFA Robust Stabilizing Leader Election(SSS’07) CaroleDelporte-Gallet(LIAFA) StéphaneDevismes(CNRS, LRI) HuguesFauconnier(LIAFA)

16. SSS’07 (+WRAS) 9th International Symposium on Stabilization, Safety, and Security of Distributed Systems 14-16 November 2007, Paris, France http://sss07.lri.fr/

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 ?

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 ALIVEthe most recently • When a process p such that Leader = p receives ALIVE from q, then Leader := qif q < p • On Time out, a process p sets Leader to p

22. Can we implement Communication-EfficientSelf-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 pperiodically 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 pperiodically 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 psuspects 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 scan volontary stop sending ALIVE

31. Phase Counter • Each process maintains in Phase[p] the number of times it looses the leadership • pperiodically 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

34. Perspectives • Communication-efficient leader election in a system with timely routing • Extend these results to other topologies and models • Robust stabilizing decision problems ?