2. 1 Revisiting Hierarchical Quorum Systems Nuno Preguiça, J. Legatheaux Martins Henry Canivel

3. 2 Purpose of Quorum Systems Used as coordination tool Data Replication Protocols Location Management Algorithms Masking Byzantine Failures Used as coordination tool Data Replication Protocols Location Management Algorithms Masking Byzantine Failures

4. 3 Quorum System Characteristics Quorum size: number of nodes that need to be contacted to form a quorum Basic: smallest majority vote Advanced: majority vote dependent on structure Grid: 2  n - 1 (size = n nodes or processes) Unstructured: n/2 + 1 (size = n) Quorum size: number of nodes that need to be contacted to form a quorum Basic: smallest majority vote Advanced: majority vote dependent on structure Grid: 2  n - 1 (size = n nodes or processes) Unstructured: n/2 + 1 (size = n)

5. 4 Quorum System Characteristics Failure probability: chance that all quorums are unavailable (I.e. the system is unusable) General: p < 0.5 Load of a system: frequency of access of each element in the system Failure probability: chance that all quorums are unavailable (I.e. the system is unusable) General: p < 0.5 Load of a system: frequency of access of each element in the system

6. 5 Quorum System Topologies Triangle Diamond Grid Hierarchical Triangle Diamond Grid Hierarchical

7. 6 Preliminary Information Simplistic probabilistic failure model Only crash failures Only in transient Simplistic probabilistic failure model Only crash failures Only in transient

8. 7 Failure probability p <= 0.5 If p > 0.5, then impossible to improve the availability when introducing new elements into quorum system p <= 0.5 If p > 0.5, then impossible to improve the availability when introducing new elements into quorum system

9. 8 Grid Quorums

10. 9 Operational Quorums Read Write Read-Write Read Write Read-Write

11. 10 Read Quorum Formed by obtaining a row-cover in the logical object on top of hierarchy an object in level i is formed by obtaining a row-cover in at least 1 object of every row of the level i-1 grid Formed by obtaining a row-cover in the logical object on top of hierarchy an object in level i is formed by obtaining a row-cover in at least 1 object of every row of the level i-1 grid

12. 11 Write Quorum Formed by obtaining a full-line in the logical object on top of hierarchy Full-line in an object in level i is formed by obtaining a full-line in at least 1 object of every row of the level i-1 grid Formed by obtaining a full-line in the logical object on top of hierarchy Full-line in an object in level i is formed by obtaining a full-line in at least 1 object of every row of the level i-1 grid

13. 12 Read-Write Quorum Combination of both read and write Full-line and row-cover Note: creates conflict when mutual exclusion is only operation necessary Combination of both read and write Full-line and row-cover Note: creates conflict when mutual exclusion is only operation necessary

14. 13 Hierarchical Quorum System Quorum created recursively from root Obtain quorum in majority of subtrees Changes: 1. Quorum size smaller than average 2. Improve availability 3. Reduce load size for system Quorum created recursively from root Obtain quorum in majority of subtrees Changes: 1. Quorum size smaller than average 2. Improve availability 3. Reduce load size for system

15. 14 Hierarchical T-grid Algorithm Obtaining Grid Quorum Obtain quorum in majority of subtrees Changes: 1. Quorum size smaller than average 2. Improve availability 3. Reduce load size for system Obtaining Grid Quorum Obtain quorum in majority of subtrees Changes: 1. Quorum size smaller than average 2. Improve availability 3. Reduce load size for system

16. 15 Hierarchical T-grid Algorithm Obtaining Grid Quorum Intersection of full-line and partial row cover Partial row-cover vs. Full row cover Full: level i object and at least 1 of every row in level i-1 Partial: sans level i Obtaining Grid Quorum Intersection of full-line and partial row cover Partial row-cover vs. Full row cover Full: level i object and at least 1 of every row in level i-1 Partial: sans level i

17. 16 Hierarchical T-grid Algorithm h-T-grid algorithm still intersects with full cover Improves failure probability by approximately 7.5-10% h-T-grid algorithm holds greater improvements for rectangular grids Failure probability = 1/3 of h-grid # lines > # columns Variable load and quorum size h-T-grid algorithm still intersects with full cover Improves failure probability by approximately 7.5-10% h-T-grid algorithm holds greater improvements for rectangular grids Failure probability = 1/3 of h-grid # lines > # columns Variable load and quorum size

18. 17 Hierarchical T-grid Two sub-triangles and a sub-grid Recursive Two sub-triangles and a sub-grid Recursive

19. 18 Hierarchical T-grid Quorum If triangle has a single line, quorum composed by element in the line If more than one line, quorum can be obtained one of three methods: 1. If A is a quorum in T1 and B is quorum in T2, A U B in triangle of level m 2. If A is a quorum in T1 and B is a row cover in G, A U B is a quorum in triangle of level m 3. If A is a quorum in T2, B is a full-line in G, A U B is a quorum in triangle in level m If triangle has a single line, quorum composed by element in the line If more than one line, quorum can be obtained one of three methods: 1. If A is a quorum in T1 and B is quorum in T2, A U B in triangle of level m 2. If A is a quorum in T1 and B is a row cover in G, A U B is a quorum in triangle of level m 3. If A is a quorum in T2, B is a full-line in G, A U B is a quorum in triangle in level m

20. 19 Hierarchical Triangle

21. 20 Hierarchical Triangle Minimizes load Easier to introduce new nodes (expand system) All quorums have the same size Minimize volume of messages passed on scaling system Minimizes load Easier to introduce new nodes (expand system) All quorums have the same size Minimize volume of messages passed on scaling system

22. 21 Final Notes Larger quorums -> larger loads Majority and HQS at top h-T-grid h-triang at bottom Larger quorums -> larger loads Majority and HQS at top h-T-grid h-triang at bottom

23. 22 Final Notes Modifications to HQS Reduce quorum size Improve availability and load Maintains stability at the least for slightly rectangular grids Introduction to triangular QS Better availability and load than grid-based Quorum size always constant and smaller than avg quorum size in grid Load almost optimal when analyzed from high availability Modifications to HQS Reduce quorum size Improve availability and load Maintains stability at the least for slightly rectangular grids Introduction to triangular QS Better availability and load than grid-based Quorum size always constant and smaller than avg quorum size in grid Load almost optimal when analyzed from high availability

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