Revisiting Hierarchical
Quorum Systems
Nuno Preguiça, J. Legatheaux Martins
1
Purpose of Quorum Systems

Used as coordination tool
Data Replication Protocols
 Location Management Algorithms
 Masking Byzantine Failures

2
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)
3
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
4
Quorum System Topologies
Triangle
 Diamond
 Grid
 Hierarchical

5
Preliminary Information

Simplistic probabilistic failure model
Only crash failures
 Only in transient

6
Failure probability

p <= 0.5

If p > 0.5, then impossible to improve the
availability when introducing new elements
into quorum system
7
Grid Quorums
8
Operational Quorums
Read
 Write
 Read-Write

9
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
10
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
11
Read-Write Quorum

Combination of both read and write

Full-line and row-cover

Note: creates conflict when mutual exclusion is
only operation necessary
12
Hierarchical Quorum System

Quorum created recursively from root


Obtain quorum in majority of subtrees
Changes:
1.
2.
3.
Quorum size smaller than average
Improve availability
Reduce load size for system
13
Hierarchical T-grid Algorithm

Obtaining Grid Quorum


Obtain quorum in majority of subtrees
Changes:
1.
2.
3.
Quorum size smaller than average
Improve availability
Reduce load size for system
14
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
15
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
16
Hierarchical T-grid


Two sub-triangles
and a sub-grid
Recursive
17
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.
2.
3.
If A is a quorum in T1 and B is quorum in T2, A
U B in triangle of level m
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
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
18
Hierarchical Triangle
19
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
20
Final Notes

Larger quorums -> larger loads
Majority and HQS at top
 h-T-grid
 h-triang at bottom

21
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
22
Comprenez-vous?
23