1. Student: Fang Hui Supervisor: Teo Yong Meng
A Control Theory Approach to Self-Stabilizing in Large Distributed System
Student: Fang Hui
Supervisor: Teo Yong Meng
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Outline
Objective
Measurement model
Dynamical analysis
Algorithm based on parameters
Conclusion
sma5510-research methodology

3. sma5510-research methodology
Objective
Find a way to describe the distributed system stability, and how to measure stability
Analyze the stability bound and finite convergence.
sma5510-research methodology

4. Stability of Distributed System
The conception of self-stabilizing distributed computation was first proposed and explored by Dijkstra in 1974.
A distributed system is self-stabilizing if,
when started from an arbitrary initial state, it is guaranteed to reach a legitimate state.
Once in a legal state, the system does not switch to an illegal state in the absence of failures.
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Assumptions
Node can only communicate with neighbors whose pointer contained in its routing table
The links and node both may fail and recover during normal operation
The recovery should be without global intervention, but system will consider the stability in global state sense
Each node will keep some extent stability
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Measurement
Global stability is accumulated by each node’s stability.
The node stability is derived from its connectivity knowledge.
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Measurement (contd)
Divides the system stability into two types:
vertex stability (considering node failure/leave) ,
edge stability (considering routing information)
G= (V, E), where | V | = n is network size
Stability distribution matrix: (D : link, w :node)
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8. Node & Global Stability
The value of node-i stability
Global stability
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Stability examples
Graph
Stability
Graph with n node connected in a line
S = ( 2/n + 3/n * (n-2) + 2/n) /n = (3n -2)/n2
Chord where each node contains log2n size finger table
S = log2n/ n
Graph with n node full connected
S = 1
sma5510-research methodology

10. Model of Dynamical System
Consider routing inconsistency
An incoming message updates or adds new routing entries (new pointer to other node). This can also be caused by node’s periodically maintanence messages besides query messages. The extra message will consume bandwidth to some extent.
The node flushes the outdated entries in its routing table, in terms of out-going message timeout or other possible way.
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Two parameters (p,q)
p: model the factor contributing to improving stability.
q: model the factor contributing to decreasing stability.
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12. Profile of node stability tendency
Max: p/(p+q)
Node stability
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When (p,q) vary
Now we consider the p and q the functions of abstract time t.
where p(t), q(t) in [0,1]
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Proved Property 2
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(p,q)-feedback
Based on above, design a feed-back system and algorithm by dynamically adjusting factor p(t) and q(t) in each step. Node stability can be maintained in certain level efficiently.
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16. Algorithm 1: achieve node stability x* in finite time
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17. Algorithm 2: achieve global stability in finite time
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18. The advantage of algorithm
No explicit node coordination after global stability requirement sent out.
Termination-detection unnecessary due to finite time.
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Conclusion
Analyze the node behavior of distributed system and give a practical evaluation on the global stability, local stability and expected convergence time.
Sort out the parameters which impact the stability dynamically, by disseminating the global stability requirement and each node reach/maintain local stability in finite time.
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Open issues
Introduce more advanced parameters to describe the global stability of system in control theory perspective.
A predefined threshold value of stability may not enough. More accuracy on the global stability also depends on the network topology, or stability distribution (specified in previous section).
Assume the stability Matrix D as a kind of stability distribution. In some areas (sub-matrix) the density of stability may be higher, while other areas with lower stability. In particular case, a sparse stability distribution may have lower fault tolerance. Some nodes with locality may also comprise of super-node. Hence we will consider the stability of local nets and the stability between two local areas.
sma5510-research methodology