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A Control Theory A Control Theory Approach to Self-Approach to Self-

Stabilizing in Large Stabilizing in Large Distributed SystemDistributed System

Student: Fang Hui Student: Fang Hui

Supervisor: Teo Yong MengSupervisor: Teo Yong Meng

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OutlineOutline

ObjectiveObjective Measurement modelMeasurement model Dynamical analysisDynamical analysis Algorithm based on parametersAlgorithm based on parameters ConclusionConclusion

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ObjectiveObjective

Find a way to describe the distributed Find a way to describe the distributed system stability, and how to measure system stability, and how to measure stabilitystability

Analyze the stability bound and finite Analyze the stability bound and finite convergence.convergence.

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Stability of Distributed Stability of Distributed SystemSystem

The conception of self-stabilizing The conception of self-stabilizing distributed computation was first proposed distributed computation was first proposed and explored by Dijkstra in 1974. and explored by Dijkstra in 1974.

A distributed system is self-stabilizing if, A distributed system is self-stabilizing if, when started from an arbitrary initial state, it is when started from an arbitrary initial state, it is

guaranteed to reach a legitimate state. guaranteed to reach a legitimate state. Once in a legal state, the system does not Once in a legal state, the system does not

switch to an illegal state in the absence of switch to an illegal state in the absence of failures. failures.

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AssumptionsAssumptions

Node can only communicate with Node can only communicate with neighbors whose pointer contained in neighbors whose pointer contained in its routing tableits routing table

The links and node both may fail and The links and node both may fail and recover during normal operationrecover during normal operation

The recovery should be without global The recovery should be without global intervention, but system will consider intervention, but system will consider the stability in global state sense the stability in global state sense

Each node will keep some extent Each node will keep some extent stability stability

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Measurement Measurement

Global stability is accumulated by Global stability is accumulated by each node’s stability. each node’s stability.

The node stability is derived from its The node stability is derived from its connectivity knowledge. connectivity knowledge.

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Measurement (contd)Measurement (contd) Divides the system stability into two types: Divides the system stability into two types:

vertex stability (considering node failure/leave) ,vertex stability (considering node failure/leave) , edge stability (considering routing information)edge stability (considering routing information)

G= (V, E), where | V | = n is network size G= (V, E), where | V | = n is network size

Stability distribution matrix: (D : link, w :node)Stability distribution matrix: (D : link, w :node)

11 1

1

n

n nn

d d

D

d d

1

2

...

w

ww

wn

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Node & Global StabilityNode & Global Stability

1

= *n

ii ij j

j

wx d w

n

1

2

...

n

x

x wX wDw

n

x

1

1( , ) =

n

ii

S D w xn

2 2 21 1 1 1

1 1 1( , ) = * * * '

n n n n

ij ij i ij ji i i i

S D w d w w d w w Dwn n n

The value of node-i stability

Global stability

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Stability examplesStability examplesGraphGraph StabilityStability

Graph with n node connected Graph with n node connected in a linein a line

S = ( 2/n + 3/n * (n-2) + 2/n) /n = (3n -2)/nS = ( 2/n + 3/n * (n-2) + 2/n) /n = (3n -2)/n22

Chord where each node Chord where each node contains logcontains log

22n size finger n size finger

tabletable

S = logS = log22n/ nn/ n

Graph with n node full Graph with n node full connected connected

S = 1S = 1

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Model of Dynamical System Model of Dynamical System

Consider routing inconsistency Consider routing inconsistency

An incoming message updates or adds new An incoming message updates or adds new routing entries (new pointer to other node). routing entries (new pointer to other node). This can also be caused by node’s This can also be caused by node’s periodically maintanence messages besides periodically maintanence messages besides query messages. The extra message will query messages. The extra message will consume bandwidth to some extent.consume bandwidth to some extent.

The node flushes the outdated entries in its The node flushes the outdated entries in its routing table, in terms of out-going message routing table, in terms of out-going message timeout or other possible way.timeout or other possible way.

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Two parameters (p,q)Two parameters (p,q)

p: model the factor contributing to p: model the factor contributing to improving stability. improving stability.

q: model the factor contributing to q: model the factor contributing to decreasing stability. decreasing stability.

(1 )dx

p x qxdt

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Profile of node stability Profile of node stability tendencytendency

Max: p/(p+q)Max: p/(p+q)

Node stability

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When (p,q) varyWhen (p,q) vary Now we consider the p and q the functions of abstract time Now we consider the p and q the functions of abstract time

t.t.

( )( )(1 ( )) ( ) ( )

dx tp t x t q t x t

dt

where p(t), q(t) in [0,1]

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Proved Property 2Proved Property 2

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(p,q)-feedback(p,q)-feedback Based on above, design a feed-back system and algorithm by Based on above, design a feed-back system and algorithm by

dynamically adjusting factor p(t) and q(t) in each step. Node dynamically adjusting factor p(t) and q(t) in each step. Node stability can be maintained in certain level efficiently. stability can be maintained in certain level efficiently.

1 (1 )t t t t t tx x p x q x

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Algorithm 1:Algorithm 1:achieve node achieve node stability x* in stability x* in finite timefinite time

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Algorithm 2: achieve global Algorithm 2: achieve global stability in finite timestability in finite time

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The advantage of The advantage of algorithmalgorithm

No explicit node coordination after No explicit node coordination after global stability requirement sent out.global stability requirement sent out.

Termination-detection unnecessary Termination-detection unnecessary due to finite time. due to finite time.

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ConclusionConclusion Analyze the node behavior of distributed Analyze the node behavior of distributed

system and give a practical evaluation on system and give a practical evaluation on the global stability, local stability and the global stability, local stability and expected convergence time.expected convergence time.

Sort out the parameters which impact the Sort out the parameters which impact the stability dynamically, by disseminating stability dynamically, by disseminating the global stability requirement and each the global stability requirement and each node reach/maintain local stability in node reach/maintain local stability in finite time.finite time.

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Open issuesOpen issues Introduce more advanced parameters to Introduce more advanced parameters to

describe the global stability of system in describe the global stability of system in control theory perspective.control theory perspective.

A predefined threshold value of stability A predefined threshold value of stability may not enough. More accuracy on the may not enough. More accuracy on the global stability also depends on the global stability also depends on the network topology, or stability distribution network topology, or stability distribution (specified in previous section).(specified in previous section).