[PPT] - Self Stabilization 1 Goals of the lecture: Self-stabili PowerPoint Presentation

SLIDE 1 Self Stabilization 1 Goals

f

the lecture: Self-stabili zation

F

ault-tolerance

Denition
f

self-stabilizing

Algo

rithm with K

state

Machines

Pro
f
Algo

rithm with 3-state Machines

Pro
f

References: Dijkstra 74, Dijkstra 86 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 2 Self Stabilization 2 F ault-tolerance

systems

which recover from faults

self-stabilization:

highly fault-tolerant

a

fault can change any data

system

view ed as consisting

f

legal and illegal states

self-stabilzation:

should reach a legal state in nite moves c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 3 Self Stabilization 3 T erminolo gy

Underlying

top

logy:

connection graph

neighb
rs
p

rivilege: b

lean

function

f
wn

state,

states
f

its neighb

rs
legal

state: application dep endent c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 4 Self Stabilization 4 Requirements

n

legal state

In

each legal state,

ne
r

mo re p rivileges

any

move from a legal state leads to a legal state

each

p rivilege p resent in at least

ne

legal state

fo

r any pair

f

legal states, there exist a sequence

f

trans- ferring moves Denition

f

self-stabilzati

n:

Rega rdless

f

initial state, and p rivilege selected each time, the system is gua ranteed to reach a legal state after a nite numb er

f

moves. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 5 Self Stabilization 5 Example: Mutual Exclusion legal state: exactly

ne

p rivilege

N+1

machines numb ered 0..N

L,S,R:

states

f

left, self, right

b
ttom

machine: machine

fo

rmat: if p rivilege then co rresp

nding

move

c

Vija y K. Ga rg Distributed Systems F all 94

SLIDE 6 Self Stabilization 6 Algo rithm I: K

state

machine (K > N ) Bottom: if (L=S) then S := S+1 mo d K

F
r
ther

machines: if (L 6= S ) then S := L

c

Vija y K. Ga rg Distributed Systems F all 94

SLIDE 7 Self Stabilization 7 Example t t t t t t B 3 3 1 4 2 4 ) t t t t t t B 4 3 1 4 2 4 ) t t t t t t B 4 4 2 4 2 4 + t t t t t t B 4 4 4 4 2 4 ( t t t t t t B 4 4 4 4 4 4 ( t t t t t t B 5 4 4 4 4 4 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 8 Self Stabilization 8 Pro

f

Lemma 0: If the system is in a legal state, then it will sta y legal. Lemma 1: A sequence

f

moves in which Bottom do es not move is nite. Lemma 2: Given any conguration, either (1) no

ther

machine has the same state as the b

ttom,
r

(2) there exists a value which is dierent from all machines. Lemma 3: With in a nite numb er

f

moves, pa rt

ne
f

Lemma 2 will b e true. Theo rem 1: Within nite numb er

f

moves, the system will reach a legal state. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 9 Self Stabilization 9 Algo rithm I I: 3-state machine Ring

f

at least 3 machines Bottom: B, No rmal: N, T

p:

T conguration view ed as a string

f

0,1,2 Bottom: if (B + 1 = R ) then B := B + 2; No rmal: if (L = S + 1)

r

(R = S + 1) then S := S + 1; T

p:

if (L = B ) and (T 6= B + 1) then T := B + 1 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 10 Self Stabilization 10 Viewing the string with a rro ws y = #

f

left-p

inting

+ 2#

f

right-p

inting

Bottom : (0) B R to B ! R y = +1 No rmal Machine: (1) L ! S R to L S ! R y = (2) L S R to L S R y = (3) L ! S R to L S R y = 3 (4) L ! S ! R to L S R y = 3 (5) L S R to L ! S R y = T

p

Machine (p rivilege also dep ends

n

B): (6) L ! T to L T y = +1 (7) L T to L T y = +1 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 11 Self Stabilization 11 Example z z z z B T 1 2 ! ! z z z z B T 1 1 2 z z z z B T 1 2 2 z z z z B T 2 2 ! z z z z B T 2 ! z z z z B T z z z z B T 1 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 12 Self Stabilization 12 Pro

f

Claim: Single a rro w implies it sta ys that w a y . Claim: string free from a rro w creates

ne

in a single move. No w sho w that if multiple a rro w then y will b e decreased in nite moves. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 13 Self Stabilization 13 Pro

f

contd Lemma 0: Bet w een t w

successive

moves

f

T

p

at least

ne

move

f

Bottom tak es place. Lemma 1: A sequence

f

moves in which Bottom do es not move is nite. Pro

f:

sucient to consider no rmal machines. (3),(4),(5) decrease numb er

f

a rro ws. (1) and (2) moves nite due to top

logy

. Theo rem: Within nite moves, there is

ne

a rro w in the string. Pro

f:b

et w een successive moves

f

b

ttom,

falsication

f

\left- most a rro w exists and p

ints

to the right" happ en in (3), (4),

r

(6). if (6) then done. If (3)

r

(4), y decreases b y 3. y can increase b y at most 2 p er move

f

Bottom, thus y is decreased b y 1. c Vija y K. Ga rg Distributed Systems F all 94