A Theoretical Investigation of Generalized Voters for Redundant Systems Class: CS791F - Fall 2005...

A Theoretical Investigation of Generalized Voters for Redundant Systems

Class: CS791F - Fall 2005

Professor : Dr. Bojan Cukic

Student: Yue Jiang


Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

1 Introduction Objective: introduction to and analysis of voting techniques in fault

tolerant systems

Related work: (1) majority voting; (2) adaptive or weighted voter; (3) median selection method

This paper: 1) Formalized majority voter 2) Generalized median voter 3) Formalized plurality voter 4) Weighted averaging voter

2 Generalized voters Formalized majority voter - select majority

Generalized median voter - select median

Formalized plurality voter - partition the set of output based on metric equality and select the

output from largest group

Weighted averaging technique - combines the output in a weighted average

Assumption: N-versions software, N is odd

2 Generalized voters-Formalized majority voter

Definition: If more than half of the version outputs agree, this common output becomes the output of the N-version structure.

Agree is not the same, i.e. output is real value.

A threshold, ε, is needed.


x1 =0.18155

x2 =0.18230

x3 =0.18130

x4 =0.18180

x5 =0.18235

ε = 0.0005

|x1- x3| = 0.00025

|x1- x4| = 0.00035

|x2- x5| = 0.00005

(x1, x3 , x4) (x2, x5)

Example 1

Result: x1 or x3 or x4


x1 = (2.1350, -1.9693, 4.3354)

x2 = (2.1340, -1.9649, 4.3281)

x3 = (2.1376, -1.9623, 4.3284)

ε = 0.0005

d2(x1, x2) = (2.1350- 2.1340)2 + [-1.9693-(- 1.9649)]2 + (4.3354- 4.3281)2

d(x1, x2) =0.0086 > ε

d(x1, x3) =0.0102 > ε

d(x2, x3) =0.0044 < ε

Example 2

Result: x2 or x3


x1 = (0, 0, 0, 1, 0, 0, 0, 0)

x2 = (0, 1, 0, 0, 0, 0, 0, 0)

x3 = (0, 0, 0, 1, 0, 0, 0, 0)

ε = 0.0005

d(x1, x2) =2 > ε

d(x1, x3) =0 = ε

d(x2, x3) =2 > ε

Example 3

Result: x1 or x3


x1 = 21338

x2 = 54106

x3 = 37722

x4 = 54106

x5 = 4954

ε = 0 {x1, x2, x4} {x3, x5}

Example 4

Result: x1 or x2 or x4

d(x1 , x2 )= | x1 - x2 | mod 32768d(x1 , x2 ) =|x1 - x2 | mod 32768 = 0 = ε

d(x1 , x3 ) = |x1 – x3 | mod 32768 = 16384 > ε

d(x1 , x4 ) = |x1 – x4 | mod 32678 = 0 = ε

d(x1 , x5 ) = |x1 – x5 | mod 32678 = 16384 > ε

d(x2 , x3 ) = 16384 > ε

d(x2 , x4 ) = 0 = ε

d(x2 , x5 ) = 16384 > ε

d(x3 , x4 ) = 16384 > ε

d(x3 , x5 ) = 0 = ε

d(x4 , x5 ) = 16384 > ε

2 Generalized voters-Generalized median voter

Select a median value from the set of N outputs.


x1 =0.18155

x2 =0.18230

x3 =0.18130

x4 =0.18180

x5 =0.18235

ε = 0.0005

|x1- x2| = 0.00075

|x1- x3| = 0.00025

|x1- x4| = 0.00035

|x1- x5| = 0.0008

|x2- x3| = 0.001

|x2- x4| = 0.0005

|x2- x5| = 0.00005

|x3- x4| = 0.0005

|x3- x5| = 0.00105 (!)

|x4- x5| = 0.00045

(x1, x2, x4)

Example 5 (1)

Result: x4

|x1- x2| = 0.00075 (!)

|x1- x4| = 0.00035

|x2- x4| = 0.0005

=>

x4


x1 = (2.1350, -1.9693, 4.3354)

x2 = (2.1340, -1.9649, 4.3281)

x3 = (2.1376, -1.9623, 4.3284)

d(x1, x2) =0.0086

d(x1, x3) =0.0102 (!)

d(x2, x3) =0.0044

Result : x2

Example 6 (2)


Example 3

x1 = (0, 0, 0, 1, 0, 0, 0, 0)

x2 = (0, 1, 0, 0, 0, 0, 0, 0)

x3 = (0, 0, 0, 1, 0, 0, 0, 0)

ε = 0.0005

d(x1, x2) =2 (!)

d(x1, x3) =0

d(x2, x3) =2

Example 7 (3)

Majority Result: x1 or x3

d(x1, x2) =2 > ε

d(x1, x3) =0 = ε

d(x2, x3) =2 > ε

Median Result: x1 or x3


x1 = 21338

x2 = 54106

x3 = 37722

x4 = 54106

x5 = 4954

ε = 0

Example 8 (4)

Median Result: x1 or x2 or x4

d(x1 , x2 ) = d(x1 , x4 ) = d(x2 , x4 ) = d(x3 , x5 ) =0

d(x1 , x3 ) = d(x1 , x5 ) = d(x2 , x3 ) =d(x2 , x5 ) = d(x3 , x4 ) = d(x4 , x5 ) = 16384

Majority Result: x1 or x2 or x4

2 Generalized voters - formalized plurality voter

Construct a partition V1, …, Vk of A where for each i the set Vi is maximal with respect to the property that for any x, y in Vi d(x,y)<= ε

If there exist a set Va from V1, …, Vk , such that |Va | > | Vi | for any Vi <> Va , randomly select an element from Va is the voter output.

Suppose N versions of software with outputs in x produce the outputs x1, x2, … xN. Let w1, w2, … wN

denote the weight. Then

Define a new element of x by

2 Generalized voters – weighted averaging voter

i=1

N

W1 =1

i=1

N

W1 XiX=

2 Generalized voters – weighted averaging voter

Weight wi can be a priori knowledge

Weight wi can be calculated dynamically, i.e. by

and where s=

a is a fixed constant for scaling.


Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

3 Formalized majority vs. formalized plurality

1. Majority: result > half; plurality not necessary, relative large.

2. Majority is a special kind of plurality.

3 Formalized majority vs. formalized plurality x1 = 0.486 x2 = 0.483 x3 = 0.530 x4 = 0.495 x5 = 0.489 x6 = 0.500 x7 = 0.481 ε =0.01

Formalized majority

{x1 , x2 , x5 , x7 } >= (N+1)/2

{x4 , x6 } {x3}

Formalized plurality

{x1 , x4 , x5 }

{x2 , x7 } {x3} {x6}

3 Formalized majority vs. generalized median

The output produced by the formalized majority voting algorithm always contain the output of generalized median voter

3 Formalized majority vs. generalized median

x1 =0.18155

x2 =0.18230

x3 =0.18130

x4 =0.18180

x5 =0.18235

ε = 0.0005

Example 1-5

Majority Result: x1 or x3 or x4

x3 Median result:

3 Formalized majority vs. formalized median

x1 = (2.1350, -1.9693, 4.3354)

x2 = (2.1340, -1.9649, 4.3281)

x3 = (2.1376, -1.9623, 4.3284)

ε = 0.0005

Example 2-6

Majority result: x2 or x3

Median result: x2

d(x1, x2) =0.0086 > ε

d(x1, x3) =0.0102 > ε

d(x2, x3) =0.0044 < ε


x1 = (0, 0, 0, 1, 0, 0, 0, 0)

x2 = (0, 1, 0, 0, 0, 0, 0, 0)

x3 = (0, 0, 0, 1, 0, 0, 0, 0)

ε = 0.0005

Example 3 - 7

Majority Result: x1 or x3

Median result: x1 or x3

d(x1, x2) =2 > ε

d(x1, x3) =0 = ε

d(x2, x3) =2 > ε


x1 = 21338

x2 = 54106

x3 = 37722

x4 = 54106

x5 = 4954

ε = 0

Example 4 - 8

Majority Result: x1 or x2 or x4 the same as Median voter

d(x1 , x2 )= | x1 - x2 | mod 32768d(x1 , x2 ) =|x1 - x2 | mod 32768 = 0 = ε

d(x1 , x3 ) = |x1 – x3 | mod 32768 = 16384 > ε

d(x1 , x4 ) = |x1 – x4 | mod 32678 = 0 = ε

d(x1 , x5 ) = |x1 – x5 | mod 32678 = 16384 > ε

d(x2 , x3 ) = 16384 > ε

d(x2 , x4 ) = 0 = ε

d(x2 , x5 ) = 16384 > ε

d(x3 , x4 ) = 16384 > ε

d(x3 , x5 ) = 0 = ε

d(x4 , x5 ) = 16384 > ε


x1 = 101 x2 = 102 x3 = 103 x4 = 104 x5 = 105 ε = 1

Majority and plurality: cannot make a decision.

Median result: x3

4 Conclusion Formalized majority voter - select majority

Generalized median voter - select median – result is contained in the formalized majority

voter

Formalized plurality voter - select relative larger output

Weighted averaging technique - dynamically combines the output in a weighted average

A Theoretical Investigation of Generalized Voters for Redundant Systems Class: CS791F - Fall 2005...

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