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A Theoretical Investigation of Generalized Voters for Redundant Systems Class: CS791F - Fall 2005 Professor : Dr. Bojan Cukic Student: Yue Jiang

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A Theoretical Investigation of Generalized Voters for Redundant Systems Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

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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

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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

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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.

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2 Generalized voters-Formalized majority voter x 1 =0.18155 x 2 =0.18230 x 3 =0.18130 x 4 =0.18180 x 5 =0.18235 ε = 0.0005 |x 1 - x 3 | = 0.00025 |x 1 - x 4 | = 0.00035 |x 2 - x 5 | = 0.00005 ( x 1, x 3, x 4 ) ( x 2, x 5 ) Example 1 Result: x 1 or x 3 or x 4

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2 Generalized voters-Formalized majority voter x 1 = (2.1350, -1.9693, 4.3354) x 2 = (2.1340, -1.9649, 4.3281) x 3 = (2.1376, -1.9623, 4.3284) ε = 0.0005 d 2 (x 1, x 2 ) = (2.1350- 2.1340) 2 + [-1.9693-(- 1.9649)] 2 + (4.3354- 4.3281) 2 d(x 1, x 2 ) =0.0086 > ε d(x 1, x 3 ) =0.0102 > ε d(x 2, x 3 ) =0.0044 < ε Example 2 Result: x 2 or x 3

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2 Generalized voters-Formalized majority voter x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = 0.0005 d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε Example 3 Result: x 1 or x 3

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2 Generalized voters-Formalized majority voter x 1 = 21338 x 2 = 54106 x 3 = 37722 x 4 = 54106 x 5 = 4954 ε = 0 {x 1, x 2, x 4 } {x 3, x 5 } Example 4 Result: x 1 or x 2 or x 4 d(x 1, x 2 )= | x 1 - x 2 | mod 32768 d(x 1, x 2 ) =|x 1 - x 2 | mod 32768 = 0 = ε d(x 1, x 3 ) = |x 1 – x 3 | mod 32768 = 16384 > ε d(x 1, x 4 ) = |x 1 – x 4 | mod 32678 = 0 = ε d(x 1, x 5 ) = |x 1 – x 5 | mod 32678 = 16384 > ε d(x 2, x 3 ) = 16384 > ε d(x 2, x 4 ) = 0 = ε d(x 2, x 5 ) = 16384 > ε d(x 3, x 4 ) = 16384 > ε d(x 3, x 5 ) = 0 = ε d(x 4, x 5 ) = 16384 > ε

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2 Generalized voters-Generalized median voter Select a median value from the set of N outputs.

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2 Generalized voters-Generalized median voter x 1 =0.18155 x 2 =0.18230 x 3 =0.18130 x 4 =0.18180 x 5 =0.18235 ε = 0.0005 |x 1 - x 2 | = 0.00075 |x 1 - x 3 | = 0.00025 |x 1 - x 4 | = 0.00035 |x 1 - x 5 | = 0.0008 |x 2 - x 3 | = 0.001 |x 2 - x 4 | = 0.0005 |x 2 - x 5 | = 0.00005 |x 3 - x 4 | = 0.0005 |x 3 - x 5 | = 0.00105 (!) |x 4 - x 5 | = 0.00045 ( x 1, x 2, x 4 ) Example 5 (1) Result: x 4 |x 1 - x 2 | = 0.00075 (!) |x 1 - x 4 | = 0.00035 |x 2 - x 4 | = 0.0005 => x4x4

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2 Generalized voters-Generalized median voter x 1 = (2.1350, -1.9693, 4.3354) x 2 = (2.1340, -1.9649, 4.3281) x 3 = (2.1376, -1.9623, 4.3284) d(x 1, x 2 ) =0.0086 d(x 1, x 3 ) =0.0102 (!) d(x 2, x 3 ) =0.0044 Result : x 2 Example 6 (2)

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2 Generalized voters-Generalized median voter Example 3 x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = 0.0005 d(x 1, x 2 ) =2 (!) d(x 1, x 3 ) =0 d(x 2, x 3 ) =2 Example 7 (3) Majority Result: x 1 or x 3 d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε Median Result: x 1 or x 3

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2 Generalized voters-Generalized median voter x 1 = 21338 x 2 = 54106 x 3 = 37722 x 4 = 54106 x 5 = 4954 ε = 0 Example 8 (4) Median Result: x 1 or x 2 or x 4 d(x 1, x 2 ) = d(x 1, x 4 ) = d(x 2, x 4 ) = d(x 3, x 5 ) =0 d(x 1, x 3 ) = d(x 1, x 5 ) = d(x 2, x 3 ) =d(x 2, x 5 ) = d(x 3, x 4 ) = d(x 4, x 5 ) = 16384 Majority Result: x 1 or x 2 or x 4

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2 Generalized voters - formalized plurality voter Construct a partition V 1, …, V k of A where for each i the set V i is maximal with respect to the property that for any x, y in V i d(x,y)<= ε If there exist a set V a from V 1, …, V k, such that |V a | > | V i | for any V i <> V a, randomly select an element from V a is the voter output.

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Suppose N versions of software with outputs in x produce the outputs x 1, x 2, … x N. Let w 1, w 2, … w N denote the weight. Then Define a new element of x by 2 Generalized voters – weighted averaging voter i=1 N W 1 =1 i=1 N W 1 X i X=

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2 Generalized voters – weighted averaging voter Weight w i can be a priori knowledge Weight w i can be calculated dynamically, i.e. by and where s= a is a fixed constant for scaling.

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A Theoretical Investigation of Generalized Voters for Redundant Systems Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

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3 Formalized majority vs. formalized plurality 1. Majority: result > half; plurality not necessary, relative large. 2. Majority is a special kind of plurality.

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3 Formalized majority vs. formalized plurality x 1 = 0.486 x 2 = 0.483 x 3 = 0.530 x 4 = 0.495 x 5 = 0.489 x 6 = 0.500 x 7 = 0.481 ε =0.01 Formalized majority {x 1, x 2, x 5, x 7 } >= (N+1)/2 {x 4, x 6 } {x 3 } Formalized plurality {x 1, x 4, x 5 } {x 2, x 7 } {x 3 } {x 6 }

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3 Formalized majority vs. generalized median The output produced by the formalized majority voting algorithm always contain the output of generalized median voter

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3 Formalized majority vs. generalized median x 1 =0.18155 x 2 =0.18230 x 3 =0.18130 x 4 =0.18180 x 5 =0.18235 ε = 0.0005 Example 1-5 Majority Result: x 1 or x 3 or x 4 x3x3 Median result:

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3 Formalized majority vs. formalized median x 1 = (2.1350, -1.9693, 4.3354) x 2 = (2.1340, -1.9649, 4.3281) x 3 = (2.1376, -1.9623, 4.3284) ε = 0.0005 Example 2-6 Majority result: x 2 or x 3 Median result: x 2 d(x 1, x 2 ) =0.0086 > ε d(x 1, x 3 ) =0.0102 > ε d(x 2, x 3 ) =0.0044 < ε

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3 Formalized majority vs. formalized median x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = 0.0005 Example 3 - 7 Majority Result: x 1 or x 3 Median result: x 1 or x 3 d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε

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3 Formalized majority vs. formalized median x 1 = 21338 x 2 = 54106 x 3 = 37722 x 4 = 54106 x 5 = 4954 ε = 0 Example 4 - 8 Majority Result: x 1 or x 2 or x 4 the same as Median voter d(x 1, x 2 )= | x 1 - x 2 | mod 32768 d(x 1, x 2 ) =|x 1 - x 2 | mod 32768 = 0 = ε d(x 1, x 3 ) = |x 1 – x 3 | mod 32768 = 16384 > ε d(x 1, x 4 ) = |x 1 – x 4 | mod 32678 = 0 = ε d(x 1, x 5 ) = |x 1 – x 5 | mod 32678 = 16384 > ε d(x 2, x 3 ) = 16384 > ε d(x 2, x 4 ) = 0 = ε d(x 2, x 5 ) = 16384 > ε d(x 3, x 4 ) = 16384 > ε d(x 3, x 5 ) = 0 = ε d(x 4, x 5 ) = 16384 > ε

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3 Formalized majority vs. formalized median x 1 = 101 x 2 = 102 x 3 = 103 x 4 = 104 x 5 = 105 ε = 1 Majority and plurality: cannot make a decision. Median result: x 3

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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

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