Fault Masking By Probabilistic Voting

Transcription

1 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) Fault Masking B Probabilistic Voting B. Bakant ALAGÖZ Abstract: In this stud, we introduced a probabilistic voter, regarding smbol probabilities in decision process besides majorit consensus. Conventional majorit voter is independent of functionalit of redundant modules. In our stud, proposed probabilistic voter is designed corresponding to functionalit of the redundant module. We tested probabilistic voter for and redundant modules with random transient errors inserted the wires and it was seen from simulation results that Multi-Modular Redundanc (M-MR) with Probabilistic Voting (PV) had been shown better availabilit performance than conventional majorit voter. Keword: Fault Tolerance, Fault Masking,Voting Algorithm. Introduction: Majorit voting is the most common voting technique used in Multi-Modular Redundanc (M-MR) for fault masking in digital sstems. [-] The most basic one is bit-bbit majorit voting, in which voter algorithm selects the most common output. In the Table, truth table of conventional bit-b-bit majorit voting for Triple Modular Redundanc (TMR) was given to demonstrate output selection from redundant modules according to majorit consensus. In this table,, and are output of modules and represents voter output. General architecture of M-MR sstem was shown in the Figure. In older studies, conventional majorit voter was seen to capable of masking single errors which occurs when one of three modules is fault. Table. Conventional bit-b-bit majorit voting [] Module Module Voter Module k Figure. Block diagram of the Multi-Modular Redundanc Conventional bit-b-bit majorit voting made its decision according to majorit consensus at outputs of redundant modules to select the most reliable results. It does not consider an information based on smbol probabilities derived from functionalit of

2 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) modules. In this paper, we proposed a bit-b-bit voting algorithm, which is rather aware of module functionalit, to increase availabilit of M-MR. This awareness relies on error probabilit of obtained result at output of the modules. In this manner, we called proposed voting algorithm as probabilistic voting. Probabilistic voting considerabl raised availabilit of M-MR in our simulations and also reduced the complexit of the voter circuitr. In the next sections, we will suggest a probabilistic voter design methodolog and then we will do performance comparison with conventional majorit voter designed for TMR, made of three redundant modules, and also MR, made of five redundant modules. Probabilistic Voting: Let member of set {,} probabilit of smbol { } be output smbols of redundant modules,,... k. E is being an error at output and E is probabilit of smbol { } being an error at output of module. For redundant module with n -bit input, logic function of n modules is defined b term in its truth table. Number of terms resulting { } at output of module is denoted b N. If smbol { } at output is an error, probabilit of this error can be expressed b N E = n () Number of terms resulting { } at output of module is denoted b N and similarl, if smbol { } at output is an error, probabilit of this error can be expressed b, N E = n () Lets expand truth table of bit-b-bit voter seen in Table to Table. Table. Generic truth table of probabilistic voting for TMR Cost for Cost for E / E / E X E / E X E E / X E / E X E E / X E E / X E / For a given particular logic function, X values in generic truth tables are determined b following equation. C C C C X = () C < C

3 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) Here, o can be calculated b following formulas, C is cost function for smbol { } and C is cost function for smbol { }. The E C = and V E C = () V Here V is number of redundant modules ielding { } smbol at output and V is number of redundant modules ielding { } smbol at output. For the TMR, ( V, V ) couples can take values given b the set of {(,),(,),(,),(,) }. Probabilistic voting selects the output smbol, which has lower cost b mean of equation (). Cost of each smbol is calculated depending onto probabilit of obtained smbol being an error ( E, E ) and number of modules producing this smbol ( V, V ) as expressed at equation (). Probabilit of obtained smbol being an error was expressed b equation () and equation () for each smbol under assumption of the uniform distributed random inputs applied to modules and under this assumption, ( E, E ) parameters is ver dependent on the functionalit of the redundant module. On the other hand, number of modules ielding a smbol reflects strength of smbol among redundant modules,,... k and cost function falls b strength of smbol. Since, lower cost means lower error probabilit and higher consensus, probabilistic voter selects the smbol that has lower cost as the most reliable output smbol. Probabilistic Voter Design Examples: In this section, we will demonstrate a design example of probabilistic voters of TMR for a given logic function. We can use Table that is a generic truth table, subjecting to E and E. Lets consider -input logic function, of which truth table was given in Table. Table. Truth table of f function a b c d f

4 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) Number of smbol { } in the table is.( N ) Number of smbol { } in the table is =.( N = ) Number of input bits is. ( n = ). Using these parameters coming from logic function, one can calculates error probabilit for smbol { } as E = / 6 and for smbol { } as E = / 6. Now, we can construct truth table of probabilistic voter for the logic function given in Table. Considering Table, produced truth table of probabilistic voter was given in the Table. Table. Truth table for Probabilistic voting Cost for {} Cost for {} (C ) (C ) /8 / /6 / /6 /6 / / /6 /6 / /6 / /8 Logic equation of probabilistic voter originated from Table can be written as following, = () Simulation Results For Probabilistic Voter: Faults on logic gates were assumed to result an error bit at the output of the gate. This error bit would be expectedl complement of correct result. In simulations, error bits satisfing desired error probabilities were inserted to wires, which is connected to outputs of fault gates. Program for this error insertion mechanism used in the simulation is seen below. Program: Error insertion to wire % rand is random number between. and. % Pe: error probabilit for wire. net: Variable for wire if (rand < Pe) net = not(net); end if In the simulation, inputs with uniform distribution were applied to fault redundant modules for each error probabilit given in a range. Outputs of fault redundant modules were connected both of conventional majorit voter and probabilistic voter. Correct results were counted for availabilit calculation defined b following equation, Cr A = (6) Tr

5 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) Cr is number of correct output and Tr is number of total output. A is the availabilit of the sstem. In the Figure, availabilit of the fault module, availabilit of TMR with conventional majorit voter and availabilit of TMR with probabilistic voter are compared for error probabilit of wires ranging. to.. Figure. Availabilit comparison Figure. Number of errors at output of sstem

6 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) It is seen from Figure and Figure, probabilistic voter exhibited superior fault masking performance than conventional bit-b-bit majorit voter. Besides, voter circuitr for probabilistic voter has lower complexit than conventional majorit voter. Probabilistic Voter Design For MR: Generic truth table of probabilistic voting for MR was given in Table. Table. Generic truth table of probabilistic voting for MR C C E / E / E X E / E X E / E / X E / E X E / E / X E / E / X E / E / X E / E X E / E / X E / E / X E / E / X E / E / X E / E / X E / E / X E E / X E / E X E / E / X E / E / X E / E / X E / E / X E / E / X E / E / X E E / X E / E / X E / E / X E / E / X E E / X E / E / X E E / X E E / X E / Let design probabilistic voter for MR with redundant module given as, f = a b c d a b c d a b c d a b c d (7) Relevant parameters for f function would be E = / 6 and E = / 6, logic equation of probabilistic voter inferred from Table can be written as following, 6

7 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) 7 = (8) Obtained results from the simulation are illustrated in the Figure and Figure. Figure. Availabilit comparison Figure. Number of errors at output of sstem

8 OncuBilim Algorithm And Sstems Labs. Vol.9, Art.No:,(9) Conclusions: Probabilistic voting regarding smbol probabilities as well as majorit consensus was seen to provide better availabilit and error masking performance. In the paper, we introduced a probabilistic voter design approach for digital sstems. Suggested probabilistic voter were demonstrated to able to have better performance in case of uniform distributed random error insertion to wires and under application of uniforml distributed random inputs. Other important point to be consider that, complexit of the probabilistic voter can be lower than conventional majorit voter circuitr. This point gains importance when high volume logic is needed to support b M-MR particularl in Register Transfer Level (RTL) designs. As a consequence, probabilistic voter designed for a specific logic function can have better availabilit-complexit performance than conventional majorit voters designed for common-use. Reference: [] S. Mitra, E.J. McCluske, Word Voter: A New Voter Design for Triple Modular Redundant Sstems,8th IEEE VLSI Test Smposium (VTS'), p. 6, () [] J.C. Lo, Single Fault Masking Logic Designs with Error Correcting Codes, International Workshop on Defect and Fault Tolerance in VLSI Sstems, p.96, (99) [] B.B. Alagoz, Hierarchical Triple-Modular Redundanc (H-TMR) Network For Fault Tolerance In Digital Sstems, Turkish Journal of Electrical Engineering and Computer Sciences, In Review, (8) [] B.B. Alagoz, Adaptive Fault Masking With Incoherence Scoring, OncuBilim Algorithm And Sstem Lab, Art. Vol.8 No:, (8) arxiv:8.86 to memor of m brother Serdar Onur Alagöz. 8