Reliability of a TMR system

Calculations and Formulae

Reliability is the probability of no failure within a given operating period. The simplest system reliability model is to assume that in a system with n modules, all the modules must work. If the module reliability is Rm with a constant-failure rate k, then the system reliability is given as

To apply TMR, all circuits A, B, and C as shown above must have equivalent logic and must have the same truth tables. In most cases, they are three replications of the same design and are identical. Using this assumption, the system reliability is given as

If all the digital circuits are independent and identical then the reliability of the TMR scheme can be determined as a function of the reliability RM of one module, assuming the voting circuit does not fail. The redundant system will function properly as long as any two modules are operational. It is assumed that failures of the three modules are independent of one another. Hence, the reliability Rtmr of the TMR scheme can be rewritten as follows in terms of the binomial theorem

The reliability of the TMR system is usually better than that suggested since the system may continue to function correctly even if two modules fail. For example, if one module has a stuck-at-1 fault on its output while another module has a stuck-at-0 fault on its output, the system still produces the correct output. Such multiple module failures that do not lead to system failures are termed as compensating module failures

If it is assumed that each module in a TMR system has passed through an extensive burn-in period and the voter is a perfect voter (Rv = 1), then Rm is an exponential function of time with a constant failure rate k, or Rm = e^kt . then we have the reliability of TMR system as:

Reference:

1.Kshirsagar, R. V., & Patrikar, R. M. (2009). Design of a novel fault-tolerant voter circuit for TMR implementation to improve reliability in digital circuits. Microelectronics Reliability,

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