Triple modular redundancy
In computing, triple modular redundancy, sometimes called triple-mode redundancy, (TMR) is a fault-tolerant form of N-modular redundancy, in which three systems perform a process and that result is processed by a majority-voting system to produce a single output. If any one of the three systems fails, the other two systems can correct and mask the fault.
Some ECC memory uses triple modular redundancy hardware (rather than the more common Hamming code), because triple modular redundancy hardware is faster than Hamming error correction software. Space satellite systems often use TMR,  although satellite RAM usually uses Hamming error correction.
Some communication systems use N-modular redundancy as a simple form of forward error correction. For example, 5-modular redundancy communication systems (such as FlexRay) use the majority of 5 samples – if any 2 of the 5 results are erroneous, the other 3 results can correct and mask the fault.
Modular redundancy is a basic concept, dating to antiquity, while the first use of TMR in a computer was the Czechoslovak computer SAPO, in the 1950s.
To use triple modular redundancy, a ship must have at least three chronometers; two chronometers provided dual modular redundancy, allowing a backup if one should cease to work, but not allowing any error correction if the two displayed a different time, since in case of contradiction between the two chronometers, it would be impossible to know which one was wrong (the error detection obtained would be the same of having only one chronometer and checking it periodically). Three chronometers provided triple modular redundancy, allowing error correction if one of the three was wrong, so the pilot would take the average of the two with closer reading (vote for average precision).
There is an old adage to this effect, stating: "Never go to sea with two chronometers; take one or three."
Mainly this means that if two chronometers contradict, how do you know which one is correct? At one time this observation or rule was an expensive one as the cost of three sufficiently accurate chronometers was more than the cost of many types of smaller merchant vessels. Some vessels carried more than three chronometers – for example, the HMS Beagle carried 22 chronometers. However, such a large number was usually only carried on ships undertaking survey work as was the case with the Beagle.
Majority logic gate
In TMR, three identical logic circuits (logic gates) are used to compute the same set of specified Boolean function. If there are no circuit failures, the outputs of the three circuits are identical. But due to circuit failures, the outputs of the three circuits may be different.
A majority logic gate is used to decide which of the circuits’ outputs is the correct output. The majority gate output is 1 if two or more of the inputs of the majority gate are 1; output is 0 if two or more of the majority gate’s inputs are 0.
The majority logic gate is a simple AND–OR circuit: if the inputs to the majority gate are denoted by x, y and z, then the output of the majority gate is
Assuming the Boolean function computed by the three identical logic gates has value 1, then: (a) if no circuit has failed, all three circuits produce an output of value 1, and the majority gate output has value 1. (b) if one circuit fails and produces an output of 0, while the other two are working correctly and produce an output of 1, the majority gate output is 1, i.e., it still has the correct value. And similarly for the case when the Boolean function computed by the three identical circuits has value 0. Thus, the majority gate output is guaranteed to be correct as long as no more than one of the three identical logic circuits has failed.
In a few TMR systems, such as the Saturn Launch Vehicle Digital Computer and functional triple modular redundancy (FTMR) systems, the voters are also triplicated. Three voters are used – one for each copy of the next stage of TMR logic. In such systems there is no single point of failure. 
However, in contrast to the relatively complicated Boolean functions computed in triplicate by the TMR system, the majority gate is a simple circuit, thus its probability of failure is significantly smaller than that of each of the three circuits computing the Boolean function. In other systems there is only a single voter—If the voter fails in such a system, then the complete system will fail. However, in a good TMR system the voter is much more reliable than the other TMR components.
Triple modular redundancy in popular culture
- The three pre-cogs in Minority Report lead to a conviction even when one disagrees.
- To rule out that a single win was "a fluke", some competitions use a two out of three falls match. This isn't true TMR, however, because the three falls are not independent of each other – each competitor knows who has most falls at any point in the competition, which influences their future actions.
- In Arthur C. Clarke's science fiction novel Rendezvous with Rama, the Ramans make heavy use of triple redundancy.
- Article about TMR with reference to TMR usage in avionics and industry
- Johnson, J. M., & Wirthlin, M. J. (2010, February). Voter insertion algorithms for FPGA designs using triple modular redundancy. In Proceedings of the 18th annual ACM/SIGDA international symposium on Field programmable gate arrays (pp. 249–258). ACM.
- David Ratter. "FPGAs on Mars"
- "Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite". Tsinghua Space Center, Tsinghua University, Beijing. Archived from the original on 2011-10-02. Retrieved 2009-02-16.
- "Actel engineers use triple-module redundancy in new rad-hard FPGA". Military & Aerospace Electronics. Retrieved 2017-04-09.
- SEU Hardening of Field Programmable Gate Arrays (FPGAs) For Space Applications and Device Characterization Archived 2011-11-25 at the Wayback Machine.
- FPGAs in Space[permanent dead link]
- Commercial Microelectronics Technologies for Applications in the Satellite Radiation Environment
- Brooks, Frederick J. (1995) . The Mythical Man-Month. Addison-Wesley. p. 64. ISBN 0-201-83595-9.
- "Re: Longitude as a Romance". Irbs.com, Navigation mailing list. 2001-07-12. Archived from the original on 2011-05-20. Retrieved 2009-02-16.
- R. Fitzroy. "Volume II: Proceedings of the Second Expedition". p. 18.
- Professor Dilip V. Sarwate, Lecture Notes for ECE 413 - Probability with Engineering Applications, Department of Electrical and Computer Engineering (ECE), UIUC College of Engineering, University of Illinois at Urbana-Champaign
- Wojciech M. Zabolotny et al. "Radiation tolerant design of RLBCS system for RPC detector in LHC experiment" Archived 2009-01-06 at the Wayback Machine.. Proceedings of SPIE. 2005.
- A.W. Krings. "Redundancy". 2007.
- Sandi Habinc. "Functional Triple Modular Redundancy (FTMR): VHDL Design Methodology for Redundancy in Combinatorial and Sequential Logic" Archived 2012-06-05 at the Wayback Machine.. 2002.
- R. E. Lyons and W. Vanderkulk. "The Use of Triple-Modular Redundancy to Improve Computer Reliability". IBM Journal. 1962.