Incomputing,triple modular redundancy, sometimes calledtriple-mode redundancy,[1] (TMR) is afault-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.
The TMR concept can be applied to many forms ofredundancy, such as software redundancy in the form ofN-version programming, and is commonly found infault-tolerant computer systems.
Space satellite systems often use TMR,[2][3] although satellite RAM usually usesHamming error correction.[4]
SomeECC memory uses triple modular redundancy hardware (rather than the more commonHamming code), because triple modular redundancy hardware is faster than Hamming error correction hardware.[5] Calledrepetition code, some communication systems use N-modular redundancy as a simple form offorward error correction. For example, 5-modular redundancy communication systems (such asFlexRay) 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 computerSAPO, in the 1950s.
The general case of TMR is calledN-modular redundancy, in which any positive number of replications of the same action is used. The number is typically taken to be at least three, so that error correction by majority vote can take place; it is also usually taken to be odd, so that no ties may happen.[6]
The 3-input 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. Thus, the majority gate is thecarry output of afull adder, i.e., the majority gate is avoting machine.[7]
The 3-input majority gate can be represented by the followingboolean equation andtruth table:
| INPUT A B C | OUTPUT Q | ||
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
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.
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.[7]
For a TMR system with a single voter of reliability (probability of working)Rv and three components of reliabilityRm, the probability of it being correct can be shown to beRTMR = Rv (3 Rm2 – 2 Rm3).[6]
TMR systems should usedata scrubbing – rewrite flip-flops periodically – in order to avoid accumulation of errors.[8]
The majority gate itself could fail. This can be protected against by applying triple redundancy to the voters themselves.[9]
In a few TMR systems, such as theSaturn Launch Vehicle Digital Computer andfunctional 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 nosingle point of failure.[10][11]
Even though only using a single voter brings a single point of failure – a failed voter will bring down the entire system – most TMR systems do not use triplicated voters. This is because the majority gates are much less complex than the systems that they guard against, so they are much morereliable.[7] By using the reliability calculations, it is possible to find the minimum reliability of the voter for TMR to be a win.[6]
To use triple modular redundancy, a ship must have at least threechronometers; two chronometers provideddual modular redundancy, allowing a backup if one should cease to work, but not allowing anyerror 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 (theerror detection obtained would be the same of having only one chronometer and checking it periodically). Three chronometers provided triple modular redundancy, allowingerror 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."[12]
Mainly this means that if twochronometers 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.[13]Some vessels carried more than three chronometers – for example,HMS Beagle carried22 chronometers.[14] However, such a large number was usually only carried on ships undertaking survey work as was the case with theBeagle.
In the modern era, ships at sea useGNSS navigation receivers (withGPS,GLONASS &WAAS etc. support) – mostly running with WAAS orEGNOS support so as to provide accurate time (and location).
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