Redundancy — TMR (triple modular redundancy), voting logic: Visual walkthrough

5.5.25 · D2 · Coding › Embedded Systems & Real-Time Software › Redundancy — TMR (triple modular redundancy), voting logic

Yeh page central TMR result — formula

$R_{TMR} = 3 R^{2} - 2 R^{3}$

— ko bilkul scratch se build karta hai. Hum shuru karte hain yeh poochh ke "ek machine ke kaam karne ka matlab kya hai?" aur end karte hain us picture ke saath ki kyun teen machines ek se behtar ho sakti hain, aur kab nahi ho sakti.

Yeh parent topic ka visual companion hai. Agar aapko plain-language mirror chahiye, toh Hinglish note padho.

Step 0 — Jis model mein hum kaam kar rahe hain

KYA. Ek hi jagah fix karo ki hamare teen-machine system ka structure kaisa dikhta hai — koi bhi algebra chhedne se pehle.

KYUN. Is page ka har formula in teen assumptions par chupke se tikaa hua hai. Inhe ek baar, upfront, likhne ka matlab hai ki baad ke kisi bhi step ko dobarah introduce nahi karna padega.

Step 1 — "Reliability $R$ " ka matlab kya hai

KYA. Us ek number par zoom in karo jis par sab kuch tikaa hai: == $R$ ==, ek single module ki reliability — probability ki ek machine us time window mein correct output deti hai jiske baare mein hum care karte hain.

KYUN. Hum "teen machines" ki baat nahi kar sakte jab tak "ek machine" ko measure nahi kar sakte. Baad ka har formula sirf is single number par arithmetic hai, isliye pehle ise pakka karna zaroori hai. Ek probability ke roop mein, $R$ $0$ se $1$ tak kahin bhi ho sakta hai inclusive: $R = 1$ matlab "kabhi fail nahi hoti", $R = 0$ matlab "hamesha fail hoti hai", aur real modules kahin beech mein hote hain.

PICTURE. Har module ko ek weighted coin samjho. Yeh GOOD (correct output) land karta hai probability $R$ se, aur BAD (galat ya silent) probability $1 - R$ se. Coin ke do slices poora circle fill karne chahiye — aur kuch nahi ho sakta.

Figure — Redundancy — TMR (triple modular redundancy), voting logic

Is poori derivation ke liye, $R$ ko $[0, 1]$ mein ek fixed known number treat karo — jaise $0.9$ . (Yeh number time ke saath kahan se aata hai, woh alag kahaani hai jo Fault Tolerance Fundamentals mein handle ki gayi hai; yahan hum ise as given lete hain.)

Step 2 — Voter kya maangta hai: "3 mein se kam se kam 2"

KYA. Game ka rule precisely state karo, aur whole-system reliability ko uska naam do. TMR system sahi answer output karta hai exactly tab jab majority sahi ho — matlab 3 mein se kam se kam 2 modules GOOD hain.

KYUN. Voter wahi value pick karta hai jis par do ya zyada modules agree karte hain. Toh ek bad module harmless hai (baaki do phir bhi majority banate hain). Lekin do bad modules akele good module ko outvote kar sakte hain. Isliye magic line "2 good" (survive) aur "1 good" (die) ke beech mein draw hoti hai.

PICTURE. "Kitne modules GOOD hain" ki ek number line: $0, 1, 2, 3$ . $2$ aur $3$ ko green shade karo (system kaam karta hai), $0$ aur $1$ ko pink shade karo (system fail hota hai). Poori derivation hai: green region ki probability add karo.

Step 3 — Probability ki teeno GOOD hain

KYA. Pehla green piece compute karo: teeno modules correct hain. (Names $M_{1}, M_{2}, M_{3}$ Step 0 mein fix kiye gaye the — chhota subscript sirf ek label hai jo pick karta hai ki teeno mein se kaun sa module hai.)

KYUN. Step 0 ke model ke anusaar, modules independently fail karte hain — ek ka marna doosron ke baare mein kuch nahi batata — aur yahi cheez hume probabilities multiply karne deti hai. Independent events ke liye, "yeh AND woh AND woh" individual chances ka product hota hai.

PICTURE. Teen GOOD coins ek row mein, $M_{1}$ , $M_{2}$ , $M_{3}$ label kiye hue. Har ek ek factor $R$ contribute karta hai; side by side rakhne se multiply ho jaata hai.

Step 4 — Probability ki exactly do GOOD hain

KYA. Doosra green piece compute karo: exactly do correct, exactly ek broken.

KYUN. Yahi woh case hai jiske liye TMR banaya gaya tha — ek machine jhooth bolti hai, doosri do use mask kar deti hain. Do cheezein is probability mein multiply hoti hain: ek specific pattern ka chance, aur un patterns ki sankhya jo "exactly ek bad hai" jaisi dikhti hain.

PICTURE. Exactly ek BAD coin hone ke teen alag-alag tarike hain: bad wala $M_{1}$ ho sakta hai, ya $M_{2}$ , ya $M_{3}$ . Har aise row ki probability $R \cdot R \cdot (1 - R)$ hai — do GOOD (har ek $R$ ) aur ek BAD (woh bacha hua slice $1 - R$ ).

Step 5 — Do green pieces ko add karo

KYA. "3 good" piece aur "exactly 2 good" piece ko total reliability mein sum karo.

KYUN. Dono cases (teeno good / exactly do good) ek saath nahi ho sakte — yeh possibility ke mutually exclusive slices hain. Un events ke liye jo kabhi overlap nahi karte, "ek YA doosra" ki probability sirf sum hoti hai. Yeh Step 2 ki payoff line hai.

PICTURE. Do green bars ek doosre ke upar stack hote hue "system works" region ko fill karte hain.

Step 6 — Famous form mein simplify karo

KYA. Sum ko clean textbook expression mein convert karo.

KYUN. Cleaner algebra par reason karna, differentiate karna, aur $R$ se compare karna aasaan hota hai. Kuch naya nahi ho raha — bas multiply karke like terms collect karo.

PICTURE. Left side ka messy expression, term by term, right side ke tidy expression mein collapse ho jaata hai.

$R_{TMR} = R^{3} + 3 R^{2} (1 - R) = R^{3} + 3 R^{2} - 3 R^{3} = 3 R^{2} - 2 R^{3}$

Step 7 — Sanity check: kya yeh actually help karta hai?

KYA. Ek achhe module, $R = 0.9$ , ko plug in karo aur TMR ko single machine se compare karo.

KYUN. Ek formula jo baseline se behtar nahi ho, woh bekar hai. Hume improvement dekhni chahiye, sirf algebra par trust nahi karna.

PICTURE. Do bars side by side: single module $0.9$ par, TMR taller $0.972$ par.

Step 8 — Dark side: jab $R < 0.5$ ho, TMR worse hota hai

KYA. Degenerate regime dikhao. Ek kharab module, $R = 0.4$ , feed karo aur dekho TMR kaise haar jaata hai.

KYUN. Contract kehta hai har case cover karo — aur yeh woh trap hai. Agar modules kaam karne se zyada fail karte hain, toh do failures (jo akele good module ko outvote kar dete hain) common outcome ban jaate hain. Redundancy reliability ko cure karne ki jagah amplify kar deta hai.

PICTURE. Do curves $y = R$ (single) aur $y = 3 R^{2} - 2 R^{3}$ (TMR) saath mein plot ki hain. Yeh exactly $R = 0.5$ par cross karte hain. Crossing ke left mein, TMR single line ke neeche hai — pink danger zone.

Ek-picture summary

Har step ek frame mein collapse ho jaata hai: teen coins → GOOD ones count karo → " $\geq 2$ " region rakho → do green slices add karo → simplify karo → single-module line se compare karo.

Recall Feynman retelling — plain words mein vapas bolo

Ek machine ko ek coin samjho jo probability $R$ se "sahi" girti hai. Ek coin, ek chance — uski reliability bas $R$ hai.

Ab teen aise coins line up karo aur ek referee (voter) ko result "sahi" call karne do jab bhi kam se kam do coins sahi girein. Jeetnе ke sirf do tarike hain: teeno sahi girein, ya exactly do girein.

Teeno sahi girna probability $R \times R \times R = R^{3}$ se hota hai, kyunki independent coins multiply karte hain. Exactly do sahi girna teen tareekon se ho sakta hai (teeno mein se koi bhi odd one out ho sakta hai), aur har tarike ki probability $R^{2} (1 - R)$ hai — do rights aur ek wrong. Toh woh piece $3 R^{2} (1 - R)$ hai.

Dono winning pieces add karo: $R^{3} + 3 R^{2} (1 - R)$ . Ise multiply out karo aur yeh tidy ho ke $3 R^{2} - 2 R^{3}$ ban jaata hai. Woh single expression TMR ki reliability hai.

Test karo: ek achha coin ( $R = 0.9$ ) deta hai $0.972$ — $0.9$ se behtar, toh tripling ne fayda diya. Ek kharab coin ( $R = 0.4$ ) deta hai $0.352$ — $0.4$ se bura, kyunki do jhoote ek sacche coin ko outvote kar dete hain. Dono curves exactly $R = 0.5$ par cross karte hain. Toh poori kahaani hai: redundancy jo pehle se hai use multiply karta hai — achhe modules aur behtar hote hain, kharab modules aur bure hote hain.

Recall Quick self-test

TMR formula factored form mein ::: $R^{2} (3 - 2 R)$ TMR single module se tabhi better hota hai jab ::: $R > 0.5$ Hum "teeno good" ke liye $R \times R \times R$ kyun multiply kar sakte hain ::: kyunki modules independently fail karte hain $R^{2} (1 - R)$ ke teen copies kahan se aate hain ::: teen tareekon se jo single bad module choose karte hain, $(2 3) = 3$ Exact crossover value of $R$ jahan TMR single module ke barabar hota hai ::: $R = 0.5$ (aur trivially $0$ aur $1$ bhi)

Connects to

Fault Tolerance Fundamentals — woh broader family jis mein yeh belong karta hai.
Common-Cause Failures — Step 3 ki independence assumption ko todata hai.
Byzantine Fault Tolerance — kya karo jab koi bad module silently fail hone ki jagah cleverly jhooth bole.
Redundancy vs. Diversity — teen identical modules bugs share karte hain; diversity ise fix karta hai.
Watchdog Timers · Error Detection Codes · Safety-Critical Systems Standards — real systems mein companion mechanisms.