Series, Parallel and k-out-of-n Systems

System reliability is simply the chance that a whole system works, figured out from the chances that its parts work. There are only two basic arrangements. In series (a chain), everything has to work, so you multiply the parts' reliabilities: $R_1R_2\cdots$. In parallel (a bank of backups), the system fails only if every part fails, so you multiply the failure chances: $1 - (1 - R_1)(1 - R_2)\cdots$. Everything else, including $k$-out-of-$n$ voting, is built from those two ideas plus the binomial you already know.

A chain and a bank. A server rack's power runs utility feed → transfer switch → UPS → rack. If any block fails, the rack goes dark. A second UPS that can carry the load alone changes the arithmetic completely. A reliability block diagram draws this logic: blocks in a chain must all work (series), blocks side by side back each other up (parallel). It describes function, not wiring: two valves piped one after the other are a parallel pair if their job is to stop flow, since either one closing does it.

Series: all must work. With independent failures, $P(\text{all work})$ is a product. The system is weaker than its weakest part, and long chains are brutal: 100 parts at 0.99 each give $0.99^{100} = 0.37$. With exponential parts, $R_s(t) = e^{-\lambda_1t}\cdots e^{-\lambda_nt} = e^{-(\lambda_1 + \cdots + \lambda_n)t}$: failure rates add, which is why electronic parts are rated in failures per billion hours and summed over a board.

Parallel: one is enough. The system fails only if every unit fails, so $1 - R_p = \prod(1 - R_i)$. Two 0.90 units give $1 - 0.1^2 = 0.99$; three give 0.999. Each added unit multiplies the failure probability by the same factor: big gains at first, then diminishing returns against cost, weight and space.

Reduce, then multiply. For a series-parallel network, replace each parallel group by one equivalent block, then multiply along the chain. Networks that don't reduce, like a bridge, yield to conditioning: split on one component working or failed, and combine the two cases with the law of total probability.

k-out-of-n: count the ways. A 2-out-of-3 voting set of sensors, 6 of 8 drives in a storage array, 3 of 4 engines: with independent identical parts, the number working is binomial, so

For 2-out-of-3, $R_{2/3} = 3R^2(1 - R) + R^3 = 3R^2 - 2R^3$, which beats a single unit only when $R > 0.5$. The same binomial sums count defective units in a batch.

Over time, plug in $R(t)$. For two active exponential units, $R(t) = 1 - (1 - e^{-\lambda t})^2 = 2e^{-\lambda t} - e^{-2\lambda t}$, and the area under it (Reliability and Hazard Functions) is $\text{MTTF} = \frac{2}{\lambda} - \frac{1}{2\lambda} = \frac{3}{2\lambda}$. The second unit adds only half a life, because while both run the first failure comes twice as fast (rate $2\lambda$). A cold standby spare that doesn't age while it waits gives $\frac{2}{\lambda}$, if the switch always works. Redundancy shines on short missions: at $\lambda t = 0.1$ an active pair fails with probability $0.009$, against $0.095$ for one unit.

Redundancy trade-offs:

• Redundancy at the component level beats redundancy at the system level: two pumps in parallel followed by two valves in parallel beats two complete pump-and-valve trains, if the cross-connections are reliable.
• Common-cause failures (one power supply, one software bug, one flood) hit every copy at once and break the independence the formulas assume. They usually set the real limit.
• Voting systems give up some reliability in exchange for catching wrong outputs, not just dead ones.
• Redundancy plus repair is far stronger than redundancy alone: fix the failed unit before its partner fails. That's the next lesson.

A common mistake. With two backup units, it's tempting to add their reliabilities. Two 0.9 units would then give 1.8, which can't be a probability. For parallel units, multiply the failure chances instead: $1 - (0.1)(0.1) = 0.99$. Watch also for using the series formula on redundant parts, and remember that all these formulas assume the parts fail independently; a shared power supply or a common flood breaks that.

Reliability block diagrams are standard in aerospace, power grids, rail signaling and data centers. Airliners fly with redundant hydraulic systems and voting flight computers, hospitals and server farms use N+1 or 2N power designs, and storage systems keep several copies of every file so that k of n survive. Engineers run these calculations to find single points of failure and to decide where one more redundant part buys the most reliability per dollar.