Two tools in one. The single-component calculator uses the exponential (constant-hazard) model to convert between failure rate ฮป and MTBF, find reliability R(t) over a mission time, and compute steady-state availability from MTBF and MTTR. The system calculator combines several component reliabilities in series or in parallel to get overall system reliability.
Reliability engineering quantifies how likely a component or system is to perform without failure, how often it fails on average, and what fraction of time it is actually up and usable. This calculator covers the three core questions: reliability R(t) โ the probability of surviving a mission of length t; MTBF โ the mean time between failures; and availability โ the long-run fraction of time the system is operational, accounting for repair time. It also combines components into series and parallel architectures to find overall system reliability.
During the useful-life (flat) portion of the bathtub curve, failures occur at a constant rate ฮป (failures per hour). Under this constant-hazard assumption, reliability follows the exponential law: R(t) = e^(โฮปt), the probability the item survives to time t with no failure. The probability it has failed by t is F(t) = 1 โ R(t).
Mean Time Between Failures is the reciprocal of the failure rate: MTBF = 1 / ฮป. So a component with ฮป = 0.0001 failures/hour has MTBF = 10,000 hours. (For non-repairable items the equivalent measure is MTTF, mean time to failure.) A useful checkpoint: when the mission time equals the MTBF, R(t) = e^(โ1) โ 0.368 โ there is only about a 37% chance of surviving one full MTBF without a failure.
Reliability measures whether something fails; availability measures whether it is up when you need it, which depends on how fast you repair it. Inherent (steady-state) availability is A = MTBF / (MTBF + MTTR), where MTTR is the Mean Time To Repair. A high-reliability machine that takes days to fix can still have poor availability; a less reliable machine that is repaired in minutes can be highly available.
Availability is often quoted in "nines": 99% โ 3.65 days of downtime per year, 99.9% โ 8.77 hours/year, 99.99% โ 52.6 minutes/year, 99.999% ("five nines") โ 5.26 minutes/year. To improve availability you can either raise MTBF (more reliable parts) or cut MTTR (faster repair, spares on hand, better diagnostics).
In a series configuration every component must work for the system to work โ they form a single chain. System reliability is the product of the individual reliabilities: Rsys = Rโ ร Rโ ร โฆ ร Rโ. Because each factor is below 1, adding components always lowers system reliability. Ten components each 99% reliable give only 0.99ยนโฐ โ 90.4% system reliability. This is why complex serial systems demand very high per-component reliability, and why reducing part count is itself a reliability strategy.
In a parallel (redundant) configuration the system works as long as at least one component works; it fails only if all of them fail. Since the components fail independently, the probability all fail is the product of their failure probabilities, so Rsys = 1 โ (1โRโ)(1โRโ)โฆ(1โRโ). Redundancy dramatically boosts reliability: two components each only 90% reliable give 1 โ (0.1)(0.1) = 99% in parallel. This is the basis of redundant power supplies, RAID arrays, and dual-engine aircraft. Real systems mix series and parallel blocks; analyze each block, then combine the block results.
Reliability R(t) is the probability of surviving a specific mission time t without failure. MTBF is the average time between failures โ a single summary number, the reciprocal of the failure rate ฮป. Availability is the long-run fraction of time the system is operational, combining how often it fails (MTBF) with how fast it is repaired (MTTR): A = MTBF/(MTBF+MTTR). You can have high MTBF but low availability if repairs are slow.
The exponential model applies during the useful-life period when the failure rate ฮป is constant (the flat bottom of the bathtub curve). A constant hazard rate mathematically produces an exponentially decaying survival probability, R(t) = e^(โฮปt). It is "memoryless": a unit that has run for years has the same chance of failing in the next hour as a brand-new one โ valid for random failures, but not for wear-out (rising hazard) or infant-mortality (falling hazard) phases.
MTBF (Mean Time Between Failures) applies to repairable systems โ the average operating time between successive failures. MTTF (Mean Time To Failure) applies to non-repairable items that are discarded on failure โ the average life until that single failure. Both equal 1/ฮป under the constant-rate model. The terms are often used loosely, but the distinction is whether the item gets repaired and returned to service.
A parallel (redundant) arrangement works as long as any one path works, so it fails only if every path fails simultaneously. Multiplying the (small) failure probabilities makes total failure very unlikely: two units each 90% reliable give 99% combined. That is why critical systems use redundant power supplies, dual controllers, RAID, and multiple engines. The trade-off is added cost, weight, and complexity.
They describe allowable downtime per year. 99% ("two nines") โ 3.65 days/year of downtime, 99.9% โ 8.8 hours/year, 99.99% โ 52.6 minutes/year, and 99.999% ("five nines") โ 5.3 minutes/year. Each additional nine cuts downtime roughly tenfold and is achieved by raising MTBF, cutting MTTR, or adding redundancy โ each progressively more expensive.