Reliability engineering studies how likely a product, machine, or system is to work without failing for a specified time. This cheat sheet helps students connect reliability formulas to real engineering decisions such as maintenance planning, part selection, and risk reduction. It is useful for understanding why failures are measured statistically instead of as exact predictions.
Grade 11-12 students can use it as a quick reference for formulas and interpretation.
Key Facts
- Reliability is the probability that a system performs its required function without failure for a stated time under stated conditions.
- For a constant failure rate, reliability is R(t) = e^(-lambda t), where lambda is the failure rate and t is time.
- Mean time between failures is MTBF = 1/lambda for repairable systems with a constant failure rate.
- Mean time to failure is MTTF = 1/lambda for nonrepairable items with a constant failure rate.
- Failure rate can be estimated by lambda = number of failures / total operating time.
- Availability for a repairable system is A = MTBF / (MTBF + MTTR), where MTTR is mean time to repair.
- For components in series, total reliability is R_total = R1 × R2 × R3 × ... × Rn.
- For two independent components in parallel redundancy, reliability is R_total = 1 - (1 - R1)(1 - R2).
Vocabulary
- Reliability
- Reliability is the probability that an item or system works correctly for a specified time under specified conditions.
- Failure rate
- Failure rate is the average number of failures per unit of operating time, often represented by lambda.
- MTBF
- Mean time between failures is the average operating time between failures for a repairable system.
- MTTR
- Mean time to repair is the average time needed to restore a failed system to working condition.
- Availability
- Availability is the fraction or probability of time that a repairable system is able to operate when needed.
- Redundancy
- Redundancy means adding backup components so the system can continue operating if one component fails.
Common Mistakes to Avoid
- Confusing MTBF with guaranteed lifetime is wrong because MTBF is an average statistical measure, not the time every unit will last.
- Using R(t) = e^(-lambda t) with inconsistent time units is wrong because lambda and t must use matching units, such as failures per hour and hours.
- Adding reliabilities in a series system is wrong because series components all must work, so their reliabilities are multiplied.
- Treating availability and reliability as the same value is wrong because availability includes repair time, while reliability usually measures failure-free operation over time.
- Ignoring independence in redundancy calculations is wrong because formulas like R_total = 1 - (1 - R1)(1 - R2) assume component failures are independent.
Practice Questions
- 1 A motor has a constant failure rate of 0.0002 failures per hour. Find its MTBF.
- 2 A sensor has lambda = 0.001 failures per hour. Use R(t) = e^(-lambda t) to find the reliability after 100 hours.
- 3 A repairable machine has MTBF = 500 hours and MTTR = 20 hours. Calculate its availability using A = MTBF / (MTBF + MTTR).
- 4 Explain why adding a backup component in parallel usually increases reliability more than adding another required component in series.
Understanding Reliability & MTBF Reference
A reliability number only has meaning when the operating conditions are clear. A phone charger used in a cool bedroom faces different stresses from one used in a hot factory. Temperature, vibration, dust, moisture, electrical load, and user handling can all change how often faults occur.
Engineers define a mission time, such as one flight, one thousand operating hours, or ten years in storage. They then ask whether the item can complete that mission.
A part may be dependable for a short mission but unsuitable for a long one. This is why test results must match the real environment as closely as possible.
The constant failure rate model is useful, but it is not true for every stage of a product's life. Many products follow a pattern called the bathtub curve. Early failures can occur because of manufacturing defects, poor assembly, or weak materials.
The middle period often has a roughly steady rate of random failures. Late failures rise as parts wear out, corrode, fatigue, or lose lubrication. A light bulb filament and a car tyre are clear wear-out examples.
Engineers may use burn-in testing to remove early weak units, then schedule replacement before wear-out becomes common. Students should notice the assumptions behind any formula before using it.
MTBF is often misunderstood as a promise that every machine will run for that exact time. It is an average based on many units and many hours of operation. If a pump has an MTBF of one thousand hours, one pump could fail very early while another lasts far longer.
The value is most useful for comparing designs under similar conditions. Maintenance teams combine it with mean time to repair.
A machine that fails rarely can still cause serious disruption if repairs take days. Spare parts, clear repair instructions, accessible components, and trained technicians can improve availability even when the failure rate stays the same.
System design creates important trade-offs. In a series system, every required component is a possible point of failure. A spacecraft power system, for example, needs working wiring, switches, batteries, controllers, and loads.
Adding more essential parts can lower the overall chance of success. Redundancy works differently. Two independent backup units can keep a function running when one unit fails.
Aircraft may have duplicate sensors or power paths for this reason. However, redundancy is not magic. If both units share the same power supply, software bug, heat source, or manufacturing defect, one event can disable both.
This is called a common-cause failure. Good designs use independent paths and test the whole system, not just each component alone.