The failure rate (故障率) measures how many times a device or system fails per unit of time.
Early failure period
Random failure period
Wear-out failure period
As parts wear out, failures increase, the failure rate climbs again and the device reaches the end of its life.
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A smaller value means failures are less likely. When failures can be treated as occurring at a roughly steady pace regardless of time, the failure rate is the reciprocal of MTBF (mean time between failures, the next term), so an MTBF of 400 hours gives a failure rate of 0.0025 failures per hour. In real equipment the failure rate shifts between first use and the end of its life, and the curve tracing that shift is called the bathtub curve (バスタブ曲線).
AnalogyIt is like counting how many flat tires your commuter bike gets in a year. The fewer flats, the lower its failure rate.
How to rememberRemember it by the units: failure rate is "failures per hour" and MTBF is "hours per failure", the same fraction flipped upside down.
Exam points
ExamThe bathtub curve (バスタブ曲線) traces how the failure rate (故障率) changes over time: it falls at first, stays flat in the middle and rises again at the end, a shape like the cross-section of a bathtub.
MTBF (mean time between failures) is the average length of normal running time from a restart after repair until the next failure.
It is found by dividing the total time the system ran normally during the operating period by the number of failures. A larger value means fewer breakdowns, so it serves as a gauge of the system's reliability (信頼性), meaning how hard it is to break.
AnalogyIt is the average number of days you can ride your bike after fixing a flat before the next flat.
How to rememberRead Between in Mean Time Between Failures as the gap between two stops, that is, the running stretch from the end of one repair to the next stop.
Exam points
ExamWhen the recorded operating time also includes repair time, first subtract the total repair time to leave only the running time, then divide by the number of failures to get MTBF.
MTTR (mean time to repair) is the average time from the moment a failure stops the system until repair is finished and it runs again.
Normal running
Failure
Repair work
Back in service
Once the system is back in service, the next running stretch begins, and both averages come from a record of this repeating cycle.
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It is found by dividing the total time spent on repairs by the number of failures. A smaller value means faster recovery, so it serves as a gauge of the system's maintainability (保守性), meaning how easy it is to fix.
AnalogyIt is the average number of hours from getting a flat until the bike is fixed and you can ride it again.
How to rememberTo Repair in Mean Time To Repair means "until it is fixed", so it measures the time spent stopped.
Exam points
ExamMTTR is found by adding up the lengths of the repair stretches in the record and dividing by the number of repairs.
Mix-upMTBF and MTTR are both divided by the number of failures, but one sums the running stretches and the other the repair stretches, and for MTBF larger is better while for MTTR smaller is better.
Availability (稼働率) is the share of the total operating time in which the system worked normally, calculated as MTBF ÷ (MTBF + MTTR).
Connection
System runs when
How to calculate availability
Example: two devices with availability 0.7
Single device
That device is running
MTBF ÷ (MTBF + MTTR)
0.7
Series
Every device is running
Product of each device's availability
0.7 × 0.7 = 0.49
Parallel
At least one device is running
1 − (probability that all devices are down at once)
1 − 0.3 × 0.3 = 0.91
It is expressed as a value from 0 to 1 (or a percentage), and the closer it is to 1, the more available the system is in the broader sense (可用性): usable whenever it is needed. Lengthening MTBF or shortening MTTR raises it. With devices in series (直列) the system runs only while every device is running, while in parallel (並列) it runs as long as any one device is running.
AnalogyIt is the share of days in a year on which your bike was in good enough shape to ride.
How to rememberThe formula has the shape B ÷ (B + R): the numerator is just the B of MTBF, and the denominator adds the R of repair to make up the whole time.
Exam points
ExamIn the availability (稼働率) formula, the numerator is the average running time and the denominator is the average of running plus repair time, so the result is the share of the total operating time spent running.
ExamAvailability depends on the ratio of MTBF to MTTR, so stretching or shrinking both by the same factor leaves its value unchanged.
ExamWhen the devices fail independently of one another, the availability of a series (直列) setup is the product of each device's availability, while for a parallel (並列) setup it is 1 minus the probability that every device is down at once. Adding devices lowers availability in series and raises it in parallel.
When failures can be treated as occurring at a steady pace regardless of time, the failure rate is the reciprocal of MTBF.
MTBF and MTTR come from an operating record of alternating running and repair stretches: add up the lengths of the relevant stretches and divide by the number of failures.
Availability is calculated as MTBF ÷ (MTBF + MTTR) and shows the share of the total operating time during which the system was running.
For a system built from several devices, availability in series is the product of each device's availability, and in parallel it is 1 minus the probability that every device is down at once.