問71
頻出度★★★
あるシステムにおいて, MTBF と MTTR がどちらも 1.5 倍になったとき, 稼働率はどうなるか。
平成23年度 特別試験 ITパスポート試験 問71
テクノロジ系システム構成要素 › システムの評価指標
This question's explanation and your answers so far.
解説
問 71
Incorrectア
The value 2/3 appears when one wrongly assumes that only the MTTR has grown 1.5 times longer and that availability is inversely proportional to MTTR. In reality the MTBF becomes 1.5 times larger at the same time, so the increase in repair time is completely offset by the longer interval between failures. In the formula as well, the 1.5 in 1.5M / (1.5M + 1.5R) cancels between numerator and denominator, so there is nothing at all that could make the availability smaller. In the example above, 0.9 does not fall to 0.6; it stays at 0.9.
Incorrectイ
The answer 1.5 times comes from mistaking availability for a quantity proportional to MTBF and overlooking that MTTR also appears in the denominator. Even if MTBF becomes 1.5 times larger, the denominator MTBF + MTTR becomes 1.5 times larger as well, so the ratio does not change. Moreover, availability is a proportion between 0 and 1 (0 to 100 percent); for a system that was at 0.9, 1.5 times that is 1.35, which would exceed 100 percent. Judging by the range of possible values alone, this option cannot hold.
Incorrectウ
The value 2 times can be derived neither from the factor of 1.5 given in the question nor from the formula availability = MTBF / (MTBF + MTTR). There is no basis for applying the same 1.5-fold change to both measures and arriving at a different factor such as 2; in the calculation, cancellation leaves a factor of 1, that is, no change. Furthermore, availability is a proportion that cannot exceed 1, so for a system whose original availability is above 0.5, a result of 2 times is impossible in the first place.
- エ
正解 The correct answer is エ. A system's availability is the proportion of the total operating time during which the system is working normally. Using the mean time between failures, MTBF (Mean Time Between Failures), and the mean time to repair, MTTR (Mean Time To Repair), it is expressed as availability = MTBF / (MTBF + MTTR). The denominator MTBF + MTTR means “the time the system ran normally plus the time spent on repair”, that is, the length of one whole failure cycle. Writing M for MTBF and R for MTTR, the availability after both become 1.5 times larger is 1.5M / (1.5M + 1.5R). Factoring 1.5 out of the denominator gives 1.5 × (M + R), so the 1.5 in the numerator and in the denominator cancel and the expression returns to M / (M + R). In other words, the result is exactly the same as the original availability. Checking with numbers: a system with MTBF = 90 hours and MTTR = 10 hours has an availability of 90 / (90 + 10) = 0.9 (90 percent); after both become 1.5 times larger, MTBF = 135 hours and MTTR = 15 hours give 135 / (135 + 15) = 135 / 150 = 0.9, which is unchanged. What this formula shows is that availability is determined not by the absolute sizes of MTBF and MTTR but only by the ratio between them. Even if the system becomes 1.5 times less prone to failure (MTBF), if at the same time it becomes 1.5 times harder to repair (MTTR), the ratio of uptime to downtime stays as it was, so in terms of availability nothing has improved. Therefore the availability does not change, and エ is correct.
ここがポイント
- The availability formula — availability = MTBF / (MTBF + MTTR). MTBF is the mean time between failures (the average time the system keeps running without failing), MTTR is the mean time to repair (the average time from a failure until the system is restored), and the denominator represents one whole failure cycle. Availability improves either by raising reliability (making MTBF longer) or by raising maintainability (making MTTR shorter).
- Availability is fixed by a ratio — availability is determined not by the absolute values of MTBF and MTTR but only by the ratio between them, so multiplying both by the same factor cancels the common factor and leaves the value unchanged. Conversely, if only one of them changes, a larger MTBF raises availability while a larger MTTR lowers it. It is also useful to note that availability is a proportion in the range 0 to 1 (0 to 100 percent), so any option whose result would exceed 1 is impossible by definition.
用語