問70
頻出度★★★
同じ装置が複数接続されているシステム構成のうち, システムが停止する可能性の最も低いものはどれか。ここで, -[ ]- は装置を表し, 並列に接続されている場合はいずれか一つの装置が動作していればよく, 直列に接続されている場合はすべての装置が動作していなければならない。
平成23年度 特別試験 ITパスポート試験 問70
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 70
Incorrectア
In the figure, ア is a series configuration of two devices connected in a single horizontal row. In a series connection the system operates only if every device is operating, so the system stops if either of the two devices fails. Its availability is a to the power of 2 and its stop probability is 1 - a^2. This is larger than 1-a, the failure probability of a single device, and far more likely to stop than the parallel configurations イ and エ. As long as the connection is in series, adding devices only lowers reliability, so this can never be the configuration with the lowest probability of stopping.
Incorrectイ
In the figure, イ is a configuration in which two devices are connected in parallel; it is indeed redundant and does not stop easily. However, its stop probability is (1-a) to the power of 2, the probability that the two devices fail at the same time, which is larger than the (1-a) to the power of 3 of エ, which requires three devices to fail at the same time. Since 1-a is less than 1, multiplying by it once more makes エ necessarily smaller. In the example with an availability of 0.9, the stop probability of イ is 0.01 while that of エ is 0.001, one tenth as much. Among parallel configurations, the more devices there are the harder it is for the system to stop, so イ with two devices does not match エ with three.
Incorrectウ
In the figure, ウ is a series configuration of three devices connected in a single horizontal row, and it is the most likely of the four to stop. In a series connection every device must be operating, so the system stops if even one of the three fails. Its availability is a to the power of 3, and its stop probability 1 - a^3 is even larger than the 1 - a^2 of ア with two devices in series. Although it uses the same three devices, it is the exact opposite of エ, whose parallel connection gives the smallest stop probability, (1-a) to the power of 3; note that in a series connection a larger number of devices acts as a decrease in reliability.
- エ
正解 The correct answer is エ. In the figure, エ is a parallel configuration of three devices: the path splits into three branches at the entrance, one device is placed on each branch, and the branches merge again at the exit. Under the condition given in the question, a parallel connection keeps the system running as long as any one device is operating, so エ stops only when all three devices fail at the same time. Let the availability of one device be a and its failure probability be 1-a, and assume that the failures of the devices are independent of each other. Then the stop probabilities of the four configurations are as follows. A series connection requires every device to be operating, so ア (2 devices in series) has an availability of a to the power of 2 and a stop probability of 1 - a^2, while ウ (3 devices in series) has an availability of a to the power of 3 and a stop probability of 1 - a^3. A parallel connection stops only when every device has failed, so イ (2 devices in parallel) has a stop probability of (1-a) to the power of 2, and エ (3 devices in parallel) has a stop probability of (1-a) to the power of 3. Because 1-a is a value between 0 and 1, the more times it is multiplied the smaller the result becomes, so (1-a)^3 < (1-a)^2 always holds. The stop probability therefore decreases in the order ウ > ア > イ > エ, and the smallest is エ. For example, if one device has an availability of 0.9 (failure probability 0.1), the stop probabilities are 0.19 for ア, 0.271 for ウ, 0.01 for イ and 0.001 for エ, which confirms numerically that エ is by far the lowest. The key point is that a parallel connection is redundancy (spare devices are prepared so that the system keeps running even if one of them breaks down) and becomes harder to stop as the number of devices increases, whereas a series connection gains more failure points as the number of devices increases and therefore stops more easily; the two have exactly opposite properties.
ここがポイント
- Availability of series and parallel systems — if the availability of one device is a, the availability of n devices in series is a multiplied by itself n times (the system works only when all of them are operating), and it falls as the number of devices increases. The availability of n devices in parallel is 1 minus (1-a) multiplied by itself n times (the system stops only when all of them fail at the same time), and it rises as the number of devices increases. If you want the probability of stopping to be as low as possible, choose the parallel configuration with the largest number of devices.
- How to think about redundancy and reliability design — a parallel connection is redundancy: spare devices are prepared so that the failure of one device does not stop the whole system, and it is a basic means of fault-tolerant design. A series connection, on the other hand, gains more single points of failure as devices are added, so its reliability falls. Availability is an indicator expressed as MTBF / (MTBF + MTTR), and the availability of the system as a whole is determined by how the configuration is put together.
用語