When introducing some kind of machine or system, the life of the product is one of the major selection points. The longer the lifespan, the longer and safer the product can be used. One of the indicators for knowing the life of such products is MTTF (mean time to failure). MTTF is the time from the start of operation until the product fails and becomes unusable, the so-called average lifespan.
However, a product with an MTTF of 10 years does not necessarily work for 10 years. In theory, it is believed that there is about a 63% chance of failure before the MTTF is reached. Why such a theory? This time, we will introduce the relationship between product reliability and MTTF.
Confidence that indicates the probability that the product is working
Why is the theory that "failure will occur with a probability of about 63% before reaching the MTTF"? An essential part of this explanation is the index of product reliability.
Reliability indicates the probability that a machine or system is operating within a unit time. If the reliability is 1, the machine/system will not fail within a unit time. Reliability can be obtained by multiplying the reliability of each system. For example, the overall reliability of a system composed of a system with a reliability of 0.8 and a system with a reliability of 0.9 is 0.8 x 0.9 = 0.72.
Reliability can also be obtained from the failure rate. In this case, confidence can be expressed as:
R=e^-λt (R is reliability, e is Napier number, λ is failure rate, t is time)
For example, to obtain the reliability of a product that fails 0.5 times a year (failure rate is 0.5) for 2 years,
R=e^(-0.5×2)=e^-1=0.37
becomes. From this formula, it can be seen that the higher the failure rate and the longer the time, the lower the reliability. The failure rate can also be expressed as 1/MTTF.
Relationship between MTTF and reliability
Based on the above, let's consider the relationship between MTTF and reliability.
For example, for a product with an MTTF of 10 years. Assume that the failure rate for this product is constant, 1/10 = 0.1. The reliability of this product after one year of operation is
R=e^-λt=e^-0.1×1=e^-0.1=0.9048=90.48%
becomes. However, as t approaches the MTTF of 10 years, the reliability drops noticeably. 81.873% at t=2, 74.082% at t=3, and at MTTF t=10
R = e^-0.1 x 10 = e^-1 = 0.36788 = 36.788%
It becomes.
As mentioned above, the reliability is the probability that the machine/system is operating within a unit time, so it means that it is operating normally only with a probability of about 37%. In other words, there is about a 63% chance of failure.
This gives the same result no matter how large the MTTF is. For example, for a product with a MTTF of 100 years and a constant failure rate of 1/100 = 0.01, the reliability at t = 100 is
R=e^-0.01×100=e^-1=0.36788=36.788%
As a result, the failure rate is also about 63%. Even if the MTTF is 1,000 years, -λt is -1 when t = 1,000, so the theoretical result shows that there is a 63% probability of failure before reaching the MTTF. From this, it can be said that no matter how long the MTTF is set, there is a good chance that it will fail before reaching the MTTF.
MTTF is often used as an index for knowing the life of a product. If you know the MTTF of a product, you can figure out the life expectancy of that product. However, according to the reliability calculations introduced in this article, theoretically, there is a 63% chance of failure before reaching the MTTF.
MTTF is an important metric for knowing the lifespan of a product, but don't take that number for granted. In particular, when selecting safety-related products, judging only by MTTF can lead to serious accidents. While using MTTF as one of the indicators, it is important to keep in mind that theoretically there is a 63% chance of failure before reaching MTTF.
In addition, there is a word "MTTFd" that is very similar to MTTF. ISO13849-1 "Safety of machinery - Safety-related parts of control systems" is one of the parameters that satisfies "PL (Performance Level)" in the scale of functional safety. represents the average time it takes. Next time, I will introduce this PL and MTTFd.