In general, the availability of a station can be characterized by two numbers: the Mean Time To Fail (MTTF) and the Mean Time to Repair (MTTR). The first number shows how often a problem (e.g., a breakdown) occurs on the station. The second number gives the average time it takes to fix a problem when it occurs. The reverse of MTTF and MTTR are respectively called failure rate and repair rate.
There are two useful ways to express the relationship between the availability d, the Mean Time To Fail (MTTF) and the Mean Time To Repair (MTTR).
Equation 1:
d = MTTF / (MTTF + MTTR) (1)
Equation 2, a simple rewriting of Equation 1:
MTTR/ MTTF = (1/d -1) (2)
To compute the overall availability of the line, d_Line, we need to know MTTF_Line and MTTR_Line of a line. We work on the general case of a line comprise of n stations.
Computing the Mean Time To Fail of the line: MTTF_Line
Since there is no intermediate buffer, the whole line will stop any time there is a failure which occurs at one of the n stations. Therefore the failure rate of the whole line is given by the some of failure rate of each line.
This means:
1/ MTTF_Line = n x 1/ MTTF (3)
or
MTTF_Line = MTTF/n (4)
This equation simply means that frequency of failures in the line will be multiplied by the number of the machines. In practice, it translate into the fact that the more machines you have the more problems you have, which is simply common sense.
Computing the Mean Time To Repair of the line: MTTR_Line
The mean time to repair of the whole line is the weight average of the mean time to repair of each machine. Since all machines are identical, they are all failling at the same failure rate of 1/MTTF, this means:
MTTR_Line = MTTR (5)
This equation means that the overall Mean Time To Repair of the line will not depend on the number of machines. This is mostly driven by the repairing squad or maintenance team performance and organization.
We may now compute d_Line, the availability of the whole line as a function of the availability d of identical stations.
From Equation (1), we may compute d_line as follows:
d_Line = MTTF_Line / (MTTF_line + MTTR_Line) (6)
By rewriting a little bit we may obtain the following:
d_Line = 1/ (1 + MTTR_Line / MTTF_Line) (7)
Now using Equations (2), (4) and (5), we may also rewrite d_Line as follows:
d_Line = 1/(1+n x( 1/d-1)) (8)
This equation means that at the end of the day the overall efficiency of your line (d_line may also be call efficiency) will solely depend on the efficiency of each machine. You may get to the same result by two ways: 1/ making sure that your frequent breakdowns, if not completely eliminated, are repaired quickly or 2/ making sure that the longest breakdowns are the least frequent .
Please note that the product of availabilities of the machines is a simple and very good approximate of this formula (Equation 8) when machines' availabilities are close to 1.
As foot-noted in the book (Measuring Operators' Performance), here is the proof of the formula that gives the overall availability of line comprised of n identical stations with no intermediate buffer. All stations are identical so are their individual availability (called d).
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Complementary Matirials
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