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Appendix 3-2

AN EXAMPLE TO SHOW THE EFFECT OF THE SHAPE FACTOR

In Appendix 3-1, we derived the plots of the failure distribution, surviving population, and hazard rates for a set of assumed data, to demonstrate the airline industry distribution of failures. In pattern E—namely, the constant hazard rate case— the value of the hazard rate is 0.015. In section 3.8 on mean availability, we discussed how the MTBF was the inverse of λ, which is the same as the hazard rate z(t) in the constant hazard case.

Thus, the MTBF = 1/0.015 = 66.7 weeks. Recall that the MTBF is the same as the scale factor η, in the constant hazard case. So η= 66.7 weeks. We are now going to use this value of η, vary the time t, and use different values of β, and see how the distribution changes as β changes.

Using expression 3.14, we compute the R(t) for the data in Appendix 3-1, namely, n= 66.7 weeks and for different values of β as t increases from 1 week to 100 weeks. From the R(t) value, we compute the cumulative failures F(t), which is = l-R(t). The F(t) values are given below.

At low values of b, the distribution of failures is skewed to the left, i.e., there are many more failures initially than toward the end of life. In our example, at the end of 10 weeks, let us see how the b value affects F(t) up to that point.

•When β =0.5, cumulative failures will be 32% of the total.

•When β =1.0, cumulative failures will be 14% of the total.

•When β =2.0, cumulative failures will be 2.2% of the total.

•When β =3.5, cumulative failures will be <0.2% of the total. we do not expect any significantfailures till about the 32nd week.

•When β =10, cumulative failures will be ~0% of the total, we do not expect any significant failures till about the 32nd week.

Also of interest is what happens after we exceed the characteristic life. In week 77, i.e., ~ 10 weeks after the characteristic life is passed,

•When β =0.5, cumulative failures will be 66%of the total.

•When β =1.0, cumulative failures will be 68% of the total.

•When β =2.0, cumulative failures will be 73% of the total.

•When β =3.5, cumulative failures willbe 80% of the total.

•When β =10, cumulative failures willbe 98% of the total.

From this sequence, you can see that the higher the β value, the more the clustering of failures towards the characteristic value, and hence the greater predictability of time of failure.

At t=66.7 weeks, for all values of β, the R(t) is the same. In other words, the shape factor does not affect the survival probability when t = scale factor.