8) Suppose a manufacturing process requires continuous use of a machine. We begi

Question

8) Suppose a manufacturing process requires continuous use of a machine. We begin with a new machine at time O. Suppose we follow the age-replacement policy: replace the machine in use when it fails or when it reaches an age of T years. This policy is useful in reducing the number of on-job failures, which tend to be more costly. Let Li be the lifetime of the ith machine and assume that Li 2 1) is a sequence of id non-negative random variables. This process produces several renewal sequences: a) Sn time of the nth failure b) Tn time of the nth planned replacement c) Un = time of the nth replacement (planned or otherwise) Suppose the machine lifetimes (in years) are iid U(2,5), and they are replaced upon failure or upon reaching age 3. Compute: the long-term rate of replacements i.e. the expected number of replacements per year as t oo; i) i) the long-term rate of planned replacements; ii) the long-term rate of machine failures.

Explanation / Answer

Solution.

We begin at Time = 0 with new machine

The replacement will occur when-

It fails or when it reached age of T years,

Li is the lifetime of the ith machine and (Li,i>1)

Hence the renewal sequence are:-

Sn = time of the nth failure

Tn= time of the nth planned replacement

Un= time of the nth replacement (planned or otherwise)

iid U(2,5) and T=3

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