Abstract
This paper is concerned with the distribution of the time between failures (TBF) under the gamma lifetime model. The exact distribution of TBF is computed in closed form. This allows interested researchers to compute the reliability of certain system models. To better visualize the distributions of TBF, its probability distribution function (pdf) is represented using matrices. Moreover, the moments of TBF is computed and given in closed form. Wolfram Mathematica cods are given in the appendix for fast and easy implementation. Finally, simulation studies for the TBF are performed together with relative tests results.
Highlights
Is computed and given in closed form
This implies that T has a gamma distribution with a positive integer shape parameter k parameter t
An important measure in reliability engineering is the mean time between failures (MTBF) of the system, which is the average time from repair to failure ([3])
Summary
The pdf of Ti is given in ([7]) as:. , where ak = (a1,a2, ,ak ) and bk = (b1,b2, ,bk 1) , k 1. The representation of the pdf of Ti as a mixture of gamma distributions makes it easy to deal with as follows. W is the diagonal matrix with {w 0,w 1, ,w (k 1)(n i )} as its diagonal entries, where w j is the wight of the j th component of the mixture representing f Tt. we sum the resulting integrals over the variables a1,a2, ,ak satisfying the constraint a k j =1 j. M be diagonal matrix whose non-zero entries are the respective components of the mixture. The proof follows by multiplying the right-hand side of (2.7) by r m and integrating term-by-term with respect to r over (0, ) and noting that j 1w j (n i ) j 1r e j m (n i )r dr = w j ( j m )!
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