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Expected demand for service parts when failure times have a Weibull distribution. (English) Zbl 0547.90031

The time between two successive failures being a random variable z with a Weibull probability density function: \[ f(z)=(\alpha\lambda )\quad (\lambda z)^{\alpha-1}\exp [-(\lambda z)^{\alpha}]. \] The function m(N,\(\tau)\) defined as the expected demand during a period time \(\tau\) of a N-parts system satisfies \[ m(1,T)=\sum^{\infty}_{j=1}c_ j(\alpha)\quad P(j,\lambda^{\alpha}T^{\alpha}) \] where P is the incomplete gamma function and \(c_ j(\alpha)\) are weighting factors. The number of service parts growing by B batches of size N/B, the expected total demand in (0,T) is: \[ M=(N/B)\quad\sum^{B}_{i=1}\quad m(1,T- t_ i), \] with ith batch entering at time \(t_ i=T_ 0(i-1)/(B-1) (i=1,...,B)\). An approximation M \(\hat A\) for M is proposed: \(\tilde M=N\lambda^{\alpha}(T-T_ 0/2)^{\alpha}\). Numerical results ”prove” the quality of this approximation, especially for certain values of the parameters \(\lambda\), \(\alpha\).
Reviewer: V.Cohen

MSC:

90B22 Queues and service in operations research
90B05 Inventory, storage, reservoirs
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References:

[1] Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions (1965), National Bureau of Standards: National Bureau of Standards Washington-DC)
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