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arXiv · 2405.11510

On some open problems in reliability theory

Geni Gupur
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Abstract

We study a stochastic scheduling on an unreliable machine with general up-times and general set-up times which is described by a group of partial differential equations with Dirac-delta functions in the boundary and initial conditions. In special case that the random processing rate of job $i,$ the random up-time rate of job $i$ and the random repair rate of job $i$ are constants, we determine the explicit expression of its time-dependent solution and give the asymptotic behavior of its time-dependent solution. Our result implies that $C_0-$semigroup theory is not suitable for this model. In general case, we determine the Laplace transform of its time-dependent solution. Next, we convert the model into an abstract Cauchy problem whose underlying operator is an evolution family. Finally, we leave some open problems.

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Geni Gupur. 2024-05-19. On some open problems in reliability theory. https://arxiv.org/abs/2405.11510

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