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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.11510 |
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| _version_ | 1866929348621107200 |
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| author | Gupur, Geni |
| author_facet | Gupur, Geni |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some open problems in reliability theory Gupur, Geni Probability Analysis of PDEs 90B25, 35A01 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. |
| title | On some open problems in reliability theory |
| topic | Probability Analysis of PDEs 90B25, 35A01 |
| url | https://arxiv.org/abs/2405.11510 |