Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior
The class of counting processes constitutes a significant part of applied probability. The classic counting processes include Poisson processes, nonhomogeneous Poisson processes, and renewal processes. More sophisticated counting processes, including Markov renewal processes, Markov modulated Poisso...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2009-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2009/219532 |
| Summary: | The class of counting processes constitutes a significant part of applied probability. The classic counting processes include Poisson processes, nonhomogeneous
Poisson processes, and renewal processes. More sophisticated counting processes,
including Markov renewal processes, Markov modulated Poisson processes, age-dependent counting processes, and the like, have been developed for accommodating
a wider range of applications. These counting processes seem to be quite different
on the surface, forcing one to understand each of them separately. The purpose
of this paper is to develop a unified multivariate counting process, enabling one to
express all of the above examples using its components, and to introduce new counting processes. The dynamic behavior of the unified multivariate counting process is
analyzed, and its asymptotic behavior as t→∞ is established. As an application,
a manufacturing system with certain maintenance policies is considered, where the
optimal maintenance policy for minimizing the total cost is obtained numerically. |
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| ISSN: | 0161-1712 1687-0425 |