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...
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| Format: | Article |
| Language: | English |
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Wiley
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 |
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| author | Ushio Sumita Jia-Ping Huang |
| author_facet | Ushio Sumita Jia-Ping Huang |
| author_sort | Ushio Sumita |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ffcd4d075338421f8f4f11688fc5d9dd |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2009-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ffcd4d075338421f8f4f11688fc5d9dd2025-08-20T03:25:54ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252009-01-01200910.1155/2009/219532219532Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic BehaviorUshio Sumita0Jia-Ping Huang1Graduate School of Systems and Information Engineering, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki 305-8573, JapanGraduate School of Systems and Information Engineering, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki 305-8573, JapanThe 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.http://dx.doi.org/10.1155/2009/219532 |
| spellingShingle | Ushio Sumita Jia-Ping Huang Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior International Journal of Mathematics and Mathematical Sciences |
| title | Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior |
| title_full | Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior |
| title_fullStr | Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior |
| title_full_unstemmed | Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior |
| title_short | Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior |
| title_sort | dynamic analysis of a unified multivariate counting process and its asymptotic behavior |
| url | http://dx.doi.org/10.1155/2009/219532 |
| work_keys_str_mv | AT ushiosumita dynamicanalysisofaunifiedmultivariatecountingprocessanditsasymptoticbehavior AT jiapinghuang dynamicanalysisofaunifiedmultivariatecountingprocessanditsasymptoticbehavior |