Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions
Abstract By studying the spectrum on the imaginary axis of the underlying operator, which corresponds to the M/G/1 retrial queueing model with general retrial times described by infinitely many partial differential equations with integral boundary conditions, we prove that the time-dependent solutio...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-10-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-018-1083-y |
| _version_ | 1828876200148205568 |
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| author | Nurehemaiti Yiming Geni Gupur |
| author_facet | Nurehemaiti Yiming Geni Gupur |
| author_sort | Nurehemaiti Yiming |
| collection | DOAJ |
| description | Abstract By studying the spectrum on the imaginary axis of the underlying operator, which corresponds to the M/G/1 retrial queueing model with general retrial times described by infinitely many partial differential equations with integral boundary conditions, we prove that the time-dependent solution of the model strongly converges to its steady-state solution. Next, when the conditional completion rates for repeated attempts and service are constants, we describe the point spectrum of the underlying operator and verify that all points in an interval in the left real line including 0 are eigenvalues of the underlying operator. Lastly, by using these results and the spectral mapping theorem we prove that the C0 $C_{0}$-semigroup generated by the underlying operator is not compact, but not eventually compact and even not quasi-compact, and it is impossible that the time-dependent solution exponentially converges to its steady-state solution. In other words, our result on convergence is optimal. |
| first_indexed | 2024-12-13T08:15:14Z |
| format | Article |
| id | doaj.art-97aa5e2352b44d02a857748ce1a510d2 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-13T08:15:14Z |
| publishDate | 2018-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-97aa5e2352b44d02a857748ce1a510d22022-12-21T23:54:07ZengSpringerOpenBoundary Value Problems1687-27702018-10-012018113110.1186/s13661-018-1083-yAsymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditionsNurehemaiti Yiming0Geni Gupur1College of Mathematics and Systems Science, Xinjiang UniversityCollege of Mathematics and Systems Science, Xinjiang UniversityAbstract By studying the spectrum on the imaginary axis of the underlying operator, which corresponds to the M/G/1 retrial queueing model with general retrial times described by infinitely many partial differential equations with integral boundary conditions, we prove that the time-dependent solution of the model strongly converges to its steady-state solution. Next, when the conditional completion rates for repeated attempts and service are constants, we describe the point spectrum of the underlying operator and verify that all points in an interval in the left real line including 0 are eigenvalues of the underlying operator. Lastly, by using these results and the spectral mapping theorem we prove that the C0 $C_{0}$-semigroup generated by the underlying operator is not compact, but not eventually compact and even not quasi-compact, and it is impossible that the time-dependent solution exponentially converges to its steady-state solution. In other words, our result on convergence is optimal.http://link.springer.com/article/10.1186/s13661-018-1083-yM/G/1 retrial queueing model with general retrial timesC 0 $C_{0}$ -semigroupTime-dependent solutionEigenvalueResolvent set |
| spellingShingle | Nurehemaiti Yiming Geni Gupur Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions Boundary Value Problems M/G/1 retrial queueing model with general retrial times C 0 $C_{0}$ -semigroup Time-dependent solution Eigenvalue Resolvent set |
| title | Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| title_full | Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| title_fullStr | Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| title_full_unstemmed | Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| title_short | Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| title_sort | asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions |
| topic | M/G/1 retrial queueing model with general retrial times C 0 $C_{0}$ -semigroup Time-dependent solution Eigenvalue Resolvent set |
| url | http://link.springer.com/article/10.1186/s13661-018-1083-y |
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