A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue
Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a sc...
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2088 |
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| author | Ayane Nakamura Tuan Phung-Duc |
| author_facet | Ayane Nakamura Tuan Phung-Duc |
| author_sort | Ayane Nakamura |
| collection | DOAJ |
| description | Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the <i>moment approach</i> to address these two limitations in an infinite-server batch service queue, which is called the M/M<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mi>X</mi></msup></semantics></math></inline-formula>/<i>∞</i> queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm. |
| first_indexed | 2024-03-11T04:13:16Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:13:16Z |
| publishDate | 2023-04-01 |
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| series | Mathematics |
| spelling | doaj.art-7b517397917b4db684ded2b1c1d39d772023-11-17T23:19:51ZengMDPI AGMathematics2227-73902023-04-01119208810.3390/math11092088A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> QueueAyane Nakamura 0Tuan Phung-Duc 1Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, JapanInstitute of Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8577, JapanSeveral studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the <i>moment approach</i> to address these two limitations in an infinite-server batch service queue, which is called the M/M<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mi>X</mi></msup></semantics></math></inline-formula>/<i>∞</i> queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm.https://www.mdpi.com/2227-7390/11/9/2088queueing modelinfinite serverasymptotic analysisweak law of large numberscentral limit theoremmoment approach |
| spellingShingle | Ayane Nakamura Tuan Phung-Duc A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue Mathematics queueing model infinite server asymptotic analysis weak law of large numbers central limit theorem moment approach |
| title | A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue |
| title_full | A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue |
| title_fullStr | A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue |
| title_full_unstemmed | A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue |
| title_short | A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M<i><sup>X</sup></i>/<i>∞</i> Queue |
| title_sort | moment approach for a conditional central limit theorem of infinite server queue a case of m m i sup x sup i i ∞ i queue |
| topic | queueing model infinite server asymptotic analysis weak law of large numbers central limit theorem moment approach |
| url | https://www.mdpi.com/2227-7390/11/9/2088 |
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