Exact and asymptotic analysis of infinite server batch service queues with random batch sizes
Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles...
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| Autor*in: |
Nakamura, Ayane [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Queueing systems - Dordrecht : Springer Science + Business Media B.V., 1986, 106(2023), 1-2 vom: 16. Dez., Seite 129-158 |
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| Übergeordnetes Werk: |
volume:106 ; year:2023 ; number:1-2 ; day:16 ; month:12 ; pages:129-158 |
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| DOI / URN: |
10.1007/s11134-023-09898-4 |
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| Katalog-ID: |
SPR054665191 |
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| 245 | 1 | 0 | |a Exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
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| 520 | |a Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. | ||
| 650 | 4 | |a Batch service |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Random batch sizes |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Infinite server queue |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Factorial moment |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Central limit theorem |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Moment approach |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Random variable version of Little’s law |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a M/M |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a / |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a queue |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a GI/M/ |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a queue |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a M/M |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a / |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a queue |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Ride-sharing |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Bus systems |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Transportation |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Phung-Duc, Tuan |0 (orcid)0000-0002-5002-4946 |4 aut | |
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10.1007/s11134-023-09898-4 doi (DE-627)SPR054665191 (SPR)s11134-023-09898-4-e DE-627 ger DE-627 rakwb eng Nakamura, Ayane verfasserin aut Exact and asymptotic analysis of infinite server batch service queues with random batch sizes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 queue (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 Phung-Duc, Tuan (orcid)0000-0002-5002-4946 aut Enthalten in Queueing systems Dordrecht : Springer Science + Business Media B.V., 1986 106(2023), 1-2 vom: 16. Dez., Seite 129-158 (DE-627)318755920 (DE-600)2014596-2 1572-9443 nnns volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 https://dx.doi.org/10.1007/s11134-023-09898-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 106 2023 1-2 16 12 129-158 |
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10.1007/s11134-023-09898-4 doi (DE-627)SPR054665191 (SPR)s11134-023-09898-4-e DE-627 ger DE-627 rakwb eng Nakamura, Ayane verfasserin aut Exact and asymptotic analysis of infinite server batch service queues with random batch sizes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 queue (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 Phung-Duc, Tuan (orcid)0000-0002-5002-4946 aut Enthalten in Queueing systems Dordrecht : Springer Science + Business Media B.V., 1986 106(2023), 1-2 vom: 16. Dez., Seite 129-158 (DE-627)318755920 (DE-600)2014596-2 1572-9443 nnns volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 https://dx.doi.org/10.1007/s11134-023-09898-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 106 2023 1-2 16 12 129-158 |
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10.1007/s11134-023-09898-4 doi (DE-627)SPR054665191 (SPR)s11134-023-09898-4-e DE-627 ger DE-627 rakwb eng Nakamura, Ayane verfasserin aut Exact and asymptotic analysis of infinite server batch service queues with random batch sizes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 queue (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 Phung-Duc, Tuan (orcid)0000-0002-5002-4946 aut Enthalten in Queueing systems Dordrecht : Springer Science + Business Media B.V., 1986 106(2023), 1-2 vom: 16. Dez., Seite 129-158 (DE-627)318755920 (DE-600)2014596-2 1572-9443 nnns volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 https://dx.doi.org/10.1007/s11134-023-09898-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 106 2023 1-2 16 12 129-158 |
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10.1007/s11134-023-09898-4 doi (DE-627)SPR054665191 (SPR)s11134-023-09898-4-e DE-627 ger DE-627 rakwb eng Nakamura, Ayane verfasserin aut Exact and asymptotic analysis of infinite server batch service queues with random batch sizes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 queue (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 Phung-Duc, Tuan (orcid)0000-0002-5002-4946 aut Enthalten in Queueing systems Dordrecht : Springer Science + Business Media B.V., 1986 106(2023), 1-2 vom: 16. Dez., Seite 129-158 (DE-627)318755920 (DE-600)2014596-2 1572-9443 nnns volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 https://dx.doi.org/10.1007/s11134-023-09898-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 106 2023 1-2 16 12 129-158 |
| allfieldsSound |
10.1007/s11134-023-09898-4 doi (DE-627)SPR054665191 (SPR)s11134-023-09898-4-e DE-627 ger DE-627 rakwb eng Nakamura, Ayane verfasserin aut Exact and asymptotic analysis of infinite server batch service queues with random batch sizes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 queue (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 Phung-Duc, Tuan (orcid)0000-0002-5002-4946 aut Enthalten in Queueing systems Dordrecht : Springer Science + Business Media B.V., 1986 106(2023), 1-2 vom: 16. Dez., Seite 129-158 (DE-627)318755920 (DE-600)2014596-2 1572-9443 nnns volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 https://dx.doi.org/10.1007/s11134-023-09898-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 106 2023 1-2 16 12 129-158 |
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Enthalten in Queueing systems 106(2023), 1-2 vom: 16. Dez., Seite 129-158 volume:106 year:2023 number:1-2 day:16 month:12 pages:129-158 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">SPR054665191</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20240207064621.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">240207s2023 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s11134-023-09898-4</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR054665191</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s11134-023-09898-4-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Nakamura, Ayane</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Exact and asymptotic analysis of infinite server batch service queues with random batch sizes</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2023</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. 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|
| author |
Nakamura, Ayane |
| spellingShingle |
Nakamura, Ayane misc Batch service misc Random batch sizes misc Infinite server queue misc Factorial moment misc Central limit theorem misc Moment approach misc Random variable version of Little’s law misc M/M misc / misc queue misc GI/M/ misc Ride-sharing misc Bus systems misc Transportation Exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
| authorStr |
Nakamura, Ayane |
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@@773@@(DE-627)318755920 |
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electronic Article |
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keep |
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aut aut |
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springer |
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true |
| illustrated |
Not Illustrated |
| issn |
1572-9443 |
| topic_title |
Exact and asymptotic analysis of infinite server batch service queues with random batch sizes Batch service (dpeaa)DE-He213 Random batch sizes (dpeaa)DE-He213 Infinite server queue (dpeaa)DE-He213 Factorial moment (dpeaa)DE-He213 Central limit theorem (dpeaa)DE-He213 Moment approach (dpeaa)DE-He213 Random variable version of Little’s law (dpeaa)DE-He213 M/M (dpeaa)DE-He213 / (dpeaa)DE-He213 queue (dpeaa)DE-He213 GI/M/ (dpeaa)DE-He213 Ride-sharing (dpeaa)DE-He213 Bus systems (dpeaa)DE-He213 Transportation (dpeaa)DE-He213 |
| topic |
misc Batch service misc Random batch sizes misc Infinite server queue misc Factorial moment misc Central limit theorem misc Moment approach misc Random variable version of Little’s law misc M/M misc / misc queue misc GI/M/ misc Ride-sharing misc Bus systems misc Transportation |
| topic_unstemmed |
misc Batch service misc Random batch sizes misc Infinite server queue misc Factorial moment misc Central limit theorem misc Moment approach misc Random variable version of Little’s law misc M/M misc / misc queue misc GI/M/ misc Ride-sharing misc Bus systems misc Transportation |
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misc Batch service misc Random batch sizes misc Infinite server queue misc Factorial moment misc Central limit theorem misc Moment approach misc Random variable version of Little’s law misc M/M misc / misc queue misc GI/M/ misc Ride-sharing misc Bus systems misc Transportation |
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Exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
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Exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
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Nakamura, Ayane Phung-Duc, Tuan |
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exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
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Exact and asymptotic analysis of infinite server batch service queues with random batch sizes |
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Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M%$^{\textit{X}}%$/%$\infty %$ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable %$\textit{X}%$, customers are served by a group. We formulate the M/M%$^{\textit{X}}%$/%$\infty %$ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that %$\textit{X}%$ has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M%$^\textit{X}%$/%$\infty %$ queue encompasses the time-based infinite server batch service queue (M/M%$^{G(x)}%$/%$\infty %$ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, %$\textit{G(x)}%$, and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M%$^{G(x)}%$/%$\infty %$ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/%$\infty %$ queue by utilizing the M/M%$^{G(x)}%$/%$\infty %$ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| score |
7.4016933 |