Sojourn-time Distribution for M/Ga/1 Queue with Batch Service of Fixed Size - Revisited

This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M / G a / 1 queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing...

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Veröffentlicht in:	Methodology and computing in applied probability Jg. 24; H. 4; S. 2897 - 2912
Hauptverfasser:	Goswami, Veena, Chaudhry, Mohan, Banik, Abhijit Datta
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2022 Springer Nature B.V
Schlagworte:	Business and Management Customer services Distribution functions Economics Eigenvalues Eigenvectors Electrical Engineering Life Sciences Mathematics and Statistics Probability Probability density functions Queues Random variables Statistics Sojourn time Distribution function Partial fractions Batch service Roots
ISSN:	1387-5841, 1573-7713
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Abstract	This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M / G a / 1 queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing in Applied Probability 20(4):1503–1514, 2018 ). We obtain two closed-form expressions for probability density functions by using the inside and outside roots of the underlying characteristic equation. Applying partial fractions and residue theorem, we determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. In illustrative examples, we compare the results obtained by both methods and find that the results match excellently.
AbstractList	This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M/Ga/1 queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing in Applied Probability 20(4):1503–1514, 2018). We obtain two closed-form expressions for probability density functions by using the inside and outside roots of the underlying characteristic equation. Applying partial fractions and residue theorem, we determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. In illustrative examples, we compare the results obtained by both methods and find that the results match excellently. This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M / G a / 1 queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing in Applied Probability 20(4):1503–1514, 2018 ). We obtain two closed-form expressions for probability density functions by using the inside and outside roots of the underlying characteristic equation. Applying partial fractions and residue theorem, we determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. In illustrative examples, we compare the results obtained by both methods and find that the results match excellently.
Author	Chaudhry, Mohan Banik, Abhijit Datta Goswami, Veena
Author_xml	– sequence: 1 givenname: Veena orcidid: 0000-0002-4260-4721 surname: Goswami fullname: Goswami, Veena email: veena_goswami@yahoo.com organization: School of Computer Applications, Kalinga Institute of Industrial Technology – sequence: 2 givenname: Mohan surname: Chaudhry fullname: Chaudhry, Mohan organization: Department of Mathematics and Computer Science, Royal Military College of Canada – sequence: 3 givenname: Abhijit Datta surname: Banik fullname: Banik, Abhijit Datta organization: School of Basic Sciences, Indian Institute of Technology
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Cites_doi	10.1007/s11134-015-9467-5 10.1016/j.peva.2004.12.002 10.1016/j.peva.2011.03.007 10.1155/S0161171292000218 10.1007/s11009-020-09828-4 10.1007/s10479-017-2476-5 10.1287/mnsc.21.7.777 10.1080/07362990500522395
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References_xml	– ident: CR18 – volume: 82 start-page: 353 issue: 3 year: 2016 end-page: 380 ident: CR16 article-title: Analysis of generalized QBD queues with matrix-geometrically distributed batch arrivals and services publication-title: Queueing Systems doi: 10.1007/s11134-015-9467-5 – volume: 63 start-page: 131 year: 2006 end-page: 145 ident: CR1 article-title: Solving the ME/ME/1 queue with state-space methods and the matrix sign function publication-title: Perform Eval doi: 10.1016/j.peva.2004.12.002 – volume: 68 start-page: 528 issue: 6 year: 2011 end-page: 549 ident: CR12 article-title: Analysis of threshold-based batch-service queueing systems with batch arrivals and general service times publication-title: Perform Eval doi: 10.1016/j.peva.2011.03.007 – volume: 18 start-page: 265 issue: 2 year: 1956 end-page: 274 ident: CR14 article-title: On limiting distributions arising in bulk service queues publication-title: J Roy Stat Soc: Ser B (Methodol) – ident: CR2 – volume: 15 start-page: 189 issue: 1 year: 1992 end-page: 194 ident: CR15 article-title: A queueing system with a fixed accumulation level, random server capacity and capacity dependent service time publication-title: Int J Math Math Sci doi: 10.1155/S0161171292000218 – ident: CR10 – ident: CR8 – volume: 23 start-page: 1551 issue: 4 year: 2021 end-page: 1579 ident: CR6 article-title: Analysis of a queueing model with batch markovian arrival process and general distribution for group clearance publication-title: Methodol Comput Appl Probab doi: 10.1007/s11009-020-09828-4 – volume: 16 start-page: 80 issue: 1 year: 1954 end-page: 87 ident: CR3 article-title: On queueing processes with bulk service publication-title: J Roy Stat Soc: Ser B (Methodol) – ident: CR27 – ident: CR23 – ident: CR21 – volume: 3 start-page: 115 issue: 1 year: 1987 end-page: 148 ident: CR4 article-title: Characterizations of generalized hyperexponential distribution functions publication-title: Stoch Model – ident: CR19 – volume: 277 start-page: 161 issue: 2 year: 2019 end-page: 196 ident: CR22 article-title: Analysis of an infinite-buffer batch-size-dependent service queue with markovian arrival process publication-title: Ann Oper Res doi: 10.1007/s10479-017-2476-5 – volume: 21 start-page: 777 issue: 7 year: 1975 end-page: 782 ident: CR20 article-title: Waiting time distribution in a poisson queue with a general bulk service rule publication-title: Manage Sci doi: 10.1287/mnsc.21.7.777 – ident: CR17 – volume: 24 start-page: 359 issue: 2 year: 2006 end-page: 365 ident: CR25 article-title: Alternative solution to a quorum queueing system publication-title: Stoch Anal Appl doi: 10.1080/07362990500522395 – volume: 17 start-page: 256 issue: 2 year: 1955 end-page: 261 ident: CR13 article-title: Waiting time in bulk service queues publication-title: J Roy Stat Soc: Ser B (Methodol) – ident: CR11 – ident: CR9 – ident: CR5 – ident: CR7 – ident: CR26 – ident: CR24
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Snippet	This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M / G a / 1 queue with a... This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M/Ga/1 queue with a fixed-size...
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SubjectTerms	Business and Management Customer services Distribution functions Economics Eigenvalues Eigenvectors Electrical Engineering Life Sciences Mathematics and Statistics Probability Probability density functions Queues Random variables Statistics
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Title	Sojourn-time Distribution for M/Ga/1 Queue with Batch Service of Fixed Size - Revisited
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