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ANALYTICALLY CLOSED-FORM SOLUTIONS FOR THE DISTRIBUTION OF A NUMBER OF CUSTOMERS SERVED DURING A BUSY PERIOD FOR SPECIAL CASES OF THE GEO/G/1 QUEUE

Published online by Cambridge University Press:  18 March 2020

M. L. Chaudhry ,
Veena Goswami Open the ORCID record for Veena Goswami [Opens in a new window]  and
Abdalla Mansur
M. L. Chaudhry
Affiliation:
Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, Kingston, Ontario, Canada K7K 7B4
Veena Goswami
Affiliation:
School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar 751 024, India E-mail: veena_goswami@yahoo.com
Abdalla Mansur
Affiliation:
Abu Dhabi Men's College, Higher Colleges of Technology, Abu Dhabi, United Arab Emirates
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Abstract

This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

Keywords

busy perioddiscrete-timegenerating functionLangrange inversion theoremnumber served in busy period
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 650 - 679
Copyright
© Cambridge University Press 2020

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