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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: [email protected]
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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References

1.Abate, J. & Whitt, W. (1992). The fourier-series method for inverting transforms of probability distributions. Queueing Systems 12(1-2): 5–87.CrossRefGoogle Scholar
2.Abate, J. & Whitt, W. (1992). Numerical inversion of probability generating functions. Operations Research Letters 12(4): 245–251.CrossRefGoogle Scholar
3.Abate, J. & Whitt, W. (1992). Solving probability transform functional equations for numerical inversion. Operations Research Letters 12(5): 275–281.CrossRefGoogle Scholar
4.Alfa, A.S. (2010). Queueing theory for telecommunications: discrete time modelling of a single node system. New York: Springer.CrossRefGoogle Scholar
5.Borel, É. (1942). Sur l'emploi du théoreme de bernoulli pour faciliter le calcul d'une infinité de coefficients. application au probleme de l'attentea un guichet. Comptes rendus de l'Académie des Sciences Paris 214: 452–456.Google Scholar
6.Consul, P.C., & Famoye, F. (2006). Lagrangian probability distributions. Birkhäuser, Boston: Springer.Google Scholar
7.Consul, P.C., & Shenton, L.R. (1972). Use of lagrange expansion for generating discrete generalized probability distributions. SIAM Journal on Applied Mathematics 23(2): 239–248.CrossRefGoogle Scholar
8.Feller, W. (1968). An introduction to probability theory and its applications, vol. 1. New York: John Wiley & Sons.Google Scholar
9.Feller, W. (2008). An introduction to probability theory and its applications, vol. 2. New York: John Wiley & Sons.Google Scholar
10.Goswami, V. (2004). Distribution of the number of customers served during a busy period in a discrete time single server queue. International Journal of Information and Management Sciences 15(3): 59–67.Google Scholar
11.Goswmi, V., & Gupta, U.C. (2002). Distribution of the number of customers served during a busy period in a discrete time Geom/Geom/1 queue. Indian Journal of Pure and Applied Mathematics 33(9): 1405–1408.Google Scholar
12.Haight, F.A. (1961). A distribution analogous to the Borel-Tanner. Biometrika 48(1/2): 167–173.CrossRefGoogle Scholar
13.Haight, F.A. (1963). Mathematical theories of traffic flow, vol. 7. New York: Academic Press.CrossRefGoogle Scholar
14.Haight, F.A. & Breuer, M.A. (1960). The Borel-Tanner distribution. Biometrika 47(1-2): 143–150.CrossRefGoogle Scholar
15.Hunter, J.J. (1983). Mathematical techniques of applied probability, vol. 2. New York: Academic Press.Google Scholar
16.Jain, J.L., Mohanty, S.G., & Böhm, W. (2016). A course on queueing models. New York: Chapman and Hall/CRC.CrossRefGoogle Scholar
17.Kemp, A.W. & Kemp, C.D. (1968). On a distribution associated with certain stochastic processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 30(1): 160–163.Google Scholar
18.Kendall, D.G. (1964). Some recent work and further problems in the theory of queues. Theory of Probability and its Applications 9(1): 1–13.CrossRefGoogle Scholar
19.Kleinrock, L. (1975). Queueing systems, vol. 1. New York, NY: Wiley.Google Scholar
20.Medhi, J. (2002). Stochastic models in queueing theory. Amsterdam: Elsevier.Google Scholar
21.Prabhu, N.U. (1960). Some results for the queue with Poisson arrivals. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 22(1): 104–107.Google Scholar
22.Prabhu, N.U. (1965). Queues and inventories. New York: John Wiley & Sons.Google Scholar
23.Stewart, W.J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton: Princeton University Press.Google Scholar
24.Takács, L. (1955). Investigation of waiting time problems by reduction to Markov processes. Acta Mathematica Hungarica 6(1-2): 101–129.Google Scholar
25.Takács, L. (1962). A generalization of the ballot problem and its application in the theory of queues. Journal of the American Statistical Association 57(298): 327–337.Google Scholar
26.Takács, L. (1967). On combinatorial methods in the theory of stochastic processes. Technical report, New York: Columbia University.Google Scholar
27.Tanner, J.C. (1953). A problem of interference between two queues. Biometrika 40(1/2): 58–69.CrossRefGoogle Scholar
28.Woodward, M.E. (1994). Burstiness of interrupted bernoulli process. Electronics Letters 30(18): 1466–1467.CrossRefGoogle Scholar
29.Woodward, M.E. (1994). Communication and computer networks: modelling with discrete-time queues. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar