Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the M/G/1 Queue. (27th August 2019)

• Record Type: Journal Article
• Authors: Chaudhry, M. L.; Goswami, Veena
• Other Names: Thavaneswaran Aera Academic Editor.
• Is Part Of: Journal of probability and statistics. Volume 2019(2019)
• Publication Date: 2019-08-27
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; Mathematical statistics; Probabilities; Periodicals; 519
• DOI: 10.1155/2019/7398658
• Languages: English
• ISSNs: 1687-952X

Abstract

Abstract : This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of theM / G / 1 queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such asM / M / 1 andM / D / 1 queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang- k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and robust due to the lucidity of the expressions.