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

• Published in: Journal of Probability and Statistics Vol. 2019; no. 2019; pp. 1 - 15
• Main Authors: Chaudhry, Mohan L., Goswami, V.
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Publishing Corporation 2019; Hindawi; Hindawi Limited; Wiley
• Subjects: Customers; Functional equations; Markov analysis; Markov processes; Poisson density functions; Queues; Queuing theory
• ISSN: 1687-952X; 1687-9538
• DOI: 10.1155/2019/7398658

This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the M/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 as M/M/1 and M/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.