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

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

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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 emtek 2113 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 read more results using our method.The derivations are very fast and robust due to the lucidity of the expressions.