Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the MG1 Queue

الملخص EN

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.