Transient and busy period analysis of the GI/G/1 queue as a Hilbert factorization problem

In this paper the authors find the waiting time distribution in the transient domain and the busy period distribution of the GI/G/1 queue. They formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. The authors achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. They obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, the authors find closed-form formulae for the first two moments of the distributions involved.