A busy period analysis of M/G/1-type queue with general retrieval time

We study the busy period of a stationary single-server system with Poisson arrivals. The customers meeting server busy join a virtual orbit (in FIFO order). The server, when becomes empty, seeks a new orbital customer during generally distributed retrieval time. %in orbit, and this seeking time is generally distributed. The analysis is based on the comparison the original system with a buffered M/G/1-type model with the first exceptional service. It is shown that the distribution of the busy period satisfies a functional equation which is similar to the well-known equation for the conventional M/G/1 system. The main result is expressed in the terms of the Laplace-Stieltjes transforms (LST) for continuous busy period length and in terms of generating functions (GF) for the integer-valued busy period length. A few analytical examples are given as well.