The Remaining Busy Period of a Finite Queue

In the M/G/1/N queuing system where no more than N < ∞ customers are allowed in the system, define the remaining busy period from state i = 0, 1, …, N as the time spent in transition from state i to state 0, i.e., until the server becomes idle for the first time. The purpose of this paper is to derive the Laplace-Stieltjes transform of the distribution of this time and the generating function of the number of customers served in this time.