In an M/G/1 queue the busy periods start with probabi¬lity q. upon the arrival of the i-th customer to find the server idle, where the q.Ts5i>l form a probability distri¬bution. Some generating functions of the queue lengths are e-valuated from where the probability generating function of the queue length at the moment of a departure (and of an ar¬rival) is found when the system works in steady state an a-nalog of the Pollaczek-Khintchine formula is given and two characterization theorems and proved.