A standard M/G~1 queueing process will be generalized in this paper under the following three assumptions. First, the assumption due to Welch [12] is postulated. (A. 1) customers who initiate a busy period have a possibly different type of service time distribution function from that of customers who do not initiate a busy period. As is stated in [12], there are many queueing situations to which such a model is immediately appropriate, and the model contains as special cases both the Pollaczek [7] and the Finch [2] models. Next, it is assumed that (A. 2) if the server once becomes idle, then the next service does not start until the number of customers in the system amounts to a fixed number k (>=1). In the case when k = l , this assumption becomes trivial. If, in the Welch model, the service time of initiating customers tends to far longer than that of non-initiating customer, then it will be generally desirable that the number of initiating points of busy periods contained in a given time interval is as few as possible. In general, when we deal with the queueing process with additional delay times or extra cost at a initiating point of each busy period, it will be useful to take the above queueing system with assumption (A. 2). Fukuta [5] has considered a generalized M/G/1 queueing process with assumptions (A. 1) and (A. 2) to find the distribution of the queue size immediately after departure. The author [11] also has obtained the transient as well as asymptotic distributions of the queue size at any time and the virtual waiting time for the same model. Thirdly, for the input process, it is assumed that (A. 3) during each busy period, customers arrive at the server according to the Poisson process with parameter ~ (-Ii-process, say), while during each idle period, customers arrive according to a renewal process (--I2-process, say).