The paper considers a discrete buffered system with infinite waiting room, one single output channel and synchronous transmission of messages from the buffer. The arrival stream of messages to the buffer is assumed to be interrupted at random time points for random length time intervals. The arrival interruptions represent a decrease in the mean arrival intensity as compared to a buffer system without arrival interruptions. They also cause the need for a whole new method of analysis, which is presented here. Time is divided into two types of time intervals: ‘A-times’, during which arrivals are possible, and ‘B-times’, during which the arrival stream is interrupted. Both types of intervals are expressed in clock time periods and may have arbitrary probability distributions, provided their probability generating functions are rational functions of the variable z. Under these circumstances, expressions are derived for the probability generating functions of the number of messages in the buffer at various time instants. These expressions contain a finite number of unknown parameters, which can only be determined by solving a generally transcendent equation for its roots. As an example of the method, the special case is treated where both A-times and B-times are geometrically distributed; explicit expressions for the probability generating functions of the buffer occupancy are obtained for this special case.