  We propose results of the investigation of properties of the random sums of
random variables. We consider the case, where the number of summands is the
first moment of an event occurrence. An integral equation is presented that
determines distributions of random sums. With the help of the obtained results
we analyse the distribution function of the time during which the Geiger-Muller
counter will not lose any particles, the distribution function of the busy
period of a redundant system with renewal, and the distribution function of the
sojourn times of a single-server queueing system.
