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We propose results of the investigation of properties of the random sums ofrandom variables. We consider the case, where the number of summands is thefirst moment of an event occurrence. An integral equation is presented thatdetermines distributions of random sums. With the help of the obtained resultswe analyse the distribution function of the time during which the Geiger-Mullercounter will not lose any particles, the distribution function of the busyperiod of a redundant system with renewal, and the distribution function of thesojourn times of a single-server queueing system.

On Distributions of One Class of Random Sums and their Applications