We propose Markov and non-Markov probabilistic models of how flows of annihilating particles interact, find the probability distribution of the number of positive applications in the model, and we present asymptotic results for the case of high intensity incoming flows. Then we study a system with non-exponential service where, using asymptotic analysis, we show that as the intensity of incoming flows grows, the probability distribution becomes Gaussian and find the parameters of the distribution. We also investigate flows of interacting particles as an infinitely linear queuing system with positive and negative applications of different systems and the probability distribution of the number of positive stationary applications in a system with exponential service is found. We also studied a case of arbitrary service by means of asymptotic analysis. We demonstrate that these systems are asymptotically equivalent.