The article presents a new approach to assessing the average response time of a cloud computing system and its dispersion. A fork-join system or a system with request splitting was chosen as a queuing model, and artificial neural networks were used as a method for estimating a variable of interest. The analysis showed that the estimates obtained were more accurate than those previously known. Besides, the proposed approach allows expanding the analysis of the cloud system to the case of a model with a non-Poisson input stream and non-exponential service time, as well as obtaining estimates for a larger number of performance indicators of the cloud system, which was not previously possible.