Tandem queueing networks are widely used to model systems where services are provided in sequential stages. In this study, we assume that each station in the tandem system operates under a general renewal process. Additionally, we assume that the arrival process for the first station is governed by a general renewal process, which implies that arrivals at subsequent stations will likely deviate from a renewal pattern. This study leverages neural networks (NNs) to approximate the steady-state distribution of the marginal number of customers at each station in the tandem queueing system, based on the external inter-arrival and service time distributions. Our approach involves decomposing each station and estimating the departure process by characterizing its first five moments and auto-correlation values, without limiting the analysis to linear or first-lag auto-correlation. We demonstrate that this method outperforms existing models, establishing it as state-of-the-art. Furthermore, we present a detailed analysis of the impact of the i^th moments of inter-arrival and service times on steady-state probabilities, showing that the first five moments are nearly sufficient to determine these probabilities. Similarly, we analyze the influence of inter-arrival auto-correlation, revealing that the first two lags of the first- and second-degree polynomial auto-correlation values almost completely determine the steady-state probabilities of a G/GI/1 queue.