We introduce a program for training deep neural networks (DNNs) to compute an approximation of the stationary distribution of the queue length for the GI/G/K queue ([5]), which has few analytic solutions. This work extends the study in [1], with its key contribution being an effective algorithm for generating samples (data) for supervised learning. The algorithm, based on matrix-analytic methods ([8]), computes the stationary distributions of PH/PH/K queues ([7], [10]), which are dense in the set of GI/G/K queues. Specifically, the program integrates the quasi birth-and-death process ([6]), the CSFP (count-server-for-phase) method ([9], [4]), and the matrix-geometric solution ([3]) to generate a large number of samples for training and validating neural networks for the GI/G/K queue. The program's effectiveness is compared against existing asymptotic methods such as simulation, heavy traffic approximations ([2]), and heuristic approaches ([11]). This work not only advances the training of neural networks for complex stochastic systems but also highlights several challenging issues—such as modeling in discrete vs. continuous time and ensuring the representativeness of training samples—that open up interesting directions for future research in neural network applications to stochastic models.