We propose a framework that integrates classical Monte Carlo simulators and Wasserstein generative adversarial networks to model, estimate, and simulate a broad class of arrival processes with general non-stationary and multi-dimensional random arrival rates. Classical Monte Carlo simulators have advantages at capturing the interpretable "physics" of a stochastic object, whereas neural-network-based simulators have advantages at capturing less-interpretable complicated dependence within a high-dimensional distribution. We propose a doubly stochastic simulator that integrates a stochastic generative neural network and a classical Monte Carlo Poisson simulator, to utilize both advantages. Such integration brings challenges to both theoretical reliability and computational tractability for the estimation of the simulator given real data, where the estimation is done through minimizing the Wasserstein distance between the distribution of the simulation output and the distribution of real data. Regarding theoretical properties, we prove consistency and convergence rate for the estimated simulator under a non-parametric smoothness assumption. Regarding computational efficiency and tractability for the estimation procedure, we address a challenge in gradient evaluation that arise from the discontinuity in the Monte Carlo Poisson simulator. Numerical experiments with synthetic and real data sets are implemented to illustrate the performance of the proposed framework.