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Current visualization systems are typically based on the concept of interactive post-processing. This decoupling of data visualization from the process of data generation offers a flexible application of visualization tools. It can also lead, however, to information loss in the visualization. Therefore, a combination of the visualization of the data generating process with the visualization of the produced data offers significant support for the understanding of the abstract data sets as well as the underlying process. Due to the application-specific characteristics of data generating processes, the task requires tailored visualization concepts. In this work, we focus on the application field of simulating biochemical reaction networks as discrete-event systems. These stochastic processes generate multi-run and multivariate time-series, which are analyzed and compared on three different process levels: model, experiment, and the level of multi-run simulation data, each associated with a broad range of analysis goals. To meet these challenging characteristics, we present visualization concepts specifically tailored to all three process levels. The fundament of all three visualization concepts is a compact view that relates the multi-run simulation data to the characteristics of the model structure and the experiment. The view provides the visualization at the experiment level. The visualization at the model level coordinates multiple instances of this view for the comparison of experiments. At the level of multi-run simulation data, the views gives an overview on the data, which can be analyzed in detail in time-series views suited for the analysis goals. Although we derive our visualization concepts for one concrete simulation process, our general concept of tailoring the visualization concepts to process levels is generally applicable for the visualization of simulation processes.