# Union Models for Model Families: Efficient Reasoning over Space and Time
## 7 Results and Discussion
### Summary and Discussion  
Thus far, this paper has explored a research question related to the performance of union models by empirically evaluating the efficiency of reasoning and analysis tasks for modeling families using union models in comparison to the use of individual models. We defined three general reasoning tasks and evaluated their performance, first using union models and then using individual models several times. We have discussed the experimental methodology, setup, and implementation and reported on the empirical results. Our experiments demonstrate the usefulness and performance gains of union models for analyzing a family of models all at once compared to individual models. Finally, we have identified several threats to validity as caveats. The speedups, whether the time for constructing the union model is considered or not, are always in favor of the union model, and these speedups generally increase as models or families grow larger. However, there is erratic behavior regarding the increase of the time speedup. For instance, the speedup for L-sized models in certain experiments was larger than the speedup of M-sized models, while in other experiments it was smaller than that for M-sized models. This behavior is due in part to the possibility of having a footprint for the creation of \(M_{U}\), which is not negligible. Moreover, in order for the behavior of the speedup to be more stable and less erratic, we may need to consider additional experimental parameters such as the ratio of variability across models and the topology of the models at hand. In addition, it is worthwhile to mention here that in certain individual experiments related to the calculation of _speedup_with_constrTime_ we encountered counter-intuitive results, specifically, where the speedup was less than one. In such experiments, a speedup value less than one suggests that the use of \(M_{U}\), when taking its construction time into account, is slower than the use of individual models. Again, these slowdowns where obtained in individual experiments only, and they were amortized by calculating the average results of fifteen runs. In terms of concrete time saved, which ranges from half a second to a few minutes depending on the experiment, the savings may not seem large at first glance; however, there are several practical implications:  
* The time savings accumulate as many analyses are performed on a same union model, especially as the union model construction time is amortized over multiple analyses;* The savings does not take into account the concrete time required when a practitioner uses a tool to open a model, perform the analysis, and save the results, which this may take many seconds per model and is prone to human error;
* The savings become essential in a context where union model construction and verification are offered _as a service_, i.e., as an online application where multiple users can concurrently upload, merge, and analyze their model families. While this paper focuses on analysis types that are language _independent_, the first author's previous work in [10; 30] explored the adaptation or _lifting_ of language-_dependent_ analyses for a given language, namely, forward and backward propagation of satisfaction values for GRL model families, again with positive performance gains. The GRL-based analysis algorithms and their lifted version were implemented using an optimizer (IBM CPLEX) with speedups up to 23 times faster when using the union model compared to individual models, and for a much lower additional memory cost. In practical terms, this means that analysis on model families is no longer limited to the verification of behavioral properties often seen in existing approaches.