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In this paper, we first carry out to our knowledge the first in-depth characterization of control-plane traffic, using a real-world control-plane trace for 37,325 UEs sampled at a real-world LTE Mobile Core Network (MCN). Our analysis shows that control events exhibit significant diversity in device types and time-of-day among UEs. Second, we study whether traditional probability distributions that have been widely adopted for modeling Internet traffic can model the control-plane traffic originated from individual UEs. Our analysis shows that the inter-arrival time of the control events as well as the sojourn time in the UE states of EMM and ECM for the cellular network cannot be modeled as Poisson processes or other traditional probability distributions. We further show that the reasons that these models fail to capture the control-plane traffic are due to its higher burstiness and longer tails in the cumulative distribution than the traditional models. Third, we propose a two-level hierarchical state-machine-based traffic model for UE clusters derived from our adaptive clustering scheme based on the Semi-Markov Model to capture key characteristics of mobile network control-plane traffic -- in particular, the dependence among events generated by each UE, and the diversity in device types and time-of-day among UEs. Finally, we show how our model can be easily adjusted from LTE to 5G to support modeling 5G control-plane traffic, when the sizable control-plane trace for 5G UEs becomes available to train the adjusted model. The developed control-plane traffic generator for LTE/5G networks is open-sourced to the research community to support high-performance MCN architecture design R&D.