document: Two-state models are often insufficient to fit complex traces, therefore we also study the approximate fitting of large M3PPs. In the single class setting, a known limitation of MMPPs is the inability to simultaneously fit many statistical descriptors due to the non-linearity of their underlying equations (Bodrog, Heindl, Horváth, & Telek, 2008; Heindl, Horváth, & Gross, 2006; Horváth & Telek, 2009). This has led to the definition of several approaches to fit complex traces by composing multiple small-sized MMPPs or MAPs using Kronecker operators (Andersen & Nielsen, 1998; Casale, Zhang, & Smirni, 2010; Horváth & Telek, 2002). These methods employ composition operators for moment fitting, offering a different trade-off between computational cost and fitting accuracy compared to fitting methods based on the EM algorithm (Breuer, 2002; Horváth & Okamura, 2013; Klemm, Lindemann, & Lohmann, 2003). In particular, the superposition operator allows one to describe a trace by the statistical multiplexing of several MMPPs, at the expense of an exponential growth of the number of states in the resulting process (Sriram & Whitt, 1986). This state space explosion is an obstacle for the application of MMPPs and MAPs to modeling real systems; for example it considerably slows down, or even renders infeasible, the numerical evaluation of queueing models by matrix geometric methods (Bini, Meini, Steffé, Pérez, & Houdt, 2012; Pérez, Velthoven, & Houdt, 2008).
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