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).
extractive_keyphrases: composition operators em algorithm fitting methods kronecker operators m3pps matrix geometric methods mmpps modeling real systems numerical evaluation of queueing models state space explosion statistical descriptors superposition operator underlying equations
abstractive_keyphrases: composing multiple small-sized mmpps or maps mmpps and maps statistical multiplexing of several mmpps study the approximate fitting of large m3pps two-state models