Fitting second-order acyclic Marked Markovian Arrival Processes

Markovian Arrival Processes (MAPs) are a tractable class of point-processes useful to model correlated time series, such as those commonly found in network traces and system logs used in performance analysis and reliability evaluation. Marked MAPs (MMAPs) generalize MAPs by further allowing the modeling of multi-class traces, possibly with cross-correlation between multi-class arrivals. In this paper, we present analytical formulas to fit second-order acyclic MMAPs with an arbitrary number of classes. We initially define closed-form formulas to fit second-order MMAPs with two classes, where the underlying MAP is in canonical form. Our approach leverages forward and backward moments, which have recently been defined, but never exploited jointly for fitting. Then, we show how to sequentially apply these formulas to fit an arbitrary number of classes. Representative examples and trace-driven simulation using storage traces show the effectiveness of our approach for fitting empirical datasets.