Fitting Methods Based on Distance Measures of Marked Markov Arrival Processes

Approximating various real-world observations with stochastic processes is an essential modelling step in several fields of applied sciences. In this chapter, we focus on the family of Markov-modulated point processes, and propose some fitting methods. The core of these methods is the computation of the distance between elements of the model family. First, we introduce a methodology for computing the squared distance between the density functions of two phase-type (PH) distributions. Later, we generalize this methodology for computing the distance between the joint density functions of \( k \) successive inter-arrival times of Markovian arrival processes (MAPs) and marked Markovian arrival processes (MMAPs). We also discuss the distance between the autocorrelation functions of such processes. Based on these computable distances, various versions of simple fitting procedures are introduced to approximate real-world observations with the mentioned Markov modulated point processes.

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