In this chapter, we present three new fitting techniques which approximate data sets with PH distributions and Markovian arrival processes, i.e., stochastic processes that are tractable. Our objective is to fit data sets that exhibit long-tailed behavior and long-range dependence into PH distributions and MAPs, respectively. In our approach, we strive for both accuracy and efficiency. We achieve these goals by developing divide-and-conquer and hierarchical fitting algorithms. The fitting algorithms leverage the use of the matrix-analytic methodology (see Chapter 3), for the performance analysis of Internet-related systems. We evaluate the accuracy of our fitting techniques not only statistically but also from the queueing systems perspective, because the intention is to use our fitted models as inputs in queueing models.
This chapter is organized as follows. In Section 4.1, we elaborate on the complexity of systems with highly variable service process, commonly encountered in Internet systems. In Section 4.2, we present a new technique for fitting long-tailed data sets into PH distributions by partitioning the data into smaller subsets of equal variability, fitting each of the subsets into PH distributions, and combining results together. In Section 4.3, we propose a similar fitting technique with the exception that the partitioning of the data is based on the expected value of each subset. In Section 4.4, we devise a technique for fitting long-range dependent data sets into MAPs. We conclude with a summary of chapter's contributions.