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Process mining provides process improvement in a variety of application domains. A primary focus of process mining is transferring information from event logs into process model. One of the issues of process mining is dealing with invisible prime tasks. An invisible prime task is an additional task in the process model to assist in showing real processes. However, a few of algorithm solves the issue. This research proposes an algorithm for dealing with invisible prime tasks. The proposed algorithm contains rules and equations utilizing probability of state transition of Coupled Hidden Markov and double time-stamped in event logs. The rules and equations are used for determining invisible prime tasks and parallel control-flows patterns. In addition to dealing with invisible prime tasks, the experiment results also show that the proposed algorithm obtains right parallel control-flow patterns from non-complete event logs. This proposed algorithm also decreases usage of the invisible prime task in A# algorithm without reducing the quality of discovered process models. It has proven with the fitness of process models obtained by the proposed algorithm are relatively high as those obtained by A# algorithm.

Existing methods, such as Graph Edit Distance (GED) and Cosine measure, still have drawback in obtaining similarity of parallel relationships by neglecting the control-flow patterns, i.e. AND, OR, and XOR. Since AND > OR > XOR, the similarity value of AND versus OR is greater than XOR versus OR and AND versus XOR. This paper proposes two new similarity methods, Tree Declarative Pattern Edit Distance (TPED) and Cosine-Tree Declarative Pattern (Cosine-TDP). They provides value to the control-flow pattern so the value of similarity can be seen more differently. The new methods utilize tree model of the declarative pattern. The results show that the proposed methods are better at differentiating parallel relationships than the existing methods, GED and Cosine measure. In obtaining AND versus OR, XOR versus OR, and AND versus XOR, TPED obtained 0.821, 0.811, and 0.78 while Consine-TDP obtained 0.834, 0.826, and 0.693. Meanwhile, GED obtained 1 for all parallel relationships whereas Cosine measure obtained 0.02, 0.08, and 0.04.

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