On State Occupancies, First Passage Times and Duration in Non-Homogeneous Semi-Markov Chains

Andreas C. Georgiou, Alexandra Papadopoulou, Pavlos Kolias, Haris Palikrousis and Evanthia Farmakioti
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Andreas C. Georgiou: Quantitative Methods and Decision Analytics Lab, Department of Business Administration, University of Macedonia, 54636 Thessaloniki, Greece
Alexandra Papadopoulou: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Pavlos Kolias: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Haris Palikrousis: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Evanthia Farmakioti: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Mathematics, 2021, vol. 9, issue 15, 1-17

Abstract: Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions. They are widely known for offering enhanced accuracy in modeling stochastic phenomena. The aim of this paper is to provide closed analytic forms for three types of probabilities which describe attributes of considerable research interest in semi-Markov modeling: (a) the number of transitions to a state through time (Occupancy), (b) the number of transitions or the amount of time required to observe the first passage to a state (First passage time) and (c) the number of transitions or the amount of time required after a state is entered before the first real transition is made to another state (Duration). The non-homogeneous in time recursive relations of the above probabilities are developed and a description of the corresponding geometric transforms is produced. By applying appropriate properties, the closed analytic forms of the above probabilities are provided. Finally, data from human DNA sequences are used to illustrate the theoretical results of the paper.

Keywords: semi-Markov modeling; occupancy; first passage time; duration; non-homogeneity; DNA sequences (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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