Probability resembles the ancient Roman God Janus since, like Janus, probability also has a face with two different sides, which correspond to the metaphorical gateways and transitions between the past and the future [...]

Markov and Semi-Markov Chains, Processes, Systems, and Emerging Related Fields

The effectiveness of naive optimization of the egress path for an active-shooter scenario

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.

https://www.mdpi.com/2227-7390/9/15/1745/pdf

Haris Palikrousis

Papers

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