In the article, the entrance probabilities and the probability distribution of the number of transitions to a state are studied to provide some answers to questions related to state occupancies for the semi Markov model. Biological sequences and Web navigation are two cases that initially seem to be different but to a certain extent they do have similarities. Two main aspects of word occurrences in biological sequences are: (a) where do they occur and (b) how many times do they occur. In Web navigation the similar questions are (a) when a node is visited and (b) how many times a node is visited. So, the theoretical results of this study are applied to model these two cases and derive distributions of word location or node occurrence and frequency of occurrences. Rewards/costs are included in the Web navigation model and analytic forms for the means, variances, and moments of total interval rewards/costs are provided.

In basketball, the offensive movements on both strong and weak sides and tactical behavior play major roles in the effectiveness of a team’s offense. In the literature, studies are mostly focused on offensive actions, such as ball screens on the strong side. In the present paper, for the first time a second-order Markov model is defined to evaluate players’ interactions on the weak side, particularly for exploring the effectiveness of tactical structures and off-ball screens regarding the final outcome. The sample consisted of 1170 possessions of the FIBA Basketball Champions League 2018–2019. The variables of interest were the type of screen on the weak side, the finishing move, and the outcome of the shot. The model incorporates partial non-homogeneity according to the time of the execution (0–24″) and the quarter of playtime, and it is conditioned on the off-ball screen type. Regarding the overall performance, the results indicated that the outcome of each possession was influenced not only by the type of the executed shot, but also by the specific type of screen that took place earlier on the weak side of the offense. Thus, the proposed model could operate as an advisory tool for the coach’s strategic plans.

In the present article we study the asymptotic behaviour of the sequence of vectors with components expectations, variances, and covariances of the state sizes of a semi-Markov system. In this respect, we transform the semi-Markov system into a Markov system with a different though equivalent state space and relate the sequence of the transition probabilities of the respective Markov system as functions of the parameters of the semi-Markov system. Also, we study the asymptotic behavior of the sequence of vectors with components variances and covariances of the duration state sizes, of the related Markov system, under perturbation of the transition probability matrices. We use the results in the study of asymptotic behavior under perturbation of the sequence of vectors with components variances and covariances of the semi-Markov system.

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.