Limiting Distribution of the Three-State Semi-Markov Model of Technical State Transitions of Ship Power Plant Machines and its Applicability in Operational Decision-Making

The article presents the three-state semi-Markov model of the process {W(t): t ≥ 0} of state transitions of a ship power plant machine, with the following interpretation of these states: s₁ – state of full serviceability, s₂ – state of partial serviceability, and s₃ – state of unserviceability. These states are precisely defined for the ship main engine (ME). A hypothesis is proposed which explains the possibility of application of this model to examine models of real state transitions of ship power plant machines. Empirical data concerning ME were used for calculating limiting probabilities for the process {W(t): t ≥ 0}. The applicability of these probabilities in decision making with the assistance of the Bayesian statistical theory is demonstrated. The probabilities were calculated using a procedure included in the computational software MATHEMATICA, taking into consideration the fact that the random variables representing state transition times of the process {W(t): t ≥ 0} have gamma distributions. The usefulness of the Bayesian statistical theory in operational decision-making concerning ship power plants is shown using a decision dendrite which maps ME states and consequences of particular decisions, thus making it possible to choose between the following two decisions: d₁ – first perform a relevant preventive service of the engine to restore its state and then perform the commissioned task within the time limit determined by the customer, and d₂ – omit the preventive service and start performing the commissioned task.