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The limit properties diffusion process in a semi-Markov environment

Chabanyuk Y., Rosa W.

Journal of Applied Mathematics and Computational Mechanics

2018

Vol. 17, nr 1

5--14

In this paper we consider the stochastic diffusion process with semi-Markov switchings in an averaging scheme. We present results and conditions on convergence to the classic diffusion process, in case with semi-Markov process perturbation is uniformly ergodic. We used small parameter scheme to get the main result.

Concept of semi-Markov process

Grabski F.

Zeszyty Naukowe Akademii Marynarki Wojennej

2016

R. 57 nr 3 (206)

25--36

This paper provides the definitions and basic properties related to a discrete state space semi-Markov process. The semi-Markov process is constructed by the so called Markov renewal process that is a special case the two-dimensional Markov sequence. The Markov renewal process is defined by the transition probabilities matrix, called the renewal kernel and an initial distribution or by another characteristics which are equivalent to the renewal kernel. The counting process corresponding to the semi-Markov process allows to determine concept of the process regularity. In the paper are also shown the other methods of determining the semi-Markov process. The presented concepts are illustrated a simple example.

Artykuł przedstawia definicje i podstawowe cechy procesu semi-Markowa dyskretnego stanu przestrzeni. Proces semi-Markova jest zbudowany przez tzw. proces odnawiania Markova, który jest specjalnym przypadkiem dwuwymiarowego ciągu Markova. Proces odnawiania Markova jest zdefiniowany przez macierz prawdopodobieństw przejściowych, zwaną jądrem odnawiania, i początkowy rozkład lub przez inne charakterystyki, które są równe jądru odnawiania. Proces obliczeniowy odpowiadający procesowi semi-Markova pozwala na określenie koncepcji regularności procesu. W artykule przedstawiono również pozostałe metody określania procesu semi-Markova. Przedstawione koncepcje są zaprezentowane na prostym przykładzie.

Application of semi-Markov processes for evaluation of diesel engines reliability with regards to diagnostics

Girtler J.

Journal of Polish CIMEEAC

2016

Vol. 11, no 1

47--53

The paper presents semi-Markov models of technical state transitions for diesel engines, useful for determination of their reliability, as a result of the conducted statistical empirical studies. Interpretation of technical states provided for this sort of engines refers to ship main engines, i.e. engines employed in propulsion systems of sea-going ships. The considerations recognize diesel engine as a diagnosed system (SDN), of which state can be identified by a diagnosing system (SDG). Both of the systems: SDN and SDG compose a diagnostic system (SD). Examples of three-state semi-Markov models were applied to demonstrate that in case of use of proper diagnosing systems (SDG) for identification of technical states of such engines as SDN, by classification of the states to the relevant class of the reference states, it is possible to make use of a Markov model to determine reliability of the engines. For developing a Markov model of state transitions for the engines, there were applied functions of the risk of damage: λ12 that causes transition from state s1 to state s2, and λ13 that causes transition from state s1 to state s3, as well as intensity functions of recovery (restitution): λ21 that causes transition from state s2 to state s1, and λ31 that causes transition from state s3 to state s1.