Abstract
We present three classical methods in the study of dynamic and stationary characteristic of processes of Markovian or Semi-Markovian type which possess points of regeneration. Our focus is on the stationary distributions and conditions of its existence and use.
The first approach is based on detailed probability analysis of time dependent passages between the states of the process at a given moment. We call this approach Kolmogorov approach.
The second approach uses the probability meaning of Laplace-Stieltjes transformation and of the probability generating functions/ Some additional arteficial excrement construction is used to show how derive direct relationships between these functions and how to find them explicitly.
The third approach obtains relationships between the stationary characteristics of the process by use of so called “equations of equilibrium”. The input flow in each state must be equal to the respective output flow from that state. In such a way no accumulations should happen on each of that states when process gets its equilibrium.
In all the illustrations of the these approaches we analyze a dynamic Marshal-Olkin reliability model with dependent components functioning in parallel. Results on this example are new.
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Notes
- 1.
The functions \(h_i(\cdot )\) in Eq. (4) are the result of application of the characteristics method to the second one of Eqs. (3). However, these functions have a clear probabilistic interpretation. The states \((i, 0)\,\,\, (i=1, 2)\) of the process Z can be considered as partially regenerative states (the state 0 is the state of full regeneration). Times of entering into these states are consequently the times of partial and full regeneration. Thus, the functions \(h_i(\cdot )\) can be considered as renewal densities of the process Z for these partial regenerative times, while the other two multipliers in formula (4) show that during time x neither failure, nor repair occurs.
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The publication has been prepared with the support of the “RUDN University Program 5–100” and funded by RFBR according to the research project No. 20-01-00575.
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Dimitrov, B., Rykov, V., Esa, S. (2020). Three Approaches in the Study of Recurrent Markovian and Semi-Markovian Processes. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks. DCCN 2020. Lecture Notes in Computer Science(), vol 12563. Springer, Cham. https://doi.org/10.1007/978-3-030-66471-8_41
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