Open AccessDEVELOPMENT OF A MULTI-MODE ACCUMULATIVE MARKOV SYSTEM
Author(s) -
Yuri Filonov,
Z. I. Nagolkina
Publication year - 2020
Publication title -
prikladnaâ geometriâ i inženernaâ grafika
Language(s) - English
Resource type - Journals
ISSN - 0131-579X
DOI - 10.32347/0131-579x.2020.99.200-207
Subject(s) - markov chain , mathematics , state space , stochastic matrix , mode (computer interface) , downtime , function (biology) , matrix (chemical analysis) , markov process , stationary distribution , interval (graph theory) , combinatorics , statistics , computer science , materials science , evolutionary biology , composite material , biology , operating system
Systems are considered that earn a resource, monetary in the economic interpretation, and use it for functioning and accumulation. The conditions for the growth of savings are sought, which simultaneously ensure the non-positivity of the governing Markov chain. The widespread inhomogeneity of the phase space is essential. In the phase space (of a general type) of such a system, a nonnegative function , a function with values 1,2,…, , an irreducible homogeneous Markov chain , n = 0,1,… are given. The values indicate the resource level of the system at time n, the values indicate the mode, which is determined by the number of service locations, devices, service lines, the number of offices, workers, etc., the type of functioning as repair, prevention, downtime work. Let be the average increments of ver the 1-st period for an arbitrary set набора , states, is the probability of transition from the state into mode is the corresponding stochastic matrix, and – is the stationary probability distribution for it. Let be the value of increased by the value of an arbitrary function ψ (x)> 0, monotone and integrable on the interval [0, ∞). In a fairly general case, based on other of the authors' works, it is argued that the condition for large X (w_i) is sufficient for the system to be acceptable in the sense that there is a convergence to 1 of the relative fraction of that time until the moment n, when the level X> C, (as n → ∞ with probability 1, for an arbitrarily large level C). That is, in a time relation that tends to 1, the functioning will take place in an ever better financial condition. Models are also considered that use the asymptotics and the model of deterioration of the system quality in a random way, but gradually, until unfavorable modes and the subsequent transition to the initial favorable mode, after the recovery mode, and further in a similar way. In this case, under the conditions of acceptability, it is not the stationary probabilities that are used, but the probabilities of system deterioration. It examines economic analogies with insurance and profitability.