Fig shows possible structures of

Fig. 1 shows possible structures of the technological modules (TMs) with intermediate storage devices (SDs) providing time redundancy (the structures operate as a part of an asynchronous automated line). The difference between the structures in Fig. 1 is due to the fact that 5 alpha reductase the same SD can work together with the preceding TM to yield production and with the subsequent TM to receive it. From a mathematical standpoint, the structures presented in Fig. 1(a) and (b) are identical, so the model shall be constructed only for the case in Fig. 1(a).
Problem setting The problem can be formulated in the following way: let us assume that the distribution functions F01(t) and F10(t) of the random variables ξ1 and η1 (which are the mean times between TM failure and recovery, respectively) are known, along with the distribution functions F03(t) and F30(t) of the random variables ξ3 and η3 (which are the mean times between SD failure and recovery, respectively). Additionally, the distribution function F12(t) of the random variable ξ2 which is the reserve time is also known. Let us make the following assumptions:
Model development The graph for the system\'s states is shown in Fig. 2. The nodes of the constructed graph (Fig. 2) are described in Table 1. The times from θ0 to θ3 spent in the states S0, S1, S2 and S3, respectively, can be found from the following expressions: where ∧ is the sign indicating the minimum of the random variables. Then the distribution functions for the times spent in the states take the forms listed in Table 1. The stationary distribution ρ(x) of the embedded Markov chain is determined by the formula where p(x, y) is the transition probability density of the embedded Markov chain. The expressions for P(x, y) have the following form:
Using system (1), let us write a system of equations for determining the stationary distribution for the embedded Markov chain ρ(x) [15–17]:
The normalization condition is as follows:
By solving system of Eqs. (2) using condition (3), we obtain: where . Let us determine the stationary distribution of a semi-Markov process. The stationary probabilities of the semi-Markov process follow the expressions: where The mean times spent in the respective states are expressed as
Considering that the MTBF distribution functions for the TM and the SD are taken to be elementary, i.e., the failure flow rate can be replaced by their mathematical expectations:
Taking into account the above-determined ρ1 and ρ3, the mean time is written as
The stationary probabilities in this case follow the expressions
The obtained expressions for the stationary distribution of the semi-Markov process (SMP) allow to determine such important system characteristics as the availability and the downtime rate:
In order to find the distribution function for the MTBF of the system as a whole, it is necessary to construct and solve a system of Markov renewal equations, while for finding the distribution function for the system\'s recovery time it is sufficient to use Eq. [18]:
However, since the S2 state is continuous, finding the distribution function of the time between failures (i.e., the time during which the system is continuously working) involves using the stationary algorithm of the phase integration of steady states [16,17] in order to move on to the system with discrete states from the system with continuous states. A graph of this system looks like the one shown in Fig. 2, but the S2 state is introduced instead of the S2 state. Then the transition probabilities shall take the form:
The stationary distribution for the embedded Markov chain (EMC) shall take the form: where The distribution function of the time the system spends in the integrated discrete S2 state has the following form:
The semi-Markov kernels for the graph of the discrete system are given in Table 2. Let us construct the Markov renewal equations (MREs) [16,17,19]: