*4.2. Calculations According to the Markov Model*

Markov processes concern exponential distributions, the most popular ones in reliability theory [11,35]. They are described by two parameters, which fully define them. The first of these is the already calculated probability matrix of interstate transitions *pij* (Table 4).

The second, important parameter is the function describing the transitions of objects between states, called the process transition intensity λ*ij*(*t*), which characterizes the rate of changes in the probability of transition *pij*(*t*) [36].

$$\lambda\_{i\bar{j}}(t) = \lim\_{\Delta t \to 0} \frac{1}{\Delta t} p\_{i\bar{j}}(t, \ t + \Delta t) \text{ for } i, \ j = 0, 1, 2, \dots \newline i \neq j. \tag{32}$$

For homogeneous Markov processes, the transition intensity is constant and equal to the inverse of the expected duration *tij* of the state *Si* before *Sj* [37]:

$$\lambda\_{i\dot{j}}(t) = \frac{1}{\mathcal{E}(t\_{i\dot{j}})},\tag{33}$$

where:

λ*ij*(*t*)—intensity of transitions from the state *i* to state *j*, *E*(*tij*)—expected duration value *tij*.

The intensities λ*ii* ≤ 0 for *i* = *j* are defined as a complement to the sum of transition intensity from state Si for *i j* to 0:

$$
\lambda\_{\vec{u}} + \Sigma\_j \,\lambda\_{\vec{u}} = 0 \tag{34}
$$

thus:

$$
\lambda\_{\vec{n}} = -\Sigma\_{\vec{j}} \,\lambda\_{\vec{n}}.\tag{35}
$$

The modules |λ*ii*| = −λ*ii* are called the exit intensities from the state *Si*.

Calculated according to the above formulas (33)–(35), the element λ*ij* of the matrix Λ of transition intensity is shown in Table 9.


**Table 9.** The transition intensity matrix of the process studied.

Then, using the relationship (36), ergodic probabilities *pj* were calculated for the Markov model in continuous time.

$$\prod^T \ast \Lambda = 0,\tag{36}$$

where:


This way, for the process studied, we obtain the following matrix Equation (37):

$$
\begin{bmatrix} p\_1 \\ p\_2 \\ p\_3 \\ p\_4 \\ p\_5 \\ p\_6 \end{bmatrix} \cdot \begin{bmatrix} -\lambda\_{11} & \lambda\_{12} & \lambda\_{13} & \lambda\_{14} & 0 & 0 \\ \lambda\_{21} & -\lambda\_{22} & \lambda\_{23} & \lambda\_{24} & 0 & 0 \\ \lambda\_{31} & 0 & -\lambda\_{33} & \lambda\_{34} & 0 & 0 \\ \lambda\_{41} & 0 & 0 & -\lambda\_{44} & p\_{45} & 0 \\ \lambda\_{51} & \lambda\_{52} & \lambda\_{53} & 0 & -\lambda\_{55} & \lambda\_{56} \\ \lambda\_{61} & 0 & \lambda\_{63} & 0 & 0 & -\lambda\_{66} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag{37}
$$

Taking into account the normalization condition: 6 *<sup>j</sup>*=<sup>1</sup> *pj* = 1, we get the limit probabilities *pj* of the system's dwelling time in the states S1–S6, which are shown in Table 10.

**Table 10.** Limit probabilities *pj* in the continuous physical time for the Markov process.


The results obtained deviate from the values determined for the semi-Markov process, disturbingly revealing that the system studied tends primarily to remain in the downtime state (S3). The state in which the production takes place (S1) comes only second and takes the value lower by over 35% in relation to calculations made according to the semi-Markov process. A comparison of the other results is presented in Table 11. The highest difference concerns state S3, and is over 534%.

**Table 11.** Comparison of results for the Markov and semi-Markov models.


The technical readiness coefficient was also calculated based on the (31), which for the Markov process amounts to 45% (*K* = 0.45)—almost half the size of what was determined according to the semi-Markov process.
