# Mathematical Models of the Processes of Operation, Renewal and Degradation of a Fleet of Complex Technical Systems with Metrological Support

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*Axioms*: Mathematical Analysis)

## Abstract

**:**

## 1. Introduction

## 2. Scientific Literature Review

## 3. Statement on the Research Problem

## 4. Materials and Methods

#### 4.1. The Classical Model

#### 4.1.1. Construction and Study of the Classical Model for Different Laws on the Distribution of Failures of the Complex Technical System

#### 4.1.2. Development of the Classical Model: The Model of False and Undetected Failures

#### 4.1.3. Development of the Classical Model: Models of Failures and Degradation of the Complex Technical System

#### 4.2. The Model of Operation of a Complex Technical System Fleet with a Fully Recoverable Resource

_{1}is a fully functional state and four states corresponding to different levels of degradation of the CTS; E

_{2}is the first group of degradation (functional state with minor deviations of the normalized metrological characteristics); E

_{3}is the second group of degradation (a state with some deviations of the metrological characteristics, from which it is possible to return to a fully functional state with small resource costs); E

_{4}is the third group degradation (a state from which it is possible to return to a fully functional state with costs associated with sufficiently resource-intensive maintenance); and E

_{5}is the fourth “heavier” group of degradation. As the degradation group number increases, returning to the state E

_{1}becomes more and more resource intensive.

#### 4.3. The Model of Operation of the Complex Technical System Fleet with a Partially Recoverable Resource

^{(2)}and third ω

^{(3)}degradation groups, in case of failure of the CTS are sent for in-depth diagnostics of the technical condition, in order to determine the feasibility of updating (replacing with a new model of the CTS) or continuing operation after repair. To simplify, some probabilistic characteristics are not indicated in Figure 10, but they can be easily restored, taking into account that the sum of the probabilities of the transitions from each vertex of the graph are equal to one. If one edge comes out of the vertex, then the corresponding transition probability is one, and if two edges come out of the vertex, and the probability of one transition is written on the graph, then the probability of the second transition is equal to the difference of one and the known probability of the first transition.

## 5. Results

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- A basic model of the CTS operation;
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- A set of CTS operation models, having different levels of degradation (for different levels of CTS degradation a different number of system states and different variants of system maintenance are used);
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- A model of false failures and undetected failures;
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- A model of CTS fleet renewal, including such renewal methods as the purchase of new CTS samples, the modernization of existing CTS samples and the development of new promising CTS samples;
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- A functional dependence model of the CTS availability factor on a number of technical parameters, organizational and technical parameters, and technological parameters of the CTS belonging to different degradation groups and different methods of CTS stock renewal.

## 6. Discussions

## 7. Conclusions

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- Management of the process of development of CTS fleets;
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- Optimization of the processes of CTS fleet functioning;
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- Identification of problematic issues in the development of CTS fleet and the formation of strategies for CTS fleet development in the presence of various constraints;
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- The solving of the problem of conditional optimization in the presence of constraints on the technological parameters of the CTS fleet development (with constraints on part of the arguments of the availability factor function);
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- Calculation of the technological and technical–economic parameters of the CTS fleet functioning and development;
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- Evaluation of the risks associated with false and undetected failures, as well as the risks associated with CTS degradation;

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- To classify the designed CTS in order to establish the requirements for their metrological support;
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- When developing plans for medium-term and long-term development of the CTS fleet.

## 8. Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Dependences of the readiness coefficients on the periodicity of the control for various distribution laws: Rayleigh’s law (1), normal law (2), exponential law (3), and Weibull’s law (4).

**Figure 4.**Dependences of the probability of an erroneous decision $(\mathsf{\alpha}+\mathsf{\beta})$, as well as the probabilities of false and undetected failures on the value of the reduced tolerance $\delta $ on $z=0.5$.

**Figure 6.**The model of a random degradation process and a scheme for the formation of a time-to-failure distribution: (

**a**) $\alpha $ distribution (fan process); (

**b**) DN distribution law; (

**c**) DM distribution law.

**Figure 8.**Graphs and subgraphs of the CTS operation model with full resource recovery: (

**a**) with the control of four degradation states; (

**b**) with the control of one degradation state (classical model).

**Figure 9.**Dependence of the readiness coefficient ${K}_{A}$ on the TIBV for technical conditions ${E}_{3}$ and ${E}_{4}$.

**Figure 10.**Graph of the model of operation, degradation and renewal of the CTS fleet with incomplete resource recovery.

**Figure 11.**Dependence of the readiness coefficient ${K}_{A}$ on the TIBV for the first and second degradation groups.

**Figure 12.**The dependence of the readiness coefficient ${K}_{A}$ on the relative operating tolerances for the controlled parameters: (

**a**) on ${\mathsf{\delta}}_{}^{(1)}$ and ${\mathsf{\delta}}_{}^{(2)}$; (

**b**) on ${\mathsf{\delta}}_{}^{(2)}$ and ${\mathsf{\delta}}_{}^{(3)}$.

**Figure 13.**Distribution of workable CTS samples by degradation levels at different values of parameters determining the rate of degradation.

**Figure 14.**Dependence of the readiness coefficient and the TIBV on the total production capacity of metrological units.

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**MDPI and ACS Style**

Khayrullin, R.; Ershov, D.; Malahov, A.; Levina, T.
Mathematical Models of the Processes of Operation, Renewal and Degradation of a Fleet of Complex Technical Systems with Metrological Support. *Axioms* **2023**, *12*, 300.
https://doi.org/10.3390/axioms12030300

**AMA Style**

Khayrullin R, Ershov D, Malahov A, Levina T.
Mathematical Models of the Processes of Operation, Renewal and Degradation of a Fleet of Complex Technical Systems with Metrological Support. *Axioms*. 2023; 12(3):300.
https://doi.org/10.3390/axioms12030300

**Chicago/Turabian Style**

Khayrullin, Rustam, Denis Ershov, Alexander Malahov, and Tatyana Levina.
2023. "Mathematical Models of the Processes of Operation, Renewal and Degradation of a Fleet of Complex Technical Systems with Metrological Support" *Axioms* 12, no. 3: 300.
https://doi.org/10.3390/axioms12030300