*2.3. Examples*

As mentioned in the Introduction, *k*-out-of-*n* models have a wide sphere of applications (see [1] and others), including the study of energy (see [11,12]), and telecommunication [13] problems. Let us focus on two examples of applying the *k*-out-of-*n* model. In the numerical analysis, we will use these examples for the special case of *n* = 6, *k* = 2.

#### 2.3.1. A Flight Module of a Tethered High-Altitude Telecommunication Platform

As an application example of the proposed *k*-out-of-*n* model, consider the model of a multi-copter flight module, which is part of the tethered high-altitude telecommunications platform [13]. The main area of its application is solving problems related to the long-term operation (tens of hours) without lowering the unmanned flight module to the ground. Therefore, unlike autonomous Unmanned Aerial Vehicles (UAVs) reliability parameters are of crucial importance for the tethered UAV-based high-altitude platforms.

A multi-rotor UAV is a system consisting of *n* rotors arranged uniformly in a circle and pairwise symmetrically with respect to the center of the circle [29]. The multi-copter may malfunction due to the failure of the propeller engines. There are various modifications of multi-rotor UAVs. The most common architectures are quad-, hexa-, and octocopters. The higher the redundancy ratio, the higher the reliability of the system. , Therefore, in practice, flight modules with 6 or 8 rotors are most often used. In this example, we consider a hexacopter as a hot standby system consisting of *n* = 6 components (rotors) that work and fail independently of each other (see Figure 1).

**Figure 1.** An unmanned hexacopter flight module of a tethered high-altitude telecommunications platform.

If the location of the failing components is not taken into account, this system fails when *k* = 2 out of 6 rotors fail.For practical use, various reliability characteristics of such a system, including those considered in the general model, are of interest.

2.3.2. An Automated System for Remote Monitoring of a Sub-Sea Pipeline

As another application example of the *k*-out-of-*n* model, we consider an automated system for remote monitoring of a sub-sea pipeline. This system has been considered in [12], where its description has been given in details. One of the main parts of this system is an Unmanned Underwater Vehicle (UUV), the structure of which is illustrated in Figure 2.

**Figure 2.** An unmanned multi-functional underwater vehicle.

The UUV consist of 6 motors, indicated by numbers 1–6, which allow it to rise, fall and move in various directions, including along the pipeline. The UUV is equipped with various devices, indicated by numbers 7–15, for receiving and transmitting information about the state of the pipe. In paper [12] the reliability function of this model has been studied in two scenarios:


However, the influence of the number of failed components on the residual lifetimes of the survived ones was not taken into account earlier. In the current paper, this model has been studied under the condition that failed components reduce the residual lifetime of surviving system's components.

#### **3. Distribution of the System's Time to Failure**
