Analytical Approach to UAV Cargo Delivery Processes Under Malicious Interference Conditions

Abstract

The instability of the geopolitical situation due to the high terrorist danger leads to the need to take into account at the planning stage the capabilities of intruders to perform UAV flight missions. A general method for analyzing the process of cargo delivery by UAVs (Unmanned Aerial Vehicles) to hard-to-reach areas during emergencies has been proposed. This method allows for the evaluation of UAV effectiveness based on the probability of successful cargo delivery within a specified time limit. The method is based on applying topological transformation techniques to stochastic networks. The cargo delivery process is modeled as a stochastic network, followed by the determination of its equivalent function and the use of Heaviside decomposition to calculate the distribution function and the expected delivery time. This presentation of the studied process for the first time made it possible to take into account the impact on the flight mission of the UAV of the destructive impact from the attacker. This approach allows the destructive effects on the UAV from malicious interference to be considered. The input data used for the analysis are parameters that characterize the properties of individual processes within the stochastic network, represented as branches, which are computed using methodologies published in earlier studies. It has been demonstrated that the resulting distribution function of the mission completion time can be accurately approximated by a gamma distribution with a level of precision suitable for practical applications. In this case, the gamma distribution parameters are determined using the equivalent function of the stochastic network. The proposed method can be used by flight planners when scheduling UAV operations in emergency zones, especially in scenarios where there is a risk of malicious interference.

1. Introduction

At present, there is a remarkable surge in the development and enhancement of unmanned aerial vehicles (UAVs). This growth can be attributed to the wide range of advantages that UAVs offer over other traditional delivery methods, particularly in terms of their flight performance and technical specifications. UAVs are becoming increasingly popular due to their ability to operate in areas where manned vehicles might face challenges, such as hard-to-reach or hazardous locations. Additionally, their capacity to deliver goods efficiently and rapidly, without the need for direct human intervention, provides a significant edge in both military and civilian applications. These advancements in UAV technology have made them an attractive alternative for tasks such as surveillance, cargo delivery, and emergency response, where speed and precision are critical.

This is due to a number of advantages that UAVs have in flight performance over other delivery vehicles. Such advantages are [1,2,3,4,5,6]:

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Stability and simplicity of UAV control provided by the application of a classical aerodynamic scheme;

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Equipped with electric motors, which greatly simplifies the operation process;

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Possibility of using such types of energy for engines that will increase the time, and therefore the range of the UAV;

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High efficiency due to low costs for moving (relocating) UAV maintenance and control bodies, no need to equip stationary long-term basing areas, low-cost repair and maintenance;

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Low cost of UAV development and operation, etc.

Relative cheapness and manufacturability determine the versatility of the field of tasks solved with the help of UAVs. The main of such tasks are reconnaissance and monitoring of the situation; monitoring of the technical condition of objects, their safety and functioning; photo and video filming of objects and places of emergency; delivery of small-sized cargo to hard-to-reach and especially dangerous emergency zones (ES), as well as delivery of mail [1,2,3].

It should be noted that along with the advantages, modern UAVs are also characterized by disadvantages, the main of which are [5,6]:

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Restrictions related to dependence on time of day and weather conditions;

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Low imitation immunity and noise immunity of UAV radio control channels;

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High sensitivity of the UAV design to mechanical damage;

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Short range of UAV remote control;

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Significant limitations on the weight and dimensions of the payload.

The need to justify the feasibility and make decisions regarding the delivery of goods using UAVs to a specific recipient, located in an area that is difficult for other transport systems to reach [4,7,8], underscores the necessity of assessing the situation. One of the most critical components of this assessment is calculating the time and reliability of UAV cargo delivery. It is important to note that the effectiveness of UAV operations is significantly influenced not only by random factors related to physical, geographical, and meteorological conditions but also by the increasing number of theft attempts on delivered cargo by malicious actors. These thefts are especially frequent in regions with unstable military–political environments or complex economic conditions.

As a result, evaluating the effectiveness of UAVs in adverse natural and antagonistic conditions, where the impact of malicious actors is significant, has become a critical area of research. This growing relevance is reflected in the large number of publications dedicated to these practical and scientific challenges.

The primary works dedicated to studying various aspects of UAV (Unmanned Aerial Vehicle) efficiency are fundamental contributions by members of the Russian Academy of Sciences (RAS) and notable professors, including M.V. Silnikov [8], G.I. Gorchitsa [7], V.S. Moiseyev [5], S.I. Makarenko [6], and others. Methods for calculating specific efficiency metrics are published in works [9,10,11,12,13,14]. However, despite the high scientific and practical significance of these works, the results presented do not provide solutions for calculating crucial indicators required for planning the cargo delivery process, such as the average time and the probability of successful UAV mission completion within a specified time.

At the same time, ref. [15] suggests methods for calculating cargo delivery times under the assumption of exponential distribution of the random component of delivery duration and without considering the search for a convenient drop-off point within the delivery area for the recipient. It is important to note that the delivery time estimation, considering the variability of the task, should account for interference by a “self-interested” or organized malicious actor. In particular, ref. [6] discusses methods and analytical tools for analyzing the effectiveness of countermeasures against UAV deployment. This includes the analysis of various forms of interference, including the use of fire and electronic warfare systems based on data from radio–electronic, optical–electronic, and acoustic intelligence.

Ref. [16] presents results that can be used to create unmanned intelligent delivery platforms with subsequent optimization of UAV swarm logistics. However, the results obtained in [16] cannot be used to plan the use of unmanned intelligent platforms in areas with unstable geopolitical and air conditions. In [17,18], an analysis of the aspects of information security and reliability of communication channels with UAVs was carried out. However, the results presented in them are not focused on determining the main characteristics of the random time of the UAV flight mission.

However, neither the approach described in [15] nor other known methods for analyzing the efficiency of UAV operations consider the possibility of intercepting and/or destroying the cargo using an air interception system deployed by a malicious actor.

In this article, using the method of topological transformation of stochastic networks, the authors have developed a model for the UAV task execution process in conditions where interference from a malicious actor is present. The capabilities of the malicious actor’s air interception system are characterized by the probabilities of detection, identification, and destruction of the UAV. These values are determined using the results provided in works [11,12,13,14].

Since the authors do not aim to analyze the methods of countering UAVs, the model considers the malicious actor’s capabilities in terms of detection probabilities by reconnaissance and the probability of UAV destruction by the countermeasures system, without specifying the types of systems used by the malicious actor. However, if necessary, these can be easily integrated by using data from publications such as [8]. Meteorological and topogeodetic conditions in the task area, as well as the flight trajectory, are taken into account when determining the characteristics of corresponding random variables, using methods described in [13,14]. Parameters that characterize the operators’ skills and the tactical methods of UAV deployment are in line with the established standards and recommendations provided in [5,8].

2. Materials and Methods

2.1. Task Statement

Let us assume that, according to the planned flight schedule and mission assignment, it is necessary to detect the user (recipient) and deliver a package to a pre-designated, hard-to-reach area located R kilometers from the UAV’s starting position, as shown in Figure 1.

Let us assume that the operator prepares for the flight and launches the UAV in a random time $t_z$ with the distribution function $Z\left(t\right)$, and the UAV flight speed is $V$ km/h. In order to more fully use the capabilities of the UAV, the user is searched for in the cargo delivery area using the tactical technique “Search with several passes” [8], while the time of one flight (pass) t_p of the user’s location area is a random variable with a distribution function B(t), and the probability of successful detection of the user and receipt of the cargo at the place of delivery is R_us. Throughout the duration of the task, the UAV can be detected by an attacker with a probability of P₀, identified in a random time t_i with a distribution function I(t), and destroyed (intercepted) with a probability of P_ubp. If the UAV has not been intercepted or destroyed, then the attacker’s air interception system searches for it again, and the UAV performs an evasion maneuver from the detection and interception means for some time t_po with the N(t) distribution function.

2.2. Assumptions and Limitations

• The UAV and its auxiliary equipment are fully operational and remain unaffected by any malicious interference until the flight mission is completed.
• The UAV to be launched is a quadcopter and does not belong to military reconnaissance or fire destruction systems.
• The mission area is designated by the flight manager, based on the UAV’s tactical and technical specifications [16] and the provided recommendations [5].
• The duty shift personnel have undergone preliminary training and are proficient in performing UAV control tasks in accordance with the established training standards [5,8].
• Values of UAV random preparation and launch time, one pass of the assigned task area, UAV identification by the intruder detection and interception subsystem, and UAV maneuvering to evade the intruder’s air interception means are independent random values.
• Since the information provided in [5,8] only characterizes the average time for completing training tasks related to UAV control calculations, it is assumed that the distribution functions of the random variables outlined in the task belong to the exponential class. The use of exponential distribution laws for the time of specific random processes is also considered acceptable, based on data published in [11,12], which demonstrated that stationary Poisson processes are sufficient for analyzing the effectiveness of UAV countermeasures. This means that the identification, re-detection, and passage processes of the UAV in the target area are also modeled as Poisson flows. Moreover, using an exponential distribution to characterize the random times of these processes significantly simplifies the integral transformations required by the modeling method employed.
• The probabilities of UAV detection, destruction (interception), and user (the recipient of the delivered cargo) detection are calculated using well-established methods [5,8,9,10,13].

Since the successful implementation of individual stages of the countermeasure process by an attacker will lead to an inevitable repetition of the stages of the UAV control process, GERT (graphical evaluation and review technique) was chosen as a modeling tool for the cargo delivery process, which proved its inherent high statistical convergence and described in detail by Alan Prisker (A.A.B. Prisker) in the NASA memorandum RM-4973 in 1966. GERT involves representing the simulated process as a stochastic network and then determining its equivalent function using the Laplace–Stieltjes transform, which is the generating function of moments for continuously distributed random variables, and determining the moments of random time for the implementation of the simulated process. It is required to determine the mathematical expectation T and the function of distribution F(t) of the time of successful delivery of cargo to the user using the UAV.

2.3. Decision

Let us consider the process of performing the cargo delivery task through the control calculation of a UAV, represented as a stochastic network (see Figure 2).

The figure indicates:

z(s), b(s), i(s) and n(s)—Laplace–Stieltjes transformation of the probability distribution functions of the random time of preparation and launch of the UAV, one pass through the designated target area of the military mission, identification of the UAV by the subsystem for detecting and intercepting an intruder, maneuvering UAVs for the purpose of evading air interception means of an attacker, respectively.

In accordance with assumption 4, these images are as follows:

$$\begin{gathered} z(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}d\left[Z(t)\right]}={\displaystyle \frac{z}{z+s}}; \\ b(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}d\left[B(t)\right]}={\displaystyle \frac{b}{b+s}}; \\ i(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}d\left[I(t)\right]}={\displaystyle \frac{i}{i+s}}; \\ n(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}d\left[N(t)\right]}={\displaystyle \frac{n}{n+s}}; \end{gathered}$$

(1)

In Formula (1):

$z={\displaystyle \frac{1}{\overline{t_z}+\frac{R}{V}}}$; $b={\displaystyle \frac{1}{\overline{t_p}}}$; $i={\displaystyle \frac{1}{\overline{t_i}}}$; $n={\displaystyle \frac{1}{\overline{t_{po}}}}$ and in these formulas $\overline{t_z}$; $\overline{t_p}$; $\overline{t_i}$ and $\overline{t_{po}}$—the average time for assembling and launching the UAV, one UAV flight along the established route in the area of the task, identification and maneuvering of the UAV in order to evade the means of air interception of the attacker.

An equivalent function of the stochastic network is obtained using the topological Mason equation for closed graphs [17,18,19]:

$$1+{\displaystyle \sum _{k=1}^K{\left(-1\right)}^k{L}_k^{(i)}(s)=0};$$

(2)

Here $L_k^{(i)}(s)$—the i-th equivalent k-th order loop function; K is the maximum order of loops included in the stochastic network.

To this end, we close the stochastic network with a fictitious (i + 1)-th branch of the first-order $L_1^{(i+1)}(s)={\displaystyle \frac{1}{Q(s)}}$

here Q(s) is the desired equivalent function) and define all loops of order k:

first-order loops, k = 1:

$$\begin{array}{l}{L}_1^{(1)}=P_0{P}_{ubp}i(s)z(s);\\ L_1^{(2)}=P_{0}i(s)n(s)\left[1-P_{ubp}\right];\\ L_1^{(3)}=b(s)\left[1-P_{us}\right].\end{array}$$

second-order loops, k = 2:

$$\begin{array}{l}{L}_2^{(1)}=P_0{P}_{ubp}i(s)z(s)P_{0}i(s)n(s)\left[1-P_{ubp}\right];\\ L_2^{(2)}=P_{0}i(s)n(s)\left[1-P_{ubp}\right]b(s)\left[1-P_{us}\right];\\ L_2^{(3)}=P_0{P}_{ubp}i(s)z(s)b(s)\left[1-P_{us}\right].\end{array}$$

In our case, there are no loops of the third and higher orders.

Solving Equation (2) with respect to Q(s), we obtain the equivalent function of the stochastic network (Figure 2):

$$Q(s)={\displaystyle \frac{z(s)\left(1-P_0\right)b(s)P_{us}}{1-L_1^{(1)}(s)-L_1^{(2)}(s)-L_1^{(3)}(s)+L_2^{(1)}(s)+L_2^{(2)}(s)+L_2^{(3)}(s)}}$$

(3)

By entering the designation $r=P_{us}b$ and substituting private images (1) in (3), we obtain:

$$Q(s)={\displaystyle \frac{rz\left(1-P_0\right)\left(i+s\right)\left(n+s\right)}{s^4+A{s}^3+B{s}^2+Cs+D}}$$

(4)

In this formula:

$$A=i+n+r+z$$

$$B=r\left(i+n+z\right)+z\left(i+n\right)+in-P_0{P}_{ubp}iz+P_{0}in\left(P_{ubp}-1\right)$$

$$C=r\left[z\left(i+n\right)+in-P_0{P}_{ubp}iz+P_{0}in\left(P_{ubp}-1\right)\right]+inz+P_{0}in\left(P_{ubp}-1\right)-P_0{P}_{ubp}inz$$

$D=rinz\left[1+P_0\left(P_{ubp}-1\right)-P_0{P}_{ubp}\right]$—denominator expansion coefficients of the equivalent function.

The physical meaning of the equivalent function Q (s) is that it is the probability of completing the process of completing a flight mission for a while less than the required delivery time of the UAV cargo. In this case, the required delivery time is an exponentially distributed random variable. That is, the physical meaning of (4) completely coincides with the physical meaning of the Laplace–Stieltjes transform. The representation of the equivalent function in the form of (4) simplifies the definition of the original Q(s) by using the Heaviside decomposition [20], which allows, if $s_i\ne s_j$, to represent the image Formula (1) as the sum of the residues in the poles of the S_i followed by the definition of the original from the image (3), i.e.,

$$f(t)={\displaystyle \sum _{i=1}^4\frac{rz\left(1-P_0\right)\left(i+s_i\right)\left(n+s_i\right)}{4s_i^3+3s_i^{2}A+2s_{i}B\hspace{0.17em}+C}}\exp (s_i,t)$$

(5)

In this formula:

S_i—are simple poles of the equivalent function Q(s).

The resulting expression is a function of the probability density, so the desired UAV task time distribution function can be defined as

$$F(t)={\displaystyle \underset{0}{\overset{t}{\int }}f(t)dt}={\displaystyle \sum _{i=1}^4\frac{rz\left(1-P_0\right)\left(i+s_i\right)\left(n+s_i\right)}{\left(4s_i^3+3s_i^{2}A+2s_{i}B+C\right)}}\left[1-\exp (s_i,t)\right]$$

(6)

In turn, the mathematical expectation of the UAV task execution time is

$$T={\displaystyle \underset{0}{\overset{\infty }{\int }}tf(t)dt}={\displaystyle \sum _{i=1}^4\frac{rz\left(1-P_0\right)\left(i+s_i\right)\left(n+s_i\right)}{\left(4s_i^3+3s_i^{2}A+2s_{i}B+C\right){\left(-s_i\right)}^2}}$$

(7)

Based on the obtained relationships (6) and (7), calculations were made, the results of which are presented in Figure 3 in the form of graphs (Figure 3).

3. Results and Discussion

The calculations were based on the assumption that a DJI MAVIC 2 PRO quadcopter from the MAVIC 2 PRO family was used for the task, with the following initial data (

Table 1. Data for experiment.

No.	Designation and Value	Units of Measure	Units of Measure Quantity Name
1	R = 20	km	Distance from the starting position to the center of the task area
2	V = 20	km/h	UAV flight speed
3	$\overline{t_z}=1$	min	Average UAV launch time (excluding assembly time)
4	$\overline{t_p}=3$	min	Average time of one flight along the assigned route in the task area
5	$\overline{t_i}=15$	s	Average time of UAV identification by the intruder detection and interception subsystem
6	$\overline{t_{po}}=10$	min	Average time of UAV evasion from the means of detection and interception of an attacker
7	$P_0=0.3\dots 0.5$		Probability of UAV detection by the intruder detection and interception subsystem
8	$P_{ubp}=0.1\dots 0.3$		Probability of interception (destruction) of a detected UAV by an attacker
9	$P_{us}=0.85$		Probability of successful user search and receipt of cargo in the task area
10	$T_{max}=0.85$	min	Maximum flight duration

):

Analysis of the Resulting Solution

1. The probability of successful completion of the flight mission within the time, which is less than 2/3 of the maximum flight duration, cannot exceed

$$P\left(t\le {\displaystyle \frac{2T_{\max }}{3}}\right)\le \underset{\begin{array}{l}{P}_0\to 0\\ P_{ubp}\to 0\\ P_{us}\to 1\end{array}}{{\displaystyle \lim }}F\left(t={\displaystyle \frac{2T_{\max }}{3}}\right)={\displaystyle \frac{r}{r-z}}\left(1-e^{-z·{\displaystyle \frac{2T_{\max }}{3}}}\right)-{\displaystyle \frac{z}{r-z}}\left(1-e^{-r·{\displaystyle \frac{2T_{\max }}{3}}}\right)$$

and is determined by the flight speed, remoteness and size of the task area, as well as the training of the operator assembling and launching the UAV.

Graphs of maximum achievable distribution functions F(t) of the time of successful completion of a combat mission with different distances of the task area from the starting position are shown in Figure 4.

It is not difficult to see that the minimum average task completion time is $T_{\min }={\overline{t}}_z+{\overline{t}}_p+{\displaystyle \frac{60R}{V}}$, which is consistent with the results presented in [15].

2. The probability of successful completion of the task depends not only on the search capabilities of the UAV, but also on its intelligence protection. When equipping a UAV with search means capable of practically guaranteed detection of the user and ensuring that he receives the cargo, the time and probability of completing the task significantly depend on the probability of detecting the UAV by means of air interception of the attacker. At the same time, the capabilities of the destruction subsystem for an attacker can be very modest. So, for example, if the probability of detection of a UAV $P_0=0.9$ and its destruction by an attacker with probability $P_{ubp}=0.1$, the probability of successful completion of the task in a time less than 2/3 of the maximum flight duration does not exceed $P\left(t\le {\displaystyle \frac{2T_{max}}{3}}\right)\le 0.11$ and the average time for successful completion of the task increases to T = 119 min.

It should be noted that the probability of UAV detection must be determined taking into account the capabilities and methods of using electronic, optical–electronic and acoustic reconnaissance by an attacker. For example, electronic reconnaissance provides detection of small-sized UAVs at a distance of up to 150 km with an accuracy of 50–150 m. However, the use of this type of reconnaissance is practically useless when controlling UAVs using fiber optic communication channels, [4,8]. Using optoelectronic reconnaissance tools, the UAV detection range in the visible range will be from 300…5000 m and will significantly depend on the meteorological and topographic–geodetic situation in the area of tasks [5,6]. At the same time, the use of the infrared range does not significantly increase the capabilities of optical–electronic intelligence. In turn, acoustic reconnaissance allows you to detect UAVs at ranges up to 1.5…2 km at altitudes up to 1000 m with extremely low accuracy of determining coordinates. The presence of shortcomings in each of these types of reconnaissance leads to the need to integrate them in order to increase the probability, range, accuracy, and reduce the time for detecting UAVs. Therefore, in order to analyze the effectiveness of the use of UAVs, it is necessary to calculate the probability of detection, taking into account the forecast for the use of these types of reconnaissance by an attacker. This shows the need to develop technologies for protecting UAVs from species and radio reconnaissance of an attacker (Figure 5).

3. Calculations show that the average task completion time significantly depends on the level of training of operators controlling UAVs, Figure 6.

It should also be noted that the impact on the probability and time of successful completion of the flight mission (Figure 6) from the probability of successful detection of the user and receipt of the cargo, determined, in turn, by the characteristics of the UAV used. So, for example, with the limited capabilities of an attacker, but a noticeable imperfection of the UAV (Rutz = 0.1), the average time for successful completion of a flight mission will increase to 51.1 min. Note that the calculation of the Rutz probability is carried out by the methods described in detail in [5,8,9,10,13].

4. In particular, with a high level of training, the average time for successful completion of the task is T = 19 min, and with a satisfactory level of training of operators, it increases to T = 28 min. In this case, the probability of successful completion of the task in a time less than 2/3 of the maximum flight duration will be $P\left(t\le {\displaystyle \frac{2T_{max}}{3}}\right)=0.8$ and $P\left(t\le {\displaystyle \frac{2T_{max}}{3}}\right)=0.63,$ respectively. Thus, the use of the proposed method allows for the assessment of UAV control trainees’ actions, not just based on individual standards, but through a more comprehensive indicator: the time required to complete a flight mission under conditions of interference from a malicious actor. Implementing this approach involves dividing the trainees into two groups. The first group performs according to the standard, while the second group actively interferes with the UAV (acting as a “simulated attacker”) using various means of suppressing the UAV, without physically destroying it.

To support this, the proposed stochastic network model should be extended with an additional branch that accounts for the time required to prepare the UAV for launch. Moreover, the standard “Flight mission for the UAV—Target Destruction” should be supplemented by a standard for the assembly and preparation of the UAV for launch. If this task demands considerable effort from the trainees, depending on the UAV model, the training group may need to be divided into three subgroups. However, the implementation of this method requires further research and adjustments to the training programs for specialists in this field.

Since this approach simulates a complex flight mission, the simulation results should be treated as part of the practical task in the final (examination) evaluation, rather than solely as an object of study for UAV piloting. Nonetheless, the implementation of this approach will necessitate some adjustments to the training programs for specialists in this profile.

5. To determine the distribution class for the time required to successfully complete the UAV’s combat mission, the first four central moments of the corresponding random variable were calculated. The results of numerous calculations showed that the obtained distribution is unimodal, not symmetrical, right-sided, with a positive kurtosis, i.e.,

$$A_s={\displaystyle \frac{m_3}{{\sigma }^3}}>0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}E={\displaystyle \frac{m_4}{{\sigma }^4}}-3>0;$$

Here $m_3;m_4;\sigma$ is the third (fourth) central moment and the standard deviation of the random task execution time) and belongs to the third class of Pearson distribution. This means that the distribution function for the time of successful task completion by the attacking UAV can be accurately approximated by the gamma distribution, as shown in Figure 7.

µ—Scale and shape parameters; σ—gamma distribution.

$$F_{\gamma }(t)={\displaystyle \underset{0}{\overset{t}{\int }}\frac{{\mu }^{\alpha }}{\mathrm{\Gamma }(\alpha )}}{x}^{\alpha -1}\exp (-\mu x)dx$$

were defined using the equivalent function (1):

$$M_1=T=-{\displaystyle \frac{d}{ds}}{\left[{\displaystyle \frac{Q(s)}{Q(0)}}\right]}_{s=0};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{M}_2={\displaystyle \frac{d^2}{d{s}^2}}{\left[{\displaystyle \frac{Q(s)}{Q(0)}}\right]}_{s=0};$$

$$\mu ={\displaystyle \frac{M_1}{M_2-M_1^2}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\alpha ={\displaystyle \frac{M_1^2}{M_2-M_1^2}}$$

The small value of the absolute error of the gamma distribution approximation of the distribution function obtained during modeling shows that the result presented in [15] is a special case of the method proposed in the work.

Given that the calculated moments and parameters of the gamma distribution are determined in the computer program in the form of simple algebraic expressions (see, for example, (7)), as well as the fact that gamma distribution is a built-in function of most object-oriented packages of computer applications, its use as a random time model for performing a flight mission UAV will allow obtaining probability values of completing a flight mission at target dates for a time close to real. This, in turn, will reduce the time for assessing the UAV application option and may somewhat simplify the task of developing special mathematical and software subsystems for planning and decision support of the automated UAV control system. Moreover, the absolute error of approximation does not exceed 3%.

4. Conclusions

The method presented in this article for analyzing the process of performing a UAV mission to deliver cargo to a hard-to-reach area in the heart of an emergency zone is based on representing the delivery process as a stochastic network. This network includes branches that characterize the key stages of the UAV flight mission. A notable feature of this method is its incorporation of the opposition from the system used for detecting and intercepting (or destroying) a malicious UAV. A distinctive feature of the method is taking into account the opposition from the system for detecting and intercepting (destroying) an attacker’s UAV.

The mathematical model underlying this method not only accounts for the technical characteristics of the UAV but also considers the operational methods used during flight missions. This highlights the importance of developing tactical techniques and specialized models for UAV deployment early in the planning phase. Furthermore, the model’s sensitivity to the properties of the attacker’s air interception system enables an evaluation of the effectiveness of protective measures designed to shield the UAV from the attacker’s reconnaissance and interception subsystems.

As the analysis of the modeling results demonstrates, the proposed method is efficient, sensitive to changes in input data, and provides results that are both logically consistent and in line with real-world UAV usage in hard-to-reach emergency zones.

In future studies, the integration of neural networks [21,22,23] could be explored to further enhance the model’s predictive power. By using neural networks, particularly deep learning techniques, it may be possible to improve the accuracy of predictions regarding UAV task completion times, even in dynamic, real-time scenarios with varying adversarial threats. Neural networks could enable the system to adapt more quickly to changes in the environment, allowing for optimized decision-making and mission execution.