#### 621.833.51 «Смешанные» вероятностные модели боя при переменных эффективных скорострельностях боевых единиц сторон

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-1-118128

On the basis of the theory of continuous Markov processes, "mixed" probabilistic models of bilateral fighting operations with exponential dependences of the effective rate of fire of the combat units of the parties on the time of battle have been developed. A numerical algorithm has been developed to calculate the main indicators of the battle of numerous groups.  It is made a comparison of the results of battle simulation using a "mixed" deterministic model with exponential dependences of effective rates of fire on the time of battle, as well as with "mixed" probabilistic models with constant effective rates of fire. The scope of these types of models applicability is established.

Чуев В.Ю., Дубограй И.В. «Смешанные» вероятностные модели боя при переменных эффективных скорострельностях боевых единиц сторон. Математическое моделирование и численные методы, 2020, № 1, с. 118-128.

#### 519.8 A "mixed" model of the confrontation of numerous groups with linear dependencies on the time of the effective speed of strikes by units of the parties

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2022-2-102113

Using the method of dynamics of averages, a "mixed" model of numerous groupings confrontation has been developed with linear dependencies on the time of effective speeds of striking by units of the parties. An algorithm is constructed that allows to investigate the course of the process and calculate its main indicators. It is established that the use of confrontation models with constant effective speed of strikes in many cases leads to significant errors in the calculation of the main indicators of the process. The influence of a preemptive strike by one of the opposing sides on the course of the progress and the final outcome are investigated.

Чуев В.Ю., Дубограй И.В. «Смешанная» модель противоборства многочисленных группировок при линейных зависимостях от времени эффективных скоростей воздействий единицами сторон. Математическое моделирование и численные методы, 2022, № 2, с. 104–115

#### 519.8 “Mixed” probabilistic models of bilateral military operations of numerous groups

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Dyakova L. N. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2017-1-91101

The purpose of this work was to develop "mixed" probabilistic models of bilateral military operations according to the theory of continuous Markov processes. In our research we obtained calculation formulas for estimating the main combat indices of groups small in number. Moreover, we developed a numerical algorithm to calculate the main combat indices of numerous groupings and made a comparison with the results of combat simulation using a deterministic model of two-way combat operations, the model being developed according to themean-value method dynamics. Findings of the research show that the correlation of the forces of the opposing sides, rather than their initial numbers, affects the errors in the mean-value method dynamics.

Chuev V., Dubogray I., Dyakova L. “Mixed” probabilistic models of bilateral military operations of numerous groups. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 91-101

#### 519.8 “Mixed” stochastic models of reciprocal hostilities for the case of one belligerent performing a pre-emptive strike

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2017-2-107123

We used the theory of continuous-time Markov processes as the basis for developing our stochastic “mixed” models of reciprocal hostilities and a numerical algorithm that makes it possible to compute main combat metrics for large forces. We show that a pre-emptive strike performed by one of the belligerents significantly affects the outcome and main metrics of combat between forces that are sufficiently similar in strength. We determine that it is not the initial numerical strengths of the two belligerents but their balance of power that affects errors shown by the method of dynamics of mean values; moreover, the errors increase with increasing the time to a pre-emptive strike.

Chuev V. Yu., Dubogray I.V. “Mixed” stochastic models of reciprocal hostilities for the case of one belligerent performing a pre-emptive strike. Маthematical Modeling and Coтputational Methods, 2017, №2 (14), pp. 107-123

#### 519.8 Modeling the conflict of interacting systems, taking into account the effect of obtaining information on the mutual state

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2023-3-125133

With the help of probabilistic methods, a model of the conflict of two interacting systems consisting of numerous structural units is proposed, taking into account the effect of delaying information about the mutual state: about the structure, number and parameters of each other's structural units. A study was made of the influence of insufficient information at a particular point in time on the outcome of the process of conflict development. It is shown that the availability of information about the state of the structural units of the opposite side can significantly increase the probability of successful development of the conflict, and with an increase in the number of units of structural units, the difference in the probability of successful development of the conflict scenario increases significantly

Чуев В.Ю., Дубограй И.В. Моделирование конфликта взаимодействующих систем с учетом эффекта получения информации о взаимном состоянии. Математическое моделирование и численные методы, 2023, № 3, с. 125–133

#### 519.8 Models of bilateral warfare of numerous groups

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2016-1-89104

Based on the theory of Markov processes the model of "poorly organized" battle was developed. Formulae for calculating its basic parameters at different initial numbers of the opposing sides were obtained. A comparison of the results of modeling a battle using probabilistic and deterministic models was performed. It was found that the dynamics model errors of the average are primarily affected by the balance of forces of the opposing sides in the beginning of the battle. It was shown that in case of military groups of similar forces the first-strike attack is of significant importance. When one of the warring parties at the beginning of the battle has a great advantage, the influence of first-strike attack is negligible. An increase in the influence of first-strike attack on the expected losses of a strong hand, and a reduction of its impact on the expected losses of the weaker party, as the number of groups involved in the fight increases proportionally, is also shown.

Chuev V., Dubogray I. Models of bilateral warfare of numerous groups. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 89-104

#### 519.8 Probabilistic model of the battle of two similar combat units against two different types

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Anisova T. L. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-2-107116

On the basis of the theory of continuous Markov processes, a model of the battle of two of the same type of combat units of the side against two of different types has been developed. The areas of application of various tactics of fighting by the side are shown. It is established that the use of the correct tactics of combat by a party can significantly increase the probability of preserving its two combat units. The developed battle model can be used to evaluate the combat effectiveness of multi-purpose weapons systems.

Чуев В.Ю., Дубограй И.В., Анисова Т.Л. Вероятностная модель боя двух однотипных боевых единиц против двух разнотипных. Математическое моделирование и численные методы. 2020. № 2. с. 107–116.

#### 519.8 Probabilistic models of bilateral combat operations with linear dependencies of effective rates of fire of combat units of the parties on the time of the battle with a preemptive strike of one of them

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2019-2-8498

On the basis of the theory of continuous Markov processes the models of bilateral military operations with linear dependences of effective rates of fire of combat units of the parties on the time of battle with a preemptive strike of one of them are developed. The algorithm allowing to calculate the main indicators of fight is developed. A comparison with the simulation results obtained on the basis of probabilistic models of combat with constant effective rates of fire and the model of the dynamics of the average with linear dependencies of effective rates of fire on the time of battle. The influence of a preemptive strike of one of the warring parties on the outcome and the main indicators of the battle is studed.

Чуев В.Ю., Дубограй И.В. Вероятностные модели двухсторонних боевых действий с линейными зависимостями эффективных скорострельностей боевых единиц сторон от времени боя при упреждающем ударе одной из них. Математическое моделирование и численные методы, 2019, № 2, с. 84–98.

#### 519.8 Probabilistic models of bilateral fighting. Comparison of results

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2021-1-6676

It is made a comparison of the results of calculating of the main indicators of the battle with the use of stochastic models developed by the authors with different dependences of the effective rate of fire of the combat units of the parties on the time of the battle. The influence of various factors on the differences in the calculations of these indicators when using these models is shown. It is established that for the solution of the majority of military -technical and military-tactical tasks it is possible to use any of the models of bilateral military operations developed by authors.

Чуев В.Ю., Дубограй И.В. Вероятностные модели двухсторонних боевых действий. Сравнение результатов. Математическое моделирование и численные методы, 2021, № 1, с. 66–76.

#### 519.8 Probability model of meeting an attack of different types of weapon

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Anisova T. L. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2018-1-9097

On the basis of the continuous Markov processes theory we have developed a probabilistic model of the two-way battle of one combat unit against two enemy units of different types. The authors have obtained calculation formulas for computing current and final states under various firing tactics of the unit. We have determined the applicability areas for different combat tactics of the unit. The study shows that the right choice of the firing tactics can considerably increase the probability of the victory in the battle. The developed model of the two-way battle may be used for estimating the combat effectiveness of the multipurpose weaponry units.

Чуев В.Ю., Дубограй И.В., Анисова Т.Л. Вероятностная модель отражения атаки разнотипных средств. Математическое моделирование и численные методы, 2018, № 1, с. 90-97

#### 519.8 Simulation of the confrontation between the two sides, taking into account redundancy

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2023-2-155163

Based on the method of dynamics of averages, a model of two parties confrontation has been developed taking into consideration the bringing up of reserves by one of the parties. It is established that timely supply of reserves can significantly affect the course of the process and its final result. It is also shown that the use of the reserve at the beginning of the action significantly increases the capabilities of a group.

Чуев В.Ю., Дубограй И.В. Моделировании противоборства двух сторон c учетом резервирования. Математическое моделирование и численные методы,2023, № 2, с. 155–163

#### 519.8 Stochastic model of combat operations of the same type of combat units against ones of different types

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2021-2-8695

On the basis of the theory of continuous Markov processes, a model of a two–way battle of two similar combat units of side X against two different types of enemy units is developed. Calculation formulas are obtained for calculating the current and final states for various tactics of fighting by the X–side. The developed model of two–way combat can be used to assess the combat effectiveness of multi-purpose weapons systems.

Чуев В.Ю., Дубограй И.В. Стохастическая модель боевых действий однотипных боевых единиц против разнотипных. Математическое моделирование и численные методы, 2021, № 2, с. 86–95.

#### 519.8 Stochastic model of repelling of attacks made by different types of means with a preemptive strike by one of the parties

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2019-1-5464

On the basis of the theory of continuous Markov processes, it was developed a stochastic model of reflection by the combat unit of the attack of two different types of enemy units with a preemptive strike of one of the opposing sides. The calculation formulas for calculating the current and final status are obtained. It is shown that the choice of the defending unit of tactics of firing does not depend on which of the opposing sides causes a preemptive strike, but it’s correct choice can significantly increase the probability of it’s victory. The model of two-way combat developed in this article can be used for estimation of the multi-purpose weapons systems combat effectiveness.

Чуев В.Ю., Дубограй И.В. Стохастическая модель отражения атаки разнотип-ных средств при упреждающем ударе одной из сторон. Математическое модели-рование и численные методы, 2019, № 1, с. 54–64.

#### 519.8 Stochastic models of bilateral hostilities of numerous groups while linear dependence of military units’ effective firing rate on the duration of the battle

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2018-2-122132

The article presents developed on the basis of the theory of Markov’s processes the models of bilateral hostilities with the linear dependence of effective firing rate of military units on the time of the battle. Developed the algorithm allows to calculate the main indicators of the battle of numerous groups. Fulfilled the comparison with the results of a battle simulation, received on the basis of probabilistic models of the battle with constant effective firing rate and the deterministic model of a combat with a linear dependency of effective firing rate on the time of the battle. The range of the last models applicability presented.

Чуев В.Ю., Дубограй И.В. Стохастические модели двухсторонних боевых тдействий многочисленных группировок при линейных зависимостях эффективнх скорострельностей боевых единиц сторон от времени боя. Математическое моделирование и численные методы, 2018, № 2, с. 122–132.

#### 519.8 Stochastic models of the two unit duel fight

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2016-2-6984

On the basis of the theory of continuous Markov processes we developed models of the two unit duel fight. We obtained computing formulas for calculating the basic fight indicators. Moreover, we found that the pre-emptive strike of one of the units participating in the fight has a significant impact on the fight outcome of the units which are similar in forces. The strike has a negligible impact, if one of the units has a significant advantage. The findings of the research show that the use of model with constant effective firing rates can lead to significant errors in the evaluation of its results. Finally, we found that the pre-emptive strike, coupled with a high degree of effective firing rate growth, can sometimes compensate for more than the double initial superiority of the opponent. We show the possibility of using approximations of the effective firing rate of the fighting units by the different functions of the fight time.

Chuev V., Dubogray I. Stochastic models of the two unit duel fight. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 69-84