К анализу системы массового обслуживания с ресурсами, функционирующей в случайном окружении

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Строится математическая модель системы, состоящей из накопителя и нескольких однородных приборов, функционирующей в случайном окружении и предоставляющей поступающим заявкам помимо обслуживания ещё и доступ к ресурсам. Случайное окружение представлено двумя независимыми марковскими процессами, управляющими поступлением заявок в систему и обслуживанием заявок. В систему поступает пуассоновский поток заявок, интенсивность поступления и объем ресурсов, необходимый заявке при обслуживании, определяются состоянием внешнего марковского процесса. Время обслуживания заявок на приборах подчинено экспоненциальному распределению. Интенсивность обслуживания и максимальный объем ресурсов системы определяются состоянием второго внешнего марковского процесса. При окончании обслуживания заявки занятые ею ресурсы возвращаются в систему. В рассматриваемой системе возможны отказы в приёме поступающих заявок из-за нехватки ресурсов, а также возможны потери уже принятых в систему заявок при изменении состояния внешнего марковского процесса, управляющего обслуживанием и предоставлением ресурсов. Построен случайный процесс, описывающий функционирование данной системы. Представлена в скалярной форме система уравнений для стационарного распределения вероятностей построенного случайного процесса. Сформулированы основные задачи для дальнейшего исследования.

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1. Introduction The mathematical model of the analysis of the functioning of modern telecommunication systems must take into account the influence of external factors, which may be realized within the framework of the queuing theory (the theory of teletraffic) [1-4] with the help of arrival and/or service processes controlled by some external random process. The application of the Markov modulated arrival process (MMAP), Markov modulated service process (MMSP)) [3, 5-9] allows us to construct not only the adequate mathematical model, but also to obtain good analytical results for different tasks [10-22]. The mathematical modeling of modern telecommunication systems when incoming applications in addition to services also require some fixed or variable volume of resources [23-29] is the actual problem. We will try to apply Markov modulated Poisson process (MMPP) theory [5-9] to construct the mathematical model of the system, that consists of a storage device and several homogeneous servers and operates in a random environment, provides incoming applications not only services, but also access to resources of the system, is being constructed. The random environment is represented by two independent Markov processes. The first of Markov processes controls the incoming flow of applications to the system and the size of resources required by each application. The service rate and the maximum amount Received 9th October, 2018. of system resources are determined by the states of the second external Markov process. The initial stages of this study were presented in [30]. The system of equations for the stationary probability distribution of the random process, describing the behavior of the system, is the main goal of this part of the research. 2. System description We will consider the queueing system ��� �2|����2|�|�|�1, �2 (according to Kendall-Basharin notation [1]), functioning in the random environment (Markov modulated Poisson arrival process and Markov modulated service process), with 1 � � < ∞ homogeneous servers and the buffer of � � ∞ capacity. The random environment is present by two-state Markov process (MP) �1(�), which control the incoming Poisson process. If the external Markov process �1(�) is in state 1 then the rate of incoming Poisson process is �1 and each arriving application requires the fixed �1 amount of system resources. If the MP �1(�) is in state 2 the each application arrives according the Poisson law with the rate �2 and requires the fixed amount of system resources of size �2. The second external two-state Markov process �2(�) controls the service process on system servers and the maximum amount of system resources. If MP �2(�) is in the state 1, then the maximum value of system resources is �1 < ∞, the service time of an application (on each of � homogeneous servers) is subject to the exponential distribution with the rate �1. If MP �2(�) is in the state 2, then the amount of system resources �2 is unlimited, the service time of an application (on each of � homogeneous servers) is subject to the exponential distribution but with the rate ��2. The transitions of Markov processes �1 and �2 from one state to another are determined by the corresponding infinitesimal matrices Λ = (��� )�,�=1,2 and � = (��� )�,�=1,2. After the end of the service each application returns to the system the resources, occupied by this application. The functioning of the system may be defined by the multidimensional random process �(�) = {�1(�), �2(�), �(�), �1(�), �2(�)}, where random process �1(�) = (�1�(�), �1� (�)) describes the number of applications with demand on �1 amount of resources (applications of the first type) on the servers (�1�(�)) and in the buffer (�1� (�)) at the time moment �. Respectively, the random process �2(�) = (�2�(�), �2� (�)) - the number of application with demand on �2 amount of resources (applications of the second type) on the servers (�2�(�)) and in the buffer (�2� (�)) at the time moment �. �(�) - the available at time � amount of system resources. If the state of the Markov process �2 is 1, then �(�) = max(0, �1 - �1�1(�)1 - �2�2(�)1), if the state of the Markov process �2 is 2 then �(�) = �2 = ∞. If the amount of the system resources �(�) at the moment of the new application arrival is less then �1 (for the first type application) or �2 (for the second type application) amount of resources needed in addition to service (i.e. �(�) < �1 or �(�) < �2), then the incoming application is lost. Also the accepted to the system applications may be dropped from the buffer due to the transition Markov chain �2 from state 2 with unlimited amount �2 of system resources to the state 1 with limited amount of resources �(�) = �1 - �1�1(�)1 - �2�2(�)1. In order to avoid downtime of servers it is supposed that the maximum value of system resources �1 < ∞ is sufficient for all servers to be occupied, that is �1 � � · max(�1, �2). The goal of this paper is to derive the system of equations for random process �(�) steady-state probability distribution. The main goals of the study as a whole are to obtain main time-probability characteristics of the system as for this general case (also for the case when the maximum values of system resources are finite, but different for all states of governing external Markov process), and for special cases of only one external governing Markov process. 3. The steady-state probability distribution. The system of equations (scalar form) The set � of states of the random process �(�) = {�1(�), �2(�), �(�), �1(�), �2(�)} may be presented as � = {(��; �� ), (��; �� ), �1(�� +�� ; �� +�� )|�2, �, �}. Here, �� and �� (0 � �� � �, �� � 0) are numbers of the first type applications on servers (��) and in the buffer (�� ); �� and �� (0 � �� � �, �� � 0) are numbers of the second type applications on servers (��) and in the buffer (�� ). It should be noted that 0 � �� + �� � �. The argument � = 1, 2 describes the state of the external Markov process �1 as well as the � = 1, 2 - the state of the Markov process �2. �1(�� + �� ; �� + �� ) = �1 - (�� + �� )�1 - (�� + �� )�2 - the current amount of the system resources in the state 1 of Markov process �2. In the case of the buffer of unlimited capacity, the entire set of states can be divided into 10 subsets corresponding to the following states: 1) the system is empty - the states {(0; 0), (0; 0), �1(0; 0), 1, 1}, {(0; 0), (0; 0), �1(0; 0), 2, 1}, {(0; 0), (0; 0), �2, 1, 2}, {(0; 0), (0; 0), �2, 2, 2}; 1. there are only applications of the first type in the system, not all servers are occupied, the buffer is empty - {(��; 0), (0; 0), �1(��; 0), 1, 1}, {(��; 0), (0; 0), �1(��; 0), 2, 1}, {(��; 0), (0; 0), �2, 1, 2}, {(��; 0), (0; 0), �2, 2, 2}, 1 � �� < �; 2. there are only applications of the first type in the system, all servers are occupied, the buffer is empty - {(�; 0), (0; 0), �1(�; 0), 1, 1}, {(�; 0), (0; 0), �1(�; 0), 2, 1}, {(�; 0), (0; 0), �2, 1, 2}, {(�; 0), (0; 0), �2, 2, 2}; 3. there are only applications of the first type in the system, all servers are occupied, the buffer is not empty - {(�; �� ), (0; 0), �1(� + �� ; 0), 1, 1}, {(�; �� ), (0; 0), �1(� + �� ; 0), 2, 1}, {(�; �� ), (0; 0), �2, 1, 2}, {(�; �� ), (0; 0), �2, 2, 2}, �� � 1; 4. there are only applications of the second type in the system, not all servers are occupied, the buffer is empty - {(0; 0), (��; 0), �1(0; ��), 1, 1}, {(0; 0), (��; 0), �1(0; ��), 2, 1}, {(0; 0), (��; 0), �2, 1, 2}, {(0; 0), (��; 0), �2, 2, 2}, 1 � �� < �; 5. there are only applications of the second type in the system, all servers are occupied, the buffer is empty - {(0; 0), (�; 0), �1(0; �), 1, 1}, {(0; 0), (�; 0), �1(0; �), 2, 1}, {(0; 0), (�; 0), �2, 1, 2}, {(0; 0), (�; 0), �2, 2, 2}; 6. there are only applications of the second type in the system, all servers are occupied, the buffer is not empty - {(0; 0), (�; �� ), �1(0; � + �� ), 1, 1}, {(0; 0), (�; �� ), �1(; � + �� ), 2, 1}, {(0; 0), (�; �� ), �2, 1, 2}, {(0; 0), (�; ��), �2, 2, 2}, �� � 1; 7. there are applications of both types in the system, not all servers are occupied, the buffer is empty - {(��; 0), (��; 0), �1(��; ��), 1, 1}, {(��; 0), (��; 0), �1(��; ��), 2, 1}, {(��; 0), (��; 0), �2, 1, 2}, {(��; 0), (��; 0), �2, 2, 2}, 1 � �� � � - 2, 1 � �� � � - 1 - ��; 8. there are applications of both types in the system, all servers are occupied, the buffer is empty - {(��; 0), (� - ��; 0), �1(��; � - ��), 1, 1}, {(��; 0), (� - ��; 0), �1(��; � - ��), 2, 1}, {(��; 0), � - ��; 0), �2, 1, 2}, {(��; 0), � - ��; 0), �2, 2, 2}, 1 � �� � � - 1; 9. there are applications of both types in the system, all servers are occupied, the buffer is not empty - {(��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 1, 1}, {(��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 2, 1}, {(��; �� ), (� - ��; �� ), �2, 1, 2}, {(��; �� ), (� - ��; �� ), �2, 2, 2}, 1 � �� � � - 1, �� + �� � 1. For the system with the buffer of finite size, three more groups of states will be introduced (the system is fully occupied by applications of only one type, the system is fully occupied by both type applications). Since the states in which the amount of resources requested by applications exceeds the amount of resources of the entire system are impossible (due to our assumptions), then conditional indicator function - the Kronecker symbol - is introduced: {︂1, �1 - (�� + �� )�1 - (�� + �� )�2 � 0, � (�1(�� + �� , �� + �� )) = 0, �1 - (�� + �� )�1 - (�� + �� )�2 (1) < 0. This indicator function will be used for the equations of transitions between the states of the groups (4), (6), (10) and for the transition from the states (3), (6) and (9) to the overlying states and for transitions from the overlying states to states of these groups. The first four equations consider the transition of the system from the zero state: (�1 + �1,2 + �1,2) � ((0; 0), (0; 0), �1(0; 0), 1, 1) = �1� ((1; 0), (0; 0), �1(1; 0), 1, 1) + + �1� ((0; 0), (1; 0), �1(0; 1), 1, 1) + �2,1� ((0; 0), (0; 0), �1(0; 0), 2, 1) + + �2,1� ((0; 0), (0; 0), �2, 1, 2) , (2) (�2 + �1,2 + �2,1) � ((0; 0), (0; 0), �1(0; 0), 2, 1) = �1� ((1; 0), (0; 0), �1(1; 0), 2, 1) + + �1� ((0; 0), (1; 0), �1(0; 1), 2, 1) + �1,2� ((0; 0), (0; 0), �1(0; 0), 1, 1) + + �2,1� ((0; 0), (0; 0), �2, 2, 2) , (3) (�1 + �2,1 + �1,2) � ((0; 0), (0; 0), �2, 1, 2) = �2� ((1; 0), (0; 0), �2, 1, 2) + + �2� ((0; 0), (1; 0), �2, 1, 2) + �2,1� ((0; 0), (0; 0), �2, 2, 2) + + �1,2� ((0; 0), (0; 0), �1(0; 0), 1, 1) , (4) (�2 + �2,1 + �2,1) � ((0; 0), (0; 0)�2, 2, 2) = �2� ((1; 0), (0; 0), �2, 2, 2) + + �2� ((0; 0), (1; 0), �2, 2, 2) + �1,2� ((0; 0), (0; 0), �2, 1, 2) + + �1,2� ((0; 0), (0; 0), �1(0; 0), 2, 1) . (5) Now consider the case where only the first type of application is present in the system and not all servers are occupied: (�1 + �1,2 + �1,2 + ���1) � ((��; 0), (0; 0), �1(��; 0), 1, 1) = = �1� ((�� - 1; 0), (0; 0), �1(�� - 1; 0), 1, 1) + �2,1� ((��; 0, )(0; 0), �1(��; 0), 2, 1) + + �2,1� ((��; 0), (0; 0), �2, 1, 2) + �1(�� + 1)� ((�� + 1; 0), (0; 0), �1(�� + 1; 0), 1, 1) + + �1� ((��; 0), (1; 0), �1(��; 1), 1, 1) , 1 � �� � � - 1, (6) (�2 + �2,1 + �1,2 + ���1) � ((��; 0), (0; 0), �1(��; 0), 2, 1) = = �1(�� + 1)� ((�� + 1; 0), (0; 0), �1(�� + 1; 0), 2, 1) + + �1� ((��; 0), (1; 0), �1(��; 1), 2, 1) + �1,2� ((��; 0), (0; 0), �1(��; 0), 1, 1) + + �2,1� ((��; 0), (0; 0), �2, 2, 2) , 1 � �� � � - 1, (7) (�1 + �1,2 + �2,1 + ���2) � ((��; 0), (0; 0), �2, 1, 2) = = �1� ((�� - 1; 0), (0; 0), �2, 1, 2) + �2,1� ((��; 0), (0; 0), �2, 2, 2) + + �1,2� ((��; 0), (0; 0), �1(��; 0), 1, 1) + �2(�� + 1)� ((�� + 1; 0), (0; 0), �2, 1, 2) + + �2� ((��; 0), (1; 0), �2, 1, 2) , 1 � �� � � - 1, (8) (�2 + �2,1 + �2,1 + ���2) � ((��; 0), (0; 0), �2, 2, 2) = �1,2� ((��; 0), (0; 0), �2, 1, 2) + + �1,2� ((��; 0), (0; 0), �1(��; 0), 2, 1) + �2(�� + 1)� ((�� + 1; 0), (0; 0), �2, 2, 2) + + �2� ((��; 0), (1; 0), �2, 2, 2) , 1 � �� � � - 1. (9) The system contains only applications of the first type, all servers are occupied, but the buffer is empty. According to the assumptions, the maximum amount of system resources is sufficient for all servers to be occupied, but it is not sufficient for arriving applications to occupy the buffer. Therefore, it is necessary to use the indicator function - verification of the existence of overlying states: (��1 + �1,2 + �1,2 + �1� (�1(� + 1, 0))) � ((�, 0), (0, 0), �1(�; 0), 1, 1) = = �1� ((� - 1, 0), (0, 0), �1(� - 1; 0), 1, 1) + �2,1� ((�, 0), (0, 0), �1(�; 0), 2, 1) + + �2,1� ((�, 0), (0, 0), �2, 1, 2) + �1� (�1(�; 1)) � ((� - 1, 1), (1, 0), �1(�; 1), 1, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(� + �1; �1))) � ((�, �), (0, �), �2, 1, 2) + �+�=1 �1 +�1 =1) + ��1� (�1(� + 1, 0)) � ((�, 1), (0, 0), �1(� + 1; 0), 1, 1) , (10) (��1 + �1,2 + �2,1 + �2� (�1(�, 1))) � ((�, 0), (0, 0), �1(�; 0), 2, 1) = = �1,2� ((�, 0)(0, 0), �1(�; 0), 1, 1) + �1� (�1(�, 1)) � ((� - 1, 1), (1, 0), �1(�; 1), 2, 1) + + �2,1� ((�, 0), (0, 0), �2, 2, 2) + ��1� (�1(� + 1, 0)) � ((�, 1), (0, 0), �1(� + 1; 0), 2, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(� + �1; �1))) � ((�, �), (0, �), �2, 2, 2) , (11) �+�=1 �1 +�1 =1) (��2 + �2,1 + �1,2 + �1) � ((�, 0), (0, 0), �2, 1, 2) = �1� ((� - 1, 0), (0, 0), �2, 1, 2) + �2,1� ((�, 0), (0, 0), �2, 2, 2) + �1,2� ((�, 0), (0, 0), �1(�; 0), 1, 1) + + �2� ((� - 1, 1), (1, 0), �2, 1, 2) + ��2� ((�, 1), (0, 0), �2, 1, 2) , (12) (��2 + �2,1 + �2,1 + �2) � ((�, 0), (0, 0), �2, 2, 2) = �1,2� ((�, 0), (0, 0), �2, 1, 2) + + �1,2� ((�, 0), (0, 0), �1(�; 0), 2, 1) + �2� ((� - 1, 1), (1, 0), �2, 2, 2) + + ��2� ((�, 1), (0, 0), �2, 2, 2) . (13) In the system there are only applications of the first type, all servers are occupied, there are also applications in the buffer. The indicator function is used to check the possibility of transition to (from) overlying states: (��1 + �1,2 + �1,2 + �1� (�1(� + �� + 1, 0))) � ((�, �� ), (0, 0), �1(� + �� ; 0), 1, 1) = = �1� ((�, �� - 1), (0, 0), �1(� + �� - 1; 0), 1, 1) + + �2,1� ((�, �� ), (0, 0), �1(� + �� ; 0), 2, 1) + �2,1� ((�, �� ), (0, 0), �2, 1, 2) + + �1� (�1(� + �� ; 1)) � ((� - 1, �� + 1)(1, 0), �1(� + �� ; 1), 1, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(� + �� + �1; �1))) � ((�, �� + �), (0, �), �2, 1, 2) + �+�=1 �1 +�1 =1) + ��1� (�1(� + �� + 1, 0)) � ((�, �� + 1), (0, 0), �1(� + �� + 1; 0), 1, 1) , �� � 1, (14) (��1 + �1,2 + �2,1 + �2� (�1(� + �� + 1, 0))) � ((�, �� )(0, 0), �1(� + �� ; 0), 2, 1) = = �1,2� ((�, �� ), (0, 0), �1(� + �� ; 0), 1, 1) + �2,1� ((�, �� ), (0, 0), �2, 2, 2) + + ��1� (�1(� + �� + 1, 0)) � ((�, �� + 1), (0, 0), �1(� + �� + 1; 0), 2, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(� + �� + �1; �1))) � ((�, �� + �), (0, �), �2, 2, 2) + �+�=1 �1 +�1 =1) + �1� (�1(� + �� ; 1)) � ((� - 1; �� + 1), (1, 0), �1(� + �� ; 1), 2, 1) , �� � 1, (15) (��2 + �2,1 + �1,2 + �1) � ((�, �� ), (0, 0), �2, 1, 2) = = �1� ((�, �� - 1), (0, 0), �2, 1, 2) + �1,2� ((�, �� ), (0, 0), �2, 1, 2) + + �1,2� (�1(� + �� ; 0)) � ((�, �� ), (0, 0), (�1(� + �� ; 0), 1, 1) + + �2� ((� - 1, �� + 1), (1, 0), �2, 1, 2) + + ��2� ((�, �� + 1), (0, 0), �2, 1, 2) , �� � 1, (16) (��2 + �2,1 + �2,1 + �2) � ((�, �� ), (0, 0), �2, 2, 2) = �2,1� ((�, �� ), (0, 0), �2, 2, 2) + + �1,2� (�1(� + �� ; 0)) � ((�, �� ), (0, 0), (�1(� + �� ; 0), 2, 1) + + �2� ((� - 1, �� + 1), (1, 0), �2, 2, 2) + ��2� ((�, �� + 1), (0, 0), �2, 2, 2) , �� � 1. (17) There are only application of the second type in the system, not all servers are occupied, the buffer is empty: (�1 + �1,2 + �1,2 + ���1) � ((0; 0), (��; 0), �1(0; ��), 1, 1) = = �1(�� + 1)� ((0; 0), (�� + 1; 0), �1(0; �� + 1), 1, 1) + + �1� ((1; 0), (��; 0), �1(1; ��), 1, 1) + �2,1� ((0; 0)(��; 0), �1(0; ��), 2, 1) + + �2,1� ((0; 0), (��; 0), �2, 1, 2) , 1 � �� � � - 1, (18) (�2 + �2,1 + �1,2 + ���1) � ((0; 0), (��; 0), �1(0; ��), 2, 1) = = �2� ((0; 0), (�� - 1; 0), �1(0; �� - 1), 2, 1) + �1� ((1; 0), (��; 0), �1(1; ��), 2, 1) + + �1(�� + 1)� ((0; 0), (�� + 1; 0), �1(0; �� + 1), 2, 1) + + �1,2� ((0; 0), (��; 0), �1(0; ��), 1, 1) + + �2,1� ((0; 0), (��; 0), �2, 2, 2) , 1 � �� � � - 1, (19) (�1 + �1,2 + �2,1 + ���2) � ((0; 0), (��; 0), �2, 1, 2) = = �2,1� ((0; 0), (��; 0), �2, 2, 2) + �1,2� ((0; 0), (��; 0), �1(0; ��), 1, 1) + + �2(�� + 1)� ((0; 0), (�� + 1; 0), �2, 1, 2) + + �2� ((1; 0), (��; 0), �2, 1, 2) , 1 � �� � � - 1, (20) (�2 + �2,1 + �2,1 + ���2) � ((0; 0), (��; 0), �2, 2, 2) = = �2� ((0; 0), (�� - 1; 0), �2, 2, 2) + �1,2� ((0; 0), (��; 0), �2, 1, 2) + + �1,2� ((0; 0), (��; 0), �1(0; ��), 2, 1) + �2(�� + 1)� ((0; 0), (�� + 1; 0), �2, 2, 2) + + �2� ((1; 0), (��; 0), �2, 2, 2) , 1 � �� � � - 1. (21) In the system there are only applications of the second type, all servers are occupied, but the buffer is empty. The indicator function is used to check the possibility of transition to (from) overlying states: (��1 + �1,2 + �1,2 + �1� (�1(1, �))) � ((0, 0), (�, 0), �1(0; �), 1, 1) = = �2,1� ((0, 0), (�, 0), �1(0; �), 2, 1) + + �2,1� ((0, 0), (�, 0), �2, 1, 2) + �1� (�1(1; �)) � (1, 0), ((� - 1, 1), �1(1; �), 1, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�1; � + �1))) � ((0, �), (�, �), �2, 1, 2) + �+�=1 �1 +�1 =1) + ��1� (�1(0, � + 1)) � ((0; 0), (�; 1), �1(0; � + 1), 1, 1) , (22) (��1 + �1,2 + �2,1 + �2� (�1(0, � + 1))) � ((0, 0), (�, 0), �1(0; �), 2, 1) = = �2� ((0, 0), (� - 1, 0), �1(0; � - 1), 1, 1) + �1,2� ((0, 0)(�, 0), �1(0; �), 1, 1) + + �1� (�1(1, �)) � (1, 0), ((� - 1, 1), �1(1; �), 2, 1) + �2,1� ((0, 0), (�, 0), �2, 2, 2) + + ��1� (�1(0, � + 1)) � ((0, 0), (�, 1), �1(0; � + 1), 2, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�1; � + �1))) � ((0, �), (�, �), �2, 2, 2) , (23) �+�=1 �1 +�1 =1) (��2 + �2,1 + �1,2 + �1) � ((0, 0), (�, 0), �2, 1, 2) = �2,1� ((0, 0), (�, 0), �2, 2, 2) + + �1,2� ((0, 0), (�, 0), �1(0; �), 1, 1) + �2� ((1, 0), (� - 1, 1), �2, 1, 2) + + ��2� ((0, 0), (�, 1), �2, 1, 2) , (24) (��2 + �2,1 + �2,1 + �2) � ((0, 0), (�, 0), �2, 2, 2) = �2� ((0, 0), (� - 1, 0), �2, 1, 2) + + �1,2� ((0, 0), (�, 0), �2, 1, 2) + �1,2� ((0, 0), (�, 0), �1(0; �), 2, 1) + + �2� ((1, 0), (� - 1, 1), �2, 2, 2) + ��2� ((0, 0), (�, 1), �2, 2, 2) . (25) In the system there are only applications of the second type, all servers are occupied, the buffer is not empty. The indicator function is used to check the possibility of transition to (from) overlying states: (��1 + �1,2 + �1,2 + �1� (�1(1, � + �� ))) � ((0, 0), (�, �� ), �1(0; � + �� ), 1, 1) = = �2,1� ((0, 0), (�, �� ), �1(0; � + �� ), 2, 1) + �2,1� ((0, 0, (�, �� ), �2, 1, 2) + + �1� (�1(1; � + �� )) � ((1, 0)(� - 1, �� + 1), �1(1; � + �� ), 1, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�1; � + �� + �1))) � ((0, �), (�, �� + �), �2, 1, 2) + �+�=1 �1 +�1 =1) + ��1� (�1(0, � + �� + 1)) � ((0, 0), (�, �� + 1), �1(0; � + �� + 1), 1, 1) , �� � 1, (26) (��1 + �1,2 + �2,1 + �2� (�1(0; � + �� + 1))) � ((0, 0)(�, �� ), �1(0; � + �� ), 2, 1) = = �2� ((0, 0), (�, �� - 1), �1(0; � + �� - 1), 1, 1) + + �1,2� ((0, 0), (�, �� ), �1(0; � + �� ), 1, 1) + �2,1� ((0, 0), (�, �� ), �2, 2, 2) + + ��1� (�1(0, � + �� + 1)) � ((0, 0), (�, �� + 1), �1(0; � + �� + 1), 2, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�1; � + �� + �1))) � ((0, �), (�, �� + �), �2, 2, 2) + �+�=1 �1 +�1 =1) + �1� (�1(1; � + �� )) � ((1, 0), (� - 1; �� + 1), �1(1; � + �� ), 2, 1) , �� � 1, (27) (��2 + �2,1 + �1,2 + �1) � ((0, 0), (�, �� ), �2, 1, 2) = �1,2� ((0, 0), (�, �� ), �2, 1, 2) + + �1,2� (�1(0; � + �� )) � ((0, 0), (�, �� ), (�1(0; � + �� ), 1, 1) + + �2� ((1, 0), (� - 1, �� + 1), �2, 1, 2) + + ��2� ((0, 0), (�, �� + 1), �2, 1, 2) , �� � 1, (28) (��2 + �2,1 + �2,1 + �2) � ((0, 0), (�, �� ), �2, 2, 2) = �2� ((0, 0), (�, �� - 1), �2, 1, 2) + + �2� ((1, 0), (� - 1, �� + 1), �2, 2, 2) + �2,1� (0, 0), ((�, �� ), �2, 2, 2) + + �1,2� (�1(0; � + �� )) � ((0, 0), (�, �� ), (�1(0; � + �� ), 2, 1) + + ��2� ((0, 0), (�, �� + 1), �2, 2, 2) , �� � 1. (29) The applications of both types are in the system, but only some (not all) servers are occupied: (�1 + �1,2 + �1,2 + (�� + ��)�1) � ((��, 0), (��, 0), �1(��; ��), 1, 1) = = �1� ((�� - 1, 0), (��, 0), �1(�� - 1; ��), 1, 1) + �2,1� ((��, 0), (��, 0), �1(��; ��), 2, 1) + + �2,1� ((��, 0), (��, 0), �2, 1, 2) + (�� + 1)�1� ((�� + 1, 0), (��, 0), �1(�� + 1; ��), 1, 1) + + (�� + 1)�1� ((��, 0), (�� + 1, 0), �1(��; �� + 1), 1, 1) , �� = 1, � - 2, �� = 1, � - 1 - ��, (30) (�2 + �1,2 + �2,1 + (�� + ��)�1) � ((��, 0), (��, 0), �1(��; ��), 2, 1) = = �2� ((��, 0), (�� - 1, 0), �1(��; �� - 1), 2, 1) + �1,2� ((��, 0), (��, 0), �1(��; ��), 1, 1) + + �2,1� ((��, 0), (��, 0), �2, 2, 2) + (�� + 1)�1� ((�� + 1, 0), (��, 0), �1(�� + 1; ��), 2, 1) + + (�� + 1)�1� ((��, 0), (�� + 1, 0), �1(��; �� + 1), 2, 1) , �� = 1, � - 2, �� = 1, � - 1 - ��, (31) (�1 + �2,1 + �1,2 + (�� + ��)�2) � ((��, 0), (��, 0), �2, 1, 2) = = �1� ((�� - 1, 0), (��, 0), �2, 1, 2) + �2,1� ((��, 0), (��, 0), �2, 2, 2) + + �1,2� ((��, 0), (��, 0), �1(��; ��), 1, 1) + (�� + 1)�2� ((�� + 1, 0), (��, 0), �2, 1, 2) + + (�� + 1)�2� ((��, 0), (�� + 1, 0), �2, 1, 2) , �� = 1, � - 2, �� = 1, � - 1 - ��, (32) (�2 + �2,1 + �2,1 + (�� + ��)�2) � ((��, 0), (��, 0), �2, 2, 2) = = �2� ((��, 0), (�� - 1, 0), �2, 2, 2) + �1,2� ((��, 0), (��, 0), �2, 1, 2) + + �1,2� ((��, 0), (��, 0)�1(��; ��), 2, 1) + (�� + 1)�2� ((�� + 1, 0), (��, 0), �2, 2, 2) + + (�� + 1)�2� ((��, 0), (�� + 1, 0), �2, 2, 2) , �� = 1, � - 2, �� = 1, � - 1 - ��. (33) The application of the first and the second types are in the system, all servers are occupied, but the buffer is empty: (��1 + �1,2 + �1,2 + �1� (�1(�� + 1, � - ��))) � ((��, 0), (� - ��, 0), �1(��; � - ��), 1, 1) = = �1� ((�� - 1, 0), (� - ��, 0), �1(�� - 1; � - ��), 1, 1) + + �2,1� ((��, 0), (� - ��, 0), �1(��; � - ��), 2, 1) + �2,1� ((��, 0), (� - ��, 0), �2, 1, 2) + + ���1� (�1(�� + 1; � - ��)) � ((��, 1), (� - ��, 0), �1(�� + 1; � - ��), 1, 1) + + (� - ��)�1� (�1(��; � - �� + 1)) � ((��, 0), (� - ��, 1), �1(��; � - �� + 1), 1, 1) + +(�-��+1)�1� (�1(��; � - �� + 1)) � ((�� - 1, 1), (� - �� + 1, 0), �1(��; � - �� + 1), 1, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�� + �1; � - �� + �1))) � ((��, �), (� - ��, �), �2, 1, 2) , �+�=1 �1 +�1 =1) �� = 1, � - 1, �� = � - ��, (34) (��1 + �1,2 + �2,1 + �2� (�1(��, � - �� + 1))) � ((��, 0), (� - ��, 0), �1(��; � - ��), 2, 1) = = �2� ((��, 0), (� - �� - 1, 0), �1(��; � - �� - 1), 2, 1) + + �1,2� ((��, 0), (� - ��, 0), �1(��; � - ��), 1, 1) + �2,1� ((��, 0), (� - ��, 0), �2, 2, 2) + + ���1� (�1(�� + 1; � - ��)) � ((��, 1), (� - ��, 0), �1(�� + 1; � - ��), 2, 1) + +(�-��+1)�1� (�1(��; � - �� + 1)) � ((�� - 1, 1), (� - �� + 1, 0), �1(��; � - �� + 1), 2, 1) + + (� - ��)�1� (�1(��; � - �� + 1)) � ((��, 0), (� - ��, 1), �1(��; � - �� + 1), 2, 1) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�� + �1; � - �� + �1))) � ((��, �), (� - ��, �), �2, 2, 2) , �+�=1 �1 +�1 =1 �� = 1, � - 1, �� = � - ��, (35) (��2 + �2,1 + �1,2 + �1) � ((��, 0), (� - ��, 0), �2, 1, 2) = = �1� ((�� - 1, 0), (� - ��, 0), �2, 1, 2) + �2,1� ((��, 0), (� - ��, 0), �2, 2, 2) + + �1,2� ((��, 0), (� - ��, 0), �1(��; ��), 1, 1) + (� - ��)�2� ((��, 0), (� - ��, 1), �2, 1, 2) + + ���2� ((��, 1), (� - ��, 0), �2, 1, 2)+(�-�� + 1)�2� ((�� - 1, 1), (� - �� + 1, 0), �2, 1, 1) , �� = 1, � - 1, �� = � - ��, (36) (��2 + �2,1 + �2,1 + �2) � ((��, 0), (� - ��, 0), �2, 2, 2) = = �2� ((��, 0), (� - �� - 1, 0), �2, 2, 2) + �1,2� ((��, 0), (� - ��, 0), �2, 1, 2) + + �1,2� ((��, 0), (� - ��, 0), �1(��; ��), 2, 1) + (� - ��)�2� ((��, 0), (� - ��, 1), �2, 2, 2) + +(�-�� + 1)�2� ((�� - 1, 1), (� - �� + 1, 0), �2, 2, 2)+ ���2� ((��, 1), (� - ��, 0), �2, 2, 2) , �� = 1, � - 1, �� = � - ��. (37) The equations for the case when both types of applications are in the system (on servers and in the buffer): (�1� (�1(�� + �� + 1, � - �� + �� )) + �1,2 + �1,2 + ��1) × × � ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 1, 1) = = �1� ((��; �� - 1), (� - ��; �� ), �1(�� + �� - 1; � - �� + �� ), 1, 1) + + �2,1� ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 2, 1) + + �2,1� ((��; �� ), (� - ��; �� ), �2, 1, 2) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�� + �� + �1; � - �� + �� + �1))) × �+�=1 �1 +�1 =1 × � ((��, �� + �), (� - ��, �� + �), �2, 1, 2) + ���1�1� (�1(�� + �� + 1; � - �� + �� )) × × � ((��; �� + 1), (� - ��; �� ), �1(�� + �� + 1; � - �� + �� ), 1, 1) + + (� - ��)�2�1� (�1(�� + �� ; � - �� + �� + 1)) × × � ((��; �� ), (� - ��; �� + 1), �1(�� + �� ; � - �� + �� + 1), 1, 1) + + (�� + 1)�2�1� (�1(�� + �� + 1; � - �� + �� )) × × � ((�� + 1; �� ), (� - �� - 1; �� + 1), �1(�� + �� + 1; � - �� + �� ), 1, 1) + + (� - �� + 1)�1�1� (�1(�� + �� ; � - �� + �� + 1)) × × � ((�� - 1; �� + 1), (� - �� + 1; �� ), �1(�� + �� ; � - �� + �� + 1), 1, 1) , �� = 1, � - 1, �� + �� � 1, (38) (�2� (�1(�� + �� , � - �� + �� + 1)) + �2,1 + �1,2 + ��1) × × � ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 2, 1) = = �2� ((��; �� ), (� - ��; �� - 1), �1(�� + �� ; � - �� + �� - 1), 2, 1) + + �1,2� ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 1, 1) + + �2,1� ((��; �� ), (� - ��; �� ), �2, 2, 2) + ∞ + �2,1 ∑︁ �+� ∏︁ (1 - � (�1(�� + �� + �1; � - �� + �� + �1))) × �+�=1 �1 +�1 =1 × � ((��, �� + �), (� - ��, �� + �), �2, 2, 2) + ���1�1� (�1(�� + �� + 1; � - �� + �� )) × × � ((��; �� + 1), (� - ��; �� ), �1(�� + �� + 1; � - �� + �� ), 2, 1) + + (� - ��)�2�1� (�1(�� + �� ; � - �� + �� + 1)) × × � ((��; �� ), (� - ��; �� + 1), �1(�� + �� ; � - �� + �� + 1), 2, 1) + + (�� + 1)�2�1� (�1(�� + �� + 1; � - �� + �� )) × × � ((�� + 1; �� ), (� - �� - 1; �� + 1), �1(�� + �� + 1; � - �� + �� ), 2, 1) + + (� - �� + 1)�1�1� (�1(�� + �� ; � - �� + �� + 1)) × × � ((�� - 1; �� + 1), (� - �� + 1; �� ), �1(�� + �� ; � - �� + �� + 1), 2, 1) , �� = 1, � - 1, �� + �� � 1, (39) (�1 + �1,2 + �1,2 + ��2) � ((��; �� ), (� - ��; �� ), �2, 1, 2) = = �1� ((��; �� - 1), (� - ��; �� ), �2, 1, 2) + �2,1� ((��; �� ), (� - ��; �� ), �2, 2, 2) + + �1,2� (�1(�� + �� ; � - �� + �� )) � ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 1, 1) + + ���1�2� ((��; �� + 1), (� - ��; �� ), �2, 1, 2) + + (� - ��)�2�2� ((��; �� ), (� - ��; �� + 1), �2, 1, 2) + + (�� + 1)�2�2� ((�� + 1; �� ), (� - �� - 1; �� + 1), �2, 1, 2) + + (� - �� + 1)�1�2� ((�� - 1; �� + 1), (� - �� + 1; �� ), �2, 1, 2) , �� = 1, � - 1, �� + �� � 1, (40) (�2 + �2,1 + �1,2 + ��2) � ((��; �� ), (� - ��; �� ), �2, 2, 2) = = �2� ((��; �� ), (� - ��; �� - 1), �2, 2, 2) + �1,2� ((��; �� ), (� - ��; �� ), �2, 1, 2) + + �1,2� (�1(�� + �� ; � - �� + �� )) � ((��; �� ), (� - ��; �� ), �1(�� + �� ; � - �� + �� ), 2, 1) + + ���1�2� ((��; �� + 1), (� - ��; �� ), �2, 2, 2) + + (� - ��)�2�2� ((��; �� ), (� - ��; �� + 1), �2, 2, 2) + + (�� + 1)�2�2� ((�� + 1; �� ), (� - �� - 1; �� + 1), �2, 2, 2) + + (� - �� + 1)�1�2� ((�� - 1; �� + 1), (� - �� + 1; �� ), �2, 2, 2) , �� = 1, � - 1, �� + �� � 1. (41) Here �1 - the probability that the first type application is taken from the buffer, �2 - the probability that the second type application is taken from the buffer. 4. Conclusions The mathematical model of the system with the allocation of resources to incoming applications and functioning in the random environment is constructed. The system of equations for steady-state probability distribution of the random process, which describes the functioning of the system, is present. The main task of future research is to present this system of equations in a matrix form and try to apply the well known matrix algorithms [6, 7, 31-33] in order to obtain the steady-state probability distribution in the analytical form. Also of interest are stationary distributions of applications of each type, the average value of the system resources, the average number of discarded (lost) applications.

×

Об авторах

Иван Сергеевич Зарядов

Российский университет дружбы народов; Институт проблем информатики Федеральный исследовательский центр «Информатика и управление» РАН

Автор, ответственный за переписку.
Email: zaryadov-is@rudn.ru

кандидат физико-математических наук, доцент кафедры прикладной информатики и теории вероятностей РУДН, старший научный сотрудник ИПИ ФИЦ ИУ РАН

ул. Миклухо-Маклая, д.6, Москва, Россия, 117198; ул. Вавилова, д. 44, кор. 2, Москва, Россия, 119333

Владимир Владимирович Цурлуков

Российский университет дружбы народов

Email: dober.vvt@gmail.com

магистр кафедры прикладной информатики и теории вероятностей РУДН

ул. Миклухо-Маклая, д.6, Москва, Россия, 117198

Кравид Виана Карвалью

Российский университет дружбы народов

Email: hilvianamat1@gmail.com

магистр кафедры прикладной информатики и теории вероятностей РУДН

ул. Миклухо-Маклая, д.6, Москва, Россия, 117198

Анна Андреевна Зайцева

Российский университет дружбы народов

Email: anna-z96@mail.ru

магистр кафедры прикладной информатики и теории вероятностей РУДН

ул. Миклухо-Маклая, д.6, Москва, Россия, 117198

Татьяна Александровна Милованова

Российский университет дружбы народов

Email: milovanova-ta@rudn.ru

кандидат физико-математических наук, старший преподаватель кафедры прикладной информатики и теории вероятностей РУДН

ул. Миклухо-Маклая, д.6, Москва, Россия, 117198

Список литературы

  1. Queueing Theory / P. P. Bocharov, C. D’Apice, A. V. Pechinkin, S. Salerno. — Utrecht, Boston: VSP, 2004. — 446 p.
  2. Basharin G. P., Gaidamaka Yu. V., Samouylov K. E. Mathematical Theory of Teletraffic and its Application to the Analysis of Multiservice Communication of Next Generation Networks // Automatic Control and Computer Sciences. — 2013. — Vol. 47, No 2. — Pp. 62–69. — doi: 10.3103/S0146411613020028.
  3. Trivedi Kishor S. Probability and Statistics with Reliability, Queuing, and Computer Science Applications, Second Edition. — John Wiley & Sons, 2016. — 830 p.
  4. Chee-Hoc N., Boon-Hee S. Queueing Modelling Fundamentals with Applications in Communication Networks. 2nd Edition. — John Wiley & Sons, 2008. — 292 p.
  5. Neuts M. F. A Versatile Markovian Point Process // Journal of Applied Probability. — 1979. — Vol. 16, No 4. — Pp. 764–779. — doi: 10.2307/3213143.
  6. Neuts M. F. Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. — Baltimore: Johns Hopkins University Press, 1981. — 352 p.
  7. Neuts M. F. Matrix-Analytic Methods in Queuing Theory // European Journal of Operational Research. — 1984. — Vol. 15, No 1. — Pp. 2–12. — doi: 10.1016/03772217(84)90034-1.
  8. Neuts M. F. Structured Stochastic Matrices of M/G/1 Type and Their Applications. — New York: Marcel Dekker Inc., 1989. — 512 p.
  9. Fisher W., Meier-Hellstern K. S. The Markov-Modulated Poisson Process (MMPP) Cookbook // Performance Evaluation. — 1993. — Vol. 18, No 2. — Pp. 149–171. — doi: 10.1016/0166-5316(93)90035-S.
  10. Andersen A., Nielsen B. A Markovian Approach for Modelling Packet Traffic with Long-Range Dependence // IEEE Journal on Selected Areas in Communications. — 1998. — Vol. 16, No 5. — Pp. 719–732. — doi: 10.1109/49.700908.
  11. Markov Models of Internet Traffic and a New Hierarchical MMPP Model / L. Muscariello, M. Mellia, M. Meo et al. // Computer Communications. — 2005. — Vol. 28, No 16. — Pp. 1835–1852. — doi: 10.1016/j.comcom.2005.02.012.
  12. Andronov A. M., Vishnevsky V. M. Markov-Modulated Continuous Time Finite Markov chain as the Model of Hybrid Wireless Communication Channels Operation // Automatic Control and Computer Sciences. — 2016. — Vol. 50, No 3. — Pp. 125–132. — doi: 10.3103/S0146411616030020.
  13. Reliability-Centric Analysis of Offloaded Computation in Cooperative Wearable Applications / A. Ometov, D. Kozyrev, V. Rykov et al. // Wireless Communications and Mobile Computing. — 2017. — Vol. 2017. — P. 9625687. — doi: 10.1155/2017/9625687.
  14. Rykov V., Kozyrev D. Analysis of Renewable Reliability Systems by Markovization Method // Lecture Notes in Computer Science / Analytical and Computational Methods in Probability Theory, ACMPT 2017. — Germany, Heidelberg, SpringerVerlag, 2017. — Pp. 210–220. — doi: 10.1007/978-3-319-71504-9 19.
  15. Rykov V., Kozyrev D., Zaripova E. Modeling and Simulation of Reliability Function of a Homogeneous Hot Double Redundant Repairable System // ECMS 2017 Proceedings / European Council for Modeling and Simulation. — Budapest, Hungary, May 23–26: European Council for Modelling and Simulation, 2017. — Pp. 701–705. — doi: 10.7148/2017-0701.
  16. Pechinkin A. P., Razumchik R. Approach for Analysis of Finite M2—M2—1—R with Hysteric Policy for SIP Server Hop-by-Hop Overload Control // Proceedings – 27th European Conference on Modelling and Simulation, ECMS 2013 / European Council for Modelling and Simulation. — Alesund, Norway, May 27–30, 2013: European Council for Modelling and Simulation, 2013. — Pp. 573–579. — doi: 10.7148/20130573.
  17. Razumchik R. Analysis of Finite MAP|PH|1 Queue with Hysteretic Control of Arrivals // International Congress on Ultra Modern Telecommunications and Control Systems and Workshops, ICUMI-2016 / IEEE Computer Society. — Lisbon, Portugal, October 18–20: IEEE Computer Society, 2016. — Pp. 288–293. — doi: 10.1109/ICUMT.2016.7765373.
  18. Razumchik R., Telek M. Delay Analysis of a Queue with Re-sequencing Buffer and Markov Environment // Queueing Systems. — 2016. — Vol. 82, No 1–2. — Pp. 7–28. — doi: 10.1007/s11134-015-9444-z.
  19. Razumchik R., Telek M. Delay Analysis of Resequencing Buffer in Markov Environment with HOQ-FIFO-LIFO Policy // Lecture Notes in Computer Science / 14th European Workshop on Computer Performance Engineering, EPEW 2017. — Vol. 10497. — Berlin, Germany, September 7–8: Springer Verlag, 2017. — Pp. 53–68. — doi: 10.1007/978-3-319-66583-2 4.
  20. Retrial Tandem Queue with BMAP-input and Semi-Markovian Service Process / V. Klimenok, O. Dudina, V. Vishnevsky, K. Samouylov // Communications in Computer and Information Science, vol. 700 / 20th International Conference on Distributed Computer and Communication Networks, DCCN 2017. — Moscow, Russian Federation, September 25–29: Springer Verlag, 2017. — Pp. 159–173. — doi: 10.1007/978-3-319-66836-9 14.
  21. Analysis of a Retrial Queue with Limited Processor Sharing Operating in the Random Environment / S. Dudin, A. Dudin, O. Dudina, K. Samouylov // Lecture Notes in Computer Science, LNCS / 15th International Conference on Wired/Wireless Internet Communications, WWIC 2017. — Vol. 10372. — St. Petersburg, Russian Federation, June 21–23: Springer Verlag, 2017. — Pp. 38–49. — doi: 10.1007/978-3-319-613826 4.
  22. The Survey on Markov-Modulated Arrival Processes and Their Application to the Analysis of Active Queue Management Algorithms / I. Zaryadov, A. Korolkova, D. Kulyabov et al. // Communications in Computer and Information Science / 20th International Conference on Distributed Computer and Communication Networks, DCCN 2017. — Vol. 700. — Moscow, Russian Federation, September 25–29: Springer Verlag, 2017. — Pp. 417–430. — doi: 10.1007/978-3-319-66836-9 35.
  23. LTE Performance Analysis Using Queuing Systems with Finite Resources and Random Requirements / V. Naumov, K. Samouylov, N. Yarkina et al. // 7th International Congress on Ultra Modern Telecommunications and Control Systems ICUMT-2015 / IEEE Computer Society. — Brno, Czech Republic, October 6–8: IEEE Computer Society, 2015. — Pp. 100–103. — doi: 10.1109/ICUMT.2015.7382412.
  24. Convolution Algorithm for Normalization Constant Evaluation in Queuing System with Random Requirements / K. Samouylov, E. Sopin, O. Vikhrova, S. Shorgin // AIP Conference Proceedings / International Conference of Numerical Analysis and Applied Mathematics 2016, ICNAAM 2016. — Vol. 1863. — Rodos Palace HotelRhodes, Greece, September 19–25: American Institute of Physics Inc., 2017. — P. 090004. — doi: 10.1063/1.4992269.
  25. Naumov V. A., Samuilov K. E., Samuilov A. K. On the Total Amount of Resources Occupied by Serviced Customers // Automation and Remote Control. — 2016. — Vol. 77, No 8. — Pp. 1419–1427. — doi: 10.1134/S0005117916080087.
  26. Samouylov K. E., Sopin E. S., Shorgin S. Ya. Queuing Systems with Resources and Signals and Their Application for Performance Evaluation of Wireless Networks // Informatika i ee Primeneniya. — 2017. — Vol. 11, No 3. — Pp. 99–105. — doi: 10.14357/19922264170311.
  27. Sopin E., Vikhrova O., Samouylov K. LTE Network Model with Signals and Random Resource Requirements // 9th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops. — Munich, Germany, November 6–8: IEEE Computer Society, 2017. — Pp. 101–106. — doi: 10.1109/ICUMT.2017.8255155.
  28. Naumov V., Samouylov K. Analysis of Multi-resource Loss System with State-dependent Arrival and Service Rates // Probability in the Engineering and Informational Sciences. — 2017. — Vol. 31, No 4. — Pp. 413–419. — doi: 10.1017/S0269964817000079.
  29. Sopin E. S., Vikhrova O. G. Probability Characteristics Evaluation in Queueing System with Random Requirements // CEUR Workshop Proceedings, vol. 1995 / 7th International Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”, ITTMM 2017. — Peoples’ Friendship University of Russia (RUDN University) Moscow, Russian Federation: CEUR-WS, 2017. — Pp. 85–92.
  30. Tsurlukov V. V. The Mathematical Model of the System with Resources and Random Environment // Proceedings of 2nd International School on Applied Probability Theory and Communications Technologies, Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation, October 23—27 / 2nd International School on Applied Probability Theory and Communications Technologies. — Moscow: RUDN University, 2017. — С. 318–320.
  31. Chakravarthy S., Alfa A. S. Matrix-Analytic Methods in Stochastic Models. — CRC Press, 1996. — P. 396.
  32. Breuer L., Baum D. An Introduction to Queueing Theory and Matrix-Analytic Methods. — Dordrecht: Springer Netherlands, 2005. — P. 286.
  33. Qi-Ming H. Fundamentals of Matrix-Analytic Methods. — New-York: Springer-Verlag New York, 2014. — P. 364.

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© Зарядов И.С., Цурлуков В.В., Карвалью К.В., Зайцева А.А., Милованова Т.А., 2018

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