Towards the Analysis of the Queuing System Operating in the Random Environment with Resource Allocation

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Abstract


The mathematical model of the system, that consists of a storage device and several homogeneous servers and operates in a random environment, and 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 incoming flow is a Poisson one, the rate of the flow and the amount of resources required for the application are determined by the state of the external Markov process. The service time for applications on servers is exponential distributed. The service rate and the maximum amount of system resources are determined by the state of the second external Markov process. When the application leaves the system, its resources are returned to the system. In the system under consideration, there may be failures in accepting incoming applications due to a lack of resources, as well as loss of the applications already accepted in the system, when the state of the external Markov process controlling the service and provision of resources changes. A random process describing the functioning of this system is constructed. The system of equations for the stationary probability distribution of the constructed random process is presented in scalar form. The main tasks for further research are formulated.


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.

Ivan S Zaryadov

Peoples’ Friendship University of Russia (RUDN University); Institute of Informatics Problems, FRC CSC RAS

Author for correspondence.
Email: zaryadov-is@rudn.ru
IPI FRC CSC RAS, 44-2 Vavilova Str., Moscow 119333, Russian Federation

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Senior Researcher of Institute of Informatics Problems of Federal Research Center “Computer Science and Control” Russian Academy of Sciences

Vladimir V Tsurlukov

Peoples’ Friendship University of Russia (RUDN University)

Email: dober.vvt@gmail.com
6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

Cravid Viana Carvalho

Peoples’ Friendship University of Russia (RUDN University)

Email: hilvianamat1@gmail.com
6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

Anna A Zaytseva

Peoples’ Friendship University of Russia (RUDN University)

Email: anna-z96@mail.ru
6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

Tatiana A Milovanova

Peoples’ Friendship University of Russia (RUDN University)

Email: milovanova-ta@rudn.ru
6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation

Candidate of Physical and Mathematical Sciences, lecturer of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

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