A Markov model of reliability of a fault-tolerant cluster has been considered, using virtualization technologies that ensure the continuity of the computational process in the event of a failure of the servers` physical resources and the impossibility of recovering from the interruption of the computational process. The probability of maintaining the system's operability under the condition of ensuring the continuity of the computing process for di erent service organization options was anilized. The mean time time to failure of such sistems was found. The purpose of the work is to increase the functional reliability of computing systems of a cluster architecture while increasing the time to failure, taking into account the requirements for ensuring the continuity of the computing process. A fault tolerance is considered as an object of study. A virtual machine is running on the cluster. The system involves launching a shadow copy of the VM on the backup server, which al-lows after the failure of the primary server to continue its implementation on the backup server.The proposed models can be used to assess the level of system reliability and are important in choosing a system con guration for certain conditions. Assessing the migration of virtual machines in the event of a failure of physical servers will allow you to calculate and evaluate the possible damage when using various models.
For cluster computing systems, especially real-time, the key is to ensure
reliability and fault tolerance while maintaining the continuity of the computing
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process. The achievement of high and stable performance indicators, reliability,
fault tolerance [
Cluster fault tolerance technology with continuous
computing
One of the e ective ways to achieve fault tolerance of computing systems and
processes is the migration of virtual resources between the physical nodes (servers)
of a computing system of a cluster architecture. In a cluster with replication
of virtual machines (VMs) on di erent physical nodes, they can migrate
between cluster nodes in the event of failure of physical resources without stopping
calculations on servers [
Virtualization allows to optimize the use of computing resources, increases
the scalability, fault tolerance and extensibility of the infrastructure, due to the
rapid redistribution of the virtual resource [
Recovery time during VM migration after failures depends on the structure
of the data storage.With shared storage for all physical nodes of the cluster, only
RAM, virtual processor registers, and VM virtual device states are transferred
during migration [
Fault tolerance ensures the continuity of the computing process (service) in
the cluster after the failure of one of the physical servers with the support of two
copies of the VM, which, in RAM, are located on di erent physical servers, so
that in case of failure of one of them, continue to work on the second. During the
functioning of the VM on the main servers, the backup copy must support the
actual copy of the RAM [
The purpose of the work is to increase the functional reliability of computing systems of a cluster architecture while increasing the time to failure, taking into account the requirements for ensuring the continuity of the computing process.
By functional reliability, we mean the ability of systems to perform the required functions, taking into account not only the operability of the resources required for their implementation, but also ensuring the necessary conditions for their implementation. Requirements to ensure the continuity of the computational process in the inadmissibility of interruptions of the reservation system at the time of recovery are proposed as conditions of operation. Thus, in the systems under consideration, recovery is possible only if it is combined with the implementation of the required functions by non-failed nodes. The system enters a non-recoverable state if it is impossible to recon gure with the activation of the required number of operability resources. For such systems, the reliability indicator is the time to failure, including taking into account violations of the continuity of the computing process, provided that the permissible recon guration time, including the costs of migration of virtual machines, is exceeded. 3
Cluster organization and options for its recovery The cluster architecture computer system contains servers (Fig.1). Each server is connected directly to one local storage device (local server storage device). In the system to ensure automatic recon guration, aimed at supporting the continuity of computing processes based on dynamic migration, pairs of physical servers of the primary and secondary are allocated in the cluster. The main server performs the required tasks critical to the continuity of the computing process. The backup server is designed to perform dynamic recon guration with ensuring the continuity of the computing process in case of possible failures of the primary server. The backup server, in addition to implementing dynamic system recon guration, performs some background tasks that are not critical to the continuity of the computational process and to the time of query execution). If the backup server fails, the background tasks that it performs may be lost or redistributed to the main server if they are performed non-priority in the background.
With the simplest implementation of a fault tolerance cluster, it is equipped with a pair of servers, one of which is designated as the main, and the second as the backup.
Fault tolerance technology involves launching a backup copy of the primary server VM on the backup server and transferring the calculations to the backup server in case of primary server or storage device failure.
Consider system options that provide (option A) and do not provide (option B) the restoration of physical nodes for states in which the continuity of the computing process is ensured during recon guration of the cluster, which allows us to select a working server and associated storage device for the implementation of the calculations.
For the options under consideration, in the event of a transition to a failure state with the impossibility of implementing the required functions at least with a minimal workable con guration, it is considered that the computational process is interrupted for a time exceeding the maximum permissible value, which entails a transition to a state of unrecoverable failure.
Let us consider cluster systems while ensuring fault-tolerant functioning with pairwise integration of physical servers into duplicated systems supporting the processes of virtual machine migration and data replication. For each pair of pairs in the cluster interacting to support dynamic recon guration of the servers (duplicated system), state and transition diagrams for a variant of organization A and B of a duplicated cluster system with recovery disciplines are shown in
Calculation of the probability of operability of duplicated systems The presented systems of di erential equations make it possible to determine the dependence of the probabilities of all states from time.
The probability of the system working while maintaining the continuity of the computing process for option A and B is de ned as: and for option B is de ned as:
P (t) = P (t) =
4 X Pi(t); i=0 3 X Pi(t): i=0
Fig. 1. Cluster model.
The results of calculating the probability of duplicated computer systems` operability provided that the computing process is continuous for options A (the curve 1) and B (the curve 2) of the maintaining process` organization are presented in Fig. 3.
The calculations were performed with the following failure rates 0 = 1:115 10 5 (1=h) , 1 = 3:425 10 6 (1=h), 2 = 2:3 10 6 (1=h), and recovery rates 0 = 0:33 (1=h) , 1 = 0:17 (1=h) , 2 = 0:33 (1=h) , 3 = 1 (1=h).
The presented dependences make it possible to evaluate the e ect on the probability of maintaining the operability of a duplicated system, restrictions on the inadmissibility of interruption of the computational process, and the impact of restoration work while maintaining the possibility of continuity of the process of performing the required functions. 5
Calculation of the probability of operability of
duplicated systems
The mean time between failures and the probability of working without failures
are related by the relation [
P (t)dt:
The average operating time to failure in accordance with the methodology of
[
For the systems under study, integrating the left and right sides of the
systems of equations (1), (2) for the models under consideration. Given that in the
presence of an absorbing state, Pi(1) = 0, we have [
Where Ti is the average time spent in working condition i when starting work
from a operable state. Mean time to failure is determined by summing Ti, for
all operational states [
T =
X Ti:
For the system under consideration, the time to failure with service discipline A : T1 = 3:891 105 hours, and B T2 = 3; 277 103 hours. 6
Conclusions The signi cance of the impact of ensuring the continuity of the computational process on duplicated systems of the cluster architecture is demonstrated. The result of the study was obtained on the basis of Markov models of reliability of a fault-tolerant cluster with the migration of virtual machines when it is impossible to recover after the interruption of the computational process. The time to the rst failure of a duplicated system with recovery and without recovery in failure states of nodes that do not violate the continuity of the computational process to perform the required functional tasks critical to the continuity of the computing process is determined.