This article proposes an approach to improving reliability and survivability based on modular arithmetic. The proposed approach makes it possible to increase the survivability of cloud storage systems, as well as reliability and fault tolerance of data storage. To increase fault tolerance in the event of a failure, the redistribution of the processed data is applied. The proposed model allows restoring the saved data in the event of failure of one or more cloud servers.
As the amount of stored data increases, more and more users are switching to cloud storage
systems. In modern conditions, the requirements for the reliability of data storage in cloud are
constantly increasing. The reliability of cloud storage [
Cloud storage survivability [
The concepts of reliability and survivability have much in common and at the same time differ significantly from each other. They are united by the principle of stability, which takes into account all the variety of factors, including various emerging failures. The stability index is a function of the reliability indicators, survivability and fault tolerance.
Differences in the concepts of reliability and survivability and reasons of normal cloud system
functioning disruption are due to significant differences in their manifestation, the nature and scale of
failures, its duration, methods of their elimination and methods of increasing fault tolerance.
Accordingly, the initial data, calculation methods, accuracy and the essence of reliability and
survivability indicators differ significantly. The first ones are well provided with statistical material,
the main influencing factors are taken into account, they are deeply developed theoretically, and they
can be used for sufficiently accurate forecasting, calculations, design and modeling. The group of
survivability indicators more reflects the qualitative picture of the behavior of the considered cloud
storage system in conditions of external influences or cascade propagation of failures [
The reliability of cloud storage systems manifests itself in the form of failures. The concept of failure is closely related to the concept of operability. Operability is the state of the cloud system, in which it is able to perform the specified functions with the parameters and requirements for the quality of the services provided. Failure - a random event that disrupts the performance of the cloud system.
A variety of approaches can be used to build a distributed system for storing and processing data. Some of them are based on cloud computing paradigms. These infrastructures have both common characteristics and fundamental differences. Using clouds for storage requires security, reliability and scalability with limited Internet bandwidth to provide fast access to distributed data and a high degree of reliability, availability and scalability.
Distributed storage can be based on multiple clouds. Typically, data is divided into several parts, which are stored in different clouds to ensure availability in the event of a failure. However, failures in distributed storage can cause inconsistencies between different copies of the same data.
Large databases can be used. In this case, to ensure high performance, data processing and analysis must be performed using parallel computing.
In the second part, the main problems of cloud storage and the uncertainty of the emerging failures are considered, in the third part, methods for increasing the reliability and survivability of cloud storage are given and the application of the residue number system for this problem is considered. The final part is about the method of increasing reliability as information backup, its advantages and disadvantages are also given.
When storing data in the cloud, situations arise in the denial of access to data, errors or deterioration in the functioning of services, and sometimes in long interruptions in their work. For these and other reasons, distributed data processing is inevitably a continuous stream of failures, errors and malfunctions. Cloud storage failure can occur slowly, over a long period of time, or in a split second.
Uncertainty can be seen as the difference between available knowledge and complete knowledge.
It can be classified in several different ways depending on their nature. [
In the implementation of the cloud storage parts, the occurrence of failure uncertainty can be qualified as follows:
Uncertainty of software failure is a limited uncertainty due to complete or partial ignorance of the conditions under which decisions must be made.
The uncertainty of a hardware failure, such as hard drives, power systems, etc., is a technical uncertainty and is a consequence of the inability to predict the exact results of solutions.
Information theory founder Claude Shannon defined information as removed uncertainty. More precisely, obtaining information is a necessary condition for removing uncertainty. Uncertainty arises in a situation of choice. The problem of reducing number of options under consideration (reducing the variety) and, as a result, the choice of one corresponding situation option from among the possible is solved in the course of removing the uncertainty. Removing uncertainty enables to make informed decisions and take action. This is the guiding role of information.
The situation of maximum uncertainty presupposes presence of several equally probable alternatives (options), i.e. neither option is preferred. Moreover, the more equally probable options are observed, the greater the uncertainty, the more difficult it is to make an unambiguous choice and the more information is required for this to be obtained. For variants, this situation is described by the following probability distribution: {
}.
The minimum uncertainty is , i.e. this is a situation of complete certainty, meaning that the choice has been made and all the necessary information has been obtained. The probability distribution for a completely certain situation looks like this: * +.
To improve the reliability of cloud systems, a mathematical apparatus, standardization, load balancing, protection from external influences and the choice of data storage schemes can be provided.
If the above methods did not give the desired result, then it is necessary to use a reservation [1216].
Let's consider the factors affecting the survivability of cloud systems [
That is why the role of number system choice for the functioning of the cloud storage system increases. The number system mainly determines the model of the system's reliability, the method of redundancy, the prevention of cascading failures and the growth of arising errors. One of the natural indicators of the qualitative measurement of survivability is the indicator preserved by the system after a fixed set of impacts.
For cloud storage systems, the main indicators of reliability are: 1. Probability of no-failure operation in the time interval from to
( ) ( ) * - ( ), where ( ) , - is distribution of failures in time to the first failure. 2. The probability of failure-free operation of the facility in the time interval from to ( ) ( ) ( ) * | 3. Density of failure distribution
( ) ( ) ( )
( ) ( )
The main practical method for improving the reliability and survivability of cloud storage systems
is redundancy. Redundancy [
Basic types of reservation: 1. Structural. 2. Temporary. 3. Functional. 4. Load.
The highest stability, reliability and survivability will be possessed by a system in which these
methods are harmoniously combined and mutually penetrate each other [
) ( ) ( )
The Residue Number System is a number system in which numbers are represented as a set of non-negative residues in coprime modulus .
Let the numbers be represented by the residues at the bases We will assume that - residues are sufficient for an unambiguous representation of the number , and , , while the working bases are .
The range of unambiguous representation for the selected modulus is equal to the product of these modulus ∏ .
Thus, in the representation of the number the residues can be discarded any
residues without compromising the uniqueness of the representation of the number , as a result of
which the RNS can control errors and it is a nonlinear code, which is called the -code [
In order to assess the ability of the RNS to control errors, we introduce the concept of the weight (| | ) of the number .
The weight (| | ) of the number will be considered equal to the number of nonzero residues. This definition of the weight of a number corresponds to the definition of the weight of the code in the Hamming metric.
In the symbol (| | ) the subscript at shows that the number is represented by such a number of residues that | | .
Obviously, the number of residues that represent the number , in this case, is equal to . The argument | | means that | | .
With this definition of the number’s weight (| | ) it is possible to calculate it both directly from and from the totality of residuals .
The concept of the weight of the number [
The distance between the points of space and is defined as the weight of the difference between and .
(| | )
To determine the correcting capabilities of the code, we calculate the average weight ̅ (| | ) of nonzero complexes. If the number is changed in the range| | , then the period for dividing the zeros is , while the number of zeros in the base will be ( ̅̅̅̅̅) Then, respectively, the number of nonzero elements in the base is equal to
The sum of the weights of the residues determined by the following expression:
̅̅̅ ∑ ( ( ) ), when is from zero to
, is ( ∑ ) The average weight ̅ (| | ) can be defined as ̅ (| | ) ( ∑ )
The more the code is adapted to error correction, the more the numbers in the code representation differ from each other, i.e. the greater the code distance. Moreover, the distance will be different between different numbers [30-33].
If we determine the average weight of the numbers forming the zero space, then the minimum code distance will never exceed the upper bound of the minimum code distance. The minimum weight of elements of the zero-code space is
{ (| | )]
With we are guaranteed to detect any error. At the same time, the use of RNS allows detecting more errors. For example, an RNS with one redundant base, the value of which is greater than any of the working ones, allows detecting 100% of single errors and 95% of double errors.
It is also possible to detect all errors of multiplicity if . This means that the RNS allows detecting errors of a given multiplicity and the multiplicity of detected errors is determined by the minimum code distance .
Let the number ( ) be distorted, and instead of the number (̃ ̃ ̃ ), such that ̃ | | . Let us calculate the difference between ̃ and determine as the magnitude of the error:
| ̃ |
Since errors in different bit digits are independent of each other, then similarly to the representation of the number , i.e. residuals ( determined as | ̃ | .
Using the accepted residuals ̃ ̃ ̃ systems: we have ̃
, which will can be represented
), which will be we calculate the number ̃ by solving the comparison ̃ ̃ ( ) ( )
To solve this system of comparisons, we use the method of orthonormal vectors of the form [
With the help of orthonormal vectors, the number ) (| |
) can be represented as where (| | ) is a function of the number rank, which for the value of the argument | | value from calculating using orthonormal vectors | | .
Having calculated the number ̃ we find the values of the residuals ̃ ̃ based on .
Syndromic components are defined as takes a ̃ of this number Let us show that the fact of an error can be determined by the syndromic components. We represent the number as
Suppose that only the residuals on the bases affected by errors. Taking this into account, we have , which are considered informational, are |
( ∑ ̃ ̃ | |
| ||∑ | | where is an error value in the -digit, is a set formed from the numbers of information bases, the residues of which are distorted.
Due to the fact that the quantity is not zero and that the excess grounds satisfy the condition of mutual simplicity, the inequality to zero of at least one syndromic component indicates the presence of errors.
Let's give an example of error detection. Let the number be represented in the RNS by the residues , when converting the number to the positional number system (during data recovery) errors may occur. Therefore, after the data recovery ̃ (̃ ̃ ̃ ) is performed, we compare the number ̃ with the number , and if ̃ , then we conclude that an error occurred during the recovery.
Let's give another example. In the generalized positional number system (GPNS), the number | | can be represented in the form ∏ (1)
( ) starting with . Since all terms except for ), solving comparisons of the form are identically zero modulo , ), therefore ) or else | Continuing the process of calculating the bit digits of the GPNS at the -th step, we get | ∑ ∏ ∏ | ( ) As a result of converting the number ( ) into the GPNS, we have the number ( ).
If | | , then will be equal to zero. This property can be used to receive data without error. The advantage of this algorithm is manifested in the fact that if our task is to establish the fact that errors have occurred, then if any is not equal to zero, the convertion process stops. This reduces the amount of computation required to detect errors.
This article proposes an approach to improving reliability and survivability based on modular arithmetic. The proposed approach makes it possible to increase the survivability of cloud storage systems, as well as to increase the reliability and fault tolerance of the data storage. To increase fault tolerance in the event of a failure, the redistribution of the processed data is applied. The introduction of low redundancy allows processing or restoring stored data in the event of a management server failure. This model allows recovering saved data in the event of a failure of one or more cloud servers. However, further research is needed to assess its efficiency in real systems. This will be the subject of our future work on a comprehensive experimental study of multipurpose optimization with real cloud providers.
Acknowledgements The reported study was funded by RFBR, project number 20-37-70023
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