In this article we describe the solution to make an estimate of allowed alternations in model components parameters that make up a data processing and control system. They reflect properties of physical and information components of the aforementioned system. This solution is derived according to the limitations of possible changes in integral property that describes the system behavior in different modes of operation. The proposed solution makes it possible to solve not only the direct problem - making a conclusion whether the vulnerability of the data processing system is negligible with respect to alternations in parameters of this systems components but the inverse as well, finding tolerable levels in alterations in systems components parameters from an acceptable level of uncertainty of targeted efficiency. The proposed solution follows known demands: inner properties of the system must be so that customer demands are fulfilled.
Data Science / V.E. Gvozdev, M.B. Guzairov, D.V. Blinova, A.S. Davlieva
Alteration of systems components parameters is the inherent property of any DPCS. Statistical uncertainty as a probability of state parameters being in tolerable intervals comprise the metric property of such alteration. On the other hand change in the parameters is the cause of statistical uncertainty of target efficiency property: v f f1(Н D ) ,
H Э f 2 (Н D ) , Here H Э is the metric uncertainty characteristic of the target efficiency;
f1() – a direct functional relationship, making a relationship v f between the metric characteristics of the uncertainty Н D and the state parameters vector components D .
f 2 () – a direct functional relationship that makes a relationship H Э between the metric characteristics of uncertainty Н D . The character f2 () is defined by the structure of the system.
From (
The initial data of the problem are: (A) Description of the system states set Si (i 1; N ) , with each state matching the characteristics of the target efficiency Эi ; (B) The same characteristics of the statistical uncertainty of the target efficiency for each states H i (Э ) ; (C) A system model that characterizes the relationships ij between the states i and j (i, j 1; N , i j) ; (D) A rule that allows us to estimate the average proportion of the time pi (i 1; N ) the system is in the i-th state; (E) Uncertainty characteristics of relations Нij ( ) ; (F) The rule for estimating the average target efficiency Э as a function of Эi and pi :
Э (Эi , pi ) ; (
H Э f 2 (Hi (Э), Hij ( ))
Note that in (
It is required: to estimate the limits on the possible values of uncertainty characteristics Hi (Э), Hij ( ) based on the limitation on the values H Э of the uncertainty characteristic of the average target efficiency.
Assumptions: (A) The apparatus of Markov processes is used as a basis for modeling the state of DPCS [12]; (B) Intervals Эi [Эi(l ) , Эi(u) ] ; ij [(ijl) ,(iju) ] are used as characteristics of the statistical uncertainty of the target efficiency H i (Э ) and relationships Hij ( ) accordingly. The index "l" corresponds to the lower limit of the interval; index "in" - the upper; (C) Linear convolution is used as an estimate of the target efficiency
Э iN1 рi Эi , (
H Э P[а(н) Э а(в) ] , where, а (l ) , а (u ) are determined by (Fig. 1):
P[0 Э а(н) ] P[а(в) Э ] 0(Э ) / 2 .
(
4. Solution for the task1
The basis solution for the task is the construction of the dependence (
Specifying a model (graph) linking the states of the system (nodes) and transitions (edges)
between states Specifying the statistical uncertainty characteristics of nodes and edges as lower and upper
limits {Эi(l ) , Эi(u) } ; {(ijl) ,(iju) } , i, j 1; N ,i j Generation arrays of random numbers corresponding to H ( ) (Э) , Hi(,j) ( ) , 1; and
i calculation on their basis an array of random values of average target efficiency H Э( ) ,
1;
H Э f 2 (Hi (Э), Нij ( )) Setting a limitations (Э ) on the variability of the statistical uncertainty index the average
0 target efficiency, determining the limits Э(l ) , Э(u) , the range of possible values Э Estimation based on the dependence H Э f2 (Hi (Э), Hij ( )) and {Э(l) , Э(u) } limitations on the values {Эi(l ) , Эi(u) } , {(ijl) ,(iju) }
In the experiment as the uncertainty characteristic of the average target efficiency H Э was the distribution density f (Э ) of the average target efficiency Э . As the uncertainty characteristics of system components Hi (Э), Hij ( ) were interval limits of possible values {Эi(l ) , Эi(u) } , {(ijl) ,(iju) } . These limits were determined by the rules: Эi(l ),(u) Эi() (1 Э ) ; (ijl),(u) (ij) (1 ) , where the sign "–" corresponds to the lower limit of the interval of possible values of the graph component characteristic; the "+" sign corresponds to the upper limit;
the index " " corresponds to the basic values of the characteristic.
Note that the interval uncertainty characteristics estimates in accordance with the principle of maximization of entropy [13, 14] can be associated a law of random variable distribution.
Fig. 3 shows the model (states graph) of the system [15, 16]. Table 1 shows the base values of average target efficiencies (Эi(b) ,i 1;4) . The base values of the transitions intensities ((ijb) ,i, j 1;4,i j) were taken to be the same and equaled ten. During the study, Э , took a different value ( Э [0;1], [0;1]) . Fig. 4 shows estimates of the distribution densities f (Э ) , corresponding to different Э , for different uncertainty distribution by the graph components: (A) The statistical uncertainty corresponds to the graph nodes, the nominal values of the transitions intensities correspond to the edges; 1 In the development of the program for conducting a statistical experiment and processing the results of the experiment, the undergraduate student of the Department of Technical Cybernetics of the Ufa State Aviation Technical University Teslenko V.V. actively participated. 3rd International conference “Information Technology and Nanotechnology 2017” 13
The constructed estimates f (Э ) became the basis for constructing H Э , for various combinations of characteristics the statistical uncertainty of the graph components. Fig. 5 shows the resulting dependencies, corresponding to different values 0(Э )
The resulting dependences H Э allow us to solve a direct problem: an estimation of uncertainty characteristics of target average efficiency H Э on the information basis on model components parameters variability; and the inverse problem: an estimation limitations on the parameters variability of graph’s nodes and edges based on the limitations on the characteristics of the target average efficiency.
An example of solving a direct problem. Given: (Э ) ; Э . It is believed that the variability of transitions intensities is 0 absent. It is required to estimate the expected uncertainty H Э . The scheme for solving the problem is shown in Fig. 6.
Nowadays DPCS play more and more of a substantial role as a vital component in systems that control complex objects. This promotes posing the problem of developing theoretical basis and development tools for managing the functional security of DPCS. One of the key tasks in solving such a problem is analyzing vulnerabilities of DPCS. They are affected by internal
Data Science / V.E. Gvozdev, M.B. Guzairov, D.V. Blinova, A.S. Davlieva properties (construction, component properties) and by external environment in which the system is operated. The allowed level of the vulnerability is determined by whether the deviation of the systems behavior from the base behavior is affecting the quality of control of a complex object.
In this article a solution is given to estimate the possible deviation in components parameters of the model OF DPCS (such parameters reflect physical and information properties of the system). This solution is based of the limitations on alternation of the integral factor that shows the behavior of the system in different modes of operation, this behavior is the target efficiency of the system. Proposed solution follows known demands: inner properties of the system must be so that customer demands are fulfilled (Kano’s model). The described solution makes it possible to solve not only the direct problem - making a conclusion whether the vulnerability of the data processing system is negligible with respect to alternations in parameters of this systems components but the inverse as well, finding tolerable levels in alterations in systems components parameters from an acceptable level of uncertainty of targeted efficiency’s index.
This work was supported by RFBR grant No. 17-07-00351 "Methodological basics of dependability assurance of transmission systems telemetry information with use of intelligent data analysis technologies".