Cyber Object State Maximal Probability Timing Obtained Through Multi-Optional Technique

Abstract

In this publication a Doctrine for the Conditional Extremization of the Hybrid-Optional Effectiveness Functions Entropy is discussed as a tool for the Cyber Object State Maximal Probability Assessments. Traditionally, most of the problems having been dealt with in this area must relate with the probabilistic problem settings. Regularly, the optimal solutions are obtained through the probability extremizations. It is shown a possibility of the optimal solutions “derivation”, with the help of a model implementing a variational principle which takes into account objectively existing parameters and components of the Markovian process. The presence of an extremum of the objective state probability is observed and determined on the basis of the proposed Doctrine with taking into account the measure of uncertainty of the hybrid-optional effectiveness functions in the view of their entropy. Such approach resembles the well known Jaynes' Entropy Maximum Principle from theoretical statistical physics adopted in subjective analysis of active systems as the subjective entropy maximum principle postulating the subjective entropy conditional optimization. The developed herewith Doctrine implies objective characteristics of the process rather than subjective individual's preferences or choices, as well as the states probabilities maximums are being found without solving a system of ordinary linear differential equations of the first order by Erlang corresponding to the graph of the process.

Keywords

• cyber hygiene
• conflict management
• global information networks
• effectiveness functions entropy
• hybrid-optional effectiveness
• multi-optionality
• optimal distribution
• variational principle
• entropy maximum principle

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1.1

Introduction

Cyber hygiene and conflict management in global information networks can be considered from the point of view of the theoretical developments for reliability [ 1 ].

The analogy of the hygiene to the maintenance procedures is very good. Therefore, the apparatus of theoretical physics related with the uncertainty measures [ 2-4 ] is quite applicable here. Thus, in the field of the Social Networking Services it is critical to take into considerations subjective entropy of preferences [ 5, 6 ]. The similar to the aircraft maintenance and repair approaches [ 7 ], combinations with the rationality of the choice behavior [ 8 ], [ 9 ], inspire global science and social science entropy research [ 10 ]. In addition, economic issues [ 11 ] in respect of risk [ 12 ], like in aviation [ 13 ], are complicated with the group decision making [ 14, 15 ].

All this initiated search for a new explanation of the described process. The presented doctrine, like developed in [ 16-30 ], is to demonstrate the possibilities of the entropy paradigm use to the variety of the problems solutions, for example discussed in works [ 31-39 ]. Mathematical means intended to be used are of the regular calculus [40]. Also, adjacent and similar formalism scientific areas, let us say mentioned in publications of [41-49], can implement the presented doctrine results. 1.2

The Problem Statement

Management of cyber incidents, warfare and conflicts are considered in terms of the mass service theory [37-39].

The considered cyber object (space) can change its states. Illustration of that is in the simplified graph (see Fig. 1).

01 1 10

20 0 12 21 02

2

Here, in Fig. 1, “0”, “1”, and “2” designate the states of the cyber object. The corresponding values of the rates ij and  ji will determine the process going on in the system.

The problem is to find the timing for the maximal values of the states probabilities, for instance of P1t , analytically and in an easier than the traditional way. The proposed is the multi-optional way. 2 2.1

Main Content

Traditional Methods

Even for the simplified (partial to Fig. 1) case, although implying the possible return of the system from the state “D” into the state of “A” without the transition into the state “F” (this transition is carried out with the parameter of 1 illustrated on the graph, see Fig. 2) the procedure is quite challenging analytically.

1 1

The corresponding, to the graph of Fig. 2, system of differential equations by Erlang will have the view of The characteristic equation for system (1) will be similarly [40]: ddPtA  1PA  1PD;  dPD  1PA  2  1PD;

 dt  ddPtF  2PD.   1  k Then, for finding two other roots from Eq. (5)  1  k 2  1  kk  11k  0 . k11  1  k 2  1  k   0 .

Reducing Eq. (7) and cancelling the similar members 11 and 11 , (1) (2) (3) (4) (5) (6) (7) The sought roots are

 k 2  k1  2  1 12  0 . where C1 ; C2 ; C3 are arbitrary constants; also is the solution of the differential equations system (1). This is the general solution of the differential equations system (1), [40].

Satisfying the condition of Eq. (6) for root k1  0 from the system of Eq. (10) where a  1 , b  1  2  1 , c  12 are corresponding coefficients of (8).

For each root ki of Eq. (2)-(5), (7), (8), namely k1 , k2 , k3 Eq. (6) and (9) we will write down the system of linear algebraic equations for 1i , 2i , 3i [40]:  1  k 1  

1 1 0  1   2  1 k 2  12  22  0  3  0; 

  0  3  0;   0  k 3  0. The system of Eq. (10) derives from an assumption of a partial solution existence PA  1ekt ;

PD  2ekt ;

PF  3ekt ; for the system of Eq. (1).

Since having three roots in the stated problem setting [40], the solution of (1):

PA1  11ek1t ; P2  12ek2t ;

A PA3  13ek3t ;

PD1  21ek1t ; PD2  22ek2t ; PD3  23ek3t ;

PF1  31ek1t ; P2  32ek2t ;

F PF3  33ek3t .

In the way of direct substitution of partial solutions (12)-(14) into equations, one can be convinced that the system of functions, similarly to [40]:

PA  C1PA1  C2PA2  C3PA3  C111ek1t  C212ek2t  C313ek3t ;  PD  C1PD1  C2PD2  C3PD3  C121ek1t  C222ek2t  C323ek3t ; PF  C1PF1  C2PF2  C3PF3  C131ek1t  C232ek2t  C333ek3t ;  (8) (9) (10) (11) (12) (13) (14) (15)  1  k11

1

. use the roots k2 and k3 , Eq. (9) of the initial quadratic equation Eq. (8).

Indeed. Equalizing Eq. (21) and (22) we get

11  12 11 1k2,3 2k2,3 1k2,3 k22,3.

And cancelling for 11 in both parts of Eq. (24) it yields Eq. (8):

12 1 2 1k2,3 k22,3  0 .

 .

1  1 1  k3 1  k2 Thus, turning back to the system of Eq. (15), we determine the unknown coefficients of the general solution of the differential equations system (1), [40], satisfying the initial conditions: t0 0 ; PA tt0 1; PD tt0  0; PF tt0  0 ; and have already known the coefficients of alpha; i.e. Eq. (18); (20); (21); (28):  1 ; PA  C111ek1t C212ek2t C313ek3t; 1 0C2 1 1k2 C3 1 k3    PD  C121ek1t C222ek2t C323ek3t;  0  0C2 C3; PF  C131ek1t C232ek2t C333ek3t;t00 0  C1 C2 k2 C3 k2 .  2 3

Substituting the values of Eq. (30) for the corresponding members into the first equation of the system of Eq. (29) we get The same result is obtained if make equal Eq. (21) and (23): 1k2,3  12 1k2,3 2k2,3 k22,3; k22,3 1 2 1k2,3 12  0 . (26) When equalling Eq. (22) and (23) it gives the same. Indeed: 2 1k2,3 k22,3  12 1k2,3 ; k22,3 1 2 1k2,3 12  0 . (27) For coefficient 2,3, from the third equation of Eq. (19) and condition (20), 3 222,3 k2,332,3  0; 2,3  2 .

3 k2,3 (24) (25) (28) (30) (31) In order to make the notations shorter let us put down the indications with the alpha symbolizations: 1  C312  C313  C313  12;

From the third equation of the system of Eq. (29) we obtain

Now, all coefficients are expressed through the given values, hence, the system of Eq. (1) is successfully solved. The Laplace integral transformation methods give the same results. For the general case described with the graph shown in Figure 1 P0t   k1ek1t  k2ek2t  k1bk12    k2 kb21 k1  k1bk12 ek1t   k2 kb21 k1 ek2t .

P1t   01 ekk11t  ke2k2t  k1ck12    k2 kc21 k1  k1ck12 ek1t   k2 kc21 k1 ek2t .

P2 t   02 ekk11t  ke2k2t  kd1k12    k2 kd21 k1  kd1k12 ek1t   k2 kd21 k1 ek2t . The values of the parameters in (35) - (37) have the mathematical expressions corresponding to the general case (see Fig. 1). Then, it has to be found the possible extreme values of the probabilities. For distinctness, let it be P1t . dP1t   dt

k101k2 k1ek1t  k2ek2t  k1  k2 kc21 k1  k1ck12 ek1t  k2  k2 kc21 k1 ek2t .(38) After equalizing (38) to zero, the needed timing is t*p  ln 01k1  c1  ln 01k2  c1 . Herein it is suggested to formulate the own concept (idea, problem, hypotheses). (32) (33) (34) (35) (36) (37) (39)

In such respect [ 1-40 ], the considered example may be given an attention to in regards with the Multi-Optional Hybrid-Effectiveness Functions Uncertainty Measure Conditional Optimization Doctrine (method, approach, concept) applicable (used, implemented) to the cyber object state maximal probability timing determination [ 17, 20, 22, 25 ].

The values can be obtained not only in the entire probabilistic way, but also in a hybrid partially probabilistic partially optional way [ 17, 20, 22, 25 ].

The essence of the doctrine (method, idea, approach, concept) is to consider the process developing in the system from the position of some hybrid optional functions distribution optimality.

Consider the options essential to the general view three state system (see Fig. 1). Objective functional, like proposed in references [ 17, 20, 22, 25 ], is as follows: i1 3 h  xF1iln xF1i t*p 3xF1iM12i  3xF1i1 , 01 i1  i1  F i  1

M12i  M M12i  ki01  c1 ,

ki01  c1 pp2  pe1  b1  c1  d1

, M  pp2  pe1  b1  c1  d1 , where x is an unknown parameter; hi  xF1i is the multi-optional hybrid functions depending upon the options effectiveness functions of F1i ; t*p 01 is the intrinsic parameter of the system and the process, which is the ratio of the timing (delivering the sought maximal value to the probability) t*p , it is unknown yet for such problem formulation and the time of t*p is going to be determined as a solution, i.e. it is not the Eq. (39) so far, however it will be, that is why the indication is the same, to the flow intensity 01 ; M12i is the algebraic addition of the initial elementary intensities matrix M formed in the style likewise from the Erlang’s system, Eq. (1), element of m12 ;  is the parameter, coefficient, function (uncertain Lagrange multiplier, weight coefficient) for the normalizing condition.

Consider an extremum existence necessary conditions for the objective functional of (40), [ 17, 20, 22, 25 ]: h  hi

h xF1i  0 ,

i 1,3 . ln xF11 t* p 01k1  c1   1  ln xF12 01 t*p  ln F11 ln F12 k2   k1 .

After that, we have got the law of subjective conservatism on one hand and on the other hand t*p  ln 02k1  d1  ln 02k2  d1 . (43) (44) (45) (46) (47) (48) (49) And finally equivalent with Eq. (39) with taking into account the roots, i.e. the second, third, and fourth expressions of the Eq. (40) and (48) results.

Now we ought to say that for the situation when the probability of P2 t  undergoes the extremum instead of the probability of P1t , the problem, due to the symmetry, has a symmetrical solution:

Conclusions

That is the system according to the developing stationary Poison flow process has the possible states optimal options related with either the system of parameters ki , 02, d1 or ki , 01, c1 (50) values for the initial moment probability of the state “0” being equaled to “1”. The proposed optional method is more compact and applicable for a cyber object state maximal probability timing determination. 33. Szafran, K.: Bezpieczeństwo lotu – zasada maksymalnej entropii. Bezpieczeństwo na

Lądzie, Morzu i w Powietrzu w XXI Wieku 1, 247–251 (2014). 34. Szafran, K., Kramarski, I.: Safety of navigation on the approaches to the ports of the republic of Poland on the basis of the radar system on the aerostat platform. International Journal on Marine Navigation and Safety of Sea Transportation 9(1), 129134 (2015). 35. Szafran, K.: Bezpieczeństwo operatora pojazdu trakcyjnego – stanowisko prób dynamicznych. Logistyka 6, 192–197 (2014). 36. Krzysztofik, I. Koruba, Z.: Mathematical model of movement of the observation and tracking head of an unmanned aerial vehicle performing ground target search and tracking.

Journal of Applied Mathematics 2014, Article ID 934250, 11 pages (2014). 37. Smirnov, N. N.: Technical operation of aircraft. Transport, Moscow, USSR (1990). 38. Gnedenko, B. V., Kovalenko, I. N.: Introduction into the mass service theory. URSS,

Moscow (2013). (in Russian) 39. Ovcharov, L. A.: Applicable problems of the theory of mass service. Machinebuilding,

Moscow, USSR (1969). (in Russian) 40. Piskunov, N. S.: Differential and Integral Calculus for Higher Engineering Educational Institutions. 13th edn. Nauka, Moscow, USSR (1985). (in Russian) 41. Zaliskyi, M., Solomentsev, O.: Method of sequential estimation of statistical distribution parameters. In: IEEE 3rd International Conference Methods and Systems of Navigation and Motion Control (MSNMC), pp. 135–138, Kyiv, Ukraine (2014). 42. Odarchenko, R., Polihenko, O., Kharlai, L., Tkalich, O.: Estimation of the communication range and bandwidth of UAV communication systems. In: IEEE 4th International Conf. on Actual Problems of UAV Developments (APUAVD), pp. 159–162, Kyiv, Ukraine (2017). 43. Odarchenko, R. S., Gnatyuk, S. O., Zhmurko, T. O., Tkalich, O. P.: Improved method of routing in UAV network. In: IEEE 3rd International Conference on Actual Problems of UAV Developments (APUAVD), pp. 294–297, Kyiv, Ukraine (2015). 44. Solomentsev, O., Zaliskyi, M., Herasymenko, T., Kozhokhina, O., Petrova, Yu.: Data processing in case of radio equipment reliability parameters monitoring. In: Advances in Wireless and Optical Communications (RTUWO 2018), Riga, Latvia, pp. 219-222 (2018). 45. Solomentsev, O., Kuzmin, V., Zaliskyi, M., Zuiev, O., Kaminskyi, Y.: Statistical data processing in radio engineering devices operation system. In: 14th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET 2018), Lviv-Slavske, Ukraine, pp. 1-4 (2018). 46. Solomentsev, O., Zaliskyi, M., Zuiev, O.: Radioelectronic equipment availability factor models. In: Signal Processing Symposium 2013 (SPS 2013), Jachranka Village, Poland, pp. 1-4 (2013). 47. O. Solomentsev, M. Zaliskyi, R. Odarchenko, S. Gnatyuk, Research of energy characteristics of QAM modulation techniques for modern broadband radio systems, Proceedings of the 2016 IEEE International Conference on Electronics and Information Technology (EIT), Odesa, Ukraine, May 23-27, 2016, рр. 14-20. 48. Solomentsev, O.V., Zaliskyi, M.Yu., Zuiev, O.V., Asanov, M.M.: Data processing in exploitation system of unmanned aerial vehicles radioelectronic equipment. In: IEEE 2nd International Conference Actual Problems of Unmanned Air Vehicles Developments (APUAVD), pp. 77-80, Kyiv, Ukraine (2013).

References

1. Dhillon , B. S. : Maintainability, maintenance, and reliability for engineers . Taylor & Francis Group, New York ( 2006 ).

2. Jaynes , E. T.: Information theory and statistical mechanics . Physical review 106(4) , 620 - 630 ( 1957 ).

3. Jaynes , E. T.: Information theory and statistical mechanics . II. Physical review 108(2) , 171 - 190 ( 1957 ).

4. Jaynes , E. T.: On the rationale of maximum-entropy methods . Proceedings of the IEEE 70 , 939 - 952 ( 1982 ).

5. Jing , Ch.: An Entropy Theory of Psychology and its Implication to Behavioral Finance . [Online]. Available at Social Science Research Network, http://ssrn.com/abstract=465280 or http://dx.doi.org/10.2139/ssrn.465280, last accessed 2014 /09/07.

6. Kasianov , V.: Subjective entropy of preferences. Subjective analysis . Institute of Aviation Scientific Publications , Warsaw, Poland ( 2013 ).

7. Kroes , M. J. , Watkins , W. A. , Delp , F. , Sterkenburg , R.: Aircraft maintenance and repair. 7th edn . McGraw-Hill , Education, New York, USA ( 2013 ).

8. Luce , R. , Krantz , D. : Conditional expected utility . Econometrica 39 , 253 - 271 ( 1971 ).

9. Luce , R. D.: Individual choice behavior: A theoretical analysis . Dover Publications , Mineola, N. Y. ( 2014 ).

10. Ma , F. C. , Lv , P. H. , Ye , M.: Study on global science and social science entropy research trend . In: IEEE 5th International Conference on Advanced Computational Intelligence (ICACI) , pp. 238 - 242 , Nanjing, Jiangsu, China ( 2012 ).

11. Silberberg , E. , Suen , W.: The structure of economics. A mathematical analysis . McGrawHill Higher Education , New York ( 2001 ).

12. Smith , D. J. : Reliability, maintainability and risk. Practical methods for engineers . Elsevier , London ( 2005 ).

13. Wild , T. W. , Kroes , M. J. : Aircraft powerplants. 8th edn. McGraw-Hill , Education, New York, New York, USA ( 2014 ).

14. Zamfirescu , C. , Duta , L. , Iantovics , B. : On investigating the cognitive complexity of designing the group decision process . Studies in Informatics and Control 19 ( 3 ), 263 - 270 ( 2010 ).

15. Zamfirescu , C. B. , Duta , L. , Iantovics , B. : The cognitive complexity in modelling the group decision process , http://ssd.valahia.ro/UICS.pdf., last accessed 2014 /09/27.

16. Goncharenko , A. : Airworthiness support measures analogy to the prospective roundabouts alternatives: theoretical aspects . Journal of Advanced Transportation 2018 , 1 - 7 ( 2018 ).

17. Goncharenko , A. : A multi-optional hybrid functions entropy as a tool for transportation means repair optimal periodicity determination . Aviation 22 ( 2 ), 60 - 66 ( 2018 ).

18. Mazin Al Hadidi, Jamil S . Al-Azzeh , R. Odarchenko , S. Gnatyuk , A. Abakumova , Adaptive Regulation of Radiated Power Radio Transmitting Devices in Modern Cellular Network Depending on Climatic Conditions, Contemporary Engineering Sciences , Vol. 9 , № 10, рр. 473 - 485 , 2016 .

19. Goncharenko , A. : Optimal controlling path determination with the help of hybrid optional functions distributions . Radio Electronics , Computer Science, Control (44) , 149 - 158 ( 2018 ).

20. Goncharenko , A. : Aeronautical and aerospace materials and structures damages to failures: theoretical concepts . International Journal of Aerospace Engineering 2018 , 1 - 7 ( 2018 ).

21. Mazin Al Hadidi, J. Samih Al-Azzeh , O. Tkalich , R. Odarchenko , S. Gnatyuk , Yu. Khokhlachova. ZigBee, Bluetooth and Wi-Fi Complex Wireless Networks Performance Increasing , International Journal on Communications Antenna and Propagation , Vol. 7 , № 1, рр. 48 - 56 , 2017 .

22. Goncharenko , A. V. : Optimal UAV maintenance periodicity obtained on the multioptional basis . In: IEEE 4th International Conference on Actual Problems of UAV Developments (APUAVD) , pp. 65 - 68 , Kyiv, Ukraine ( 2017 ).

23. Goncharenko , A. V. : Relative pseudo-entropy functions and variation model theoretically adjusted to an activity splitting . In: 9th International Conference on Advanced Computer Information Technologies (ACIT' 2019 ), pp. 52 - 55 , Ceske Budejovice , Czech Rep . ( 2019 ).

24. R. Odarchenko , S. Gnatyuk , T. Zhmurko , O. Tkalich , Improved Method of Routing in UAV Network , Proceedings of the 2015 IEEE 3rd International Conference on Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD) , Kyiv, Ukraine, October 13-15 , Vol. 1 , 2015 , рр. 294 - 297 .

25. Goncharenko , A. V. : Multi-optional hybrid effectiveness functions optimality doctrine for maintenance purposes . ,” In: IEEE 14th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET- 2018 ), pp. 771 - 775 , Lviv-Slavske, Ukraine ( 2018 ).

26. Goncharenko , A. V. : An entropy model of the aircraft gas turbine engine blades restoration method choice . In: 8th International Conference on Advanced Computer Information Technologies (ACIT' 2018 ), pp. 2 - 5 , Ceske Budejovice , Czech Republic ( 2018 ).

27. I. Parkhomey, S. Gnatyuk , R. Odarchenko , T. Zhmurko et al, Method For UAV Trajectory Parameters Estimation Using Additional Radar Data , Proceedings of the 2016 4th International Conference on Methods and Systems of Navigation and Motion Control , Kyiv, Ukraine, October 18-20 , 2016 , рр. 39 - 42 .

28. S. Gnatyuk , A. Okhrimenko , M. Kovtun , T. Gancarczyk , V. Karpinskyi , Method of Algorithm Building for Modular Reducing by Irreducible Polynomial , Proceedings of the 16th International Conference on Control, Automation and Systems , Oct. 16 - 19 , Gyeongju, Korea, 2016 , рр. 1476 - 1479 .

29. Goncharenko , A. V. : Navigational alternatives, their control and subjective entropy of individual preferences . In: IEEE 3rd International Conference on Methods and Systems of Navigation and Motion Control (MSNMC) , pp. 99 - 103 , Kyiv, Ukraine ( 2014 ).

30. Al-Azzeh J.S. , Al Hadidi M. , Odarchenko R. , Gnatyuk S. , Shevchuk Z. , Hu Z. Analysis of self-similar traffic models in computer networks , International Review on Modelling and Simulations , № 10 ( 5 ), pp. 328 - 336 , 2017 .

31. Kasjanov , V. , Szafran , K. : Some hybrid models of subjective analysis in the theory of active systems . Transactions of the Institute of Aviation 3 ( 240 ), 27 - 31 ( 2015 ).

32. Pagowski , Z. , Szafran , K. : “Ground effect” inter-modal fast sea transport . The International Journal on Marine Navigation and Safety of Sea Transportation 8 ( 2 ), 317 - 320 ( 2014 ).