DISPERSION OF THE NUMBER OF FAILURES IN MODELS OF PROCESSES OF RESTORATION OF TECHNICAL AND INFORMATION SYSTEMS . OPTIMIZATION PROBLEMS

In this work, for several models of recovery processes, dispersion formulas for the number of failures are obtained, depending both on the recovery functions of the considered model of the recovery process and on the recovery functions (average number of failures) of other models. Considering the formulas for the average and variance of the number of failures, the problem statements are given on the organization of the recovery process in which the minimum variance is achieved with a given limit on the average number of failures, or so that there is the smallest average number of failures with a given dispersion limit. The formulation tasks resemble Markowitz’s well-known task of forming a portfolio of securities, where the average makes sense of income, risk variance. The solution of the formulated problems is obtained for a simple recovery process with an exponential distribution of operating time, and for this case the Chebyshev inequality and the formula for the coefficient of variation are written. The developed mathematical apparatus is intended for use in the formulation and solution of various optimization problems of information and computer security, as well as in the operation of technical and information systems, software and hardware-software information protection when failures, threats of attacks, and security threats of a random nature occur.

1. Borovkov A.A. Teoriya veroyatnostey/ A.A. Borovkov. -M.: Librokom, -2009. -652 s.

2. Baykhel't F. Nadezhnost' i tekhnicheskoe obsluzhivanie. Matematicheskiy podkhod: per. s angl./ F. Baykhel't, P. Franken. -M.: Radio i svyaz', -1988. -392 s.

3. Vaynshteyn I.I. Protsessy i strategii vosstanovleniya s izmenyayushchimisya funktsiyami raspredeleniya v teorii nadezhnosti/ I.I. Vaynshteyn. -Krasnoyarsk: SFU, -2016. -189 s

4. Vaynshteyn I.I. O modelyakh protsessov vosstanovleniya v teorii nadezhnosti/ I.I. Vaynshteyn, V.I. Vaynshteyn, E.A. Veysov// Voprosy matematicheskogo analiza: sb. nauch. tr./red. V. I. Polovinkin. -IPTs KGTU. Krasnoyarsk. -2003. -V. 6. -S.78-84.

5. Vaynshteyn V.I. Chislennoe nakhozhdenie funktsii vosstanovleniya dlya odnoy modeli protsessa vosstanovleniya/ V.I. Vaynshteyn, E.A. Veysov, O.O Shmidt//Vychislitel'nye tekhnologii. -Novosibirsk. -2005. -№10. -S. 4-9.

6. Bulinskaya E.V. Asimptoticheskoe povedenie nekotorykh stokhasticheskikh sistem khraneniya/ E.V. Bulinskaya, A.I. Sokolova// Sovremennye problemy matematiki i mekhaniki, -2015. -C.37-62.

7. Vaynshteyn V.I. Funktsii vosstanovleniya pri raspredelenii narabotok elementov tekhnicheskikh sistem kak smes' n funktsiy raspredeleniya//Sovremennye naukoemkie tekhnologii, -Moskva. -2018. -№6. –S.44-49.

8. Markowits Harry M. Portfolio Selection // Journal of Finance. 1952. 7. № 1 pp. 71-91

9. Kasimov Yu.F. Osnovy teorii optimal'nogo portfelya tsennykh bumag/ Yu.F. Kasimov. M: Informatsionno-izdatel'skiy dom «Filin"», -1998. -144 s.

10. Babeshko L.O. Matematicheskoe modelirovanie finansovoy deyatel'nosti/ L.O. Kasimov. -M.: «Kio-Rus», -2013. -212 s.