Translated from the journal
«Teor. Imovirn. Mat. Stat.»,
2010, Vyp. 82, P.9-12 (in Ukrainian)
About I.N.Kovalenko's selected works
V.S.Korolyuk
Institute of Mathematics
National Academie of Sciences of Ukraine
In 50–60s I was the informal head of some I.N.Kovalenko's works when he was a student of the mechanical-mathematical faculty of Т.G. Shevchenko’s KNU and afterwards a post-graduate student of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR. However, once the guidance was official when we, together with E.L.Yushchenko, supervised practical work of the 4th grade students of the mechanical-mathematical faculty on computer "MESM" at the Computer Centre of the Academy of Sciencesof former Ukrainian SSR. These students had attended my special course on address programming (the first in our country!), and each of them was given this or that problem to be solved on the electronic computer.
In 1954 B.V.Gnedenko's post-graduate students V.S.Mikhalevich, А.V.Skorohod and I were directed to the M.V.LomonosovMoscowStateUniversity for practical study. Having returned to Kyiv as quite formed scientists in 1956 we actively carried out both our own research and guidance of students and later on post-graduate students. Many problemsthat were stated by me and V.S.Mikhalevich to young men originated from our Moscow head A.N.Kolmogorov.
Many talented, devoted to science employees have concentrated at the Probability Theory Department headed by B.V.Gnedenko (called by probabilists simply as the Teacher those days): L.A.Kaluzhnin, I.I.Gikhman, O.S.Parasyuk, Yu.M.Berezansky and others. Igor Kovalenko fulfilled term papers at L.A.Kaluzhnin and I.I.Gikhman, and on the fifth year – a degree work at V.S.Mikhalevich. A part of this work [1] consisted in the solution of the following problem by A.N.Kolmogorov. Let the sequential statistical control of itemsbe made, each of thembeing defective with an unknown probability. It is required to develop such a method of the sequential analysis which would guarantee the given errors of the first and second kind at and and atthe same time would minimize the average number of observations at an intermediate value.
At the postgraduate department of theInstitute of Mathematics of the Academy of Sciencesof the Ukrainian SSR, where I.N.Kovalenko studied in 1957 – 1960, B.V.Gnedenko offered two research problems to his disciples: queues with impatient customers and reliability of redundant systems with a general distribution function of repairtime of an element. Within the framework of the first problem S.M. Brodi investigated systems with «- waiting» and «-sojourn» in which the waiting time and accordingly the sojourn time of the call in the system is limited by a constant. I offered to Igor Kovalenko to construct such a random process which would generalize both models – with «-waiting» and «-sojourn». He had very quickly solved that problem [2], having used a modification of known L.Takács method. I.N.Kovalenko's approach has been included in some known monographs (J.W.Cohen, T. Saaty, etc.).
The second of B.V.Gnedenko's mentioned problems goes back to B.A.Sevastyanov's well-known result on theinvariance of the distribution of number of the busy lines in a telephone system with losses M/G/m/0 with respect to service time distribution function under specified average time. [Earlier a French mathematician R. Fortet has established such invariance for an absolutely continuous distribution function of service time. B.V.Gnedenko has offered to post-graduate students T.P. Maryanovich, T.I.Nasirova and V.N.Yaroshenko to investigate systems’ models interesting to the theory of reliability using the same method and they have succeeded; however in certain cases the method did not work properly. Then I.N.Kovalenko [3] has found a necessary and sufficient invariance condition for a quite general system with losses. Robert Fortet, who visited our institute, invited Igor on the teletraffic congress (Paris, 1961), but the young employee hadn’t succeeded in going there, nevertheless, I.N.Kovalenko's criterion has received wide applications, especially in works of the Soviet and West German scientists.
In the sixties at the Institute of mathematics I started with my disciple D.V.Gusak toexplore actively applications of factorization identities to studying various random walks. Having got involved in our research I.N.Kovalenko [4] found factorization representation of the distribution function of the size of jump for random walk with the step accepting values of both signs. This result has been noted by A.N.Kolmogorov at one of conferences.
In the late fifties at Kyiv High Engineer Radio Technical School of Air Defense B.V.Gnedenko gave a course of lectures on queueing theory. Employees of our department gave consultations to scientific employees of this and other institutions that allowed us to get to the bottom of specificity of operations research problems. The book [5] published after lectures is an interesting attempt of authors to interpret queueing systems as models of military operations research.
At the turn of 50s and 60s many prominent applied mathematicians were interested in application of queueing systems to research defense problems: N.P.Buslenko, E.S.Ventsel, L.A.Ovcharov, N.A.Shishonok, B.P.Kredentser, M.M.Lastovchenko and others. It is not surprising that having felt big bent for such research work at the beginning of 1962 I.N.Kovalenko moved to Moscow and started working at a defenseinstitutes where N.P.Buslenko was the deputy chief for science.
At the turn of centuries an outstanding scientist L. Takács in the recommendation given to Igor Kovalenko wrote: «It is impossible even to list all theorems proved by Kovalenko». Needless to say, I also do not set as a purpose to review all of his works. Instead I will mention only some of his results; nevertheless, the versatility of his scientific interests and long-term creative activity of my old disciple will be seen.
In the sixties I.N.Kovalenko introduced and investigated piecewise linear Markovian processes (PLMP) that generalized the Cox’s processes (D. Cox, 1955) with supplementary variables. In B.V.Gnedenko and I.N.Kovalenko's well-known book «Introduction to the queueing theory» (last edition: [6]) PLMP were taken as basic statements about all queueing systems that cannot be described by discrete Markovian or semi-Markovian processes. In the book [7] PLMP and so-called piecewise linear units represent itself as models of complex systems with discrete events.
I.N.Kovalenko's monographs [8,9] are devoted to the approximation methods of calculation of reliability. Among the books written by him in the co-authorship the most known is [10].
I will only mention one result of I.N.Kovalenko concerning the theory of highly reliable systems. For certainty I’ll consider queueing system − the -channel system with waiting places, with the Poisson input with hazard rateand distribution function of repair time. An elementary approach to the analysis of a flow of losses of such a system consists in the decomposition of hazard rate of nonmonotonous losses and other characteristics in powers of a small parameterunder assumption that isa fixed function. It is much more difficult to investigate a case when tends to zero and thus also changes in such a way that only the value of some «small functional» remains fixed. In such a statement the most subtle results have been obtained by A.D. Solovyov (Moscow) and disciples see, e.g., [11]. In work [12] I.N.Kovalenko applied the wit probability approach which allowed him to improve the estimates of A.D. Solovyov’s school. Thus, for parameter (see above) the estimate
,
isderived where
.
The idea of statistical analysis of highly reliable systems by means of probability interpretation and simulating the coefficients of theexpansion of reliability characteristic in powers of a small parameter belongs to I.N.Kovalenko [13]. His disciple N.Yu.Kuznetsov investigated more general methods of variance reduction in statistical simulation, see [10]. V.D.Shpak [14] developed a method of fastsimulation of semi-Markovian systems. A.N.Nakonechny [15] investigated a general scheme of simulating rare events depending ona smallparameter and also the corresponding optimization of system’s parameters.
I.N.Kovalenko's participation in studying methods of analysis of reliability of systems with incomplete information has begun with article [16]. Later on L.S.Stoykova developed a method to estimate functionals, connected to reliability; her latest work: [17].
With the help of I.N.Kovalenko a cycle of works on queueing systems with repeated calls was carried out – I will specify only [18,19]. In 1997 I.N.Kovalenko [20] confirmed D.Kendall's well-known hypothesis from stochastic geometry.
To conclude the article I’ll mention probability combinatorial calculus where I.N.Kovalenko and his followers A.A.Levitskaya, V.I.Masol, M.N.Savchuk have brought a considerable contribution. All has started with the work [21], in which I.N.Kovalenko confirmed A.N.Kolmogorov's hypothesis on Boolean determinant of large dimension.
For the problem’s current state of art see [22].
References
1.Kovalenko I.N. On a class of optimal decision functions for a binomial family of distributions // Teor. Veroyatn. Primen. – 1959. – 4. – P. 101-105 (in Russian).
2.Kovalenko I.N. Some queueing problems with restrictions // Theor. Probab. Appl. – 1961. –6. – P. 204-208.
3.Kovalenko I.N. On the insensitivity of the steady-state distribution of states of a queueing system to the form of the service time distribution // Probl. Peredachi Inform. – 1962. – 11. – P. 147-151 (in Russian).
4.Kovalenko I.N. On the limit distribution of the first jump // Teor. Veroyatn. Primen.– 1960. – 5. – P.469-472 (in Russian).
5.Gnedenko B.V., Kovalenko I.N.Lectures on Queueing Theory. – Kyiv: KVIRTU, 1963. – Issues 1 to 3 (in Russian).
6.Gnedenko B.V., Kovalenko I.N.Introduction to Queueing Theory. – Moscow: URSS, 2005. – 397 p. (in Russian).
7.Buslenko N.P., Kalashnikov V.V., Kovalenko I.N. Lectures on Complex Systems Theory. – Moscow: Sovetskoe Radio, 1973. – 440 p. (in Russian).
8.Kovalenko I.N. Investigations on the Reliability of Complex Systems. – Kyiv: Naukova Dumka, 1975 (in Russian).
9.Kovalenko I.N. Analysis of Rare Events in Estimating the Efficiency and Reliability of Systems. – Moscow: Sovetskoe Radio, 1980. – 208 p.(in Russian).
10.Kovalenko I.N., Kuznetsov N.Yu., Pegg Ph.A. Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications. – Chichester: Wiley, 1997. – 303 p.
11.Gnedenko D.B., Solovyov A.D. The estimation of the reliability of complex repairable systems // Tekhn. Kibern. – 1975. – 13,No. 3. – P. 89-96 (in Russian).
12.Kovalenko I.N. Estimation of the intensity of the flow of nonmonotone failures in the queueing system// Ukr. Math. J. – 2000.– 52, No.9. – P. 1396-1402 (in Russian).
13.Kovalenko I.N. Evaluation of characteristics of highly reliable systems by analytical-statistical method // Electron. Modelirovanie. – 1980. – 2, No. 4. – P. 5-8 (in Russian).
14.Shpak V.D. Analytical-statistical estimates for non-stationary reliability and efficiency characteristics of semi-Markov systems // Kibernetika. – 1991. – No. 3. – P. 103-107 (in Russian).
15.Kovalenko I.N., Nakonechnyi A.N. Approximate Evaluation and Optimization of the Reliability. – Kyiv: Naukova Dumka, 1989. – 183 P. (in Russian).
16.Barzilovich E.Yu., Kashtanov V.A., Kovalenko I.N. On minimax criterions in problems of reliability theory // Tekhn. Kibern. – 1971. – No.3. – P. 87-98 (in Russian).
17.Stoykova L.S. Generalized Tchebyshev inequalities and their application in mathematical reliability theory (short review) // Kibernetika i Sistemny Analiz. – 2010. − No.3 (in Russian).
18.Koba O.V., Kovalenko I.M. The ergodicity condition for an retrial queue with nonlatticed orbit cycle distribution // Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. – 2004. –No. 8. – P. 70-77 (in Ukrainian).
19.Koba E.V. On a retrial queueing system with FIFO queueing discipline // Theory of Stochastic Processes. – 2002. – 24, No.8. – Р. 201-207.
20.Kovalenko I.N. Proof of David Kendall’s conjecture concerning the shape of large random polygons // Cybern. Syst. Anal. – 1997. –33, No.4. – P. 461-467.
21.Kovalenko I.N. On the limit distribution of the number of solutions of a random system of linear equations in the class of Boolean functions //Teor. Veroyatn. Primen.– 1967. – 12. – P.51-64 (in Russian).
22.Levitskaya A.A.Systems of random equations over finite algebraic structures // Kibernetika i Sistemny Analiz. – 2005. − No. 1. – P. 82-116 (in Russian).
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