Geodesic
Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 3, pp. 525-540.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper proposes a method for constructing models based on the analysis of birth and death processes with linear growth in semimartingale terms. Based on this method, stochastic models of simple just-in-time systems (analyzed in the theory of productive systems) and windows of vulnerability (widely discussed in risk theory) are considered. The main results obtained in the work are presented in terms of the average values of the time during which the processes reach zero values. At the same time, they are considered and used in the study of assessment models for local times of the processes. Here, simple Markov processes with a linear growth of intensities (perhaps, depending on time) are analyzed. At the same time, the obtained and used estimates are of theoretical interest. Thus, for example, the average value of the stopping time, at which the process reaches zero, depends on functions such as the harmonic number and the remainder term for the logarithmic function in the Taylor theorem. As the main result, the method of mathematical modeling of just-in-time systems and windows of vulnerability is proposed. The semimartingale description method used here should be considered as the first step of such a modeling, since, being a trajectory method, it allows diffusion (including non-Markov processes) generalizations when constructing stochastic models of windows of vulnerability and just-in-time. In the theoretical part of the article, we formulate statements for the average values of the local time and the stopping times when the birth and death processes reach a given value. This allows us to uniformly present estimates for the models of the just-in-time system and for windows of vulnerability, the result for which is given in the form of a limit theorem. The main results are formulated as theorems and lemmas. The proofs use semimartingale methods.
Keywords: modeling, process of birth and death, stopping time, compensator, intensity, counting process, trajectory, local time, just-in-time, window of vulnerability.
Mots-clés : martingale 
@article{VSGTU_2019_23_3_a7,
     author = {A. A. Butov and A. A. Kovalenko},
     title = {Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {525--540},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2019_23_3_a7/}
}
TY  - JOUR
AU  - A. A. Butov
AU  - A. A. Kovalenko
TI  - Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2019
SP  - 525
EP  - 540
VL  - 23
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2019_23_3_a7/
LA  - en
ID  - VSGTU_2019_23_3_a7
ER  - 
%0 Journal Article
%A A. A. Butov
%A A. A. Kovalenko
%T Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2019
%P 525-540
%V 23
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2019_23_3_a7/
%G en
%F VSGTU_2019_23_3_a7
A. A. Butov; A. A. Kovalenko. Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 3, pp. 525-540. http://geodesic.mathdoc.fr/item/VSGTU_2019_23_3_a7/

[1] Rajasooriya S. M., Tsokos C. P., Kaluarachchi P. K., “Stochastic modelling of vulnerability life cycle and security risk evaluation”, Journal of information Security, 7:4 (2016), 269–279 | DOI | MR

[2] Kaluarachchi P. K., Tsokos C. P., Rajasooriya S. M., “Cybersecurity: a statistical predictive model for the expected path length”, Journal of information Security, 7:3 (2016), 112–128 | DOI | MR

[3] Kaluarachchi P. K., Tsokos C. P., Rajasooriya S. M., “Non-homogeneous stochastic model for cyber security predictions”, Journal of information Security, 9:1 (2018), 12–24 | DOI

[4] Butov A. A., Kovalenko A. A., “Stochastic models of simple controlled systems just-in-time”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:3 (2018), 518–531 | DOI | Zbl

[5] Sugimori Y., Kusunoki K., Cho F., Uchikawa S., “Toyota production system and kanban system materialization of just-in-time and respect-for-human system”, Int. J. Prod. Res., 15:6 (1977), 553–564 | DOI

[6] Yavuz M., “Fuzziness in JIT and Lean Production Systems.”, Production Engineering and Management under Fuzziness, Studies in Fuzziness and Soft Computing, 252, Springer, Berlin, Heidelberg, 2010, 59–75 | DOI | Zbl

[7] Killi S., Morrison A., “Just-in-Time Teaching, Just-in-Need Learning: Designing towards Optimized Pedagogical Outcomes”, Universal Journal of Educational Research, 3:10 (2015), 742–750 | DOI

[8] Pape T., Bolz C. F., Hirschfeld R., “Adaptive just-in-time value class optimization for lowering memory consumption and improving execution time performance”, Science of Computer Programming, 140 (2017), 17–29 | DOI

[9] Chakrabarty R., Roy T., Chaudhuri K., “A production-inventory model with stochastic lead time and JIT set up cost”, Int. J. Oper. Res., 33:2 (2018), 161–178 | DOI | MR

[10] Föllmer H., “Random fields and diffusion processes”, École d'Été de Probabilités de Saint-Flour XV–XVII. 1985–87, Lecture Notes in Mathematics, 1362, Springer, Berlin, Heidelberg, 1988, 101–203 | DOI | MR

[11] Jacod J., Protter P., “Time Reversal on Lévy Processes”, Ann. Probab., 16:2 (1988), 620–641 | DOI | MR | Zbl

[12] Elliott R. J., Tsoi A. H., “Time reversal of non-Markov point processes”, Ann. Inst. Henri Poincaré, Probab. Stat., 26:2 (1990), 357–373 https://eudml.org/doc/77383 | MR | Zbl

[13] Privault N., Zambrini J.-C., “Markovian bridges and reversible diffusion processes with jumps”, Ann. Inst. Henri Poincaré, Probab. Stat., 40:5 (2004), 599–633 | DOI | MR | Zbl

[14] Conforti G., Léonard C., Murr R., Roelly S., “Bridges of Markov counting processes. Reciprocal classes and duality formulas”, Electron. Commun. Prob., 20 (2015), 18, 12 pp. | DOI | MR | Zbl

[15] Longla M., “Remarks on limit theorems for reversible Markov processes and their applications”, J. Stat. Plan. Inference., 187 (2017), 28–43 | DOI | MR | Zbl

[16] Ho L. S. T., Xu J., Crawford F. W., et al., “Birth/birth-death processes and their computable transition probabilities with biological applications”, J. Math. Biol., 76:4 (2018), 911–944 | DOI | MR | Zbl

[17] Butov A. A., “Some estimates for a one-dimensional birth and death process in a random environment”, Theory Probab. Appl., 36:3 (1991), 578–583 | DOI | MR | Zbl

[18] Butov A. A., “Martingale methods for random walks in a one-dimensional random environment”, Theory Probab. Appl., 39:4 (1994), 558–572 | DOI | MR | Zbl

[19] Butov A. A., “Random walks in random environments of a general type”, Stoch. Stoch. Reports, 48 (1994), 145–160 | DOI | MR | Zbl

[20] Butov A. A., “On the problem of optimal instant observations of the linear birth and death processes”, Stat. Probab. Lett., 101 (2015), 49–53 | DOI | MR | Zbl

[21] Dellacherie C., Capacités et processus stochastiques, Springer-Verlag, Berlin, 1972, ix+155 pp | DOI | MR