Analytical and computational methods in probability theory and its applications (ACMPT-2017). Proceedings of the International Scientific Conference
We study a single-channel queuing system with an arbitrary distribution of the duration of service requirements, on the input of which there are n Poisson processes. The requirements of the various processes come in dierent queues. The task is to determine the rule for selecting service requirements and to determine the optimal strategy for establish- ing dynamic priorities.
The functionals constructed on trajectories of the controlled semi- Markov process with a finite set of states are investigated. Theorems of functionals’ structure (dependences on the probability measures defining the Markov homogeneous randomized strategy of control) and of structure of probability measures on which the extremum of these functionals is reached, are formulated. Examples are given.
We propose deterministic and stochastic models of clock synchronization in nodes of large distributed network locally coupled with a reliable external exact time server.
We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the center-of-mass coordinate system.
We construct a mathematical model of anti-virus protection of local area networks. The model belongs to the class of regenerative processes. To protect the network from the external attacks of viruses and the spread of viruses within the network we apply two methods: updating antivirus signatures and reinstallings of operating systems (OS). Operating systems are reinstalled in the case of failure of any of the computers (non- scheduled emergent reinstalling) or at scheduled time moments. We consider a maximization problem of an average unit income. The cumulative distribution function (CDF) of the scheduled intervals between complete OS reinstallings is considered as a control. We prove that the optimal CDF has to be degenerate, ie, it is localized at a point τ τ
We study the problem of parameters estimating if there is a slight deviation between the parametric model and real distributions. The estimator is based on suboptimal testing of builded by a special way nonparametric hypotheses. It is proposed a natural for this problem risk function. We found that the risk function has an exponential decrease to the mean number of observations. Numerical results of a comparative analysis our risk function behaviour for proposed estimator and some another estimators are outlined. We give remarks how to apply this results to machine learning methods.
This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.