Analytical and Computational Methods in Probability Theory

The mathematical model of cloud computing system based on the queuing system with the splitting of the incoming queries and synchronization of services is considered. The queuing system consists of a single buffer and N servers ($$N>2$$N>2), service times are independent and exponentially distributed. The incoming query enters the system as a whole and only before service is divided into subqueries, each subquery is served by its device. The servers with parts of the same query are considered to be employed as long as the query is not serviced as a whole: the query is handled only when the last of it is out and a new query may be served only when there are enough free servers (the response time is the maximum of service times of all parts of this query). Expressions for the stationary performance characteristics of the system are presented.

This study mainly concerned with the K-capacity queueing system with recurrent input and two heterogeneous servers. Interarrival times are independent and have an arbitrary distribution. There are two servers and server k has an exponential distribution with parameter $$\mu _k$$μk. Arriving customers choose server from the empty servers with equal probability. At an arrival time the customer joins the queue when both servers are busy. In addition an arrival leaves without having service when the system capacity is achieved. The defined system is represented by semi-Markov process and embedded Markov chain is obtained. Steady-state probabilities are found and loss probability is calculated by analyzing stream of overflows. Moreover loss probabilities are computed numerically for the queueing systems where the interarrival times are assumed as exponential, Erlang and deterministic distribution.

A closed network consists of several multi-servers with n customers. Service requirements of customers at a multi-server have a common cdf. State parameters of the network: for each multi-server empirical measure of the age of customers being serviced and for the queues the numbers of customers in them, all multiplied by $$n^{-1}$$n-1.Our objective: asymptotics of dynamics as $$n\rightarrow \infty $$n→∞. The asymptotics of dynamics of a single multi-server and its queue with an arrival process as the number of servers $$n\rightarrow \infty $$n→∞ is currently studied by famous scientists K. Ramanan, W. Whitt et al. Presently there are no universal results for general distributions of service requirements — the results are either for continuous or for discrete time ones; the same for the arrival process. We establish the asymptotics for a network in discrete time, find its equilibrium and prove convergence as $$t\rightarrow \infty $$t→∞.Motivation for studying such models: they represent call/contact centers and help to construct them effectively.

A class of systems with several non–ordinary Poisson input flows is studied. It is assumed that the flows are conflicting which means they cannot be served simultaneously. A service device carries out control function also. A probabilistic model for the class of the systems is constructed. Easily verifiable conditions of stationarity are determined analytically for two control algorithms: a cyclic algorithm for the homogeneous flows and a feedback algorithm with threshold priority and prolongations for the flows that differs in priority and intensity. A computer simulation model is described. Some examples of determining the quasi-optimal values of the control system parameters are given.

In this paper, we study a finite capacity Markovian multi-server queuing system with discouraged arrivals, reneging, and retention of reneging customers. The transient state probabilities of the queuing system are obtained by using a computational technique based on the 4th order Runge- Kutta method. With the help of the transient state probabilities, we develop some important measures of performance of the system, such as time-dependent expected system size, time-dependent expected reneging rate, and time-dependent expected retention rate. The transient behavior of the system size probabilities and the expected system size is also studied. Further, the variations in the expected system size, the expected reneging rate, and the expected retention rate with respect to the probability of retaining a reneging customer are also studied. Finally, the effect of discouraged arrivals in the same model is analyzed.

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 different queues. The task is to determine the rule for selecting service requirements and to determine the optimal strategy for establishing dynamic priorities. We consider a case $$n=2$$n=2.To this end, a controlled semi-Markov process is defined, on the trajectories of which a functional is constructed that determines the quality of management and takes into account the number of lost requirements, the number of serviced requirements, the time of the requirement stay in the system, and so on. An algorithm for determining the optimal strategy is formulated.

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, i.e., it is localized at a point $$\tau $$τ.

A control process for conflict flows of nonhomogeneous arrivals is considered. A mathematical model of a control system with variable structure is constructed and studied. Recurrence relations are found for the states of the server and the queues length. Recurrence relations are also obtained for one-dimensional probability distributions for the vector Markovian sequence of the system states in one step and in the number of steps equal to the number of the server’s basic states. We propose an iterative-majorant method that allows to find easily verifiable necessary conditions for the stationary probability distribution existence.

This paper deals with a queuing system $$GI^{\nu }|M|1|\infty $$GIν|M|1|∞, i. e., single server queue with general renewal arrivals, exponentially distributed service times and infinite number of waiting positions. The purpose is to find the steady-state results in terms of the probability-generating functions for the number of customers in the queue.

A class of priority queueing systems with non-zero switchover times is considered. Some performance characteristics such as distributions of busy periods, conditions of stationarity, traffic coefficients, distribution of queue length, probabilities of the system’s state, etc. are presented. Numerical algorithms for their modelling are developed.

In the paper, the retrial queueing system of MMPP/M/1 type is considered. The process of the number of calls in the system is analyzed. The method for the approximate calculation of the first and the second moments is suggested. We propose the method of the discrete gamma approximation based on obtained moments. The numerical analysis of the obtained results for different values of the system parameters is provided. Comparison of the distributions obtained by simulation and the approximate ones is presented.

The paper contains research of the inventory management model with On/Off control. We study mathematical model of inventory management system under following conditions: the rate of input product flow is a constant, the random part of demand is modeled as Poisson process with a piecewise-constant intensity. Firstly, some property of stationary probability density function of inventory level accumulated in system is developed. Then explicit expression for a stationary distribution of the inventory level is obtained for Phase-type distributions of purchase values of demands. For case arbitrary distribution of demands purchases values Fourier transform of the stationary probability density function is determined. Finally, the obtained results are discussed with illustrative numerical examples.

A c–server queueing system providing service to two types of customers, say, Type 1 and Type 2 to which customers arrive according to a marked Poisson process is considered. A Type 1 customer has to receive service by one of c servers while a Type 2 customer may be served by a Type 1 customer (with probability p) who is available to act as a server soon after getting own service or by one of c servers. Upon completion of a service a free server will offer service to a Type 1 customer on a FCFS basis. However, if there is no Type 1 customer waiting in the system, that server will serve a Type 2 customer if one of that type is present in the queue. The service time is exponentially distributed for each category. We consider preemptive service discipline. Condition for system stability is established. Crucial system characteristics are computed.

A single server retrial queueing-inventory is considered in which customers join directly to the orbit according to a Markovian arrival process (MAP). Service time of customers are independent and identical distributed phase-type distributed (PH) random variables. Inter retrial times are exponentially distributed with parameter $$n\eta $$nη when n customers are in the orbit. Unsuccessful retrial customers tend to leave the system (impatience) with positive probability. In addition we also introduce search of orbital customers for next service with state dependent probability, immediately on current service completion. This system is shown to be always stable. We compute the long run system state probability. Under certain stringent conditions we prove that a particular case has a product form solution. We get explicit solutions to some retrial queueing models.

The problem of performance evaluation of a wireless sensor node with energy harvesting is considered. It is reduced to analysis of the stationary distribution of a single-server queueing system to which the Marked Markovian Arrival process of customers and energy units arrives. The buffer for the customers has an infinite capacity while the buffer for energy accumulation is the finite one. Energy is required for providing service of a customer. If energy is not available, service is postponed. To account possible fluctuation of the system parameters, it is assumed that the system operates in a random environment which is defined by the finite state continuous-time Markov chain. Such fluctuations are possible, e.g., due to the change of the signal’s generation rate in the sensor node or the change of energy harvesting rate depending on weather conditions. Under the fixed state of the random environment, the rates of arrivals of energy and customers and service rates are constant while they can change their values at the moments of the jumps of the random environment. Customers in the buffer may be impatient and leave the system after the exponentially distributed amount of time. The stationary distribution of the system states and the main performance measures of the system are calculated.

In this paper we consider dynamics of complex systems using random neural networks with an infinite number of cells. The Cauchy problem for singular perturbed infinite order systems of stochastic differential equations which describes the random neural network with infinite number of cells is studied.

Considered system consists of three renewable components that are connected in parallel. The components are described by continuous time independent alternating processes. The sojourn times in the operative state for all components have exponential distributions. The sojourn times in the failed state have arbitrary absolute continuous distributions. All sojourn times are independent. The system is working at time t if at least one component is working. We consider a problem of computation of system reliability on given time interval for the known initial states of the components. Non-stationary and stationary regimes are considered.

Markovization method is used for heterogeneous double redundant hot standby renewable reliability system analysis. The time dependent, stationary and quasi-stationary probability distributions for the system are calculated.

New models were developed in actuarial sciences during the last two decades. They include different notions of insurance company ruin (bankruptcy) and other objective functions evaluating the company performance. Several types of decision (such as dividends payment, reinsurance, investment) are used for optimization of company functioning. Therefore it is necessary to be sure that the model under consideration is stable with respect to parameters fluctuation and perturbation of underlying stochastic processes. The aim of the paper is description of methods for investigation of these problems and presentation of recent results concerning some insurance models.

We study the life annuity insurance model when simple investment strategies (SISs) of the two types are used: risky investments and risk-free ones. According to a SIS of the first type, the insurance company invests a constant positive part of its surplus into a risky asset while the remaining part is invested in a risk-free asset. A risk-free SIS means that the whole surplus is invested in a risk-free asset. We formulate and study some associated singular problems for linear integro-differential equations (IDEs). For the case of exponential distribution of revenue sizes, we state that survival probabilities as the functions of the initial surplus (IS) are unique solutions of the corresponding problems. Using the results of computational experiments, we conclude that in the region of small sizes of IS the risky SIS may be more effective tool for increasing of the survival probability than risk-free one.

We derive asymptotic approximation of high risk probability (ruin probability) for multidimensional aggregated reliability index which is a linear combination of single independent indexes, whose reliability functions (distribution tails) behave like Weibull tails.

The paper deals with the sensitivity analysis of reliability and performance measures for a multi-server queueing system where customers at the head of the queue retries to occupy a server in exponentially distributed time. The servers can differ in service and reliability characteristics. We have proved the insensitivity of the mean number of customers in the system to the type of allocation policy for equal service rates and confirmed a weak sensitivity in a general case of unequal service rates. A further sensitivity analysis is conducted to investigate the effect of changes in system parameters on a reliability function, a distribution of the number of failures of a server and a maximum queue length during a life time.

We first give historical remarks about the forgotten univariate Teissier model. We introduce symmetric and asymmetric bivarite versions of the Teissier distribution and outline basic properties. The corresponding copula is obtained and applications are discussed.

The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy-power inequality for the weighted entropy and discuss connections with weighted Lieb’s splitting inequality. The concepts of rates of the weighted entropy and information are also discussed.

The Shannon Noiseless coding theorem (the data-compression principle) asserts that for an information source with an alphabet $$\mathcal {X}=\{0,\ldots ,\ell -1\}$$X={0,…,ℓ-1} and an asymptotic equipartition property, one can reduce the number of stored strings $$(x_0,\ldots ,x_{n-1})\in \mathcal {X}^{n}$$(x0,…,xn-1)∈Xn to $$\ell ^{nh}$$ℓnh with an arbitrary small error-probability. Here h is the entropy rate of the source (calculated to the base $$\ell $$ℓ). We consider further reduction based on the concept of utility of a string measured in terms of a rate of a weight function. The novelty of the work is that the distribution of memory is analyzed from a probabilistic point of view. A convenient tool for assessing the degree of reduction is a probabilistic large deviation principle. Assuming a Markov-type setting, we discuss some relevant formulas and examples.

It is well known that the classical Lindeberg condition is sufficient for validity of the central limit theorem. It will be also a necessary if the summands satisfy the infinite smallness condition (Feller’s theorem). The limit theorems for the distributions of the sums of independent random variables which do not use the infinite smallness condition were called non-classical.In this paper a non-classical version of Lindeberg-Feller’s theorem is given. The exact bounds for the Lindeberg, Rotar characteristics using the difference of the distribution of sum of independent random variables and a standard normal distribution are established. These results improve Feller’s theorem.

Let the vertices of a complete q-ary tree be assigned independent random marks having uniform distribution on a finite alphabet. We consider pairs of identically marked embeddings of a given subtree template. An asymptotic formula for the expectation of the number of such pairs is obtained and the Poisson limit theorem for this number is proposed.

We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice long-time behavior. We are interested in limit probability laws related to the steady state viewed from the center-of-mass coordinate system.This paper is the extended version of the talk prepared for the International Conference ACMPT-2017 dedicated to the 90th birth anniversary of Aleksandr Dmitrievich Solov$${}'\!$$′ev.

We propose deterministic and stochastic models of clock synchronization in nodes of large distributed network locally coupled with a reliable external exact time server.

The goal is to modify the known method of mirror descent (MD) in convex optimization, which having been proposed by Nemirovsky and Yudin in 1979 and generalized the standard gradient method. To start, the paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in continuous time. In particular, in the Euclidean case, the heavy ball method by Polyak is realized. It is noted that the new method does not use additional averaging of the points. Then, a discrete algorithm of inertial MD is described and the upper bound on error in objective function is proved. Finally, inertial MD randomized algorithm for finding a principal eigenvector of a given stochastic matrix (i.e., for solving a well known PageRank problem) is treated. Particular numerical example illustrates the general decrease of the error in time and corroborates theoretical results.

Earlier for support of information security of network interactions a control of network connections by means of meta data was suggested. Meta data contain information on admissible interactions of tasks and positions of applications for their solution in a distributed network.This security mechanism can be attacked. To prevent found vulnerabilities an extension of meta data was built. It was necessary to consider stochastic elements in extended meta data. Control of network interconnections based on stochastic meta data was investigated.

The paper contains several results on the existence of limits for the first two moments of the popular model in the population dynamics: continuous-time branching random walks on the multidimensional lattice $$\mathbb Z^d$$Zd, $$d\ge 1$$d≥1, with immigration and infinite number of initial particles. Additional result concerns the Lyapunov stability of the moments with respect to small perturbations of the parameters of the model such as mortality rate, the rate of the birth of $$(n-1)$$(n-1) offsprings and, finally, the immigration rate.

We consider a class of additive functionals of ergodic semi-Markov processes and prove that their associated Markov renewal processes have a martingale decomposition representation. This leads to two main results, a functional central limit theorem for the additive functionals of semi-Markov processes and a functional almost sure limit theorem for their corresponding empirical processes, called an almost sure functional central limit theorem.

In this paper we have two main goals. One of them is to construct stochastic processes associated with a class of systems of semilinear parabolic equations which allow to obtain a probabilistic representations of classical solutions of the Cauchy problem for systems from this class. The second goal is to reduce the Cauchy problem solution of a PDE system to solution of a closed system of stochastic relations, prove the existence and uniqueness theorem for the correspondent stochastic system and apply it to develop algorithms to construct numerically the required solution of the PDE system.

The well-known Leibniz Criterion or alternating series test of convergence of alternating series is generalized for the case when the absolute value of terms of series are “not absolutely monotonously” convergent to zero. Questions of accuracy of the estimation for the series remainder are considered.

We have developed a quantum-semiclassical approach for calculation of transition probabilities in few-dimensional quantum systems. In this approach the problem is reduced to the Schrödinger-like equation for some degrees of freedom which integrated symphoniously with the classical equations describing the remaining part. This approach was successfully applied for treating self-ionization of hydrogen-like ions in magnetic fields, break-up of some halo nuclei and for excitation and stripping of helium ions by protons. Here we present the method application to calculation of ionization and excitation/deexcitation of helium ions by slow antiprotons. The calculated cross sections are important for experimental investigations in antiproton physics. Moreover, the considered case is very perspective as an object for investigation of quantum measurements. Actually, the charge-exchange channel, dominant in collisions with protons, is absent in our case and all possible quantum communication channels are accurately described in our approach.

The properties of the homogeneity tests of Smirnov, Lehmann-Rosenblatt, Anderson-Darling, k-sampling tests of Anderson-Darling and Zhang have been studied. Models of limiting distributions for k-sampling Anderson-Darling test under various numbers of compared samples have been presented. Power ratings have been obtained. Comparative analysis of the power of the homogeneity tests has been performed. The tests have been ordered in terms of power relative to various alternatives. Recommendations on the application of tests have been given.

Computationally efficient algorithm realizing exact computation of decomposable statistics distributions for multinomial scheme is described. The algorithm is based on the embedding the scheme into nonhomogeneous Markov chain. It was used to compute the Pearson statistics distribution and distributions of some statistics for the random allocation of particles into cells. Comparisons of exact numerical values of distribution functions of statistics with usually used approximations from corresponding limit theorems show that exact tail probabilities may be considerably larger than that of approximating distributions.

The paper is devoted to construction of parsimonious (small-parametric) models of high-order Markov chains and to computer algorithms for statistical inferences on parameters of these models.

We introduce a novel approach to estimate the parameters of a mixture of two distributions. The method combines a grid approach with the method of moments and can be applied to a wide range of two-component mixture models. The grid approach enables the use of parallel computing and the method can easily be combined with resampling techniques. We derive the method for the special cases when the data are described by the mixture of two Weibull distributions or the mixture of two normal distributions, and apply the method on gene expression data from 409 $$ER+$$ER+ breast cancer patients.

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.

We study statistical experiments with random change of time, which transforms a discrete stochastic basis in a continuous one. The adapted stochastic experiments are studied in continuous stochastic basis in the series scheme. The transition to limit by the series parameter generates an approximation of adapted statistical experiments by a diffusion process with evolution.The average intensity parameter of renewal times are estimated in three different cases: the Poisson renewal process, a stationary renewal process with delay and the general renewal process with Weibull-Gnedenko renewal time distribution.