Probabilistic micromechanical model for two-dimensional slow particulate flow: Dry friction

A probabilistic model for the slow shearing motion of spherical particles dominated by dry inter-particle frictional forces is proposed. The particles can move by sliding, rolling, solid body rotation and jumping following the criterion of minimum energy dissipation. The continuous deformation of particle assembly is regarded as a Markov process in discrete time and continuous space. The components of the contact angle between two neighboring particles are taken as the primary random variables. The coefficients of the probabilistic governing equation are determined from the rate of energy dissipation and micromechanical analysis of particle motion. The inter-particle sliding and rolling friction laws are input data. The constitutive relationships, apparent viscosity and internal friction angle of the particulate material as a whole are obtained by integration. The validity of the proposed model is verified by comparing the model predictions to analytical solutions and available experimental results.

Using the control problem of a two-state Markov process in discrete time as an example, we consider the basic stages concerning the application of theory of conditional Markov processes to synthesize optimal algorithms of the control of stochastic systems. It is assumed that the control changes the statistical properties of the states of a controlled plant. The numerical method for solving the problem and the results of solving particular example are presented. The special features of the solution of this problem compared to the well-known problem in continuous time are discussed.

We considered the issues associated with methods for solving the problems of representation of continuous Markov processes in discrete-time by the example of radar signals, which are subject to the simultaneous effect of additive and multiplicative noise described by stochastic differential equations. We considered the algorithm, which allows us to carry out computer simulation of fading amplitude, phase and information parameter of the signal in discrete observation time using the example of a continuous frequency-modulated signal, the amplitude of which adheres to the Nakagami distribution that is typical for radar signals under simultaneous effect of additive and multiplicative noise. It is shown that with the help of computer simulation it is possible to create not only optimal algorithms, but also their corresponding optimal structural diagrams, which makes it possible to elevate the processing of radar signals to higher standards.

A Markov process in discrete time with a finite state space is controlled by choosing the transition probabilities from a given convex family of distributions depending on the present state. The immediate cost is prescribed for each choice and it is required to minimise the average expected cost over an infinite future. The paper considers a special case of this general problem and provides the foundation for a general solution. The main result is that an optimal policy exists if each state of the system can be reached with positive probability from any other state by choosing a suitable policy.

The rapid development of UAV technologies and the wide access to them by various consumers give rise to a new class of threats. To neutralize them, complex UAV protection systems are created, which include passive and active measuring systems. As a passive system, a wireless sensor network is used, which allows to determine location of UAV with high accuracy based on TDOA measurements. However, in real conditions, abnormal (rough) measurement results may appear that lead to divergence of Kalman filtering algorithms for UAV movement parameters.On the basis of a mathematical apparatus of the mixed Markov processes in discrete time, a quasi-optimal algorithm of adaptive estimation of UAV movement parameters based on the TDOA-measurement of the sensor network in the presence of abnormal measurement errors was synthesized. The efficiency analysis of obtained algorithm and its comparison with algorithm of Kalman filtering are performed using the statistical simulation..

Extended piecewise Markov processes in discrete time

To increase the detection efficiency of moving targets masked by interfering reflections, radars are widely used, in which target spatial coordinates and also range rate are measured. When solving the radar data processing problems, the presence of target range rate measurements increases the target tracking efficiency in the presence of false plots, and can significantly improve the tracking quality, especially with intensive target maneuvering. In the work on the mathematical apparatus basis of the mixed Markov processes in discrete time, a quasi-optimal adaptive algorithm for tracking maneuvering targets in a difficult jamming environment for a multifunctional radar with range rate measurement is developed. Using statistical computer modeling, a comparative analysis of tracking algorithms with and without measurement of range rate in the same interference environment at different data rates is performed.

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting “MRCA age” process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright–Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on ℝ++×ℝ++ with intensity measures of the form λ⊗μ where λ is Lebesgue measure, and μ (the “family lifetime measure”) is an arbitrary, absolutely continuous measure satisfying μ((0, ∞))=∞ and μ((x, ∞))<∞ for all x>0. Special cases of this construction describe the time evolution of the MRCA age in (1+β)-stable continuous state branching processes conditioned on nonextinction—a particular case of which, β=1, is Feller’s continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist. Moreover, we introduce a new family of Markov processes that stands in a relation with respect to the general (1+β)-stable continuous state branching process and its conditioned version that is similar to the one between the family of Bessel-squared diffusions and the unconditioned and conditioned Feller continuous state branching process.

In this paper, we study the classification problem of discrete time and continuous time Markov processes with a tree structure. We first show some useful properties associated with the fixed points of a nondecreasing mapping. Mainly we find the conditions for a fixed point to be the minimal fixed point by using fixed point theory and degree theory. We then use these results to identify conditions for Markov chains of M/G/1 type or GI/M/1 type with a tree structure to be positive recurrent, null recurrent, or transient. The results are generalized to Markov chains of matrix M/G/1 type with a tree structure. For all these cases, a relationship between a certain fixed point, the matrix of partial differentiation (Jacobian) associated with the fixed point, and the classification of the Markov chain with a tree structure is established. More specifically, we show that the Perron-Frobenius eigenvalue of the matrix of partial differentiation associated with a certain fixed point provides information for a complete classification of the Markov chains of interest.

This work investigates the computation of finite abstractions of Stochastic Max-Plus-Linear (SMPL) systems and their formal verification against general bounded-time linear temporal specifications. SMPL systems are probabilistic extensions of discrete-event MPL systems, which are widely employed for modeling engineering systems dealing with practical timing and synchronization issues. Departing from the standard existing approaches for the analysis of SMPL systems, we newly propose to construct formal, finite abstractions of a given SMPL system: the SMPL system is first re-formulated as a discrete-time Markov process, then abstracted as a finite-state Markov Chain (MC). The derivation of precise guarantees on the level of the introduced formal approximation allows us to probabilistically model check the obtained MC against bounded-time linear temporal specifications (which are of rather general applicability), and to reliably export the obtained results over the original SMPL system. The approach is practically implemented on a dedicated software and is elucidated and run over numerical examples.

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter, time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuous-time stochastic processes where the evolution is a flow through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.

Absorbing sets within the state space of a general discrete-time Markov process have been shown to play a crucial role for the model-checking of model specifications over an infinite time horizon. Infinite-horizon specifications such as probabilistic invariance, reachability and reach-avoid have formally been characterized and computed based on this structural property of the model. This paper further extends these recent results by investigating stability properties of the model over absorbing sets. Theoretical results are developed to study attractivity properties of such sets, and the outcomes are applied over a case study.

Comparison of ITRF2014 station coordinate input time series of DORIS, VLBI and GNSS

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.

Reduced-order [formula omitted] filtering for linear systems with Markovian jump parameters