The laplace transform for the distribution of the lower bound functional in a semi-Markov walk process with a delay screen at zero

A jump process of a semi-Markov walk is constructed. The Laplace transform for the distribution of the first instant of crossing the zero level is found with the proviso that the random walk adheres to a compound Laplace distribution.     

Laplace transform of the distribution of the semi-Markov walk process with a positive drift, negative jumps, and a delay screen at zero

In this paper, we investigate the process of semi-Markov random walk with a positive drift, negative jumps, and a delay screen at zero. Explicit forms are found of the Laplace transform in time, the Laplace-Stieltjes transform by the state (phase) of the conditional and unconditional distributions of the process states, and the explicit form of the Laplace-Stieltjes transform of the ergodic distribution of the process.     

Investigation of some probabilistic characteristics of one class of semi-Markov wandering with delaying screens

The process of semi-Markov wandering with delaying screens “b” and “a” (a > b > 0) is constructed by the given sequence of independent and identically distributed random vectors (ξ _i , η _i ), i ≥ 1. The integral equation for the Laplace transform by time and the Laplace-Stieltjes transform by the phase of its conditional distribution is derived. If the wandering occurs by a complicated Laplace distribution, the ergodic distribution of the process and its moments are found. Then, the integral equation for the generating function of the conditional distribution of the number of process steps at which it firstly reaches the level a is derived. When the wandering occurs by the simple Laplace distribution, its generating functions and moments are found.     

A parallel algorithm for random walk construction with applicationto the Monte Carlo solution of partial differential equations

Random walks are widely applicable in statistical and scientific computations. In particular, they are used in the Monte Carlo method to solve elliptic and parabolic partial differential equations (PDEs). This method holds several advantages over other methods for PDEs as it solves problems with irregular boundaries and/or discontinuities, gives solutions at individual points, and exhibits great parallelism. However, the generation of each random walk in the Monte Carlo method has been done sequentially because each point in the walk is derived from the preceding point by moving one grid step along a randomly selected direction. A parallel algorithm for random walk generation in regular as well as irregular regions is presented. The algorithm is based on parallel prefix computations. The communication structure of the algorithm is shown to ideally fit on a hypercube of n nodes, where n is the number of processors     

On the generation of random numbers with at choice distribution

This paper deals with a method supplying truly random numbers in cycle-free sequences of any length and with a specified statistical distribution as desired. The method is based on an appropriate randomness-conserving rearrangement of truly random numbers delivered by a random process. Here the radioactive decay of nuclei is employed as the random process that produces a basic set of truly random numbers with a Poisson distribution. However, any other random process can be used with the method. The paper contains the theory and some essential points of programming for a computer.     

Filtration of a random process in a statistically uncertain linear stochastic differential system

Minimax filtration of a process in a stochastic linear differential system with uncertain perturbation intensities for dynamics and observation models is studied. The filter is optimized by an integral quality criterion. Minimax filtering equations are derived from the solution of the dual optimization problem. A numerical filter designing method is described and its convergence is proved. Results of numerical experiments are given.Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 59–71.Original Russian Text Copyright © 2005 by Miller, Pankov.This work was supported by the Russian Foundation for Basic Research, project no. 02-01-00361.     

A model of finite-step random walk with absorbent boundaries

This paper proposes a model of finite-step lattice random walk with absorbent boundaries. We address a problem of optimal stop for this model, which is defined as the absorbent boundary value with maximum profit. Compared with many existing optimal stop investigations in the random process, our study only considers the small-sample behaviour (i.e., small number of steps behaviour) and does not consider the limit behaviour of the walk. The optimal stop time is given based on classical probability computation. Since the small-sample is more practical and common than the large-sample in many real world problems, the result obtained in this paper may provide some useful guidelines for real applications associated with the finite-step random walk such as the stock market and gambling games.     

Laplace-Stieltjes transform of the distribution of the first moment of crossing the level a(a > 0) by a semi-Markovian random walk with positive drift and negative jumps

Using a sequence of independent equally distributed two-dimensional random variables, the process of a semi-Markovian random walk with positive drift and negative jumps is constructed. The Laplace-Stieltjes transform of the distribution of the first moment of crossing the level a(a > 0) is found.     

Time averaged distribution of a discrete-time quantum walk on the path

The discrete time quantum walk which is a quantum counterpart of random walk plays important roles in the theory of quantum information theory. In the present paper, we focus on discrete time quantum walks viewed as quantization of random walks on the path. We obtain a weak limit theorem for the time averaged distribution of our quantum walks.     

Testing the random walk hypothesis through robust estimation of correlation

This paper uses Monte Carlo simulations to examine the properties of the conventional Pearson and some of the most well-known robust to outliers estimators of correlation in the presence of general heteroskedasticity. We show that the tests of a random walk based on the Pearson autocorrelation coefficient, including the Lo and MacKinlay [1988. Stock market prices do not follow random walks: evidence from a simple specification test. Rev. Financial Studies 1, 41-66] robust form of the variance-ratio test, can be unreliable in the presence of some forms of conditional heteroskedasticity. As an alternative to the Pearson autocorrelation coefficient, we propose the median coefficient of autocorrelation. Our simulation results show that, in contrast to the Pearson autocorrelation coefficient, the median coefficient of autocorrelation is robust to conditional heteroskedasticity. When applied to exchange rate returns, the variance-ratio test based on the median autocorrelation coefficient provides stronger evidence against the random walk hypothesis compared with the Lo and MacKinlay [1988. Stock market prices do not follow random walks: evidence from a simple specification test. Rev. Financial Studies 1, 41-66] robust variance-ratio test.     

Total cumulative work until failure of a system modelled by a finite semi-Markov process

Let Y = (Y_t , t ≥ 0) be a semi-Markov reliability model of a system whose finite state space S is partitioned as S = W ∪ R ∪ F, where W, R and F stand for the sets of working, repair, and failed system states respectively. The system characteristic considered in this paper is the cumulative operational time until failure, C, defined as the time spent by Y in the set of working states W until failure, i.e. until absorption into F. A method is described here for the computation of the cumulative distribution function of C. It is based on approximating the continuous-time model by a discrete-parameter semi-Markov process for which the cumulative distribution function of the discrete-time counterpart of C can be computed by a recurrence relation. The reliability model of a two-unit parallel system with sequential preventive maintenance is examined numerically by the present method and a comparison is made with results from simulation.     

Optimal TAL-based registration with cell-based central policy in mobile cellular networks: a semi-Markov process approach

The Journal of Supercomputing - LTE networks consist of tracking areas (TAs) or a group of cells, while several TAs constitute a TA list (TAL). The LTE network adopts TAL-based registration, where,...     

Filtering of a semi-Markov queueing system with retrials

A recursive differential equation for the conditional probability of the number of waiting customers is derived for the M_wrM|G|1 queue with continuous monitoring of the server state.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 169–173, July–August, 1991.     

Control of a group of systems whose communication channels are assigned by a semi-Markov process

This technical note is concerned with the problem of medium access constraint for a group of networked systems. The scheduling of each subsystem is defined by a stochastic protocol, which can be modelled by a semi-Makov chain with a time-varying transition probability matrix. The resulting closed-loop nonlinear systems are a semi-Markovian jump system with delay. Sufficient conditions for exponential mean-square stability of the resulting closed-loop systems are derived via a Lyapunov–Krasovskii method. Based on the stability criterion, the controller gain of each subsystem is designed. A simulation example is used to demonstrate the effectiveness of proposed method.