semi-Markovian Random Walk Research Articles

Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero

On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of the level zero.

On the stationary characteristics of the extended model of type (s,S) with Gaussian distribution of summands

In this study, an extended model of type (s,S) is considered and a semi-Markovian random walk with Gaussian distribution of summands, which mathematically describes this model, is constructed. Moreover, under some weak assumptions the ergodicity of the process is discussed. In addition, characteristic function of the ergodic distribution of the process is expressed by means of appropriating a boundary functional S N . Using this relation, the exact formulas for the first four moments of ergodic distribution are obtained and the asymptotic expansions are derived for up to three terms, as β=S−s→∞. Moreover, approximations are offered for the coefficients of these asymptotic expansions, which are sufficiently satisfied of the needs of applications. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter β.

Explicit Formulae for the Queue Length Distribution of Batch Arrival Systems

A G θ I/G/1-type batch arrival system is considered. Explicit formulae for the distribution of queue length both at the fixed time t and as t → ∞ are obtained. The study is based on the generalization of Korolyuk's method for semi-markov random walks.

Asymptotic expansions for the moments of the Gaussian random walk with two barriers

In this study, the semi-Markovian random walk ( X( t)) with a normal distribution of summands and two barriers in the levels 0 and β>0 is considered. Moreover, under some weak assumptions the ergodicity of the process is discussed and the characteristic function of the ergodic distribution of the process X( t) is expressed by means of appropriating one of a boundary functional S N . Using this relation, the exact formulas for the first four moments of ergodic distribution are obtained and the asymptotic expansions are derived with three terms for the one's, as β→∞. Finally, using the Monte Carlo experiments, the degree of accuracy of obtained approximate formulas to exact one's have been tested.

Semi-Markov Random Walk in Poisson Approximation Scheme

We present a compound Poisson with drift approximation in series scheme of a semi-Markov random walk obtained by the superposition of two independent renewal processes.

Upper Bounds for Exponential Moments of Hitting Times for Semi-Markov Processes

Necessary and sufficient conditions for the existence of exponential moments for hitting times for semi-Markov processes are found. These conditions and the corresponding upper bounds for exponential moments are given in terms of test-functions. Applications to hitting times for semi-Markov random walks and queuing systems illustrate the results.

The Virtual Waiting Time for the Batch Arrival Queueing Systems

In the article the G η I/G/1-type batch arrival system with infinite waiting-room is considered. The explicit formulae for the distribution of the virtual waiting time at any fixed moment t and as t → ∞ are obtained. The study is based on generalization of Korolyuk's method for semi-markov random walks.

Application of the Superposition of Renewal Processes to the Study of Batch Arrival Queues

The Gθ/G/1-type batch arrival system is considered. We deal with non-steady-state characteristics of the system like the first busy period and the first idle time, the number of customers served on the first busy period. The study is based on a generalization of Korolyuk's method which he developed for semi-Markov random walks.

On the semi-Markovian random walk with two reflecting barriers

In this paper, the semi-Markovian random walk with two reflecting barriers is constructed mathematically and non-stationary distribution functions of it are expressed by means of the probability characteristics of renewal process {T n } and random walk {Y n } without barriers. In particular, when the time between two jump instants has exponential or Erlang distribution, explicit formulae are obtained for non-stationary distribution functions of the process. Moreover, explicit expressions are given for expected value, variance and moment generating function of the first reflection moment, an important boundary functional, of the process from lower reflecting barrier.

A Model of Inventory Control. II

A model of inventory control is considered. It is described by a semi-Markovian random walk with a negative drift at an angle of α (0^{\circ} \lt \alpha \lt 90^{\circ}), with positive random jumps, delays, and absorbing screens at zero and at i>a > 0. The Laplace transformation is found for the distribution of the first moment of the stoking up of a warehouse, and its first and second moments are explicitly obtained.

Semimarkov walk on the composition of two renewal processes

For a semi-Markov random walk on the composition of two renewal processes under the assumption that the positive jumps of the walk are distributed by an exponential law, an analytic expression is obtained for the generating function of the times of reaching level zero.