Abstract
In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v1 and v2. We consider a special type of boundary which can trap the particle for a random time. We found closed-form expressions for the stationary distribution of the position of the particle not only for the alternating Markov process but also for a broad class of semi-Markov processes.
Highlights
In this paper we study the stationary distribution of a one-dimensional random motion performed with two velocities, where the random times separating consecutive velocity changes perform an alternating Markov process
We assume that the particle moves on the line in the following manner: At each instant it moves according to one of two velocities, namely v1 > 0 or v2 < 0 Starting at the position x0 ∈ the particle continues its motion with velocity v1 > 0 during random time τ1, where τ1 is an exponential random variable with parameter λ1, the particle moves with velocity v2 < 0 during random time τ2, where τ2 is an exponential distributed random variable with parameter λ2
The two-state continuous time random walk has been studied by many researchers for the Markov case and only a few have studied for non-Markovian processes [10]
Summary
In this paper we study the stationary distribution of a one-dimensional random motion performed with two velocities, where the random times separating consecutive velocity changes perform an alternating Markov process. The sojourn times of this process are exponentially distributed random variables. We consider a generalization of these results for semi-Markov processes, i.e., when the random variables τ1 and τ2 are different from exponential. This paper is divided in two main parts, namely the Markov case and the generalization to the semi-Markov modeling. In the first part of this paper, consists on finding the stationary distribution of the well-known telegrapher process on the line with delays in reflecting boundaries. We find the stationary distribution of a more general continuous time random walk when the sojourn times are generally distributed
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