The paper studies the distribution of the location, at time $t$, of a particle moving $U$ time units upwards, $V$ time units downwards, and $W$ time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of $U$, $V$ and $W$ are absolutely continuous. The velocities are $v=+1$ upwards, $v=-1$ downwards and $v=0$ during idle periods. Let $Y^+(t)$, $Y^-(t)$ and $Y^0(t)$ denote the total time in $(0,t)$ of movements upwards, downwards and no movements, respectively. The exact distributions of $Y^+(t)$ is derived. We also obtain the probability law of $X(t)=Y^+(t)-Y^-(t)$, which describes the particle's location at time $t$. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

GENERALIZED TELEGRAPH PROCESS WITH RANDOM DELAYS

Di Crescenzo, Antonio;Martinucci, Barbara;
2012

Abstract

The paper studies the distribution of the location, at time $t$, of a particle moving $U$ time units upwards, $V$ time units downwards, and $W$ time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of $U$, $V$ and $W$ are absolutely continuous. The velocities are $v=+1$ upwards, $v=-1$ downwards and $v=0$ during idle periods. Let $Y^+(t)$, $Y^-(t)$ and $Y^0(t)$ denote the total time in $(0,t)$ of movements upwards, downwards and no movements, respectively. The exact distributions of $Y^+(t)$ is derived. We also obtain the probability law of $X(t)=Y^+(t)-Y^-(t)$, which describes the particle's location at time $t$. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/3138360
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