GENERALIZED TELEGRAPH PROCESS WITH RANDOM DELAYS

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).