Generalized telegraph process with random delays

TLDR: In this article, the authors studied 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).

Abstract: In this paper we study 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 distribution 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).