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

#### 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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2012
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/3138360`
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