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Generalized Telegraph Process with Random Delays

  • Daoud Bshouty (a1), Antonio Di Crescenzo (a2), Barbara Martinucci (a2) and Shelemyahu Zacks (a3)

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).

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Copyright

Corresponding author

Postal address: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. Email address: daoud@tx.technion.ac.il
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, I-84084 Fisciano (SA), Italy.
∗∗∗ Email address: adicrescenzo@unisa.it
∗∗∗∗ Email address: bmartinucci@unisa.it
∗∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu

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Dedicated to Marcel Neuts on the occasion of his 75th birthday, in admiration of his most profound contributions to research and applications of stochastic processes.

Footnotes

References

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