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Generalized Telegraph Process with Random Delays

Part of:Special processesStochastic processes

Published online by Cambridge University Press:  04 February 2016

Daoud Bshouty,
Antonio Di Crescenzo,
Barbara Martinucci and
Shelemyahu Zacks
Daoud Bshouty*
Affiliation:
Technion - Israel Institute of Technology
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Barbara Martinucci*
Affiliation:
Università di Salerno
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
∗ 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.
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, I-84084 Fisciano (SA), Italy.
∗∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
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Abstract

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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).

Keywords

Telegraph processdelayed telegraphalternating renewalexponential distributiondamped processcompound Poisson processhypergeometric series

MSC classification

Primary:60G55: Point processes
Secondary:60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)60G40: Stopping times; optimal stopping problems; gambling theory

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012, pp. 850 - 865
Copyright
© Applied Probability Trust 

Footnotes

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.

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