Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T19:59:32.492Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized integrated telegraph processes and the distribution of related stopping times

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

S. Zacks
S. Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

Keywords

Telegraph processalternating renewal processcompound Poissonupper and lower crossing times

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K15: Markov renewal processes, semi-Markov processes 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 497 - 507
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Di Crescenzo, A. (2000). On Brownian motions with alternating drifts. In Cybernetics and Systems 2000 (15th Europ. Meeting Cybernetics Systems Res.), ed. Trappi, R., Austrian Society for Cybernetics, Vienna, pp. 324–329.Google Scholar
Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.Google Scholar
Di Crescenzo, A., and Pellerey, F. (2000). Stochastic comparison of wear processes characterized by random linear wear rates. In Papers from the 2nd Internat. Conf. Math. Methods Reliability (Bordeaux, July 2000), Vol. 1 (abstracts), eds. Nikulin, M. and Limnios, N., Birkhäuser, Boston, MA, pp. 339–342.Google Scholar
Di Crescenzo, A., and Pellerey, F. (2002). On prices’ evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171–184.Google Scholar
Foong, S. K. (1992). First-passage time, maximum displacement, and Kac's solution of the telegrapher equation. Phys. Rev. A 46, R707–R710.Google Scholar
Foong, S. K., and Kanno, S. (1994). Properties of the telegrapher's random process with or without a trap. Stoch. Process. Appl. 53, 147–173.Google Scholar
Masoliver, J., and Weiss, G. H. (1992). First passage times for a generalized telegrapher's equation. Phys. A 183, 537–548.Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.CrossRefGoogle Scholar
Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators. Stoch. Equat. 3, 9–21.Google Scholar
Perry, D., Stadje, W., and Zacks, S. (1999a). Contributions to the theory of first-exit times of some compound process in queuing theory. Queuing Systems 33, 369–379.Google Scholar
Perry, D., Stadje, W., and Zacks, S. (1999b). First exit times for increasing compound processes. Commun. Statist. Stoch. Models 15, 977–992.CrossRefGoogle Scholar
Perry, D., Stadje, W., and Zacks, S. (2002). Hitting and ruin probabilities for compound Poisson processes and the cycle maximum of the M/G/1 queue. Stoch. Models 18, 553–564.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.CrossRefGoogle Scholar
Stadje, W., and Zacks, S. (2003). Upper first-exit times of compound Poisson processes revisited. Prob. Eng. Inf. Sci. 17, 459–465.Google Scholar
Zacks, S., Perry, D., Bshouty, D., and Bar-Lev, S. (1999). Distributions of stopping times for compound Poisson processes with positive jumps and linear boundaries. Commun. Statist. Stoch. Models 15, 89–101.Google Scholar