Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T07:19:27.007Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME RESULTS FOR SKIP-FREE RANDOM WALK

Published online by Cambridge University Press:  19 August 2010

Mark Brown ,
Erol A. Peköz  and
Sheldon M. Ross
Mark Brown
Affiliation:
Department of Mathematics, City College, CUNY, New York, NY E-mail: [email protected]
Erol A. Peköz
Affiliation:
School of Management, Boston University, Boston, MA 02215, E-mail: [email protected]
Sheldon M. Ross
Affiliation:
Department of Industrial and System Engineering, University of Southern California, Los Angeles, CA 90089, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A random walk that is skip-free to the left can only move down one level at a time but can skip up several levels. Such random walk features prominently in many places in applied probability including queuing theory and the theory of branching processes. This article exploits the special structure in this class of random walk to obtain a number of simplified derivations for results that are much more difficult in general cases. Although some of the results in this article have appeared elsewhere, our proof approach is different.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 4 , October 2010 , pp. 491 - 507
Copyright
Copyright © Cambridge University Press 2010

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

1.Asmussen, S. (2003). Applied probability and queues, 2nd ed.New York: Springer.Google Scholar
2.Brown, M. & Shao, Y.S. (1987). Identifying coefficients in the spectral representation for first passage time distributions. Probability in the Engineering and Informational Sciences 1: 69–74.CrossRefGoogle Scholar
3.Feller, W. (1968). An introduction to probability theory and its applications. New York: John Wiley and Sons.Google Scholar
4.Jagers, P. & Lageras, A. (2008). General branching processes conditioned on extinction are still branching processes. Electronic Communications in Probability 13: 540–547.CrossRefGoogle Scholar
5.Kendall, D.G. (1956). Deterministic and stochastic epidemics in closed populations. In Proceedings of the third Berkeley symposium on mathematical statistics and probability, Vol. 4. Berkeley: University of California Press, 149–165.Google Scholar
6.Fill, J.A.On hitting times and fastest strong stationary times for skip-free chains. Journal of Theoretical Probability 23: 587–600.Google Scholar
7.Grimmett, G.R. & Stirzaker, D.R. (2001). Probability and random processes, 3rd ed.Oxford University Press.Google Scholar
8.Karlin, S. & Taylor, H.M. (1981). A second course in stochastic processes. New York: Academic Press.Google Scholar
9.Latouche, G. & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia, PA: SIAM/Alexandria, VA: American Statistical Association.Google Scholar
10.Latouche, G., Jacobs, P.A., & Gaver, D.P. (1984). Finite Markov chain models skip-free in one direction. Naval Research Logistics Quarterly 31(4): 571–588.CrossRefGoogle Scholar
11.Waugh, W.A. O'N. (1958). Conditioned Markov processes. Biometrika 45(1/2), 241–249.CrossRefGoogle Scholar
12.Ross, S.M. (2006). Intrduction to probability models, 9th ed.San Diego, CA: Academic Press.Google Scholar
13.Ross, S. & Peköz, E.A. (2007). A second course in probability. Boston, MA: Probability Bookstore. Available form www.ProbabilityBookstore.comGoogle Scholar
14.van der Hofstad, R. & Keane, M. (2008). An elementary proof of the hitting time theorem. American Mathematical Monthly Vol. 115(8): 753–756.CrossRefGoogle Scholar