Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T03:41:05.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison Results for Markov-Modulated Recursive Models

Published online by Cambridge University Press:  27 July 2009

Nicole Bäuerle  and
Ulrich Rieder
Nicole Bäuerle
Affiliation:
Department of Mathematics vii, University of Ulm, D-89059 Ulm, Germany
Ulrich Rieder
Affiliation:
Department of Mathematics vii, University of Ulm, D-89059 Ulm, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a general discrete-time stochastic recursive model that is influenced by an external Markov chain. Our aim is to investigate the effect that the transition matrix of the external process has on the system states of the model. To answer this question, we use new stochastic ordering concepts. Especially interesting are the results for infinite-stage Markov-modulated models. We illustrate our main results by three applications: an inventory model, a consumption model, and a queueing model for a time division multiplexing system.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 11 , Issue 2 , April 1997 , pp. 203 - 217
Copyright
Copyright © Cambridge University Press 1997

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. (1989). Risk theory in a Markovian environment. Scandinavian Actuarial Journal, 69–100.CrossRefGoogle Scholar
2.Bäuerle, N. (1994). Stochastic models with a Markovian environment. Ph.D. dissertation, University of Ulm, Ulm (in German).Google Scholar
3.Bäuerle, N. (1997). Monotonicity results for MR/GI/1 queues. Journal of Applied Probability 34 (to appear).Google Scholar
4.Brandt, A.Franken, P. & Lisek, B. (1990). Stationary stochastic models. Chichester: Wiley.Google Scholar
5.Chang, C., Chao, X. & Pinedo, M. (1990). Integration of discrete-time correlated Markov processes in a TDM system. Probability in the Engineering and Informational Sciences 4: 29–56.CrossRefGoogle Scholar
6.Chang, C., Chao, X. & Pinedo, M. (1991). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Advances in Applied Probability 23: 210–228.CrossRefGoogle Scholar
7.Chang, C.Chao, X.Pinedo, M. & Shanthikumar, J. (1991). Stochastic convexity for multidimensional processes and its applications. IEEE Transactions on Automatic Control 36: 1347–1355.CrossRefGoogle Scholar
8.Doob, J. (1964). Stochastic processes. New York: Wiley.Google Scholar
9.Khamisy, A. & Sidi, M. (1992). Discrete-tirne priority queues with two-state Markov modulated arrivals. Communications in Statistics–Stochastic Models 8: 337–357.CrossRefGoogle Scholar
10.Li, S.-Q. & Mark, J.W. (1988). Performance trade-offs in an integrated voice/data service TDM system. Performance Evaluation 9: 51–64.Google Scholar
11.Loynes, R. (1962). The stability of a queue with non-independent interarrival and service times. Proceedings of the Cambridge Philosophical Society 10: 497–520.CrossRefGoogle Scholar
12.Marshall, A. & Olkin, l.. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press.Google Scholar
13.Meester, L. & Shanthikumar, J. (1993). Regularity of stochastic processes. Probability in the Engineering and Informational Sciences 7: 343–360.Google Scholar
14.Reinhard, J. (1984). On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. Astin Bulletin XIV: 23–43.Google Scholar
15.Rolski, T. (1989). Queues with nonstationary inputs. Queueing Systems 5: 113–130.Google Scholar
16.Shaked, M. & Shanthikumar, J. (1994). Stochastic orders and their applications. New York: Academic Press.Google Scholar
17.Stoyan, D. (1983). Comparison methods for queues and other stochastic models. Chichester: Wiley.Google Scholar
18.Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics 97. New York: Springer.CrossRefGoogle Scholar
19.Szekli, R.Disney, R. L. & Hur, S. (1994). MR/GI/l queues with positively correlated arrival stream. Journal of Applied Probability 31: 497–514.CrossRefGoogle Scholar
20.Tchen, A. (1980). Inequalities for distributions with given marginals. Annals of Probability 8 (4): 814–827.Google Scholar