Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T12:06:16.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependencies in Markovian networks

Part of: Multivariate analysis Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

H. Daduna  and
R. Szekli
H. Daduna*
Affiliation:
Hamburg University
R. Szekli*
Affiliation:
Wrocław University
*
*Postal address: Institute of Mathematical Stochastics, Hamburg University, Bundesstrasse 55, 20146 Hamburg, Germany.
**Postal address: Mathematical Institute, Wrocław University, P1. Grunwaldzki 2/4, 50–384 Wrocław, Poland.
Get access Rights & Permissions [Opens in a new window]

Abstract

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

Keywords

QUEUEING NETWORKSGENERALIZED BIRTH–DEATH PROCESSESMONOTONICITYCORRELATION INEQUALITIESORDERINGPOINT PROCESSES

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 226 - 254
Copyright
Copyright © Applied Probability Trust 1995 

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.)

Footnotes

Research carried out while this author held an Alexander von Humboldt Fellowship at Hamburg University.

References

Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag. New York.CrossRefGoogle Scholar
Brown, T. C. and Nair, M. G. (1988) A simple proof of the multivariate random time change theorem for point processes. J. Appl. Prob. 25, 210–214.CrossRefGoogle Scholar
Chang, C. S., Chao, X., Pinedo, M. and Shanthikumar, J. G. (1991) Stochastic convexity for multidimensional processes and its applications. IEEE Trans. Autom. Control 36, 1341–1355.CrossRefGoogle Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks: A Markov Renewal Approach. The Johns Hopkins University Press, London.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. (1967) Association of random variables with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Foley, R. D. and Kiessler, P. C. (1989) Positive correlations in a three node Jackson queueing network. Adv. Appl. Prob. 21, 241–242.CrossRefGoogle Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.CrossRefGoogle Scholar
Harris, T. G. (1977) A correlation inequality for Markov processes in partially ordered spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
Joag-Dev, K. and Proschan, F. (1983) Negative association of random variables with applications. Ann. Statist. 11, 286–295.CrossRefGoogle Scholar
Kamae, T., Krengel, U. and O'Brien, G. C. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
Kanter, M. (1985) Lower bounds for the probability of overload in certain queueing networks. J. Appl. Prob. 22, 429–436.CrossRefGoogle Scholar
Kelly, F. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kwiecinski, A. and Szekli, R. (1991) Compensator conditions for stochastic ordering of point processes. J. Appl. Prob. 28, 751–761.CrossRefGoogle Scholar
Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.CrossRefGoogle Scholar
Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York.CrossRefGoogle Scholar
Lindqvist, B. M. (1988) Association of probability measures on partially ordered spaces. J. Multivariate Anal. 26, 111–132.CrossRefGoogle Scholar
Massey, W. A. (1987). Stochastic ordering for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.CrossRefGoogle Scholar
Massey, W. A. (1989) Stochastic ordering for birth-death-migration processes. Preprint.Google Scholar
Mcnickle, D. (1991) Lagged queue-length correlations in two-node networks. Queueing Systems 8, 97–104.CrossRefGoogle Scholar
Nachbin, L. (1965) Topology and Order. Van Nostrand, Princeton.Google Scholar
Ridder, A. A. (1987) Stochastic Inequalities for Queues. Thesis.Google Scholar
Serfozo, R. (1989) Poisson functionals of Markov processes and queueing networks. Adv. Appl. Prob. 21, 595–611.CrossRefGoogle Scholar
Serfozo, R. (1992) Reversibility and compound birth-death and migration processes. In Queueing and Related Models , pp. 65–90. Oxford University Press.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. John Wiley, Berlin.Google Scholar
Van Der Wal, J. (1986) Monotonicity of the throughput of a closed exponential queueing network in the number of jobs. OR Spectrum 11, 97–100.CrossRefGoogle Scholar
Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.CrossRefGoogle Scholar