Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T20:23:19.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Impact of Routeing on Correlation Strength in Stationary Queueing Network Processes

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Hans Daduna  and
Ryszard Szekli
Hans Daduna*
Affiliation:
Hamburg University
Ryszard Szekli*
Affiliation:
Wrocław University
*
∗Postal address: Department of Mathematics, Hamburg University, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: daduna@math.uni-hamburg.de
∗∗Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.

Keywords

Product form networkconcordance orderingspace-time correlationspectral gapasymptotic variancestochastic monotonicityMCMCPeskun ordering

MSC classification

Secondary: 60K25: Queueing theory 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 846 - 878
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Alberti, P. M. and Uhlmann, A. (1982). Stochasticity and Partial Order. VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
[2] Bäuerle, N. and Rolski, T. (1998). A monotonicity result for the workload in Markov-modulated queues. J. Appl. Prob. 35, 741–747.Google Scholar
[3] Daduna, H. and Szekli, R. (1995). Dependencies in Markovian networks. Adv. Appl. Prob. 27, 226–254.Google Scholar
[4] Daduna, H. and Szekli, R. (2006). Dependence ordering for Markov processes on partially ordered spaces. J. Appl. Prob. 43, 793–814.Google Scholar
[5] Gordon, W. J. and Newell, G.F. (1967). Closed queueing networks with exponential servers. Operat. Res. 15, 254–265.CrossRefGoogle Scholar
[6] Harris, T. G. (1977). A correlation inequality for Markov processes in partially ordered spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
[7] Hu, T. and Pan, X. (2000). Comparisons of dependence for stationary Markov processes. Prob. Eng. Inf. Sci. 14, 299–315.Google Scholar
[8] Iscoe, I. and McDonald, D. (1994). Asymptotics of exit times for Markov Jump processes I. Ann. Appl. Prob. 22, 372–397.Google Scholar
[9] Iscoe, I. and McDonald, D. (1994). Asymptotics of exit times for Markov Jump processes II. Applications to Jackson networks. Ann. Appl. Prob. 22, 2168–2182.Google Scholar
[10] Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[11] Jackson, J. R. (1963). Jobshop-like queueing systems. Manag. Sci. 10, 131–142.Google Scholar
[12] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
[13] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorisation and Its Applications. Academic Press, New York.Google Scholar
[14] Liggett, T. M. (1985). Interacting Particle Systems, (Fundamental Principles Math. Sci. 276). Springer, Berlin.Google Scholar
[15] Lorek, P. (2007). Speed of convergence to stationarity for stochastically monotone Markov chains. , Mathematical Institute, University of Wrocław.Google Scholar
[16] Li, H. and Shaked, M. (1994). Stochastic convexity and concavity of Markov processes. Math. Operat. Res. 19, 477–493.Google Scholar
[17] Li, H. and Xu, S. H. (2000). Stochastic bounds and dependence properties of survival times in a multicomponent shock model. J. Appl. Prob. 37, 1020–1043.Google Scholar
[18] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
[19] Neal, R. M. (2004). Improving asymptotic variance of {MCMC estimators: Non-reversible chains are better}. Tech. Rep. 0406 Department of Statistics, University of Toronto.Google Scholar
[20] Peskun, P. H. (1973). Optimum {Monte-Carlo} sampling using Markov chains. Biometrika 60, 607–612.Google Scholar
[21] Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.Google Scholar
[22] Szekli, R., Disney, R. L. and Hur, S. (1994). {MR/GI/1} queues with positively correlated arrival streams. J. Appl. Prob. 31, 497–514.Google Scholar
[23] Tierney, L. (1998). A note on Metropolis–{Hastings} kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.Google Scholar