Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-06T08:12:02.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Correlation formulas for Markovian network processes in a random environment

Part of: Markov processes Special processes

Published online by Cambridge University Press:  24 March 2016

Hans Daduna  and
Ryszard Szekli
Hans Daduna*
Affiliation:
Hamburg University
Ryszard Szekli*
Affiliation:
Wrocław University
*
* Postal address: Department of Mathematics, University of Hamburg, 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.

We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.

Keywords

Product-form networkspace-time correlationspectral gapasymptotic variancePeskun ordering

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60K37: Processes in random environments
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 176 - 198
Copyright
Copyright © Applied Probability Trust 2016 

References

[1]Balsamo, S. and Marin, A. (2013). Separable solutions for Markov processes in random environments. Europ. J. Operat. Res. 229, 391–403. CrossRefGoogle Scholar
[2]Balsamo, S., de Nitto Personé, V. and Onvural, R. (2001). Analysis of Queueing Networks with Blocking. Kluwer, Boston, MA. Google Scholar
[3]Chen, M.-F. (2004). From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, River Edge, NJ. Google Scholar
[4]Daduna, H. and Szekli, R. (2008). Impact of routeing on correlation strength in stationary queueing network processes. J. Appl. Prob. 45, 846–878. Google Scholar
[5]Daduna, H. and Szekli, R. (2013). Correlation formulas for Markovian network processes in a random environment. Preprint 2013-05, Department of Mathematics, University of Hamburg. Google Scholar
[6]Daduna, H. and Szekli, R. (2015). Correlation formulas for networks with finite capacity. Preprint, Department of Mathematics, University of Hamburg (in preparation). Google Scholar
[7]Economou, A. (2005). Generalized product-form stationary distributions for Markov chains in random environments with queueing applications. Adv. Appl. Prob. 37, 185–211. Google Scholar
[8]Ignatiouk-Robert, I. and Tibi, D. (2012). Explicit Lyapunov functions and estimates of the essential spectral radius for Jackson networks. Preprint. Available at http://arxiv.org/abs/1206.3066. Google Scholar
[9]Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 5, 518–521. Google Scholar
[10]Liggett, T. M. (1989). Exponential L 2 convergence of attractive reversible nearest particle systems. Ann. Prob. 17, 403–432. CrossRefGoogle Scholar
[11]Lorek, P. and Szekli, R. (2015). Computable bounds on the spectral gap for unreliable Jackson networks. Adv. Appl. Prob. 47, 402–424. Google Scholar
[12]Mira, A. and Geyer, C. J. (1999). Ordering Monte Carlo Markov chains. Tech. Rep., School of Statistics, University of Minnesota. Google Scholar
[13]Perros, H. G. (1990). Approximation algorithms for open queueing networks with blocking. In Stochastic Analysis of Computer and Communication Systems, North-Holland, Amsterdam, pp. 451–498. Google Scholar
[14]Peskun, P. H. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60, 607–612. CrossRefGoogle Scholar
[15]Sauer, C. (2006). Stochastic product form networks with unreliable nodes: analysis of performance and availability. Doctoral Thesis, Department of Mathematics, University of Hamburg. Google Scholar
[16]Sauer, C. and Daduna, H. (2003). Availability formulas and performance measures for separable degradable networks. Econom. Quality Control 18, 165–194. Google Scholar
[17]Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9. CrossRefGoogle Scholar
[18]Van Dijk, N. M. (1988). On Jackson's product form with 'jump-over' blocking. Operat. Res. Lett. 7, 233–235. CrossRefGoogle Scholar
[19]Van Dijk, N. M. (2011). On practical product form characterizations. In Queueing Networks (Internat. Ser. Operat. Res. Manag. Sci. 154), Springer, New York, pp. 1–83. Google Scholar
[20]Van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Prob. Math. Statist. 65, 37–43. Google Scholar
[21]Zhu, Y. (1994). Markovian queueing networks in a random environment. Operat. Res. Lett. 15, 11–17. CrossRefGoogle Scholar