### Correlation formulas for Markovian network processes in a random environment

#### Abstract

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.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 176-198.

Dates
First available in Project Euclid: 8 March 2016

https://projecteuclid.org/euclid.aap/1457466161

Mathematical Reviews number (MathSciNet)
MR3473573

Zentralblatt MATH identifier
1337.60250

#### Citation

Daduna, Hans; Szekli, Ryszard. Correlation formulas for Markovian network processes in a random environment. Adv. in Appl. Probab. 48 (2016), no. 1, 176--198. https://projecteuclid.org/euclid.aap/1457466161

#### References

• Balsamo, S. and Marin, A. (2013). Separable solutions for Markov processes in random environments. Europ. J. Operat. Res. 229, 391–403.
• Balsamo, S., de Nitto Personé, V. and Onvural, R. (2001). Analysis of Queueing Networks with Blocking. Kluwer, Boston, MA.
• Chen, M.-F. (2004). From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, River Edge, NJ.
• Daduna, H. and Szekli, R. (2008). Impact of routeing on correlation strength in stationary queueing network processes. J. Appl. Prob. 45, 846–878.
• 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.
• Daduna, H. and Szekli, R. (2015). Correlation formulas for networks with finite capacity. Preprint, Department of Mathematics, University of Hamburg (in preparation).
• Economou, A. (2005). Generalized product-form stationary distributions for Markov chains in random environments with queueing applications. Adv. Appl. Prob. 37, 185–211.
• 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.
• Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 5, 518–521.
• Liggett, T. M. (1989). Exponential $L_2$ convergence of attractive reversible nearest particle systems. Ann. Prob. 17, 403–432.
• Lorek, P. and Szekli, R. (2015). Computable bounds on the spectral gap for unreliable Jackson networks. Adv. Appl. Prob. 47, 402–424.
• Mira, A. and Geyer, C. J. (1999). Ordering Monte Carlo Markov chains. Tech. Rep., School of Statistics, University of Minnesota.
• 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.
• Peskun, P. H. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60, 607–612.
• Sauer, C. (2006). Stochastic product form networks with unreliable nodes: analysis of performance and availability. Doctoral Thesis, Department of Mathematics, University of Hamburg.
• Sauer, C. and Daduna, H. (2003). Availability formulas and performance measures for separable degradable networks. Econom. Quality Control 18, 165–194.
• Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.
• Van Dijk, N. M. (1988). On Jackson's product form with `jump-over' blocking. Operat. Res. Lett. 7, 233–235.
• 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.
• Van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Prob. Math. Statist. 65, 37–43.
• Zhu, Y. (1994). Markovian queueing networks in a random environment. Operat. Res. Lett. 15, 11–17.