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Department of Mathematics

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Markov processes with lattice ordered state space: Theory of order and dependence, with applications

This is a joint project with Ryszard Szekli. (Wroclaw University). The project is funded by DAAD (Germany) and KBN (Poland) in 2005–06.
The class of monotone Markov processes plays an important role in many applications, e.g. interacting particle systems, queueing networks, reliability structures. Moreover a satisfying theory for positive correlation of Markovian processes has up to now only been developed for monotone Markov processes.
The theory of dependence and order is develpoped up to now mainly for linearly ordered state spaces or products of linearly ordered spaces. In this project we intend to investigate Markov processes with nonlinearly ordered state spaces, especially with lattice ordered state space. The planned subtopics of the project encompass:
Dependence ordering for monotone Markov processes, isotone differences ordering for Markov processes with state spaces of product form, dependence ordering for Markov processes without assuming monotonicity, and applications in reliability ordering for complex systems with internal dependencies, and dependencies in stochastic networks.
Cooperation with Rafal Kulik, Pavel Lorek, Christian Malchin, Cornelia Sauer, Ryszard Szekli, Kersten Tippner.

Some recent articles concerning Markov processes with lattice ordered state space: Theory of order and dependence, with applications :

  1. Dependence ordering for queueing networks with breakdown and repair (with Cornelia Sauer, Rafal Kulik, Ryszard Szekli)
    Probability in the Engineering and Informational Sciences 20, 575–594, 2006
    also available as: Isotone differences ordering for unreliable Markovian queueing networks, Preprint No.2005–04, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2005.
  2. Dependence ordering for Markov processes on partially ordered spaces (with Ryszard Szekli)
    Journal of Applied Probability 43, 793–814, 2006
    also available as: Report No. 16 2004/2005, fall, Institut Mittag-Leffler, 2005.
  3. Impact of routing on correlation strength in stationary queueing network processes (with Ryszard Szekli)
    Journal of Applied Probability 45 (3), 846–878, 2008