Non–product form models and processes associated with product form networks

This is a joint project with Ryszard Szekli. (Wroclaw University). The project is funded by DAAD (Germany) and KBN (Poland) in 2003–04.
We start from a class of stochastic networks (networks of queues) and their describing stochastic processes which exhibit a product form stationary distribution. Extending this class of networks by attaching inventories to the service nodes of the network and considering the nodes to be unreliable due to external random influences, we arrive at non-standard models, which in some cases exhibit a new form of product structure for the equilibrium and in other cases deviate from the product form class completely. Our aim is to compute in both cases steady states for the (supplemented) network processes and to derive performance quantities of the systems in an explicite form.
Further topics are stochastic orderings in these class of systems and the structure of dependencies over time and space of the processes, as well as the asymptotic behaviour of the processes. In cases where no explicit quantitative measures are available we prove easy to apply bounds by comparing different systems.
Cooperation with Rafal Kulik, Pawel Lorek, Christian Malchin, Cornelia Sauer, Maike Schwarz, Ryszard Szekli, Kersten Tippner.

Some recent articles concerning Non-product form models and processes associated with product form networks :

1. Modeling networks with unreliable servers (with Cornelia Sauer)
   Proceedings of the 2. MMB–Arbeitsgespräch Leistungs-, Zuverlässigkeits und Verlässlichkeitsbewertung von Kommunikationsnetzen und verteilten Systemen, Berichte des Fachbereichs Informatik der Universität Hamburg No. 242, 83 – 90, Hamburg 2002.
2. Degradable networks with general up and down time distributions. (with C. Sauer)
   in: Buchholz, P.; Lehnert, R.; Pioro, M. (eds.): MMB & PGTS 2004,
   Proceedings of the 12th GI/ITG Conference on Measuring, Modelling and Evaluation of Computer and Communications Systems and 3rd Polish-German Teletraffic Symposium, 185 – 194, 2004
   Extended Version: Preprint No.2004–04, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2004.
3. M/M/1 – systems with different inventory management policies and lost sales
   (with Maike K. Schwarz, Cornelia Sauer, Rafal Kulik, Ryszard Szekli)
   to appear in: Queueing Systems and Their Applications
   also available as: Preprint No.2003–07, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003.
4. An invariance property of conditional sojourn time distributions in cyclic networks of queues (with Christian Malchin)
   Operations Research Letters 33, 1–8, 2004
   Preliminary version: Preprint No.2003–06, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003.
5. Exponential queueing networks with an attached inventory under (r,Q)- or (r,S)-policy (with M. Schwarz and C. Sauer)
   Preprint No.2003–09, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003.
6. Availability formulas and performance measures for separable degradable networks (with C. Sauer)
   Economic Quality Control, 18, 165 – 194, 2003
7. On the structure of roundtrip time distributions in discrete time networks (with Christian Malchin)
   Preprint No.2004–03, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2004.
8. 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.
9. Queueing systems with inventory management with random lead times and with backordering (with M. Schwarz )
   Mathematical Methods of Operations Research 64, 383–414, 2006
   also available as: Preprint No.2005–03, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2005.
10. 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, submitted.
11. Product form models for queueing networks with an inventory (with M. Schwarz and C. Sauer)
    Stochastic Models 23, 627 – 663, 2007; also available as extended version:
    Exponential queueing networks with an attached inventory under (r,Q)- or (r,S)-policy (with M. Schwarz and C. Sauer)
    Preprint No.2003–09, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003.