Hans Daduna

Research interests and projects


Last revision March 2, 2017
My research interest are in the field of stochastic processes and their applications, especially in computer science and operations research.

Most of my work is on stochastic networks (networks of queues), their asymptotic behaviour, steady state distributions, busy period analysis for single servers and subnetworks, sojourn time distribution for individual customers, and computational algorithms for performance evaluation of networks.
A survey of some of these topics can be found in
Stochastic networks with product form equilibrium, in: Shanbhag, D.N.; Rao, C.R. (eds.): Handboof of Statistics, Vol. 19, Chapter 11 (p. 309 - 364) Elsevier Sciences B.V., 2001

Parallel to this work on stochastic networks in the classical continuous time framework is research with the aim to develop a product form calculus for stochastic networks in discrete time. A survey of parts of this research is the book
Queueing Networks with Discrete Time Scale
Explicit Expressions for the Steady State Behavior of Discrete Time Stochastic Networks
; Lecture Notes in Computer Science 2046, Springer Heidelberg 2001

More recent are research projects on queueing systems and networks in a random environment, stochastic orders and dependence ordering for Markov processes and stochastic networks, reliability of queueing networks (Performability), the integration of queueing and inventory management, and on optimization of stochastic networks and of locally interacting multidimensional Markov processes.

Complete list of publications.

Research projects:

  1. Queueing systems in a random environment
  2. Integrated models for production and inventories. Supply chains
  3. Local stabilization of large unstable networks
  4. Discrete time queueing networks
  5. Non-product form models and processes associated with product form networks
  6. Optimal control of systems with random influences:
    Operations Research models
  7. Performability:
    Reliability and Performance Analysis of Stochastic Networks
  8. Stochastic orders, monotonicty and correlation in stochastic networks
  9. Passage time and sojourn time analysis for individual customers in stochastic networks
  10. Markov processes with lattice ordered state space: Theory of order and dependence, with applications

Queueing systems in a random environment

The investigation in this project is motivated by the observation that many complex networks are typically subject to influences from an external environment. The impact of the evolving environment on the network is typically by changing service capacities (upgrading and/or degrading, breakdown, repair) or by changing the paths of the travelling customers when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. We are especially interested in separable models, i.e., although there is the important bidirectional interaction of the network's nodes with the environment, asymptotically and in steady state the environment and the network (or a single node) behave stochastically as if they are independent (with respect to one-dimensional time marginals).
Recent applications, where nodes are influenced by their environment are mobile and ad-hoc networks or sensor networks. Here the environment of a typical node is composed of the geographical surroundings and the behaviour of the node's neighboured nodes. In this example the behaviour of the referenced node influences the behaviour of the environment.
Collaboration with Ruslan Krenzler (Hamburg University), Sonja Otten (Hamburg University), Ryszard Szekli (Wroclaw University.
Some recent articles concerning Queueing systems in a random environment:
  1. Loss systems in a random environment - steady state analysis (with Ruslan Krenzler)
    Queueing Systems 80, 127 - 153, 2015 (DOI) 10.1007/s11134-014-9426-6, also available as
    Preprint No.2012-04, Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2012, pdf-file.
  2. Jackson networks in non-autonomous random environments (with Sonja Otten and Ruslan Krenzler)
    Advances in Applied Probability 48, 315 - 331, 2016
  3. Networks of queues in a random environment: Survey of product form results
    In: Wolfinger, Bernd E. ; Heidtmann, Klaus-D. (eds): Proc. MMBnet 2015, Berichte des Fachbereichs Informatik der Universität Hamburg, 302, pages 1--17, 2015
  4. Modeling and Performance Analysis of a Node in Fault Tolerant Wireless Sensor Networks (with Ruslan Krenzler)
    in: Fischbach, K., Krieger, U. (eds.) Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance, Lecture Notes in Computer Science Volume 8376, Springer 2014, pp 73-87
  5. Correlation formulas for Markovian network processes in a random environment (with Ryszard Szekli)
    Advances in Applied Probability 48 (1), 176 - 198, 2016; doi:10.1017/apr.2015.12; also available as
    Preprint No.2013-05, Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2013 pdf-file.

Integrated models for production and inventories. Supply chains

Queueing theory and inventory theory are fields of Operations Research with different methodologies to optimize e.g. production processes and inventory control. In classical Operations Research queueing theory and inventory theory are often considered as disjoint areas of research. On the other side, the emergence of complex supply chains calls for integrated queueing-inventory models, which are the focus of our present research. We consider a supply chain consisting of production systems (servers) at several locations, each with a local inventory, and a supplier network of workstations, which manufactures items of possibly different kinds (raw material), to replenish the local inventories.
Recent applications, where
Collaboration with Nha-Nghi de la Cruz (Hamburg University), Ruslan Krenzler (Hamburg University), Rafal Kulik (Wroclaw University), Pawel Lorek (Wroclaw University), Christian Malchin (Hamburg University), Sonja Otten (Hamburg University), Cornelia Sauer (Hamburg University), Maike Schwarz (Hamburg University), Ryszard Szekli (Wroclaw University), Kersten Tippner (Hamburg University)
Some recent articles concerning Integrated models for production and inventories. Supply chains:
  1. Monotonicity of Base Stock Policies (with Nha-Nghi de la Cruz)
    Operations Research Letters 44, 186 - 190, 2016,
  2. Models for integrated production-inventory systems: Steady state and cost analysis (with Sonja Otten and Ruslan Krenzler)
    Int. Journal of Production Research, 54 (20), 6174 - 6191, 2016, DOI:10.1080/00207543.2015.10826692015,
  3. 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, ps-file.
  4. 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, submitted ps-file.
  5. M/M/1 Queueing systems with inventory
    (with Maike K. Schwarz, Cornelia Sauer, Rafal Kulik, Ryszard Szekli)
    Queueing Systems and Their Applications 54, 55 - 78, 2006
    also available as: Preprint No.2003-07, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003 ps-file.

Local stabilization of large unstable networks

The investigation in this project is motivated by the following observation: In large and complex networks, unstable subnetworks often occur due to local overload or due to non-availability of nodes or links, but nevertheless, there co-exist regions where other subnetworks stabilize locally. Due to the local instability the describing Markovian network processes are generically non-ergodic. Nevertheless it can be seen that locally, i.e., for some subset of nodes the local processes, which are not Markovian of their own, converge to a well defined stochastic limit.
Additionally, we investigate the impact of unreliability of nodes on the stabilization of networks or subnetworks.
Recent applications, where nodes or links may be not available for some time, are mobile and ad-hoc networks or sensor networks. The emergence of wireless ad-hoc networks, which are built of varying sets of mobile users with wireless communication capabilities without relying on a pre-existing infrastructure, introduces problems concerning availability of transmission nodes in the vicinity of a user. On the other hand, in such networks, overload emerges from too many users entering a region and applying for transmission.
Collaboration with Jennifer Mylosz (University of Hamburg) and Bernd Heidergott (Vrije Universiteit Amsterdam).
Some recent articles concerning Local stabilization of large unstable networks:
  1. Non-ergodic Jackson networks with infinite supply - local stabilization and local equilibrium analysis (with Jennifer Sommer and Bernd Heidergott)
    Journal of Applied Probability 53(4), 1125 - 1142, 2016
  2. On the behavior of stable subnetworks in non-ergodic networks with unreliable nodes (with Jennifer Mylosz)
    Computer Networks 53 (8), 1249-1263, 2009
  3. On the behavior of stable subnetworks in non-ergodic networks: A quasi-stationary approach in a two-node system (with Jennifer Mylosz)
    in: Wolfinger, B. E. and Heidtmann, K.-D. (eds.): Leistungs-, Zuverlässigkeits und Verlässlichkeitsbewertung von Kommunikationsnetzen und verteilten Systemen, 6. GI/NTG-Workshop MMBnet 2011
    Berichte des Fachbereichs Informatik der Universität Hamburg No. 298, 99-106, Hamburg 2011.

Discrete time queueing networks

The aim of this project is to develop a product form calculus for such networks similar to the BCMP and Kelly networks in continuous time. This will provide us with a versatile class of discrete time network models where the equilibrium behaviour can be explicitly computed. Having this at hand, the main performance characteristics can be derived as well.
Collaboration with Ruslan K. Chornei (Inter-Regional Academy of Personnel Management, Kiew, Ukraina), Bernadette Desert (Hamburg), Marten Holst (Hamburg), Pavel S. Knopov (Kiew, National Academy of Sciences of the Ukraina), Christian Malchin (University of Hamburg) Victor Pestien (University of Miami), Subramanian Ramakrishnan (University of Miami), and Kersten Tippner (Hamburg).

Part of this research programm is the project Stochastic networks in discrete time: Analysis of performance and availability, funded by the DFG (German Reseach Foundation) in 2005-06 under DA 774/1
Some recent articles concerning Discrete time queueing networks:
  1. Some results for steady-state and sojourn time distributions in open and closed linear networks of Bernoulli servers with state- dependent service and arrival rates
    Performance Evaluation 30 (1997), 3 -18
  2. The cycle time distribution in a cycle of Bernoulli servers in discrete time
    Mathematical Methods in Operations Research 44, 295 - 332, 1996
  3. Discrete time queueing networks: Recent developments Tutorial Lecture Notes,
    in : Tutorials Performance 96 (Lecture Notes), 163 - 204, IFIP, LRC--EPFL, Lausanne, 1996
  4. Networks of queues in discrete time
    Lecture Notes, Hamburg 1997
  5. Discrete time analysis of a state dependent tandem with different customer types
    In Lecture Notes in Computer Science 1337, Springer Heidelberg 2001:
    Foundations of Computer Science, Potential--Theory--Cognition. Christian Freksa, Matthias Jantzen, Rüdiger Valk (eds.), 287 -296, Springer Berlin 1997
  6. The joint distribution of sojourn times for a customer traversing a series of queues : The discrete time case
    Queueing Systems and Their Applications 27, 297 -323, 1998
  7. Controlled Markov fields with finite state space on graphs ( with Ruslan K. Chornei and Pavel S. Knopov)
    Stochastic Models 21, 847-874, 2005
    also available: Preprint No.2000-08, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2000, ps-file
  8. Queueing Networks with Discrete Time Scale
    Explicit Expressions for the Steady State Behavior of Discrete Time Stochastic Networks
    Lecture Notes in Computer Science 2046, Springer Heidelberg 2001
  9. Asymptotic throughput in discrete-time cyclic networks with queue-length-dependent service rates
    (with Subrahmanian Ramakrishnan and Victor Pestien)
    Stochastic Models, 19, 483 - 506, 2003
    Preliminary version: Preprint No.2001-04, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2001, ps-file.
  10. Discrete time tandem networks of queues - Effects of different regulation schemes for simultaneous events
    (with Bernadette Desert)
    Performance Evaluation 47, 73 - 104, 2002
  11. 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, submitted ps-file.
  12. Asymptotic throughput in discrete-time cyclic networks with queue-length-dependent service rates
    (with Subrahmanian Ramakrishnan and Victor Pestien)
    Stochastic Models, 19, 483 - 506, 2003
    Preliminary version: Preprint No.2001-04, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2001, ps-file.
  13. Availability and performance analysis in a discrete time tandem network with product form steady state (with Christian Malchin)
    in: German, R.; Heindl, A. (eds.): Proceedings of the GI/ITG Conference on Measuring, Modelling and Evaluation of Computer and Communications Systems , 2006, 381 -398, 2006
    Extended Version: Preprint No.2005-06, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2005 ps-file.
  14. Discrete time queueing networks with product form steady state: Availability and performance analysis in an integrated model (with Christian Malchin)
    Queueing Systems and Their Applications 65, 385 - 421, 2010 2010; also available as
    Preprint No.2006-02, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2006 ps-file.
  15. Discrete time queueing networks with product form steady state
    in: Boucherie, R. J. and van Dijk, N. (eds) Queueing Networks: A Fundamental Approach, 269 - 312, Springer, New York, 2010
  16. Bottleneck analysis of discrete time networks: Sojourn times (with Christian Malchin)
    Preprint No.2007-03, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2007, submitted ps-file.
  17. Customer oriented performance measures for packet transmission in a ring network with blocking (with Marten Holst)
    in: (eds.): Proceedings of the GI/ITG Conference on Measuring, Modelling and Evaluation of Computer and Communications Systems , 2008, 223 -236, 2008


Stochastic orders, monotonicty and correlation in stochastic networks

This is a joint project with Ryszard Szekli. (Wroclaw University). The project is funded by the Alexander-von-Humboldt-Stiftung, DAAD (Germany), and KBN (Poland). We consider stochastic networks exhibiting in steady state a product form equilibrium, i.e., the nodes of the networks behave asymptotically at a fixed time instant as if they are spatially independent. In the transient phase of such systems and in steady state when considering joint distributions of the network processes over time strong correlations in space and time occur. The aim of the project is to characterize these correlation and to obtain bounds for the transient probabilities of the networks.
Collaboration with Lars Peter Saul.
Some recent articles concerning Stochastic orders, monotonicty and correlation in stochastic networks :
  1. Dependencies in Markovian networks (with R.Szekli)
    Adv.Appl.Prob. 27, 226 - 254 1995
  2. A queueing theoretical proof of increasing property for Polya-frequency functions (with R.Szekli)
    Statistics and Probabilty Letters 26, 233 - 242, 1996
  3. On the correlation of sojourn times in open networks of exponential multiserver queues (with R. Szekli)
    Queueing Systems and Their Applications 34, 169-181, 2000
  4. On the correlation structure of closed queueing networks (with Ryszard Szekli)
    Stochastic Models, 20, 1 - 30, 2004
    Preliminary version: Monotonicity and dependence properties of sojourn and cycle times in closed networks
    Preprint No.2000-05, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2000 ps-file.
  5. Dependence structure of sojourn times via partition separated ordering (with Ryszard Szekli)
    Operations Research Letters 31, 462 - 472, 2003
    Preliminary version: Preprint No.2002-03, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2002, ps-file .
  6. 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, ps-file or pdf-file.
  7. Impact of routing on correlation strength in stationary queueing network processes (with Ryszard Szekli)
    Journal of Applied Probability 45 (3), 846-878 2008
  8. Availability in large networks: Global characteristics from local unreliability properties (with Lars Peter Saul)
    to appear in: Schmitt, J. (ed.): Proceedings of the 16th International GI/ITG Conference on Measurement, Modelling and Evaluation of Computing Systems and Dependability and Fault Tolerance, Lecture Notes in Computer Science 7201, 1-15, Springer Berlin 2012


Passage time and sojourn time analysis for individual customers in stochastic networks

The sojourn times or passage times of customers in stochastic networks represent (e.g.) the end-to-end-delay in telecommunication networks, the lead times in production systems, waiting and service times in computer systems, life times in population models,... . From a theoretical point of view the stochastic behaviour of these times is up to now not well understood. Being interested in equilibrium results as a first distributional characterisation of sojourn times I investigate the steady state passage time distribution and the joint distribution of a customer's sojourn times at the successive nodes of a path when traversing the network. Parts of the research are funded by the Alexander-von-Humboldt-Stiftung, DAAD (Germany), and KBN (Poland).
Collaboration with Rolf Schassberger (Technical University of Braunschweig), Stephan Meyer (Hamburg), Christian Malchin (Hamburg), Onno Boxma (Technical University of Eindhoven), Ryszard Szekli.
Some recent articles concerning Passage time and sojourn time analysis for individual customers in stochastic networks :
  1. The cyclic queue and the tandem queue (with Onno Boxma)
    Queueing Systems 77, 275 - 295, 2014; DOI 10.1007/s11134-013-9380-8, also available as
    Preprint No.2013-01, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2013 pdf-file.
  2. Weak convergence limits for closed cyclic networks of queues with multiple bottleneck nodes (with Ole Stenzel)
    Journal of Applied Probability 49 (1), 60-83, 2012; also available as
    Preprint No.2009-05, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2009 ps-file.
  3. Weak convergence limits for sojourn times in cyclic queues under heavy traffic (with Christian Malchin and Ryszard Szekli)
    Journal of Applied Probability 45, 1 -14, 2008; also available as
    Preprint No.2007-04, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2007 ps-file.
  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, ps-file.
  5. Dependence structure of sojourn times via partition separated ordering (with Ryszard Szekli)
    Operations Research Letters 31, 462 - 472, 2003
    Preliminary version: Preprint No.2002-03, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2002, ps-file .
  6. Conditional job observer property for multitype closed queueing networks
    Journal of Applied Probability 39 (2002) p. 865-881
  7. On the correlation of sojourn times in open networks of exponential multiserver queues (with R. Szekli)
    Queueing Systems and Their Applications 34, 169-181, 2000
  8. Individual customer's behaviour in networks with state-dependent arrival rates (with S. Meyer)
    Queueing Systems and Their Applications 32 (1999) p. 351-362
  9. Sojourn time distributions in non-product-form queueing networks
    in : Dshalalow,J.(ed.):Frontiers of queueing: Models and Applications in Science and Engineering, Chapter 8, CRC Press, Boca Raton, 1996
  10. Delay time distributions and adjusted transfer rates for Jackson networks (mit R.Schassberger)
    AEU-Archiv fuer Elektronik und Uebertragungstechnik 47 (Special Issue on Teletraffic Theory and Engineering in Memory of Felix Pollaczek) 342-348 1993
  11. On network flow equations and splitting formulas for sojourn times in in queueing networks
    J.Appl.Math.and Stoch.Analysis 4, 111-116 1991
  12. Sojourn times in queueing networks ( with O.J.Boxma )
    in: Takagi,H.(ed): Stochastic analysis of computer and communication systems, 401-450 North-Holland, Amsterdam 1990
  13. Sojourn times in queueing networks with multiserver nodes (with R.Schassberger)
    J. Appl. Prob. 24, 511-521 1987


Optimal control of systems with random influences:
Operations Research models

This is a joint project with Pavel S. Knopov (V. M. Glushkov Institute of Cybernetics, Academy of Science of the Ukraina). The project is funded by the DFG (Germany).
(1) We consider stochastic models from the field of Operations Research (queueing systems, inventory sytems, repair and renewal sytems) subject to an external (open loop) control by a decision maker. The aim is to find simple optimal or nearly optimal control policies which minimize the long range costs (maximize the long range reward). The common feature of the models is the contiuous state space of the describing system processes; so the variable (observation) where the control decision is based on is continuous.
(2) Stochastic networks are random fields on graphs, when considered at a fixed time instant. Incoprorating the space-time variabilty of such systems leads to time dependent random fields. If the systems are controlled by decision makers which act at the nodes and make their decisions on the basis of local information only, and if the space-time behavior of the system carries Markov structures in time and space, we obtain controlled Markov processes with locally interacting components. Our interest is in finding optimal strategies inside the class of local strategies and in proving conditions which guarantee that these locally optimal strategies are even globally optimal.
Collaboration with Ludmila P. Tur (Kiew) and Ruslan K. Chornei (Kiew).
Some recent articles concerning Optimal control of systems with random influences: Operations Research models :
  1. Optimal admission control for M/D/1/K queueing systems (with Pavel S. Knopov )
    Mathematical Methods of Operations Research 50 (1999), p. 91-100
  2. Optimal strategies for an inventory systems with cost functions of general form (with Pavel S. Knopov and Ludmila P. Tur )
    Cybernetics and System Analysis 35, 602 - 618, 1999
    (Translation from Kibernetica i Sistemyi Analiz 4' 99, 106-123, 1999)
  3. Local control of interacting Markov fields on graphs with compact state space (with Ruslan K. Chornei and Pavel S. Knopov)
    Kibernetica i Sistemyi Analiz 2001, No. 3 p. 62 - 77, also available as
    Preprint No.2000-09, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2002, submitted - ps-file.
  4. Stochastic games for distributed players on graphs (with Ruslan K. Chornei and Pavel S. Knopov)
    Mathematical Methods of Operations Research 60, 279-298, 2004
    Preliminary version: Preprint No.2002-06, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2002, ps-file.
  5. Controlled semi-Markov fields with graph-structured compact state space (with Ruslan K. Chornei and Pavel S. Knopov)
    Theory of Probability and Mathematical Statistics 69, 39-53, 2004
    translation from: Teor. Ymovirnost. Matem. Statist. 69, 42-47, 2003, (in Ukrainian)
    also available as Preprint No.2002-08, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2002, ps-file.
  6. Controlled Markov fields with finite state space on graphs ( with Ruslan K. Chornei and Pavel S. Knopov)
    Stochastic Models 21, 847-874, 2005
    also available: Preprint No.2000-08, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2000, ps-file
  7. Control of Spatially Structured Random Processes and Random fields with Applications ( with Ruslan K. Chornei and Pavel S. Knopov)
    Springer, New York, 2006


Performability:
Reliability and Performance Analysis of Stochastic Networks

This is a joint project with Cornelia Sauer.
We consider stochastic networks (networks of queues) where nodes or the servers at the nodes are degradable, i.e., the nodes may be subject to breakdown or perturbations, which decrease their capacity. Following this, repair mechanisms and repair strategies are introduced. Our aim is to find network models where the performance behaviour of the network and the breakdown and repair mechanism are incorporated into a unified Markovian system description, which nevertheless admits an explicit steady state.
Collaboration with Jennifer Mylosz (University of Hamburg).
Some recent articles concerning Performability: Reliability and Performance Analysis of Stochastic 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. Separable networks with unreliable servers (with Cornelia Sauer)
    in Charzinski, J; Lehnert, R.; Tran-Gia, P. (eds.): Providing Quality of Service in Heterogeneous Environments (Proceedings of the 18 ITC, Berlin), Teletraffic Science and Engineering Vol 5b, 821 - 830, Elsevier 2003.
    Extended version: BCMP networks with unreliable servers (with Cornelia Sauer)
    Preprint No.2003-01, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2003, ps-file.
  3. Availability formulas and performance measures for separable degradable networks (with C. Sauer)
    Economic Quality Control, 18, 165 - 194, 2003
  4. 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 ps-file.
  5. 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 ps-file.
  6. Discrete time queueing networks with product form steady state: Availability and performance analysis in an integrated model (with Christian Malchin)
    Preprint No.2006-02, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, University of Hamburg 2006, submitted ps-file.
  7. On the behavior of stable subnetworks in non-ergodic networks with unreliable nodes (with Jennifer Mylosz)
    Computer Networks 53 (8), 1249-1263, 2009


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 ps-file.
  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 ps-file.
  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, ps-file.
  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, submitted ps-file.submitted
  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, submitted ps-file.
  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 ps-file or pdf-file.
  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 ps-file.
  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 ps-file.
  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, submitted ps-file.


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, submitted ps-file.
  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, ps-file or pdf-file.
  3. Impact of routing on correlation strength in stationary queueing network processes (with Ryszard Szekli)
    Journal of Applied Probability 45 (3), 846-878, 2008



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