Bernd Heidergott                                               


Affiliation                 Vrije Universiteit
Faculty of Economics and Business Administration
Department of Econometrics and Operations Research

Research fellow          Tinbergen Institute and EURANDOM.

E-mail: bheidergott [at] feweb [dot] vu [dot] nl
Phone: 0031 +20 +5986016
Fax: 0031 +20 +5986020

    Bernd Heidergott has received the Best Lecturer Award of the faculty of Economics and Business Administration of the VU for the academic year 2008/2009.

Index: Research Activities, Publications, Career, Grants and Fellowships and Teaching

Research Activities:

My main current research directions are as follows:

Gradient Estimation is concerned with providing unbiased gradient estimators for stochastic networks. We have worked on the main approaches in this area such as infinitesimal perturbation analysis (IPA), smoothed perturbation analysis (SPA), finite perturbation analysis (FPA), rare perturbation analysis (RPA), score function (SF) and phantom estimators (originating from measure-valued differentiation). In case studies comparing the performance of the estimators we could show that phantom estimators have the potential to outperform single-run gradient estimators like IPA, SPA of SF (which goes against the common belief in the simulation community). Recently we obtained first analytical results supporting this finding. This is joint work with F. Vázquez-Abad.

Differentiation theory for Markov chains seeks sufficient conditions for differentiability of performance characteristics of Markov chains and tries to obtain closed-form analytical expressions for the derivatives. The research in this area is the driving force in our attempt to unify gradient estimation and to develop a deeper understanding of stochastic optimization. We have intensively worked on this topic and could show that MVD in deed allows for a unified approach to gradient estimation. A recent breakthrough result is that we could show existence of derivatives of a particular class of Markov chains (phase-homogeneous random walks) under substantially weaker conditions than known in the literature. Furthermore, we have established an important link between differentiation theory and Banach space theory. This is joint work with A. Hordijk. An overview on the state-of-the-art in MVD can be found here (pdf file) .


Taylor series expansions are a powerful tool in performance analysis. By evaluating a finite number of higher-order derivatives, Taylor series allow to obtain the performance characteristic as a function of the parameter of interest. Key question are that of (1) convergence of the (finite) Taylor series to the true performance function and that of (2) establishing bound on the remainder term (i.e., the error made by approximating the true performance function via a finite Taylor polynomial). We have established results on (1) for max-plus linear systems (see below) and Markov chains. We achieved a breakthrough result on (2) by showing that for finite state-space Markov chains the series can be efficiently computed and a sharp bound on the error term can be obtained. This is joint work with A. Hordijk.

Max-plus algebra is an algebraic approach to discrete event dynamic systems, like queuing networks, that are prone to synchronization. Max-plus algebra has been the main theme of the European research project ALAPEDES. Based on the weak differentiation approach, we developed a calculus of weak differentiation for max-plus-linear stochastic systems. Based on this framework, we established Taylor series expansions for max-plus linear systems. Together with G. J. Olsder and J. van der Woude we have written a book on deterministic max-plus algebra. Our work on stochastic max-plus algebra and Taylor series expansions has been published in a monograph. Recent work is concerned with ergodic theory of stochastic max-plus linear stochastic systems. For more on max-plus algebra please visit the max-plus web portal. Slides of a tutorial held on max-plus algebra at the IFORS 2002 conference in Edinburgh, UK, can be found here (pdf file).


For Publications form 2002 onwards go here.

Working Papers:

Perturbation Analysis of inhomogeneous Markov processes (with Haralambie Leahu, Andreas Loepker and Georg Pflug), Working Paper, 2012. (pfd-file)

Publications until 2001 are listed below:

Train Movement Analysis at Railway Stations: Procedures and Evaluation of Wakob's Approach (with A. de Kort, R. van Egmond and G. Hooghiemstra)

Studies in Transport Series No. S99/1, TRAIL Research School,

ISBN: 90-407-1855-5 (1999). (contact publisher)

Sample-Path Analysis for Stochastic Networks (in German)

Forschungsergebnisse zur Informatik 10, Verlag Dr. Kovac,

ISBN: 978-3-86064-086-9 (1996). (contact publisher)


Weak Differentiation and Gradient Estimation for Discrete Event Processes (with  F. Vázquez-Abad).
In Discrete Event Systems: Analysis and Control, pages 433-440, Kluwer,

ISBN: 0-7923-7897-0 (2000)

GSMP with discrete lifetime distributions.
In Lecture Notes in Control and Information Sciences vol. 199, pages 456-462. Springer, ISBN: 0-387-19896-2 (1994).


Journal Papers:

Towards a control theory for transportation networks (with R. de Vries).
Discrete Event Dynamic Systems, vol. 11, pages 371-398, 2001.

A differential calculus for random matrices with applications to (max,+)-linear stochastic systems.
Mathematics of Operations Research
, vol. 26, pages 679-699, 2001.

Option pricing via Monte Carlo Simulation: A weak derivative approach.
Probability in Engineering and Informational Sciences, vol. 15, pages 335-349, 2001.

A weak derivative approach to optimization of threshold parameters in a multi-component maintenance system.
Journal of Applied Probability, vol. 38, pages 386-406, 2001.

Analyzing sojourn times in queuing networks: a structural approach.
Mathematical Methods of Operations Research, vol. 52, pages 115-132, 2000.

A characterization for (max,+)-linear queuing systems.
Queuing Systems: Theory and Applications, vol. 35, pages 237-262, 2000.

Customer-oriented finite perturbation analysis for queuing networks.
Discrete Event Dynamic Systems, pages 201-232, 2000.

Optimization of a single-component maintenance system: a smoothed perturbation analysis approach.
European Journal of Operations Research
, pages 181-190, 1999.

Infinitesimal perturbation analysis for queuing networks with general service time distributions
Queuing Systems: Theory and Applications, pages 43-58, 1999.

The zero utility principle for scale families of risk distributions ( with D. Pfeifer).
Deutsche Gesellschaft fürVersicherungsmathematik, pages 711-722, 1996.

Sensitivity analysis of a manufacturing workstation using perturbation analysis techniques.
International Journal of Production Research, vol. 33, pages 611-622, 1995.

Conferences with Refereed Proceedings:

Sensitivity analysis of joint characteristics of (max,+)-linear queuing networks.
IFAC satellite workshop on (max,+) algebra, Prague, Czech Republic, pages 221-226, 2001.

Bounding the Coupling Time of (Max,+)-Linear Systems (with Soto y Koelemeijer).
Transport, Infrastructure and Logistics, 6th TRAIL annual congress, The Hague, the Netherlands, 2000.

Weak differentiation and gradient estimation for discrete event processes (with  F. Vázquez-Abad).
Discrete Event Systems: Analysis and Control, Kluwer Academic Publisher, proceedings of the WODES (WODES'00), Gent, Belgium, pages 433-440, Kluwer, Boston, 2000.

Optimization of synchronization constraints via weak derivatives.
International Workshop on DES (WODES'98), Cagliari, Italy, pages 261-266. IEE, London, 1998.

Modeling and Analysis of queuing processes at railway stations: a case study (with R.-J. van Egmond, A. de Kort, and G. Hooghiemstra).
Transport, Infrastructure and Logistics, 4th TRAIL annual congress, The Hague, the Netherlands, 1998.

An overview of waiting time approximations for single server queues (with R.-J. van Egmond, A. de Kort, and  G. Hooghiemstra).
Transport, Infrastructure and Logistics, 4th TRAIL annual congress, The Hague, the Netherlands, 1998.

A stochastic minimal headway model for trains (with R.-J. van Egmond, A. de Kort, and G. Hooghiemstra).
Transport, Infrastructure and Logistics, 4th TRAIL annual congress, The Hague, the Netherlands, 1998.

GSMP with discrete lifetime distributions.
Lecture Notes in Control and Information Sciences, vol. 199, Springer, London, pages 456-462, 1994.

Sensitivitätsanalyse eines Fertigungssystems mit Hilfe der infinitesimalen Perturbationsanalyse (Sensitivity analysis for a manufacturing system using IPA).
ASIM91, Hagen, Germany, 1991.

Infinitesimal perturbation analysis: an overview.
Methods of OR, vol. 64, Vienna, Austria, pages 163-172, 1990.

Perturbation analysis, concept of an implementation.
SCS European Simulation Multiconference, Erlangen-Nürnberg, Germany, pages 192-197, 1990.



from 9/2002 until today Department of Econometrics and Operations Research, Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam

from 3/2001 to 8/2002 Operations Research and Statistics, Department of Mathematics and Computer Science, Division of Mathematics, TU Eindhoven

from 8/1999 to 2/2001 EURANDOM

from 4/1997 to 7/1999 DIAM, Department of Information Technology and Systems

from 5/1996 to 3/1997 Econometric Institute, Faculty of Economics, Erasmus University Rotterdam

from 3/1992 to 4/1992 Center of Mathematical Statistics and Stochastic Processes, Department of Mathematics, University of Hamburg (Ph.D. student)


Grants and Fellowships:

STW grant for research project Modeling and Analysis of Operations in Railway Networks: the Influence of Stochasticity. This is a joint project together with F.M. Dekking, I. Hansen and L. Meester, which will run from October 2003 to September 2007.

Deutsche Forschungsgemeinschaft (DFG) Fellowship, from August 1, 1999, to July 31, 2001.

Verein zur Förderung der Versicherungswissenschaft in Hamburg , Fellowship from October 1, 1995, to May 31, 1996.

Wilhelm-Blaschke-Stiftung, travel grants, June 1994 and July 1995.



At the Vrije Universiteit Amsterdam I am mainly involved in teaching mathematics and statistics for economists. In addition, I am teaching a course on Convex Analysis and Optimization for econometricians.