Arie Hordijk

Ph.D. Amsterdam, 1974

Stochastic Operations Research

personal homepage: http://www.math.leidenuniv.nl/~hordijk/

research group homepage: http://www.math.leidenuniv.nl/~popov/or/

Research Interests and Selected Publications

Ergodicity of Markov Chains

A type of strong ergodicity for which the convergence is in operator norm, is studied. Sufficient and necessary conditions are derived. A main tool is the construction of a linear Lyapunov function for the corresponding fluid model. Applications to stochastic networks, e.g. Jackson and polling networks, are included.

A.A. Borovkov, A. Hordijk: On normed ergodicity of Markov chains. MI-report Leiden University (2000).

Markov Decision Chains and Markov Games

Recently the most selective optimality criterion, i.e. Blackwell optimality, is studied. The theory is developed for a finite, a countable and a Borel state space. Applications to the optimal control of queueing networks play an important role. Also MDC?s with a constrained policy space are studied. Markov games in which there are more than one controller are also contained in this research. The worst-case analysis for admission and service control, which is a special application of Markov games, is included.

E. Altman, A. Hordijk, F.M. Spieksma: Contraction conditions for average and discount optimality in countable state Markov games with unbounded rewards. Math.Oper.Res. 22 (1997), 588-618.

A. Hordijk, J.A. Loeve, J.J.W. Tiggelman: Analysis of a finite-source model with no state information. Math.Meth.Oper.Res. 47 (1998), 317-336.

A. Hordijk, A.A. Yushkevich: Blackwell optimality in the class of all policies in Markov decision chains with a Borel state space and unbounded rewards. Math.Meth.Oper.Res. 50 (1999), 421-448.

Control of Fluid Networks

Fluid models are important performance tools for modern telecommunication networks in which there are events on different time-scales. Fluid models are also relevant as approximation of discrete-time queueing networks. In this research the optimal server control in multiclass fluid networks is studied. These are the fluid models which correspond to the server control in multiclass queueing networks. The structure of optimal policies in the fluid network are analysed and asymptotically optimal policies for the queueing network are constructed.

A. Gajrat, A. Hordijk: Fluid approximation of a controlled multiclass tandem network. Queueing Systems 35 (2000) 349-380.

A. Gajrat, A. Hordijk: Optimal server control in multiclass fluid networks. MI-report Leiden University (2000).

Discrete-event Control of Stochastic Networks

This research is focused on a wide class of control (or of optimization) problems over sequences of integer numbers. The so-called multimodular functions play an important role. Many performance measures in queueing networks turn out to be multimodular. This includes admission control to networks, routing control into networks, and service assignment problems. The combinatorial notion of a regular sequence and the unbalance of a nonregular sequence as a measure of its deviation from the regular sequence are studied and applied to the open-loop admission control in FIFO-stochastic event graphs.

E. Altman, B. Gaujal, A. Hordijk: Multimodularity, convexity and optimization properties. Math.Oper.Res. 25 (2000) 324-347.

E. Altman, B. Gaujal, A. Hordijk: Admission control in stochastic event graphs. IEEE Automatic Control 45 (2000) 854-867.

E. Altman, B. Gaujal, A. Hordijk: Balanced sequences and optimal routing. Journal of the ACM 47 (2000) 752-775.

A. Hordijk, D.A. van der Laan: Periodic routing to parallel queues with bounds on the average waiting time. MI-report Leiden University (2000).

Performance Analysis

A. Hordijk: Comparison of queues with different discrete-time arrival processes. Prob.Eng.Inf.Sci. 15 (2001) 1-14.

A. Hordijk, Z. Liu, D. Towsley: Smoothing effect of the superposition of homogeneous sources in networks. Journal Appl.Prob. 37 (2000).

A. Hordijk, A. Shwartz: Performance bounds for queues via generating functions. IEEE Automatic Control 46 (2001).

Large Deviations

The problem of finding the best asymptotically optimal policy leads to a large deviations analysis of SN. A tandem controllable network has been analyzed. Also the open problem of computing the large deviations bounds for sample paths close to the origin for the transient, face-homogeneous random walk in the quarter plane has been partially solved. Applications are made to SN that model queueing networks with coupled processors.