Designs of optimal switching feedback decentralized control policies for fluid queueing networks Authors V. Feoktistova Department of Mathematics and Mechanics Saint Petersburg University A. Matveev Department of Mathematics and Mechanics Saint Petersburg University E. Lefeber Department of Mechanical Engineering Eindhoven University of Technology J. E. Rooda Department of Mechanical Engineering Eindhoven University of Technology Open Access Original Article

First Online: 11 April 2012 Received: 12 September 2010 Accepted: 26 March 2012 DOI :
10.1007/s00498-012-0086-y

Cite this article as: Feoktistova, V., Matveev, A., Lefeber, E. et al. Math. Control Signals Syst. (2012) 24: 477. doi:10.1007/s00498-012-0086-y
Abstract
The paper considers standard fluid models of multi-product multiple-server production systems where setup times are incurred whenever a server changes product. We consider a general approach to the problem of optimizing the long-run average cost per unit time that consists of first determining an optimal steady state (periodic) behavior and then to design a feedback scheduling protocol ensuring convergence to this behavior as time progresses. In this paper, we focus on the latter part and introduce a systematic approach. This approach gives rise to protocols that are cyclic and distributed: the servers do not need information about the entire system state. Each of them proceeds basically from the local data concerning only the currently served queue, although a fixed finite number of one-bit notification signals should be exchanged between the servers during every cycle. The approach is illustrated by simple instructive examples concerning polling systems, single server systems with processor sharing scheme, and the re-entrant two-server manufacturing network with non-negligible setup times introduced by Kumar and Seidman. For the last network considered in the analytical form, some cases of optimal steady-state (periodic) behavior are first recalled. For all examples, based on the desired steady state behavior and using the presented theory, we designed simple distributed feedback switching control laws. These laws not only give rise to the required behaviors but also make them globally attractive, irrespective of the system parameters and initial state.

Keywords
Hybrid dynamical systems
Optimal switched control
Control of networks
Fluid models
Queueing
This work was supported by the Netherlands Organization for Scientific Research (NWO-VIDI grant 639.072.072; NWO visitors grant 040.11.131), the Russian Foundation for Basic Research (grant 09-08-00803), the Russian Federal Program (grant N 1.1-111-128-033) and the Russian Federal Program “Research and Teaching Cadres” (contract N 16.740.11.0042).

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