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https://hdl.handle.net/2142/20921
Description
Title
Stability of queueing networks
Author(s)
Down, Douglas Graham
Issue Date
1995
Doctoral Committee Chair(s)
Meyn, Sean P.
Department of Study
Electrical and Computer Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Operations Research
Language
eng
Abstract
In this thesis, the stability of queueing networks is studied. The use of test functions is a unifying thread. Tools are provided to construct appropriate test functions for complex networks, and the structure of such test functions is examined for specific network models. The analysis of queueing networks is performed in a manner that progresses in increasing complexity for increasingly complex networks.
Single class networks are considered first. The particular form studied is open generalized Jackson networks with general arrival streams and general service time distributions. Assuming that the arrival rate does not exceed the network capacity and that the service times possess conditionally bounded second moments, stability is deduced by bounding the expected waiting time for a customer entering the network. For Markovian networks convergence of the total work in the system is obtained, as well as convergence of the mean queue size and mean customer delay, to a unique finite steady state value.
Acyclic multiclass networks are the next topic. Once again, assuming that the arrival rate does not exceed the network capacity, stability of the network is deduced using the tools of ergodic theory. The distributions of the process are shown to converge to a unique steady state value, and under appropriate moment conditions, the convergence takes place at an exponential rate.
The final topic is general re-entrant lines. In this case, piecewise linear test functions are developed for the analysis of both queueing networks and their associated fluid models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid model is stable and, hence, that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a different appropriate LP admits a solution, then the network model is transient.
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