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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2017.tde-16102017-154842
Document
Author
Full name
Mark Andrew Gannon
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Iambartsev, Anatoli (President)
Belitsky, Vladimir
Fontes, Luiz Renato Goncalves
Lebensztayn, Élcio
Pechersky, Eugene Abramovich
Title in Portuguese
Passeios aleatórios em redes finitas e infinitas de filas
Keywords in Portuguese
Alcance zero
Exclusao simples
Forma produto
Probabilidade estacionaria
Processo de Markov em tempo continuo
Rede de filas
Rede de Jackson
Reversibilidade
Sistema interagente de particulas
Abstract in Portuguese
Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel.
Title in English
Random walks in finite and infinite queueing networks
Keywords in English
Continuous-time Markov process
Interacting particle system
Jackson network
Product form
Queueing network
Reversibility
Simple exclusion
Stationary probability
Zero-range
Abstract in English
A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.
 
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Publishing Date
2017-11-16
 
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