报告题目：Nonlinear Markov Processes of GI/G/1 TypeMean-Field Equations
Large-Scale Networks under Weak Interactions
This talk provides a simple overview for our research onnonlinear Markov processes of GI/G/1 type, which are established by analysis of non-Markovsupermarket modelsand non-Markovwork-stealing models. To set up the mean-field equations, we apply the supplementary variable method and the RG-factorizations to discussing several important examples, such as, M/G/1-supermarket model, GI/M/1-supermarket model, GI/G/1-supermarket model; MAP/G/1-supermarket model, GI/PH/1-supermarket model. As an interesting generalization of ordinary Markov chains of GI/G/1 Type (see Neuts (1981, 1989)), the abovesupermarket models may result in various nonlinear Markov processes of GI/G/1 type through the mean-field equations, where some new interpretation is given for the R-, U- and G-measures, therefore, the algorithms for R-, U- and G-measures are renewed in a novel framework.
We use a macroscopic analysis ofthemean-field theory from the statistical physics,and set upthenonlinear Markov processeswithblockstructurefrom some practical systems such as thesupermarket models andwork-stealing models,in whichtheirdynamic systems perform many new and interesting characteristics.Based on this,stability and metastability ofthelarge-scalestochastic networks will be interesting in our present work. We indicate that the metastability can be analyzed by the entropy approximation, and more generally, information theory of queueing networks and large deviations of Markov processes.
Our work focuses onnumericalcomputationof large-scale stochastic networks,includingperformance analysis,nonlinear adaptivecontrol, metastability and Lyapunov technique, entropy approximation and information theory. To that end, we develop some interesting mathematical modeling and analysis, including operator semigroup, mean-field method, martingale limits, density-dependent jump Markov processes, and specifically, matrix-analytic methods.
Finally, we list some promising open problems in our future study.