Stochastic Networks: Control and Performance

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The aim of this project is to study mathematical problems associated with the control and performance analysis of stochastic networks. The network models being considered are heterogeneous, may have complex feedback mechanisms, and allow for stochastic variability in arrivals, service times and routing. Buffers in the network enable jobs to be stored that cannot be served immediately. In addition, the models with control allow for dynamic sequencing and alternate routing of jobs. Since the complexity of these network models usually precludes exact analysis, the focus is on approximate models. Two levels of approximation are being considered, namely fluid models (first order approximations) and diffusion models (second order approximations). Investigating the interplay between these models is an important feature of the research. The two main topics are (i) dynamic control through sequencing and alternate routing, and (ii) performance analysis for processor sharing networks. With regard to topic (i), some authors have successfully used diffusion control problems as formal tools for generating good control policies for some specific network models. However, there are few rigorous analyses of the performance of such policies. The PI is developing a systematic approach to finding and interpreting solutions of diffusion control problems and to analyzing the performance of the policies generated in this manner. Regarding topic (ii), the processor sharing discipline is an example of a service discipline for which a natural state descriptor involves a measure-valued process (to keep track of the residual service times of all jobs in the network). The PI is studying fluid and diffusion approximations of processor sharing networks with the aims of understanding the dynamics of such networks, obtaining measures of performance, and developing general tools for studying measure-valued processes associated with network models. Stochastic networks are used as models for complex manufacturing, telecommunications and computer systems. A challenging problem for such networks is to design controls that are simple to implement and yet are near optimal in an appropriate sense. Motivated by such problems, a number of mathematical questions associated with controlling and analyzing the performance of stochastic networks are being studied under this grant. Since the complexity of stochastic network models usually precludes exact analysis, the focus is on approximate models with a hierarchical structure. As with recent work on the performance analysis of some networks, the interplay between the levels in this hierarchy is an important feature of the research.