Description
- Abstract:
- Large-scale stochastic networks arise in a variety of real world applications such as telecommunications, service systems, computer networks, health care, and biological systems. Such networks are typically too complex and not amenable to exact analysis. However, it is often possible to provide insight into network performance, in both the transient and equilibrium regimes, through tractable approximations. Finite-dimensional Markov processes have been used extensively in the literature to describe the scaling limits of stochastic networks under simplifying assumptions such as exponential service distributions. However, statistical analysis shows that the service distribution is typically non-exponential. For networks with general service distributions, the dimension of the system descriptor typically grows with size of the network, and hence any scaling limit is inherently infinite dimensional. Thus, a significant challenge is to find a suitable representation of the network that leads to a tractable scaling limit. This Thesis has focused on developing a mathematical framework for the analysis of infinite-dimensional scaling limits of many-server stochastic networks. Specifically, we have focused on two rather different problems for two kinds of large-scale stochastic networks -- a many-server queueing model (also known as the GI/GI/N model) and a randomized load balancing model, both in the presence of general service distributions. For the many-server queueing model, we show that the limit of normalized fluctuations of the network can be captured by a non-standard Stochastic Partial Differential Equation (SPDE). We also establish ergodicity of the solution to this SPDE via a novel construction of an asymptotic (equivalent) coupling, as traditional Harris recurrent techniques are often inapplicable in the infinite-dimensional setting. Our result resolves an open problem on a many-server queueing model originally raised by Halfin and Whitt in 1981. For the randomized load-balancing network, we introduce a new representation using interacting measure-valued processes, and show that the mean transient behavior of the network can be approximated by a countably infinite coupled system of partial integro-differential equations. This provides a significant extension of the ODE method used in the case of an exponential service distribution.
- Notes:
- Thesis (Ph.D. -- Brown University (2016)
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Citation
Aghajani, Mohammadreza,
"Infinite-Dimensional Scaling Limits of Stochastic Networks"
(2016).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0P55KWK
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....