Description
- Abstract:
- Stochastic networks arise in a variety of real world applications including telecommunications, service systems such as call centers, computer networks, health care services and biological systems. Leaving aside some very simple examples, it is usually infeasible to perform an exact analysis of these networks. A useful alternative is instead to identify a suitable approximation that provides insight into performance and can be shown to be accurate in a relevant asymptotic regime. Networks in which jobs or customers are processed at nodes with multiple servers arise in many applications, and tend to be harder to analyze than networks where each node is processed only by a single server. Moreover, when the service times of jobs are generally distributed, the analysis becomes even more challenging. For example, the dimension of a Markovian state descriptor of a many-server network with general service distributions typically grows with the size of the network. Thus, a common Markovian state description that is applicable for a network of any size must necessarily be infinite-dimensional. This thesis analyzes the equilibrium behavior of two different types of scaling limits of two classes of many-server stochastic networks. In both cases, the scaling limits are infinite-dimensional and require the development of new techniques for their analysis. The first part of the thesis considers a many-server parallel network, where each server has its own queue, to which jobs arriving with independent and identically distributed (i.i.d.) service times are routed immediately on arrival using the shortest- of-d randomized load balancing algorithm. Such models are of interest in hash tables in data switches, parallel computing, wireless networks, etc. The focus here is on the so-called hydrodynamic limit, which describes the mean behavior of the state dynamics, as the number of servers goes to infinity. For a general class of service distributions, the hydrodynamic limit was characterized by Aghajani and Ramanan as the unique solution to a countable coupled system of deterministic measure-valued processes. This thesis shows that the hydrodynamic limit has an invariant state, which is a probability measure that describes the distribution of jobs in queues, and also provides a characterization of it in terms of a sequence of recursive fixed point problems. The latter is also shown to enable approximate computation of the invariant state, which is proved to be unique when d=2. Furthermore, under an additional finite moment condition on the service distribution and an a priori exponential tail decay assumption on the invariant state, it is shown that any invariant state of the hydrodynamic equations of a sub-critical network with finite mean queue length exhibits double exponential tail decay. This result takes an important step towards identifying the class of service distributions for which the power of two choices holds, beyond the case of service distributions with decreasing hazard rate considered by Bramson, Lu and Prabhakar. Numerical evidence is provided to support the analytical results. The second model is a many-server queue with reneging often denoted as the G/GI/N+GI queue, in which there are N servers and a common queue to which jobs arrive with i.i.d. service and patience times and wait to be processed till a server becomes free. Jobs depart a system either on completion of service or by reneging from the queue when the time waiting in the queue exceeds the patience time. Under suitable assumptions, a functional central limit theorem for the state dynamics is established in the so-called Halfin-Whitt asymptotic regime. The limit is characterized as the unique solution to a pair of coupled stochastic partial differential equations with a nonlinear boundary condition having a non-standard form. It is also shown that the limit process describing the total number in system is an Itô diffusion. A suitable function-valued representation of the state process plays an important role in the analysis for this model. The function-valued representation can be viewed as a more tractable reduced version of the measure-valued representation that still retain relevant information about the model, including limiting fluctuations of the queue length and virtual waiting time. In particular, our work allows the study of the overloaded or supercritical regime, where the asymptotic mean arrival rate exceeds the mean service time, which is of particular interest in applications with reneging. The work here extends previous work on both many-server queues without reneging as well as queues with reneging, which have been restricted to fluctuations of the queue length in the critical regime, where the asymptotic mean arrival rate equals the mean service time.
- Notes:
- Thesis (Ph. D.)--Brown University, 2021
Citation
Agarwal, Pooja,
"Equilibrium Behavior of Infinite-Dimensional Scaling Limits of Many-Server Stochastic Networks"
(2021).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:57947mwe/
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....