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- All Subjects: Stein's method
- Creators: Zhang, Junshan
Description
This dissertation studies load balancing algorithms for many-server systems (with N servers) and focuses on the steady-state performance of load balancing algorithms in the heavy traffic regime. The framework of Stein’s method and (iterative) state-space collapse (SSC) are used to analyze three load balancing systems: 1) load balancing in the Sub-Halfin-Whitt regime with exponential service time; 2) load balancing in the Beyond-Halfin-Whitt regime with exponential service time; 3) load balancing in the Sub-Halfin-Whitt regime with Coxian-2 service time.
When in the Sub-Halfin-Whitt regime, the sufficient conditions are established such that any load balancing algorithm that satisfies the conditions have both asymptotic zero waiting time and zero waiting probability. Furthermore, the number of servers with more than one jobs is o(1), in other words, the system collapses to a one-dimensional space. The result is proven using Stein’s method and state space collapse (SSC), which are powerful mathematical tools for steady-state analysis of load balancing algorithms. The second system is in even “heavier” traffic regime, and an iterative refined procedure is proposed to obtain the steady-state metrics. Again, asymptotic zero delay and waiting are established for a set of load balancing algorithms. Different from the first system, the system collapses to a two-dimensional state-space instead of one-dimensional state-space. The third system is more challenging because of “non-monotonicity” with Coxian-2 service time, and an iterative state space collapse is proposed to tackle the “non-monotonicity” challenge. For these three systems, a set of load balancing algorithms is established, respectively, under which the probability that an incoming job is routed to an idle server is one asymptotically at steady-state. The set of load balancing algorithms includes join-the-shortest-queue (JSQ), idle-one-first(I1F), join-the-idle-queue (JIQ), and power-of-d-choices (Pod) with a carefully-chosen d.
When in the Sub-Halfin-Whitt regime, the sufficient conditions are established such that any load balancing algorithm that satisfies the conditions have both asymptotic zero waiting time and zero waiting probability. Furthermore, the number of servers with more than one jobs is o(1), in other words, the system collapses to a one-dimensional space. The result is proven using Stein’s method and state space collapse (SSC), which are powerful mathematical tools for steady-state analysis of load balancing algorithms. The second system is in even “heavier” traffic regime, and an iterative refined procedure is proposed to obtain the steady-state metrics. Again, asymptotic zero delay and waiting are established for a set of load balancing algorithms. Different from the first system, the system collapses to a two-dimensional state-space instead of one-dimensional state-space. The third system is more challenging because of “non-monotonicity” with Coxian-2 service time, and an iterative state space collapse is proposed to tackle the “non-monotonicity” challenge. For these three systems, a set of load balancing algorithms is established, respectively, under which the probability that an incoming job is routed to an idle server is one asymptotically at steady-state. The set of load balancing algorithms includes join-the-shortest-queue (JSQ), idle-one-first(I1F), join-the-idle-queue (JIQ), and power-of-d-choices (Pod) with a carefully-chosen d.
ContributorsLiu, Xin (Author) / Ying, Lei (Thesis advisor) / Maguluri, Siva Theja (Committee member) / Wang, Weina (Committee member) / Zhang, Junshan (Committee member) / Arizona State University (Publisher)
Created2019
Description
This dissertation presents a novel algorithm for recovering missing values of co-evolving time series with partial embedded network information. The idea is to connect two sources of data through a shared low dimensional latent space. The proposed algorithm, named NetDyna, is an Expectation-Maximization algorithm, and uses the Kalman filter and matrix factorization approaches to infer the missing values both in the time series and embedded network. The experimental results on real datasets, including a Motes dataset and a Motion Capture dataset, show that (1) NetDyna outperforms other state-of-the-art algorithms, especially with partially observed network information; (2) its computational complexity scales linearly with the time duration of time series; and (3) the algorithm recovers the embedded network in addition to missing time series values.
This dissertation also studies a load balancing algorithm, the so called power-of-two-choices(Po2), for many-server systems (with N servers) and focuses on the convergence of stationary distribution of Po2 in the both light and heavy traffic regimes to the solution of mean-field system. The framework of Stein’s method and state space collapse (SSC) are used to analyze both regimes.
In both regimes, the thesis first uses the argument of state space collapse to show that the probability of the state being far from the mean-field solution is small enough. By a simple Markov inequality, it is able to show that the probability is indeed very small with a proper choice of parameters.
Then, for the state space close to the solution of mean-field model, the thesis uses Stein’s method to show that the stochastic system is close to a linear mean-field model. By characterizing the generator difference, it is able to characterize the dominant terms in both regimes. Note that for heavy traffic case, the lower and upper bound analysis of a tridiagonal matrix, which arises from the linear mean-field model, is needed. From the dominant term, it allows to calculate the coefficient of the convergence rate.
In the end, comparisons between the theoretical predictions and numerical simulations are presented.
This dissertation also studies a load balancing algorithm, the so called power-of-two-choices(Po2), for many-server systems (with N servers) and focuses on the convergence of stationary distribution of Po2 in the both light and heavy traffic regimes to the solution of mean-field system. The framework of Stein’s method and state space collapse (SSC) are used to analyze both regimes.
In both regimes, the thesis first uses the argument of state space collapse to show that the probability of the state being far from the mean-field solution is small enough. By a simple Markov inequality, it is able to show that the probability is indeed very small with a proper choice of parameters.
Then, for the state space close to the solution of mean-field model, the thesis uses Stein’s method to show that the stochastic system is close to a linear mean-field model. By characterizing the generator difference, it is able to characterize the dominant terms in both regimes. Note that for heavy traffic case, the lower and upper bound analysis of a tridiagonal matrix, which arises from the linear mean-field model, is needed. From the dominant term, it allows to calculate the coefficient of the convergence rate.
In the end, comparisons between the theoretical predictions and numerical simulations are presented.
ContributorsHairi, FNU (Author) / Ying, Lei (Thesis advisor) / Wang, Weina (Committee member) / Zhang, Junshan (Committee member) / Zhang, Yanchao (Committee member) / Arizona State University (Publisher)
Created2020