scholarly journals Uniform Polynomial Rates of Convergence for A Class of Lévy-Driven Controlled SDEs Arising in Multiclass Many-Server Queues

2019 ◽  
pp. 1-20
Author(s):  
Ari Arapostathis ◽  
Hassan Hmedi ◽  
Guodong Pang ◽  
Nikola Sandrić
Keyword(s):  
Rates Of Convergence ◽  
Many Server Queues

A new view of the heavy-traffic limit theorem for infinite-server queues

1991 ◽  
Vol 23 (01) ◽  
pp. 188-209 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Infinite Server ◽  
Open Networks ◽  
Heavy Traffic Limit ◽  
Many Server Queues ◽  
Finite Set ◽  
Deterministic Service ◽  
Infinite Server Queues ◽  
Service Times

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

A new view of the heavy-traffic limit theorem for infinite-server queues

1991 ◽  
Vol 23 (1) ◽  
pp. 188-209 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Infinite Server ◽  
Open Networks ◽  
Heavy Traffic Limit ◽  
Many Server Queues ◽  
Finite Set ◽  
Deterministic Service ◽  
Infinite Server Queues ◽  
Service Times

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

Optimal Rates of Convergence for Deconvolving a Density

1988 ◽  
Author(s):  
Raymond J. Carroll ◽  
Peter Hall
Keyword(s):  
Rates Of Convergence ◽  
Optimal Rates Of Convergence

scholarly journals Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021 ◽  
Author(s):  
Felix Herold ◽  
Daniel Hug ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Hyperbolic Space ◽  
Central Limit ◽  
Hausdorff Measure ◽  
Rates Of Convergence ◽  
Limit Problem ◽  
Dimensional Hausdorff Measure ◽  
The Central Limit Theorem ◽  
Order Quantities

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Integral Equations ◽  
Waiting Time ◽  
Queue Length ◽  
Joint Distribution ◽  
Waiting Times ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Many Server Queues ◽  
The Many ◽  
Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

Distributed learning with indefinite kernels

2019 ◽  
Vol 17 (06) ◽  
pp. 947-975 ◽  
Author(s):  
Lei Shi
Keyword(s):  
Large Scale ◽  
Substantial Reduction ◽  
Computation Time ◽  
Distributed Learning ◽  
Rates Of Convergence ◽  
Regression Problem ◽  
Data Set ◽  
Regularization Scheme ◽  
Original Algorithm ◽  
Indefinite Kernel

We investigate the distributed learning with coefficient-based regularization scheme under the framework of kernel regression methods. Compared with the classical kernel ridge regression (KRR), the algorithm under consideration does not require the kernel function to be positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods. The distributed learning approach partitions a massive data set into several disjoint data subsets, and then produces a global estimator by taking an average of the local estimator on each data subset. Easy exercisable partitions and performing algorithm on each subset in parallel lead to a substantial reduction in computation time versus the standard approach of performing the original algorithm on the entire samples. We establish the first mini-max optimal rates of convergence for distributed coefficient-based regularization scheme with indefinite kernels. We thus demonstrate that compared with distributed KRR, the concerned algorithm is more flexible and effective in regression problem for large-scale data sets.

scholarly journals $$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

2021 ◽  
Vol 19 (2) ◽  
Author(s):  
Moritz Moeller ◽  
Tino Ullrich
Keyword(s):  
Random Function ◽  
Additional Assumption ◽  
Main Tool ◽  
Singular Values ◽  
Rates Of Convergence ◽  
Concentration Inequality ◽  
Worst Case ◽  
Mercer Kernels ◽  
Finite Trace ◽  
Complex Valued

AbstractIn this paper we study $$L_2$$ L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $$D \subset \mathbb {R}^d$$ D ⊂ R d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into $$L_2$$ L 2 ) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in n, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.

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