Finite Blocking Research Articles

Periodic points on the regular and double n-gon surfaces

Using the transfer principle of Lie theory, we classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

Generalizations of the Eierlegende–Wollmilchsau

We classify a natural collection of GL(2,R)-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmuller space. This classification is the key input in subsequent work of the authors that classifies high rank invariant subvarieties, and in subsequent work of the first author that classifies invariant subvarieties with Lyapunov spectrum as degenerate as possible and not consisting of torus covers. We also derive applications to the complex geometry of Teichmuller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.

Marked points on translation surfaces

We show that all GL(2,R) equivariant point markings over orbit closures of translation surfaces arise from branched covering constructions and periodic points, completely classify such point markings over strata of quadratic differentials, and give applications to the finite blocking problem.

GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.

Evaluation of ZnS:6LiF and ZnO:6LiF Scintillation Neutron Detectors Readout With SiPMs

It is generally accepted that the count rate capability of ZnS: <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> LiF neutron detectors is strongly limited due to the long afterglow of the scintillator. However, realized as an array of individual sensitive elements (pixels) with independent signal readout and processing, such detectors can be operated with blocking times down to 1 μs and promising counting rates up to 100 kHz/pixel. In this paper, we investigate how the long emission of the ZnS scintillator influences the detector performance apart from requiring a finite blocking time to suppress the probability of multiple counts. We also study the potential of the ZnO: <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> LiF scintillator with lower afterglow as a possible replacement of ZnS: <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> LiF.

Conveyor Merges in Zone Picking Systems: A Tractable and Accurate Approximate Model

Sequential zone picking systems are popular conveyor-based picker-to-parts order picking systems that divide the order picking area in work zones. When designing a zone picking system, it is important to know whether the throughput capability of the system can meet customer demand. However, the performance and maximum throughput capability of a zone picking system is largely determined by congestion and blocking that occur at the various conveyor merges in the system. In this paper we develop an analytical model to study the impact of conveyor merges in sequential zone picking systems. Because of finite buffers, blocking, recirculation, and merging, the resulting queueing model does not have a product-form stationary queue-length distribution which makes exact analysis practically infeasible. Therefore, we develop an approximate solution by using an aggregation technique and matrix-geometric methods to study the throughput capability of the system. The model is suitable to support rapid design of complex zone picking systems, in terms of number and length of zones, input and output buffer capacities, and storage allocation of products to zones to meet prespecified performance targets. Comparison of the approximation results to simulation show that for a wide range of parameters the mean relative error in the system throughput is typically less than 5%. The model accurately predicts the loss in throughput due to congestion and blocking at the merges, and can be used to allocate input and output buffer spaces to maximize the throughput capability of the system.

Everything is illuminated

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the SL(2,R)-action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert-Schmoll-Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture. Our results crucially rely on the recent breakthrough results of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, and on related results of Wright.

Switchless Elastic Rate Node (SERANO) Architecture: A Universal Node for Optical Grooming and Adaptive Networking

The switchless elastic rate node (SERANO) is an elastic optical network block architecture facilitating reduction in network cost and blocking due to spectral fragmentation while extending the capacity-reach product in translucent optical networks. We present an analytical mathematical method for minimizing the number of transceivers that are solely used to construct this block, albeit a finite blocking probability. Moreover, based on a realistic traffic matrix and making use of a networking planning tool, we deduce a SERANO's block dimension for BT's national backbone network. Finally, we estimate the conditions under which the cost of a SERANO block is lower to the corresponding electronic switch for providing important networking functions like 3R regeneration, defragmentation, and network reoptimization.

Generic absence of finite blocking for interior points of Birkhoff billiards

Let $x$ and $y$ be points in a billiard table $M=M(\sigma)$ in$\mathbb{\mathbb{R}}^{2}$ that is bounded by a curve $\sigma$. Weassume $\sigma\in\Sigma_{r}$ with $r\geq2$, where $\Sigma_{r}$is the set of simple closed $C^{r}$ curves in $\mathbb{R}^{2}$ withpositive curvature. A subset $B$ of $M\setminus\{x,y\}$ is calleda blocking set for the pair $(x,y)$ if every billiard pathin $M$ from $x$ to $y$ passes through a point in $B$. If a finiteblocking set exists, the pair $(x,y)$ is called secure in$M;$ if not, it is called insecure. We show that for $\sigma$in a dense $G_{\delta}$ subset of $\Sigma_{r}$ with the $C^{r}$topology, there exists a dense $G_{\delta}$ subset $\mathcal{\mathcal{R}=R}(\sigma)$of $M(\sigma)\times M(\sigma)$ such that $(x,y)$ is insecure in$M(\sigma)$ for each $(x,y)\in\mathcal{R}$. In this sense, for thegeneric Birkhoff billiard, the generic pair of interior points isinsecure. This is related to a theorem of S. Tabachnikov [24]that $(x,y)$ is insecure for all sufficiently close points$x$ and $y$ on a strictly convex arc on the boundary of a smoothtable.

Cyclic Queueing Networks With Subexponential Service Times and Finite Buffers

In this technical note, we consider closed tandem queueing networks with finite buffers and blocking. We assume that at least one station has subexponential service time distribution. We analyze this system under communication blocking and manufacturing blocking rules. Our objective is to derive expressions for the tail asymptotics of transient cycle times and waiting times. Furthermore, we study under which conditions on system parameters these tail asymptotics also hold for their stationary counter parts. Finally, we provide numerical examples to understand the convergence behavior of the tail asymptotics.

Real analytic metrics on S 2 with total absence of finite blocking

If (M, g) is a Riemannian manifold and \({(x,y) \in M \times M,}\) then a set \({P \subset M \backslash \{x,y\}}\) is said to be a blocking set for (x, y) if every geodesic from x to y passes through a point of P. If no pair (x, y) in M × M has a finite blocking set, then (M, g) is said to be totally insecure. We prove that there exist real analytic metrics h on S 2 such that (S 2, h) is totally insecure. Previously known examples of totally insecure metrics on S 2 were only C 1.

A dense G-delta set of Riemannian metrics without the finite blocking property

A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) in M x M, there exists a dense G-delta set of C-infinity Riemannian metrics on M such that (x,y) fails to have the finite blocking property for every g in that set.

Finite blocking property versus pure periodicity

AbstractA translation surface 𝒮 is said to have the finite blocking property if for every pair (O,A) of points in 𝒮 there exists a finite number of ‘blocking’ points B1,…,Bn such that every geodesic from O to A meets one of the Bis. 𝒮 is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that 𝒮 admits a cylinder decomposition in such directions). We will prove that the finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus two: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum such surfaces form a set of null measure. In Appendix A, we prove that completely periodic translation surfaces form a set of null measure in every stratum.

Performance measurement and lumped parameter modeling of single server flow lines subject to blocking: An effective process time approach

The present paper extends the so-called effective process time (EPT) approach to single server flow lines with finite buffers and blocking. The power of the EPT-approach is that it quantifies variability in workstation process times without the need to identify each of the contributing disturbances, and that it directly provides an algorithm for the actual computation of EPTs. It is shown that EPT-realizations can be simply obtained from arrival and departure times of lots, by using sample path equations. The measured EPTs can be used for bottleneck analysis and for lumped parameter modeling. Simulation experiments show that for lumped parameter modeling of flow lines with finite buffers, in addition to the mean and variance, offset is also a relevant parameter of the process time distribution. A case from the automotive industry illustrates the approach.

Blocking light in compact Riemannian manifolds

We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat metrics. Using entropy considerations, we verify this conjecture amongst metrics with nonpositive sectional curvatures. Using the same approach, K. Burns and E. Gutkin have independently obtained this result. Additionally, we show that compact quotients of Euclidean buildings have the finite blocking property. On the positive curvature side, we conjecture that compact Riemannian manifolds with the same blocking properties as compact rank one symmetric spaces are necessarily isometric to a compact rank one symmetric space. We include some results providing evidence for this conjecture.

Optimal control of processing times in single-stage discrete event dynamic systems with blocking

This note concerns an optimal control problem in single-stage discrete event dynamic systems with finite buffers and blocking. The system is modeled as a deterministic queue, slated to process a finite sequence of jobs. Each job is characterized by its arrival time, service time, and due date, and has associated with it a cost function that penalizes short service times, buffer times, and lateness of completion times with respect to the due date. The sequencing (order) of the jobs at the server is given, and the variable parameter consists of the jobs' service times. Even though the cost function associated with each job is assumed to be convex, the aggregated cost functional is not convex. Therefore, much of the analysis focuses on a decomposition of the problem into a finite sequence of reduced-order convex programming problems which can be solved one at a time. This approach has been investigated in the past, but the present note provides an analysis under the most general and realistic assumptions considered to date.

Performance analysis of multi-server tandem queues with finite buffers and blocking

Performance analysis of multi-server tandem queues with finite buffers and blocking

A Counter-Example to the Theorem of Hiemer and Snurnikov

A planar polygonal billiard $\P$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\P$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$'s. As a counter-example to a theorem of Hiemer and Snurnikov, we construct a family of rational billiards that lack the finite blocking property.

A review on queueing network models with finite capacity queues for software architectures performance prediction

A review is carried out on how queueing network models with blocking have been applied so far into the performance evaluation and prediction of Software Architectures (SA). Queueing network models with finite capacity queues and blocking have recently been introduced and applied as more realistic models of systems with finite capacity resources and population constraints. Queueing network models have been often adopted as models for the evaluation of software performance. Starting from our own experience, we observe the need of a more accurate definition of the performance models of SA to capture some features of the communication systems. We consider queueing networks with finite capacity and blocking after service (BAS) to represent some synchronization constraints that cannot be easily modeled with queueing network models with infinite capacity queues. We investigate the use of queueing networks with blocking as performance models of SA with concurrent components and synchronous communication. Queueing theoretic analysis is used to solve the queueing network model and study the synchronous communication and performance of concurrent software components. Our experience is supported by other approaches that also propose the use of queueing networks with blocking. Directions for future research work in the field are included.

Mean value analysis of product form solution queueing networks with repetitive service blocking

Queueing network models with finite capacity queues and blocking are used to represent systems with resource constraints, such as production, communication and computer systems. Various blocking mechanisms have been defined in the literature to represent the different behaviours of real systems with limited resources. Queueing networks with blocking have a product form solution under special constraints, for different blocking mechanisms. In this paper we present a Mean Value Analysis for the computation of performance measures in product form solution queueing networks with repetitive service blocking. Basic to the derivation of this algorithm are recursive expressions for the performance indices that are a non trivial generalisation of those derived for the Mean Value Analysis of product form queueing networks without blocking. In this paper we give a formal derivation of several recursive relations as well as details on their implementation. A few basic examples are evaluated with the techniques discussed in this paper to show the advantages of this approach.