Workload and busy period for M/GI/1 with a general impatience mechanism

The paper deals with the workload and busy period for the $M/GI/1$ M / G I / 1 system with impatience under FCFS discipline. The customers may become impatient during their waiting for service with generally distributed maximal waiting times and also during their service with generally distributed maximal service times depending on the time waited for service. This general impatience mechanism was originally introduced by Kovalenko (1961) and considered by Daley (1965), too. It covers the special cases of impatience on waiting times as well as impatience on sojourn times, for which Boxma et al. (2010, 2011) gave new results and outlined special cases recently. Our unified approach bases on the vector process of workload and busy time. Explicit representations for the LSTs of workload and busy period are given in case of phase-type distributed impatience.     

Join the shortest queue among k parallel queues: tail asymptotics of its stationary distribution

We are concerned with an $M/M$ -type join the shortest queue ( $M/M$ -JSQ for short) with $k$ parallel queues for an arbitrary positive integer $k$ , where the servers may be heterogeneous. We are interested in the tail asymptotic of the stationary distribution of this queueing model, provided the system is stable. We prove that this asymptotic for the minimum queue length is exactly geometric, and its decay rate is the $k$ th power of the traffic intensity of the corresponding $k$ server queues with a single waiting line. For this, we use two formulations, a quasi-birth-and-death (QBD for short) process and a reflecting random walk on the boundary of the $k+1$ -dimensional orthant. The QBD process is typically used in the literature for studying the JSQ with two parallel queues, but the random walk also plays a key roll in our arguments, which enables us to use the existing results on tail asymptotics for the QBD process.     

A Helly-type theorem for intersections of orthogonally starshaped sets in \mathbb R^d

Let $\mathcal K$ be a finite family of orthogonal polytopes in $\mathbb R^d$ such that, for every nonempty subfamily $\mathcal K^\prime $ of $\mathcal K, \cap \{K : K$ in $\mathcal K^\prime \}$ , if nonempty, is a finite union of boxes whose intersection graph is a tree. Assume that every $d + 1$ (not necessarily distinct) members of $\mathcal K$ meet in a (nonempty) staircase starshaped set. Then $S \equiv \cap \{ K : K$ in $\mathcal K\}$ is nonempty and staircase starshaped.     

Subsets of full measure in a generic submanifold in \mathbb C ^n are non-plurithin

In this paper we prove that if $I\subset M $ is a subset of measure $0$ in a $C^2$ -smooth generic submanifold $M \subset \mathbb C ^n$ , then $M \setminus I$ is non-plurithin at each point of $M$ in $\mathbb C ^n$ . This result improves a previous result of A. Edigarian and J. Wiegerinck who considered the case where $I$ is pluripolar set contained in a $C^1$ -smooth generic submanifold $M \subset \mathbb C ^n$ (Edigarian and Wiegernick in Math. Z. 266(2):393–398, 2010). The proof of our result is essentially different.     

Solving convex optimization with side constraints in a multi-class queue by adaptive c\mu rule

We study convex optimization problems with side constraints in a multi-class \(M/G/1\) queue with controllable service rates. In the simplest problem of optimizing linear costs with fixed service rate, the \(c\mu \) rule is known to be optimal. A natural question to ask is whether such simple policies exist for more complex control objectives. In this paper, combining the achievable region approach in queueing systems and the Lyapunov drift theory suitable to optimize renewal systems with time-average constraints, we show that convex optimization problems can be solved by variants of adaptive \(c\mu \) rules. These policies greedily re-prioritize job classes at the end of busy periods in response to past observed delays in each job class. Our method transforms the original problems into a new set of queue stability problems, and the adaptive \(c\mu \) rules are queue stable policies. An attractive feature of the adaptive \(c\mu \) rules is that they use limited statistics of the queue, where no statistics are required for the problem of satisfying average queueing delay in each job class.     

On the existence of Evans potentials

It is shown that, for every noncompact parabolic Riemannian manifold $X$ and every nonpolar compact $K$ in  $X$ , there exists a positive harmonic function on $X\setminus K$ which tends to $\infty $ at infinity. (This is trivial for $\mathbb{R }$ , easy for  $\mathbb{R }^2$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space  $X$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set  $K$ , there is a symmetric (positive) Green function for $X\setminus K$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$ , and the line segments $\{n\}\times [0,1], n=0,1,2,\dots $ ).     

A Question from a Famous Paper of Erdős

Given a convex body $K$ K , consider the smallest number $N$ N so that there is a point $P\in \partial K$ P ∈ ? K such that every circle centred at $P$ P intersects $\partial K$ ? K in at most $N$ N points. In 1946 Erd?s conjectured that $N=2$ N = 2 for all $K$ K , but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $N=\infty $ N = ∞ and that there are convex bodies for which $N = 6$ N = 6 .     

Contact structures on M \times S^2

We show that if a manifold $M$ admits a contact structure, then so does $M \times S^2$ . Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if $M$ admits a contact structure then so does $M \times T^2$ .     

The pure virtual braid group is quadratic

If an augmented algebra $K$ over $\mathbb Q $ is filtered by powers of its augmentation ideal $I$ , the associated graded algebra $gr_I K$ need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper, we give a sufficient criterion (called the PVH Criterion) for $gr_I K$ to be quadratic. When $K$ is the group algebra of a group $G$ , quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for $G$ . Thus, the PVH Criterion also implies the existence of such a universal finite type invariant for the group $G$ . We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

The Chevalley–Eilenberg Complex and Deformation Quantization in Presence of Two Branes

In this note, we prove that, for a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K$ of characteristic 0 which contains $\mathbb C$ , the Chevalley–Eilenberg complex $\mathrm U(\mathfrak g)\otimes \wedge(\mathfrak g)$ , which is in a natural way a deformation quantization of the Koszul complex of $\mathrm S(\mathfrak g)$ , is A _∞-quasi-isomorphic to the deformation quantization of the A _∞-bimodule $K=\mathbb K$ provided by the Formality Theorem in presence of two branes (Calaque et al., Comput Math 147(01):105–160, 2011).     

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \mathbb R ^d$ be a smooth convex set and let $\mathcal{P }_{\lambda }$ be a Poisson point process on $\mathbb R ^d$ of intensity ${\lambda }$ . The convex hull of $\mathcal{P }_{\lambda }\cap K$ is a random convex polytope $K_{\lambda }$ . As ${\lambda }\rightarrow \infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{\lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .     

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$ -modules. Assume $n$ is a nonnegative integer and that the tensor product $M\otimes _{R}N$ is an $(n+c)$ th syzygy of some finitely generated $R$ -module. If ${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$ , then both $M$ and $N$ are $n$ th syzygies of some finitely generated $R$ -modules.     

Large deviations for the empirical mean of an M/M/1 queue

Let $(Q(k):k\ge 0)$ be an $M/M/1$ queue with traffic intensity $\rho \in (0,1).$ Consider the quantity $$\begin{aligned} S_{n}(p)=\frac{1}{n}\sum _{j=1}^{n}Q\left( j\right) ^{p} \end{aligned}$$ for any $p>0.$ The ergodic theorem yields that $S_{n}(p) \rightarrow \mu (p) :=E[Q(\infty )^{p}]$ , where $Q(\infty )$ is geometrically distributed with mean $\rho /(1-\rho ).$ It is known that one can explicitly characterize $I(\varepsilon )>0$ such that $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)<\mu \left( p\right) -\varepsilon \big ) =-I\left( \varepsilon \right) ,\quad \varepsilon >0. \end{aligned}$$ In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n^{1/(1+p)}}\log P\big (S_{n} (p)>\mu \big (p\big )+\varepsilon \big )=-C\big (p\big ) \varepsilon ^{1/(1+p)}, \end{aligned}$$ where $C(p)>0$ is obtained as the solution of a variational problem. We discuss why this phenomenon—Weibullian right tail asymptotics rather than exponential asymptotics—can be expected to occur in more general queueing systems.     

Stability analysis of GI/GI/c/K retrial queue with constant retrial rate

We consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has \(c\) identical servers and can accommodate up to \(K\) jobs (including \(c\) jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. We establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that we obtained have clear probabilistic interpretation.     

On the structure of co-Kähler manifolds

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_\varphi $ , where $K$ is a Kähler manifold and $\varphi $ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\overline{M}$ of the form $\overline{M} \cong K \times S^1$ , where $\cong $ is equivariant diffeomorphism with respect to an action of $S^1$ on $M$ and the action of $S^1$ on $K \times S^1$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $S^1, K$ and are translations on the $S^1$ factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.     

Evolution of spacelike surfaces in \mathrm{AdS}_3 by their Lagrangian angle

We study spacelike hypersurfaces $M$ in an anti-De Sitter spacetime $N$ of constant sectional curvature $-\kappa , \kappa >0$ that evolve by the Lagrangian angle of their Gauß maps. In the two dimensional case we prove a convergence result to a maximal spacelike surface, if the Gauß curvature $K$ of the initial surface $M\subset N$ and the sectional curvature of $N$ satisfy $|K|<\kappa $ .     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

Toric surfaces,K-stability and Calabi flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$ . Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $ \int _{\partial P} u d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$ . Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson’s $M$ -condition holds for $u$ . As an application, we show that if $(X,P)$ is (analytic) relative $K$ -stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.     

Local circular law for random matrices

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point \(z\) away from the unit circle. More precisely, if \( | |z| - 1 | \ge \tau \) for arbitrarily small \(\tau > 0\) , the circular law is valid around \(z\) up to scale \(N^{-1/2+ {\varepsilon }}\) for any \({\varepsilon }> 0\) under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.