Polling models with multi-phase gated service

In this paper we introduce and analyze a new class of service policies called multi-phase gated service. This policy is a generalization of the classical single-phase and two-phase gated policies and works as follows. Each customer that arrives at queue i will have to wait K _i ??1 cycles before it receives service. The aim of this policy is to provide an interleaving scheme to avoid monopolization of the system by heavily loaded queues, by choosing the proper values of interleaving levels K _i . In this paper, we analyze the effectiveness of the interleaving scheme on the queueing behavior of the system, and consider the problem of identifying the proper combination of interleaving levels ${\underline{K}}^{*}=(K_{1}^{*},\ldots,K_{N}^{*})$ that minimizes a weighted sum of the mean waiting times at each of the N queues. Obviously, the proper choice of the interleaving levels is most critical when the system is heavily loaded. For this reason, we explore the framework developed in Queueing Syst. 57, 29?C46 (2007) to obtain closed-form expressions for the asymptotic waiting-time distributions in heavy traffic, and use these expressions to derive simple heuristics for approximating the optimal interleaving scheme ${\underline{K}}^{*}$ . Numerical results with simulations demonstrate that the accuracy of these approximations is extremely high.     

Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product

where the mass points a_k belong to [−1,1], M_k,i0, i=0,…,N_k−1, and M_k,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants M_k,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in .     

Uniform persistence for Lotka–Volterra-type delay differential systems

Consider the uniform persistence (permanence) of models governed by the following Lotka–Volterra-type delay differential system:

where each r_i(t) is a nonnegative continuous function on [0,+∞), r_i(t)0, each a_i0 and τ_ij^k(t)t, 1i,jn, 0km.In this paper, we establish sufficient conditions of the uniform persistence and contractivity for solutions (and global asymptotic stability). In particular, we extend the results in Wang and Ma (J. Math. Anal. Appl. 158 (1991) 256) for a predator–prey system and Lu and Takeuchi (Nonlinear Anal. TMA 22 (1994) 847) for a competitive system in the case n=2, to the above system with n2.     

Some new sums related to D. H. Lehmer problem

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1 ? a _i ? p ? 1 such that a ₁ a ₂ … a _k ≡ 1 mod p and 2 | a _i + ā _i + 1, i = 1, 2, …, k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it.     

A Stochastic Directional Convexity Result and Its Application in Comparison of Queues

Second order properties of queues are important in design and analysis of service systems. In this paper we show that the blocking probability of M/M/C/N queue is increasing directionally convex in (λ,?μ), where λ is arrival rate and μ is service rate. To illustrate the usefulness of this result we consider a heterogeneous queueing system with non-stationary arrival and service processes. The arrival and service rates alternate between two levels (λ₁,μ₁) and (λ₂,μ₂), spending an exponentially distributed amount of time with rate cα _i in level i, i=1,2. When the system is in state i, the arrival rate is λ _i and the service rate is μ _i . Applying the increasing directional convexity result we show that the blocking probability is decreasing in c, extending a result of Fond and Ross [7] for the case C=N=1.

     

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ _i the (i + 1)-th phase starts, with probability p _i the customer feedback to the tail of the queue and with probability 1 − θ _i − p _i  = q _i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p _k feedback to the tail of the queue and with probability 1 − p _k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.     

Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs

We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

     

On subspace arrangements of type

Let denote the subspace arrangement formed by all linear subspaces in given by equations of the form

₁x_i1=₂x_i2==_kx_ik,

where 1i₁<<i_kn and (₁,…,_k){+1,−1}^k.Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In a previous work on a larger class of subspace arrangements by Björner and Sagan (J. Algebraic Combin. 5 (1996) 291–314) the topology of the intersection lattice turned out to be a particularly interesting and difficult case.We prove in this paper that Pure(Π_n,k^±) is shellable, hence that Π_n,k^± is shellable for k>n/2. Moreover, we prove that unless i≡n−2 (mod k−2) or i≡n−3 (mod k−2), and that is free abelian for i≡n−2 (mod k−2). In the special case of Π_2k,k^± we determine homology completely. Our tools are generalized lexicographic shellability, as introduced in Kozlov (Ann. Combin. 1 (1997) 67–90), and a spectral sequence method for the computation of poset homology first used in Hanlon (Trans. Amer. Math. Soc. 325 (1991) 1–37).We state implications of our results on the cohomology of the complements of the considered arrangements.     

Structure of random 312‐avoiding permutations

We evaluate the probabilities of various events under the uniform distribution on the set of 312‐avoiding permutations of . We derive exact formulas for the probability that the i^th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312‐avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 599–631, 2016     

Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f([`(x)]) = ?<sub>i = 1</sub><sup>k</sup> h<sub>i</sub> ([`(x)]) ·g_i ([`(x)])f(\bar x) = \sum\nolimits_{i = 1}^k {h_i (\bar x) \cdot g_i (\bar x)} , where each h _i is a polynomial that depends on only ρ linear functions, and each g _i is a product of linear functions (when h _i = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When max _i (deg(h _i · g _i )) = d we say that f is computable by a ΣΠΣ(k; d;ρ) circuit. We obtain the following results.

Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace

Summary The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.The following principle governs the passage from results in the plane to results in space:Let B be a three-dimensional Brownian motion, and P ₁, ..., P _k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B ⁱ (1ik), defined respectively as the orthogonal projections of B on P _i (1ik), take place jointly, and the corresponding limit variables are independent.     

An asymptotic version of a conjecture by Enomoto and Ota

In 2000, Enomoto and Ota [J Graph Theory 34 (2000), 163–169] stated the following conjecture. Let G be a graph of order n, and let n₁, n₂, …, n_k be positive integers with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} {{n}}_{{{i}}} = {{n}}\end{eqnarray*}. If σ₂(G)≥n+ k?1, then for any k distinct vertices x₁, x₂, …, x_k in G, there exist vertex disjoint paths P₁, P₂, …, P_k such that |P_i|=n_i and x_i is an endpoint of P_i for every i, 1≤i≤k. We prove an asymptotic version of this conjecture in the following sense. For every k positive real numbers γ₁, …, γ_k with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} \gamma_{{{i}}} = {{1}}\end{eqnarray*}, and for every ε>0, there exists n₀ such that for every graph G of order n≥n₀ with σ₂(G)≥n+ k?1, and for every choice of k vertices x₁, …, x_k∈V(G), there exist vertex disjoint paths P₁, …, P_k in G such that \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} |{{P}}_{{{i}}}| = {{n}}\end{eqnarray*}, the vertex x_i is an endpoint of the path P_i, and (γ_i?ε)n<|P_i|<(γ_i + ε)n for every i, 1≤i≤k. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 37–51, 2010     

Graph decompositions without isolated vertices III

Let G be a graph of order n, and n = Σ^k_i=1 a_i be a partition of n with a_i ≥ 2. In this article we show that if the minimum degree of G is at least 3k−2, then for any distinct k vertices v₁,…, v_k of G, the vertex set V(G) can be decomposed into k disjoint subsets A₁,…, A_k so that |A_i| = a_i,v_iisA_i is an element of A_i and “the subgraph induced by A_i contains no isolated vertices” for all i, 1 ≥ i ≥ k. Here, the bound on the minimum degree is sharp. © 1997 John Wiley & Sons, Inc.     

Multiplicative Property of the Digital Fundamental Group

Recent papers have partially discussed the multiplicative or the non-multiplicative property of the digital fundamental group. Thus, the paper studies a condition of which the multiplicative property of the digital fundamental group holds. Precisely, for two digital spaces with k _i -adjacencies of Z^n_i\mathbf{Z}^{n_{i}} , denoted by (X _i ,k _i ), i∈{1,2}, using the L _HS- or L _HC-property of the digital product (or Cartesian product of digital spaces) with k-adjacency (X ₁×X ₂,k), a k-homotopic thinning of the digital product, and various properties from digital covering and digital homotopy theories, we provide a method of calculating the k-fundamental group of the digital product. Furthermore, the notion of HT-(k ₀,k ₁)-isomorphism is established and used in calculating the k-fundamental group of a digital product. Finally, we find a condition of which the multiplicative property of the digital fundamental group holds. This property can be used in classifying digital spaces from the view points of digital homotopy theory, mathematical morphology, and digital geometry.     

Path-Bicolorable Graphs

In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V, E) is P _k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., color classes) V ₁ and V ₂ such that for every induced P _k (a path with exactly k − 1 edges and k vertices) in G, the two colors alternate along the P _k , i.e., no two consecutive vertices of the P _k belong to the same color class V _i , i = 1, 2. Obviously, a graph is bipartite if and only if it is P ₂-bicolorable. We give a structural characterization of P ₃-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P ₄-bicolorable graphs in terms of forbidden subgraphs.     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

Designs with mutually orthogonal resolutions and decompositions of edge‐colored graphs

Two resolutions R and R^′ of a combinatorial design are called orthogonal if |R_i∩R|≤1 for all R_i∈R and R∈R^′. A set Q={R¹, R², …, R^d} of d resolutions of a combinatorial design is called a set of mutually orthogonal resolutions (MORs) if the resolutions of Q are pairwise orthogonal. In this paper, we establish necessary and sufficient conditions for the asymptotic existence of a (v, k, 1)‐BIBD with d mutually orthogonal resolutions for d≥2 and k≥3 and necessary and sufficient conditions for the asymptotic existence of a (v, k, k?1)‐BIBD with d mutually orthogonal near resolutions for d≥2 and k≥3. We use complementary designs and the most general form of an asymptotic existence theorem for decompositions of edge‐colored complete digraphs into prespecified edge‐colored subgraphs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 425–447, 2009     

Mixed covering arrays on graphs

Covering arrays have applications in software, network and circuit testing. In this article, we consider a generalization of covering arrays that allows mixed alphabet sizes as well as a graph structure that specifies the pairwise interactions that need to be tested. Let k and n be positive integers, and let G be a graph with k vertices v₁,v₂,…, v_k with respective vertex weights g₁ ≤ g₂ ≤ … ≤ g_k. A mixed covering array on G, denoted by , is an n × k array such that column i corresponds to v_i, cells in column i are filled with elements from ?_gi and every pair of columns i,j corresponding to an edge v_i,v_j in G has every possible pair from ?_gi × ?_gj appearing in some row. The number of rows in such array is called its size. Given a weighted graph G, a mixed covering array on G with minimum size is called optimal. In this article, we give upper and lower bounds on the size of mixed covering arrays on graphs based on graph homomorphisms. We provide constructions for covering arrays on graphs based on basic graph operations. In particular, we construct optimal mixed covering arrays on trees, cycles and bipartite graphs; the constructed optimal objects have the additional property of being nearly point balanced. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 393–404, 2007