Erlang loss queueing system with batch arrivals operating in a random environment

We consider the BMAP/PH/N/0$\text{<italic>BMAP</italic>}/\text{<italic>PH</italic>}/N/0$ queueing system operating in a finite state space Markovian random environment. Disciplines of partial admission, complete rejection and complete admission are analyzed. The stationary distribution of the system states is calculated. The loss probability and other main performance measures of the system are derived. The Laplace–Stieltjes transform of the sojourn time distribution of accepted customers is obtained. Illustrative numerical examples are presented. They show effect of an admission strategy, a correlation in an arrival process, a variation of a service process. Poor quality of the loss probability approximation by means of more simple models utilization is illustrated.     

An approximation algorithm to the k-Steiner Forest problem

Given a graph G$G$, an integer k$k$, and a demand set D={(_s1,_t1),…,(_sl,_tl)}$D=\{(s_1,t_1),\dots ,(s_l,t_l)\}$, the k$k$-Steiner Forest problem finds a forest in graph G$G$ to connect at least k$k$ demands in D$D$ such that the cost of the forest is minimized. This problem was proposed by Hajiaghayi and Jain in SODA’06. Thereafter, using a Lagrangian relaxation technique, Segev et al. gave the first approximation algorithm to this problem in ESA’06, with performance ratio O(n^2/3logl)$O(n^{2/3}logl)$. We give a simpler and faster approximation algorithm to this problem with performance ratio O(n^2/3logk)$O(n^{2/3}logk)$ via greedy approach, improving the previously best known ratio in the literature.     

The complexity of the zero-sum 3-flows

A zero-sum k-flow for a graph G   is a vector in the null-space of the 0,1$0,1$-incidence matrix of G   such that its entries belong to {±1,?,±(k−1)}$\{\pm 1,?,\pm (k-1)\}$. Akbari et al. (2009) [5] conjectured that if G is a graph with a zero-sum flow, then G   admits a zero-sum 6-flow. (2,3)$(2,3)$-semiregular graphs are an important family in studying zero-sum flows. Akbari et al. (2009) [5] proved that if Zero-Sum Conjecture is true for any (2,3)$(2,3)$-semiregular graph, then it is true for any graph. In this paper, we show that there is a polynomial time algorithm to determine whether a given (2,3)$(2,3)$-graph G   has a zero-sum 3-flow. In fact, we show that, there is a polynomial time algorithm to determine whether a given (2,4)$(2,4)$-graph G with n   vertices has a zero-sum 3-flow, where the number of vertices of degree four is O(log?n)$O(\log ?n)$. Furthermore, we show that it is NP-complete to determine whether a given (3,4)$(3,4)$-semiregular graph has a zero-sum 3-flow.     

Algorithms for a class of infinite permutation groups

Motivated by the famous 3n+1$3n+1$ conjecture, we call a mapping from Z$\mathbb{Z}$ to Z$\mathbb{Z}$residue-class-wise affine   if there is a positive integer m$m$ such that it is affine on residue classes (mod m$m$). This article describes a collection of algorithms and methods for computation in permutation groups and monoids formed by residue-class-wise affine mappings.     

Tag systems and Collatz-like functions

Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have proved to be very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus important instruments for studying limits or boundaries of solvability and unsolvability. Although there are some results on tag systems and their limits of solvability and unsolvability, there are hardly any that consider both   the shift number v$v$ and the number of symbols μ$\mu$. This paper aims to contribute to research on limits of solvability and unsolvability for tag systems, taking into account these two parameters. The main result is the reduction of the 3n+1$3n+1$-problem to a surprisingly small tag system. It indicates that the present unsolvability line–defined in terms of μ$\mu$ and v$v$–for tag systems might be significantly decreased.     

A rounding algorithm for approximating minimum Manhattan networks

For a set T$T$ of n$n$ points (terminals) in the plane, a Manhattan network   on T$T$ is a network N(T)=(V,E)$N\left(T\right)=(V,E)$ with the property that its edges are horizontal or vertical segments connecting points in V⊇T$V\supseteq T$ and for every pair of terminals, the network N(T)$N\left(T\right)$ contains a shortest _l1$l_1$-path between them. A minimum Manhattan network   on T$T$ is a Manhattan network of minimum possible length. The problem of finding minimum Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan [J. Gudmundsson, C. Levcopoulos, G. Narasimhan, Approximating a minimum Manhattan network, Nordic Journal of Computing 8 (2001) 219–232. Proc. APPROX’99, 1999, pp. 28–37] and its complexity status is unknown. Several approximation algorithms (with factors 8, 4, and 3) have been proposed; recently Kato, Imai, and Asano [R. Kato, K. Imai, T. Asano, An improved algorithm for the minimum Manhattan network problem, ISAAC’02, in: LNCS, vol. 2518, 2002, pp. 344–356] have given a factor 2-approximation algorithm, however their correctness proof is incomplete. In this paper, we propose a rounding 2-approximation algorithm based on an LP-formulation of the minimum Manhattan network problem.     

Hamiltonian circuit and linear array embeddings in faulty k-ary n-cubes

In this paper, we investigate the fault-tolerant capabilities of the k-ary n-cubes for even integer k with respect to the hamiltonian and hamiltonian-connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F   be a faulty set with nodes and/or links, and let k?3$k?3$ be an odd integer. When |F|?2n-2$\left|F\right|?2n-2$, we show that there exists a hamiltonian cycle in a wounded k-ary n  -cube. In addition, when |F|?2n-3$\left|F\right|?2n-3$, we prove that, for two arbitrary nodes, there exists a hamiltonian path connecting these two nodes in a wounded k-ary n-cube. Since the k-ary n  -cube is regular of degree 2n$2n$, the degrees of fault-tolerance 2n-3$2n-3$ and 2n-2$2n-2$ respectively, are optimal in the worst case.     

Non-associative Gröbner bases,finitely-presented Lie rings and the Engel condition,II

We give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n$n$-Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t$t$-generator Lie rings that satisfy the n$n$-Engel condition, for (t,n)=(2,3),(3,3),(4,3),(2,4)$(t,n)=(2,3),(3,3),(4,3),(2,4)$.     

Stratification associated with local b-functions

The local b$b$-function _bf,p(s)$b_{f,p}\left(s\right)$ of an n$n$-variate polynomial f∈C[x]$f\in \mathbb{C}\left[x\right]$ (x=(_x1,…,_xn)$x=(x_1,\dots ,x_n)$) at a point p∈Cⁿ$p\in {\mathbb{C}}^n$ is constant on each stratum of a stratification of Cⁿ${\mathbb{C}}^n$. We propose a new method for computing such a stratification and _bf,p(s)$b_{f,p}\left(s\right)$ on each stratum. In the existing method proposed in Oaku (1997b), a primary ideal decomposition of an ideal in C[x,s]$\mathbb{C}[x,s]$ is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in C[x]$\mathbb{C}\left[x\right]$ defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.     

On finding short resolution refutations and small unsatisfiable subsets

We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k  . We show that both problems are complete for the class W[1]$\mathrm{W}[1]$, the first level of the W-hierarchy of fixed-parameter intractable problems. Our results remain true if restricted to 3-SAT instances and/or to various restricted versions of resolution including tree-like resolution, input resolution, and read-once resolution. Applying a metatheorem of Frick and Grohe, we show that, restricted to classes of sets of clauses of locally bounded treewidth, the considered problems are fixed-parameter tractable. For example, the problems are fixed-parameter tractable for planar CNF formulas.     

Simplifying the weft hierarchy

We give simple, self-contained proofs of the basic hardness results for the classes W[t]$W[t]$ of the weft hierarchy. We extend these proofs to higher levels of the hierarchy and illuminate the distinctions among its classes. The anti-monotone collapse at W[1,s]$W[1,s]$ and the normalization of weft-t$t$ formulas arise as by-products of the proofs.     

Properties and performance modelling of finite buffer M/G/1/K networks

Finite buffer, single-server queueing systems and networks are difficult to analyze since the length of time a customer spends in the system does not follow the Markovian property. A two-moment approximation schema is developed for the probability distribution of M/G/1/K systems and extended to the analysis of M/G/1/K   queueing networks. The general purpose of this paper is to develop a flexible and practical transform-free approach for computing the probability distribution and performance measures of the system as well as identify the underlying properties of these systems. It is shown that for most performance measures, a sigmoid or S-shaped curve with an inflection point at ρ=1$\rho =1$ appears as K→∞$K\to \infty$. This has direct implications for the analysis and optimization of such systems. The performance modelling of the M/G/1/K queueing networks of general topologies along with extensive numerical results accompany the paper along with the linear concave performance measures for these systems.     

Fast simulation of self-similar and correlated processes

Simulations with long-range dependent or self-similar input processes are hindered both by the slowness of convergence displayed by the output data and by the high computational complexity of the on-line methods for generating the input process. In this paper, we present optimized algorithms for simulating efficiently the occupancy process of a M/G/∞$\infty$ system, which can be used as a sequential pseudo-random number generator of a broad class of self-similar and correlated sample-paths. We advocate the use of this approach in the simulation toolbox, as a simple method to overcome the drawbacks of other synthetic generators of Gaussian self-similar time series. Our approach to fast simulation of the M/G/∞$\infty$ model is the decomposition of the service time distribution as a linear combination of deterministic and memoryless random variables, plus a residual term. Then, the original M/G/∞$\infty$ system is replaced by a number of parallel, independent, virtual and easier to simulate M/G/∞$\infty$ subsystems, the dynamics of which can be replicated sequentially or in parallel too. We report the results of several experiments demonstrating the substantial improvements attainable with this decomposition.     

Accurate 2d finite element calculations for hydrogen in magnetic fields of arbitrary strength

Recent observations of hundreds of hydrogen-rich magnetic white dwarf stars with magnetic fields up to 10⁵  T (10³  MG) have called for more comprehensive and accurate databases for wavelengths and oscillator strengths of the H atom in strong magnetic fields for all states evolving from the field-free levels with principal quantum numbers n≤10$n\le 10$. We present a code to calculate the energy eigenvalues and wave functions of such states which is capable of covering the entire regime of field strengths B=0$B=0$ T to B∼10⁹$B\sim {10}^9$ T. We achieve this high flexibility by using a two-dimensional finite element expansion of the wave functions in terms of B$B$-splines in the directions parallel and perpendicular to the magnetic field, instead of using asymptotically valid basis expansions in terms of spherical harmonics or Landau orbitals. We have paid special attention to the automation of the program such that the data points for the magnetic field strengths at which the energy of a given state are calculated can be selected automatically. Furthermore, an elaborate method for varying the basis parameters is applied to ensure that the results reach a pre-selected precision, which also can be adjusted freely. Energies and wave functions are stored in a convenient format for further analysis, e.g. for the calculation of transition energies and oscillator strengths. The code has been tested to work for 300 states with an accuracy of better than 10⁻⁶ Rydberg across several symmetry subspaces over the entire regime of magnetic field strengths.     

Queueing network analysis on dynamic lot streaming

Lot streaming is a technique used to split a production lot into several smaller transfer batches. In this way, processing a product in a multistage serial production system by overlapping production between stages is available. As a result, production can be accelerated. In this paper, a modified N-policyM/G/1$N\text{-}\mathrm{policy}M/G/1$ queueing system with a non-reliable server is proposed. With the relevant parameters of this queueing system, the problem of lot streaming in dynamic situations can be solved. To obtain the optimal transfer batch size, a cost objective function including transport cost, repair cost, turning cost, and holding cost is formulated. Further, in order to illustrate the usefulness of the proposed methods, numerical examples are solved. On the basis of the results of these examples, some important conclusions are drawn.     

Accelerating knowledge-based energy evaluation in protein structure modeling with Graphics Processing Units

Evaluating the energy of a protein molecule is one of the most computationally costly operations in many protein structure modeling applications. In this paper, we present an efficient implementation of knowledge-based energy functions by taking advantage of the recent Graphics Processing Unit (GPU) architectures. We use DFIRE, a knowledge-based all-atom potential, as an example to demonstrate our GPU implementations on the latest NVIDIA Fermi architecture. A load balancing workload distribution scheme is designed to assign computations of pair-wise atom interactions to threads to achieve perfect or near-perfect load balancing in the symmetric N$N$-body problem in DFIRE. Reorganizing atoms in the protein also improves the cache efficiency in Fermi GPU architecture, which is particularly effective for small proteins. Our DFIRE implementation on GPU (GPU-DFIRE) has exhibited a speedup of up to ∼$\sim$150 on NVIDIA Quadro FX3800M and ∼$\sim$250 on NVIDIA Tesla M2050 compared to the serial DFIRE implementation on CPU. Furthermore, we show that protein structure modeling applications, including a Monte Carlo sampling program and a local optimization program, can benefit from GPU-DFIRE with little programming modification but significant computational performance improvement.