Numerical analysis for bulk-arrival queueing systems: Root-finding and steady-state probabilities inGI r /M/1 queues

Queueing theorists have presented, as solutions to many queueing models, probability generating functions in which state probabilities are expressed as functions of the roots of characteristic equations, evaluation of the roots in particular cases being left to the reader. Many users have complained that such solutions are inadequate. Some queueing theorists, in particular Neuts [6], rather than use Rouché's theorem to count roots and an equation-solver to find them, have developed new algorithms to solve queueing problems numerically, without explicit calculation of roots. Powell [7] has shown that in many bulk service queues arising in transportation models, characteristic equations can be solved and state probabilities can be found without serious difficulty, even when the number of roots to be found is large. We have slightly modified Powell's method, and have extended his work to cover a number of bulk-service queues discussed by Chaudhry et al. [1] and a number of bulk-arrival queues discussed in the present paper.     

完备格上基于Sup-T合成算子的矩阵的平方根

对于完备格L上给定的|I|×|I|的矩阵R,若存在|I|×|I|的L上的矩阵S满足S⊙S=R,则称S为R的平方根,其中I表示指标集|I|的基数,⊙在本文中指的是sup-T合成算子并且T是无限∨分配的保序的算子。本文给出了完备格上基于sup-T合成算子的矩阵平方根存在的充要条件以及相应的理论上的算法求解所有的平方根。     

A basic family of iteration functions for polynomial root finding and its characterizations

Let p(x) be a polynomial of degree n?2 with coefficients in a subfield K of the complex numbers. For each natural number m?2, let L_m(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are . For i=1,2, let L_m^i)(x) be the matrix obtained from L_m(x) by deleting its first i rows and its last i columns. L₁⁽¹⁾(x)≡1. Then, the function B_m(x)=x−p(x) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, B_m(x) coincides with Newton's and Halley's, respectively. The function B_m(x) is a member of S(m,m+n−2), where for any M?m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑_i=m^Mγ_i(x)(θ−x)ⁱ, with γ_i(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gⁱ(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is . For B_m(x) we obtain . If all roots of p(x) are simple, B_m(x) is the unique member of S(m,m + n − 2). By making use of the identity , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of B_m(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schröder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p^(j) with f^(j) in g as well as in γ_m(θ).     

Two New Sesquiterpenes from Ligularia Duciformis

ＡｐｒｅｖｉｏｕｓｐｈｙｔｏｃｈｅｍｉｃａｌｓｔｕｄｙｏｆＬｉｇｕｌａｒｉａｄｕｃｉｆｏｒｍｉｓｄｅａｌｔｗｉｔｈｔｈｅｉｓｏｌａｔｉｏｎｏｆｃｏｎｉｆｅｒｙｌａｌｃｏｈｏｌｄｅｒｉｖａｔｉｖｅｓ［１,２］．Ｉｎｏｒｄｅｒｔｏｆｉｎｄｓｅｓｑｕｉｔｅ...     

Quantifying vegetation biomass impacts on vehicle mobility

Soil impacts on vehicle mobility are well known; however, most data are for bare soil or the type and amount of vegetation is not documented. This study summarizes results from experiments to quantify the effect of above ground and below ground vegetation biomass on vehicle performance. Soil–vegetation combinations of three soils and three grasses were used. The vegetation was tested at various growth stages and was also subjected to stressors such as trafficking, burning, and cutting. Vegetation measurements included above ground (leaves and shoots) and below ground (root) biomass weights, lengths, diameters and surface area parameters. The soils were characterized for size distribution, moisture, density and terrain strength for each test condition. Vehicle traction and motion resistance were measured for each soil–grass combination using the CRREL Instrumented Vehicle. Results showed an increase in net traction biomass in sandy soils. For clay soils above ground biomass generally increased resistance while increased root diameter clearly decreased resistance. This study represents the first measurements quantifying the impacts of specific biomass parameters on vehicle mobility. The results will serve to guide new experimental methods, improve datasets, and develop physics-based models for years to come.     

Computing roots of polynomials on vector processing machines

The Durand and Kerner algorithm for the computation of roots of polynomials has not been og reat use up to now. The reason is the existence of more efficient methods for computers ofthe SISD type (sequential). Actually vector processing machines such as CRAY-1 or CDC CYBER 205 must bring Durand's algorithm back to honour because of its possibility of extensive vectorization. However, as the method has its maximum efficiency for a polynomial with no multiple root a criterion using Vignes' permutation-perturbation method is given to know whether the roots are all distinct or not. The optimum criterion for stopping the iterations is shown to be vectorizable and some useful properties of the method are given. Numerical examples are considered.     

On the cost of computing roots of polynomials

Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods. This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n ³ log₂(n/ε)). This is a worst case analysis and holds for all polynomials without exception. This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim Foundation.     

The closure of the set of roots of strongly connected reliability polynomials is the entire complex plane

The strongly connected reliabilityscRel(D,p) of a digraph D is the probability that the spanning subgraph of D consisting of the operational arcs is strongly connected, given that the vertices always operate, but each arc is independently operational with probability p∈[0,1]. We show that the closure of the set of roots of strongly connected reliability polynomials is the whole complex plane.