Capacity planning of a perinatal network with generalised loss network model with overflow

Recent literature shows that the arrival and discharge processes in hospital intensive care units do not satisfy the Markovian property, that is, interarrival times and length of stay tend to have a long tail. In this paper we develop a generalised loss network framework for capacity planning of a perinatal network in the UK. Decomposing the network by hospitals, each unit is analysed with a GI/G/c/0 overflow loss network model. A two-moment approximation is performed to obtain the steady state solution of the GI/G/c/0 loss systems, and expressions for rejection probability and overflow probability have been derived. Using the model framework, the number of required cots can be estimated based on the rejection probability at each level of care of the neonatal units in a network. The generalisation ensures that the model can be applied to any perinatal network for renewal arrival and discharge processes.     

Some problems in finite dams with an application to insurance risk

Summary This paper considers a finite dam in continuous time fed by inputs, with a negative exponential distribution, whose arrival times form a Poisson process; there is a continuous release at unit rate, and overflow is allowed. Various results have been obtained by appropriate limiting methods from an analogous discrete time process, for which it is possible to find some solutions directly by determinantal methods.First the stationary dam content distribution is found. The distribution of the probability of first emptiness is obtained both when overflow is, and is not allowed. This is followed by the probability the overflow before emptiness, which is then applied to determine the exact solution for an insurance risk problem with claims having a negative exponential distribution. The time-dependent content distribution is found, and the analogy with queueing theory is discussed.     

Oja神经网络模型的有限逸时性

O ja连续型全反馈神经网络模型可以有效计算实对称矩阵的主特征向量,该网络的动态行为由描述其模型的微分方程所决定,详细研究了O ja动力系统的稳定性问题.对于非正定实对称矩阵最大特征根为零,且至少有一特征根为负的情形,证明了从单位球外出发的解并不一定必然导致有限逸时,完善了O ja模型计算实对称矩阵主特征向量的收敛性结果,数值实验结果进一步验证了理论分析的正确性.     

The static stochastic knapsack problem with normally distributed item sizes

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.     

Finite dams with double level of release

We considered a finite dam with discrete additive input and double level of release. If the current dam content is not greater than a certain boundM, the release is one unit unless the dam is empty; and if the current dam content is greater thanM, the release isr (? 1) units provided it is available, otherwise the whole content will be withdrawn. We derive all the expressions of the distributions of first emptiness with and without overflow, the distributions of emptiness with and without overflow, the time dependent distributions of dam content with and without overflow, and the distributions of overflow times and quantities. IfM is equal to the dam capacity, the results are reduced to the case of unit release; and ifM=0, the results are reduced to the case of releaser.     

Harmonic symmetrization of convex sets and of Finsler structures,with applications to Hilbert geometry

David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain Ω$\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of Ω$\Omega$, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological   weak Finsler metric, in which the unit ball in each tangent space is naturally identified with the domain Ω$\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure   on Ω$\Omega$, and the unit ball of the Hilbert metric at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.     

Stochastic Models for the Amount of Overflow in a Finite Dam with Random Inputs,Random Outputs,and Exponential Release Policy

In this article, we discuss finite dam models to study the expected amount of overflow in a given time. The inputs into the dam are taken as random and there are two types of outputs—one is random and the other is deterministic which is proportional to the content of the dam. The master equation for the expected amount of overflow is an one dimensional equation with separable kernel. For this class of master equation, the integral equation for the expected amount of overflow has been transformed exactly into ordinary differential equation with variable coefficients. The imbedding method is used to study the expected amount of overflow in a given time without emptiness in this period. We also consider the model for the expected amount of overflow in a given time with any number of emptiness of the dam in this period. The results are derived in the form of a third order differential Equation for the Laplace transformation function for the expected overflow. The closed form analytical solutions are obtained in terms of beta functions and degenerate hyper-geometric functions of two variables.     

Attitude control of rigid spacecraft with disturbance generated by time varying exosystems

This paper is concerned with the problem of attitude control and disturbance rejection of rigid spacecraft in the presence of parameter uncertainty. It is assumed that the external disturbance is generated by some time varying exosystems. The unit quaternion is used as the kinematic variables since it is free of singularity. An internal model and an adaptive control law are proposed. The parameter uncertainty caused by the unknown inertia matrix is handled by combining the semi-tensor product and adaptive control method. The asymptotical stability of the closed-loop system is given via backstepping and Lyapunov analysis. Finally, an illustrative example is provided to show the effectiveness of the proposed approach.     

Large deviations analysis of the generalized processor sharing policy

In this paper we consider a stochastic server (modeling a multiclass communication switch) fed by a set of parallel buffers. The dynamics of the system evolve in discrete-time and the generalized processor sharing (GPS) scheduling policy of [25] is implemented. The arrival process in each buffer is an arbitrary, and possibly autocorrelated, stochastic process. We obtain a large deviations asymptotic for the buffer overflow probability at each buffer. In the standard large deviations methodology, we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. We view the problem of finding a most likely sample path that leads to an overflow as an optimal control problem. Using ideas from convex optimization we analytically solve the control problem to obtain both the asymptotic exponent of the overflow probability and a characterization of most likely modes of overflow. These results have important implications for traffic management of high-speed networks. They extend the deterministic, worst-case analysis of [25] to the case where a detailed statistical model of the input traffic is available and can be used as a basis for an admission control mechanism. This revised version was published online in June 2006 with corrections to the Cover Date.     

2-D digital filter realization without overflow oscillations

A novel criterion for the elimination of overflow oscillations in 2-D state-space digital filters described by the Roesser model employing two’s complement overflow arithmetic is presented. The criterion takes the form of linear matrix inequality (LMI) and, hence, is computationally tractable. The criterion is a generalization and improvement over an earlier criterion. An example shows the effectiveness of the new criterion.     

Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square

We prove new optimal bounds for the error of numerical integration in bivariate Besov spaces with dominating mixed order r$r$. The results essentially improve on the so far best known upper bound achieved by using cubature formulas taking points from a sparse grid. Motivated by Hinrichs’ observation that Hammersley type point sets provide optimal discrepancy estimates in Besov spaces with mixed smoothness on the unit square, we directly study quasi-Monte Carlo integration on such point sets. As the main tool we prove the representation of a bivariate periodic function in a piecewise linear tensor Faber basis. This allows for optimal worst case estimates of the QMC integration error with respect to Besov spaces with dominating mixed smoothness up to order r<2$r<2$. The results in this paper are the first step towards sharp results for spaces with arbitrarily large mixed order on the d$d$-dimensional unit cube. In fact, in contrast to Fibonacci lattice rules, which are also practicable in this context, the QMC methods used in this paper have a proper counterpart in d$d$ dimensions.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Rare-Event Simulation of Non-Markovian Queueing Networks Using a State-Dependent Change of Measure Determined Using Cross-Entropy

A method is described for the efficient estimation of small overflow probabilities in nonMarkovian queueing network models. The method uses importance sampling with a state-dependent change of measure, which is determined adaptively using the cross-entropy method, thus avoiding the need for a detailed mathematical analysis. Experiments show that the use of rescheduling is needed in order to get a significant simulation speedup, and that the method can be used to estimate overflow probabilities in a two-node tandem queue network model for which simulation using a state-independent change of measure does not work well.     

Application of auxiliary Markov chains to start-up demonstration tests

We use auxiliary Markov chains to derive probabilistic results for five types of start-up demonstration tests, with start-ups that are Markovian of a general order. Four of the tests are based on consecutive (or total) successful start-ups and consecutive (or total) failures; the fifth has two rejection criteria. For each test type, we obtain the probability of the test ending with acceptance of the unit, the probability distribution and moments of the number of start-ups in the test, the probability of acceptance (or rejection) of the equipment in a specified number of trials, and the conditional distribution of the number of start-ups in the test given that the unit is accepted or rejected. Numerical examples are given. Though the results are for these specific types of start-up demonstration tests, the method of derivation may be used for tests with other stopping criteria, and in other situations as well.     

基于离散事件仿真的妇产科孕妇建卡接收规则优化

孕妇在怀孕九周后会选择一家医院申请建立孕妇档案卡(简称建大卡),管理者需决定是否接收。一旦该医院做出接收决策,孕妇将在该院接受妊娠期内的产检并在该医院分娩。孕妇在妊娠期内所需的医疗资源的多样化、尤其是对关键资源即病床需求时间及服务时间不确定性,增加了决策的难度,从而导致妇产科资源利用率分布不均衡,影响医疗服务质量。为了解决该问题,本文首先建立动态规划模型来最小化病床的空闲、加床和转院的惩罚成本。由于动态规划模型非常复杂,难以用解析方法来获取最优控制策略,因此本文建立离散事件仿真系统,以病床的空闲、加床和转院的惩罚成本最小化为目标,采用广域搜索对多种阈值控制策略进行参数优化。数值实验表明,本文提出的阈值策略比医院目前采用的阈值策略有显著改进,进一步进行了敏感度分析,发现随着到达率逐渐增大,期望成本首先逐渐降低然后在某一水平上波动,加床成本的上升对于期望成本的影响更大。     

Valid constraints for the Point Packing in a Square problem

We consider the problem of placing n$n$ points in the unit square in such a way as to maximize their minimum pairwise distance m$m$. Starting from two properties of the optimal solution presented by Locatelli and Raber in [Discrete Applied Mathematics 122 (1–3) (2002) 139–166], and using the known theoretical lower and upper bounds, we derive some constraints for tightening the original formulation of the problem.     

A hierarchical model for the location of perinatal facilities in the municipality of Rio de Janeiro

We present a 3-level hierarchical model for the location of maternal and perinatal health care facilities in Rio de Janeiro. Relaxations and heuristics are developed for this model and computational results are given for problems available in the literature, for networks ranging from 10 to 400 vertices. The quality of the solutions produced by the procedures we developed do not differ significantly among themselves. The model is also applied to a case study corresponding to real 1995 data of the municipality of Rio de Janeiro.     

Estimating Mahler measures using periodic points for the doubling map

The Mahler measure m(P)$m\left(P\right)$ of a polynomial P$P$ is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P)$m\left(P\right)$ which we illustrate with several examples.