Martingale Methods for Analysing Single-Server Queues

In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system.     

Heavy-Traffic Limits for Loss Proportions in Single-Server Queues

We establish heavy-traffic stochastic-process limits for the queue-length and overflow stochastic processes in the standard single-server queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related single-server models with finite waiting room, such as the finite dam, satisfy the same heavy-traffic stochastic-process limits. As a consequence, we obtain heavy-traffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavy-traffic limits for the steady-state loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavy-traffic limits for the steady-state blocking probabilities.     

A Unified Approach to the Proportional Relation for Discrete-Time Single-Server Queues

It is well known that a simple relation called proportional relation holds for some queueing models, that is, the stationary queue length distribution of one system can be expressed as the product of a constant and the distribution of another system which is different only in the buffer capacity. Recently, the proportional relation has been verified for various discrete-time single-server queues with correlated arrivals, where it has been also shown that the proportional constant can be expressed in terms of the distribution of one system. This implies that the stationary queue length distribution of one system can be completely expressed in terms of the distribution of the other system. In this paper, we consider a generalized model of discrete-time single-server queue, which covers all previous ones, and give a simple and unified proof to the proportional relation as well as the expression of the proportional constant.     

Reduced-Load Equivalence for Queues with Gaussian Input

In this note, we consider a queue fed by a number of independent heterogeneous Gaussian sources. We study under what conditions a reduced load equivalence holds, i.e., when a subset of the sources becomes asymptotically dominant as the buffer size increases. For this, recent results on extremes of Gaussian processes [6] are combined with de Haan theory. We explain how the results of this note relate to square root insensitivity and moderately heavy tails.The research was supported by the Netherlands Organization for Scientific Research (NWO) under grant 631.000.002.This revised version was published online in June 2005 with corrected coverdate     

Heavy Traffic Limits for Queues with Many Deterministic Servers

Queueing Systems - Consider a sequence of stationary GI/D/Nqueues indexed by N↑∞, with servers' utilization $1 - \beta /\sqrt N ,\beta &;gt;0$ . For such queues we show that...     

Asymptotics for Some Proximal-like Method Involving Inertia and Memory Aspects

Given a Hilbert space H and a closed convex function Φ:\(H\rightarrow{\mathbb{R}} \cup \{+\infty\}\), we consider the inertial proximal algorithm

$ x_{n+1}-x_n-\alpha_n(x_n-x_{n-1})+\beta_n\partial\Phi(x_{n+1})\ni 0, \qquad \qquad \qquad \qquad({\mathcal{A}}) $

where (α _n ) and (β _n ) are nonnegative sequences. The notation \(\partial \Phi\) stands for the subdifferential of Φ in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to \(\inf\Phi\) as n→?+?∞. When Φ has a unique minimum, the question of the convergence of (x _n ) is solved. In the case of multiple minima, it is proved that if \(\left(\prod_{k=1}^n \alpha_k\right)\not \in l^1\) and if a suitable geometric condition on the set argmin Φ is fulfilled, then non stationary sequences of (A) cannot converge.

     

Stability and Asymptotic Behaviour for the Numerical Solution of a Reaction--Diffusion Model for a Deterministic Diffusive Epidemic

The aim of this paper is to outline the numerical solution ofa reaction—diffusion system describing the evolution ofan epidemic in an isolated habitat. The model we consider isdescribed by two weakly coupled semi-linear parabolic equationsand we introduce a finite difference scheme for its numericalsolution. We study the behaviour of the exact solution by meansof the numerical scheme. We show the positivity, the decreaseand the decay to extinction of the numerical solution. Finallywe report the results of the numerical tests; in these simulationswe observe that the asymptotic behaviour of the reaction-diffusionsystem is the same as that of the associated ODE system (Kermack—McKendrickmodel).     

Convergence of Numerical Approximation for Jump Models Involving Delay and Mean-Reverting Square Root Process

The mean-reverting square root process with jump has been widely used as a model on the financial market. Since the diffusion coefficient in the model does not satisfy the linear growth condition and local Lipschitz condition, we can not examine its properties by traditional techniques. To overcome the difficulties, we develop several new techniques to examine the numerical method of jump models involving delay and mean-reverting square root. We show that the numerical approximate solutions converge to the true solutions. Finally, we apply the convergence to examine a path-dependent option price and a bond in the financial pricing.     

几种基于散乱数据拟合的局部插值方法

本文首先针对散乱数据拟合的Shepard方法,结合截断多项式、B样条基函数和指数函数来构造其权函数,使新的权函数具有更高的光滑度和更好的衰减性,并且其光滑性和衰减性可以根据实际需要自由调节,从而提高了曲面的拟合质量．同时还给出一种类似的局部插值方法．另外,本文还基于多重二次插值,结合多元样条的思想,给出了两个局部插值算法．该算法较好地继承了多重二次插值曲面的性质,从而保证了拟合曲面具有好地光顺性和拟合精度．曲面整体也具有较高的光滑性．     

生成SAS数据集的几种方法

高惠璇，生成ＳＡＳ数据集的几种方法，数理统计与管理，１９９８，１７（３），４９～５４．本文介绍生成ＳＡＳ数据集几种不同的方法。     

数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.     

关于一些数值求积公式的渐近性

该文给出了一些数值求积公式的渐近性质,这些公式包括求积分的矩形法则、梯形法则和抛物线法则,包含于余项中的中介点的位置当积分区间的长度趋于零时被确定,对应于该法则的校正公式被得到,它们具有较高的代数精度,我们也进行了一些数值试验,得到较满意的数值结果。     

Existence and Regularity of Solutions for Some Partial Integrodifferential Equations Involving the Nonlocal Conditions

The aim of this work is to prove some results about the existence and regularity of solutions for some partial integrodifferential equations with nonlocal conditions. We suppose that the linear part has a resolvent operator in the sens given by Grimmer. The non linear part is assumed to be continuous and Lipschitzian with respect to the second argument.     

Some Existence Results for Vector Quasivariational Inequalities Involving Multifunctions and Applications to Traffic Equilibrium Problems

Some existence results for vector quasivariational inequalities with multifunctions in Banach spaces are derived by employing the KKM-Fan theorem. In particular, we generalize a result by Lin, Yang and Yao, and avoid monotonicity assumptions. We also consider a new quasivariational inequality problem and propose notions of weak and strong equilibria while applying the results to traffic network problems.     

Some Eigenfunction Methods for Computing a Numerical Fourier Transform

A number of methods for calculating the Fourier transform ofa function given numerically are studied. These methods exploitthe fact that the Hermite functions are eigen-functions of theFourier transform. The transforms of four types of functionsare considered: (i) functions of the form p(x) exp (–x²/2),where p(x) is a polynomial, (ii) functions with bounded support.(iii) rapidly decreasing functions, and (iv) functions whosetransform has bounded support. In each case algorithms for calculatingthe transformed function are derived. Error estimates are madein two of the cases and results of numerical experiments presentedin an appendix.     

A Numerical Approach for Solving Some Convex Maximization Problems

We are concerned with concave programming or the convex maximization problem. In this paper, we propose a method and algorithm for solving the problem which are based on the global optimality conditions first obtained by Strekalovsky (Soviet Mathematical Doklady, 8(1987)). The method continues approaches given in (Journal of global optimization, 8(1996); Journal of Nolinear and convex Analyses 4(1)(2003)). Under certain assumptions a convergence property of the proposed method has been established. Some computational results are reported. Also, it has been shown that the problem of finding the largest eigenvalue can be found by the proposed method.     

Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time

Journal of Theoretical Probability - In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process...     

具有奇异项的拟线性椭圆型方程组无穷多弱解的存在性

本文在一定条件下讨论了一类具有奇异项的,被两个pLaplacian算子控制的拟线性椭圆型方程组Dirichlet问题无穷多弱解的存在性.     

Deterministic Mathematical Modelling for the Spatial Allocation of Multi-Categorical Resources: with an Application to Real Health Data

This paper presents some mathematical formulations of deterministic non-linear optimization models for planning the spatial distribution of public service facilities and their utilization. The modelling is performed as a function of multi-categorical resource types and the consumer's zone of residence over a large geographical domain. The mathematical solution to the deterministic model, its parameter estimation by log-linear regression, and some preliminary results of simulation for a Massachusetts hospital database are presented.