Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

Some Truncated Bivariate Distributions

Truncated versions of the bivariate generalized Pareto, bivariate inverted dirichlet and the bivariate Pearson type VII distributions are introduced. Unlike the un-truncated versions, these possess finite moments of all orders and could therefore be better models for certain practical situations. Explicit expressions for the moments are derived for each of the truncated distribution.     

Some Truncated Distributions

Long-tailed distributions arise in many areas of the sciences. These distributions, however, suffer from the weakness of not having finite moments of all orders and this weakness has restricted their use. In this note, we introduce truncated versions of five of the most commonly known long-tailed distributions—which possess finite moments of all orders and could therefore be better models. Explicit expressions for the moments are derived for each of the truncated distributions. Several applications are illustrated using real data.     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

Explicit expressions for moments of gamma order statistics

Explicit closed form expressions are derived for the moments of order statistics from the gamma and generalized gamma distributions. The expressions involve the Lauricella functions of type A and type B. The usefulness of the result is illustrated through two quality control data sets.     

Limit distributions of multivariate kurtosis and moments under Watson rotational symmetric distributions

The limit distribution of Mardia's measure of sample multivariate kurtosis is derived for a wide class of multivariate distributions which includes both the family of elliptical and Watson rotational symmetric distributions. Explicit expressions are given for the higher-order moments of Watson rotational symmetric distributions. The problem of constructing approximate confidence intervals for the kurtosis parameter is also discussed.     

The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds

In this paper, we consider a perturbed risk model with two independent classes of risks under multiple thresholds in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations with boundary conditions for the expected discounted penalty function. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

The maximum surplus before ruin for two classes of perturbed risk model

In this paper, we consider the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

Multivariate reliability modelling based on dependent dynamic shock models

In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.     

Simulation of random vectors from three-dimensional spherically symmetric stable distributions

A simple approximate algorithm of sampling random vectors from three-dimensional symmetric stable distributions based on analytical approximation of inverse distribution functions is proposed. Explicit expressions for simulating the stable distributions with indices α=0.2(0.1) 1.8 are presented. The errors of the algorithm do not exceed 1–2% in the probability density function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

一类基于进入过程保险风险模型的破产概率(英文)

In this note,one kind of insurance risk models with the policies having multiple validity times are investigated.Explicit expressions for the ruin probabilities are obtained by using the martingale method.As a consequence,the obtained probability serves as an upper bound for the ruin probability of a newly developed entrance processes based risk model.     

The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions

This article examines the magnetohydrodynamic (MHD) flow of non-Newtonian nanofluid in a pipe. The temperature of the pipe is assumed to be higher than the temperature of the fluid. In particular two temperature dependent viscosity models, have been considered. The nonlinear partial differential equations along with the boundary conditions are first cast into a dimensionless form and then the equations are solved by homotopy analysis method (HAM). Explicit analytical expressions for the velocity field, the temperature distribution and nano concentration have been derived analytically. The effects of various physical parameters on velocity, temperature and nano concentration are discussed by using graphical approach.     

N策略带启动时间的Geom/Geom/1工作休假排队

考虑N策略带启动时间的Geom/Geom/1工作休假排队,服务员在休假期间并未完全停止工作而是以较低的速率为顾客服务.运用拟生灭链和矩阵几何解方法,给出了该模型的稳态队长的分布和等待时间的概率母函数,并证明了队长和等待时间的条件随机分解结构.     

The discrete time Geom/Geom/1 queue with multiple working vacations

In this paper, we study a discrete time Geom/Geom/1 queue with multiple working vacations. Using the quasi birth and death chain and matrix-geometric solution method, we give distributions for the number of customers in system and the waiting time of a customer and their stochastic decomposition structures, and obtain distributions of the additional number of customers and additional delay. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, by numerical examples, we analyze the effect of the parameters on the expected queue length and sojourn time.     

The Maximum Surplus Distribution before Ruin in an Erlang（n） Risk Process Perturbed by Diffusion

In this paper, we consider the distribution of the maximum surplus before ruin in a generalized Erlang(n) risk process (i.e., convolution of n exponential distributions with possibly different parameters) perturbed by diffusion. It is shown that the maximum surplus distribution before ruin satisfies the integro-differential equation with certain boundary conditions. Explicit expressions are obtained when claims amounts are rationally distributed. Finally, the surplus distribution at the time of ruin and the surplus distribution immediately before ruin are presented.     

The Gerber-Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber-Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber-Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber-Shiu functions are established, respectively. And explicit results for the Gerber-Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.     

On the Expected Present Value of Total Dividends in a Risk Model with Potentially Delayed Claims

In this paper, we consider a risk model in which two types of individual claims, main claims and by-claims, are defined. Every by-claim is induced by the main claim randomly and may be delayed for one time period with a certain probability. The dividend policy that certain amount of dividends will be paid as long as the surplus is greater than a constant dividend barrier is also introduced into this delayed claims risk model. By means of the probability generating functions, formulae for the expected present value of total dividend payments prior to ruin are obtained for discrete-type individual claims. Explicit expressions for the corresponding results are derived for K n claim amount distributions. Numerical illustrations are also given.     

Large-deviation analysis of Burgers turbulence with white-noise initial data

The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity is a white-noise function. Explicit representations for the probability distribution functions (PDFs) of the velocity and the distance between two successive shocks are found in terms of the solution to a PDE system. By using these expressions and large-deviation theory, tight bounds are found for the tails of these distributions and for that of the shock-strength distribution. These bounds give the exact rates of decay for the tails. © 1998 John Wiley & Sons, Inc.     

变系数模型中的逐元B-Spline估计法

给出了一种用于估计变系数模型中未知函数的逐元B-Spline方法,建立了估计量的局部渐近偏差,方差和渐近正态分布,开发了一种快速选择估计量窗宽的方法,通过Monte Carlo模拟研究了估计量的有限样本性质.     

General collision branching processes with two parameters

A new class of branching models, the general collision branching processes with two parameters, is considered in this paper. For such models, it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states. Regularity and uniqueness criteria are firstly established. Explicit expressions are then obtained for the extinction probability vector, the mean extinction times and the conditional mean extinction times. The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established. The mean global holding time is also obtained. It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases. This work was partially supported by National Natural Science Foundation of China (Grant No. 10771216), Research Grants Council of Hong Kong (Grant No. HKU 7010/06P) and Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry of China (Grant No. [2007]1108)