THE ERGODICITY FOR BI-IMMIGRATION BIRTH AND DEATH PROCESSES IN RANDOM ENVIRONMENT

The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Q（θ） with birth rate λ 〈 death rate μ, the following results are proved, （1） there is an unique q-process in random environment, P^-（θ＊（0）;t） = （p^-（θ^＊（0）;t,i,j）,i,j ≥ 0）, which is ergodic, that is, lim t→∞（θ^＊（0）;t,i,j） = π^-（θ^＊（0）;j） ≥0 does not depend on i ≥ 0 and ∑j≥0π （θ＊（0）;j） = 1, （2） there is a bi-immigration birth and death process in random enjvironment （X^＊ = {X^＊,t ≥ 0},ε^＊ = {εt,t ∈ （-∞, ∞）}） with random transition matrix P^-（θ^＊ （0）;t） such that X^＊ is a strictly stationary process.

Finite family trees of continuous time birth and death processes for evaluating the transmitting speed of information on communication networks

A finite random graph generated by continuous time birth and death processes with exponentially distributed waiting times was investigated, which is similar to a communication network in daily life. The vertices are the living particles, and directed edges go from mothers to daughters. The size of the communication network was studied. Furthermore, the probability of successfully connecting senders with receivers and the transmitting speed of information were obtained.

A Stochastic SIVS Epidemic Model Based on Birth and Death Process

A new stochastic epidemic model, that is, a general continuous time birth and death chain model, is formulated based on a deterministic model including vaccination. We use continuous time Markov chain to construct the birth and death process. Through the Kolmogorov forward equation and the theory of moment generating function, the corresponding population expectations are studied. The theoretical result of the stochastic model and deterministic version is also given. Finally, numerical simulations are carried out to substantiate the theoretical results of random walk.

Explicit criteria on separation cutoff for birth and death chains

The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which depends only on the birth and death rates. Furthermore, we present two ways to estimate moments of the fastest strong stationary time and then give another but equivalent criterion explicitly.

Unified Characteristic Numbers and Solutions of Equations for Birth and Death Processes with Barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.

Kinetics of aggregation growth with competition between catalyzed birth and catalyzed death

An aggregation growth model of three species A, B and C with the competition between catalyzed birth and catalyzed death is proposed. Irreversible aggregation occurs between any two aggregates of the like species with theconstant rate kernels In（n = 1,2, 3）. Meanwhile, a monomer birth of an A species aggregate of size k occurs under the catalysis of a B species aggregate of size j with the catalyzed birth rate kernel K（k, j） = Kkj^v, and a monomer death of an A species aggregate of size k occurs under the catalysis of a C species aggregate of size j with the catalyzed death rate kernel L（k, j） = Lkj^v, whcre v is a parameter reflecting the dependence of the catalysis reaction rates of birth and death on the size of catalyst aggregate. The kinetic evolution behaviours of the three species are investigated by the rate equation approach based on the mean-field theory. The form of the aggregate size distribution of A species ak （t） is found to be dependent crucially on the competition between the catalyzed birth and death of A species, as well as the irreversible aggregation processes of the three species： （i） In the v 〈 0 case, the irreversible aggregation dominates the process, and ak（t） satisfies the conventional scaling form; （2） In the v ≥ 0 casc, the competition between the catalyzed birth and death dominates the process. When the catalyzed birth controls the process, ak（t） takes the conventional or generalized scaling form. While the catalyzed death controls the process, the scaling description of the aggregate size distribution breaks down completely.

Potentials and the Distributions of the Last Exit Times of Birth and Death Processes

0 .Introduction The mathematical eqnivalenoe of Brownian切otion and olaosioal poten七ialtheory has great imPulsed the study of Potentials of Markov Prooesse

Exchange-Driven Growth with Birth Rate Less Than Death

We further study the kinetic behavior of the exchange-driven growth withbirth and death for the case of birth rate kernel being less than that of death based on themean-Geld theory. The symmetric exchange rate kernel is K(k,j) = K′(k,j) = Ikj~v, and the birth anddeath rates are proportional to the aggregate's size. The long time asymptotic behavior of theaggregate size distribution a_k(t) is found to obey a much unusual scaling law with an exponentiallygrowing scaling function Φ(x) = exp(x).

Method for Measuring Residual Stresses Induced by Boring in Internal Surface of Tube and Its Validation with XRD Method

Residual stresses can have a strong effect on the usability of machined parts,and the X-ray diffraction(XRD)measuring equipment,which is commonly used to measure residual stresses,is very expensive.This paper presents a method of measuring the residual stresses induced by boring in the internal surface of a tube with much cheaper equipment.The method,called the strain-based method is mainly based on the strains measured on the external surface of the tube.It is proposed on the basis of the very long tube assumption.The finite element method(FEM)analysis is thus used to validate the length of the tube.Guided by the FEM results,an appropriate length of the tube is chosen,and the residual stresses are obtained from both the strain-based method and the XRD method.Stress profiles obtained from both two methods are compared.The comparison result indicates that the profiles of the two methods agree well with each other.Therefore,it can be concluded that the accuracy of the strain-based method is high enough,and it can be applied to residual stress measurement in practice.

Elasto-Plastic FEM Analysis of Residual Stress in Spun Tube

The residual stress distribution of Hastelloy C corrosion-resistant alloy tubes after power spinning was simulated with the elasto-plastic finite element method combining with the element birth and death technique, the influences of spinning parameters on the distribution of the residual stress were investigated in detail, and the formation mechanism of residual stress during tube spinning was discussed. Based on the calculation of the residual stress, the reasons for annealing cracks on the spun tube during interpass heat treatment were explored. The simulation results and the characteristics of annealing cracks show that the circumferential residual tensile stress is a main factor to cause the annealing cracks.

The First- Passage Time Moment of the GNP Diffusion Process to a Determined Value

This paper describes an accurate method of approximating the moments of the first-passage time for the birth and death Gross National Product GNP diffusion process when the GNP is a determined value or constant absorbing barrier. This was done by approximating the differential equations by equivalent difference equations.

EFFECT OF VELOCITY ON DUCTILITY UNDER HIGH VELOCITY FORMING

The ring expansion procedures over various forming velocities are calculated with ANSYS software in order to show the effect of forming velocity on ductility of rate insensitive materials. Ring expansion procedures are simplified to one-dimensional tension by constraining the radial deformation, with element birth and death method, fracture problem of circular ring are considered. The calculated results show that for insensitive materials of 1060 aluminum and 3A21 aluminum alloy, fracture strain increases corresponding to the increase of forming velocity. This trend agrees well with experimental results, and indicates inertia is the key factor to affect ductility; With element birth and death methods, fracture problems can be solved effectively. Experimental studies on formability of tubular workpieces are also conducted, experimental results show that the formability of 1060 aluminum and 3A21 aluminum alloy under electromagnetic forming is higher than that under quasistatic forming, according to the characteristics of electromagnetic forming, the forming limit diasrams of the two materials tube are also built respectively, this is very important to promote the development of electromagnetic forming and guide the engineering practices.

THE M/PH/1 QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.

Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. PerronFrobenius theory ensures the existence of a corresponding positive eigenvector ψ. The goal of the paper is to give bounds on the amplitude max ψ/ min ψ. Two approaches are proposed: One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo(2014).

Siegmund Duality for Continuous Time Markov Chains on Z_+~d

For the continuous time Markov chain with transition function P（t） on Z d ＋ , we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P - （t）. If Q, the q-matrix of P（t）, is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegnmnd dual of some Q-function is also given.