The explicit solution to the Poisson equation corresponding to the Q-matrix of a single birth process is obtained,thus the explicit inverse(if exists)is presented directly.As an application,inspired by the inverse pow...The explicit solution to the Poisson equation corresponding to the Q-matrix of a single birth process is obtained,thus the explicit inverse(if exists)is presented directly.As an application,inspired by the inverse power method,combining the explicit inverse with CollatzWielandt formula,a powerful approximation theorem for the maximal eigenpair corresponding to the Q-matrix of a single birth process is presented.Different from the classical acceleration method using some fixed shift in the iteration,the shift in each iteration step is varying and the sequence formed by these shifts is strictly monotone and increases to the eigenvalue needed,which effectively reduces the number of iterations.Some examples are studied to illustrate the power of these results.展开更多
Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems （including uniqueness, recurrenc...Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems （including uniqueness, recurrence, ergodicity, exponential ergodicity, strong ergodicity, as well as extinction probability, etc.） for the processes are presented.展开更多
The single birth process is a Markov chain, either time-continuous or time-discrete, valuedin the non-negative integers: the system jumps with positive rate from k to k + 1 but not tok +j for all j 2 (this explains th...The single birth process is a Markov chain, either time-continuous or time-discrete, valuedin the non-negative integers: the system jumps with positive rate from k to k + 1 but not tok +j for all j 2 (this explains the meaning of 'single birth') . However, there is no restrictionfor the jumps from k to k - j(1 j< k). This note mainly deals with the uniqueness problemfor the time-continuous processes with an extension: the jumps from k to k + 1 may also beforbidden for at most finite number of k. In both cases (time-continuous or -discrete), thehitting probability and the first moment of the hitting time are also studied展开更多
We obtain sufficient criteria for central limit theorems （CLTs） for ergodic continuous-time Markov chains （CTMCs）. We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we pr...We obtain sufficient criteria for central limit theorems （CLTs） for ergodic continuous-time Markov chains （CTMCs）. We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.展开更多
Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci...Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci is nondecreasing with respect to i and a（i） ＋ u（i） - ci ≥ 0 （i≥ 0）, then VarX（t）-EX（t）≥-X（0）e^-2ct,t≥0,or （ii） if there exists a constant u（i） - c≥ 0 such that b（i）-a（i）＋ci is non-increasing with respect to i and a（i）＋u（i）-ci≤0（i≥0）,then VarX（t） - EX（t） ≤ -X（0）e^-2c,t ≥ 0 Hereb（i） = qii＋1, a（0） = 0, a（i） = ∑j=^ijqii-j （i≥ 1）, u（0） = u（1） =0 and u（i） = 1/2∑j=^ij（j - 1）qii-j （i ≥ 2） . This result covers the results for birth-death processes obtained in [7].展开更多
This paper aims at two problems which exist in most of repairable spare part demand models at present: the exponential distribution as the basic assumption and one typical distribution corresponding to a model. A gene...This paper aims at two problems which exist in most of repairable spare part demand models at present: the exponential distribution as the basic assumption and one typical distribution corresponding to a model. A general repairable spare part demand model built on quasi birth-and-death process is developed. This model assumes that both the operational time of the unit and the maintenance time of the unit follow the continuous time phase type distributions. The first passage time distribution to be out of spares, the first mean time to be out of spares, and an algorithm to get the minimal amount of spares under certain restrictions are obtained. At the end of this paper, a numerical example is given.展开更多
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 rand...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 rateλ〈death rateμ,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.展开更多
At first, the concept of bridge reliability is given, followed with its mathematic model. Then, based on the analysis about the mechanism of the damage and repair of bridges, and the state diversion of bridge network,...At first, the concept of bridge reliability is given, followed with its mathematic model. Then, based on the analysis about the mechanism of the damage and repair of bridges, and the state diversion of bridge network, the state diversion process is proved to be birth-and-death process. In the end, the state diversion balance equation of bridge network is built, and the evaluation model of wartime bridge reliability is got. The model is used in a certain example, and it is proved to be precise and credible.展开更多
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 t...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.展开更多
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...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.展开更多
In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type in...In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type inequality and a criteria for a is also presented.展开更多
By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumu...By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when s is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the ＇Hawk-Dove＇ game and the ＇Coordinate＇ game are used to illustrate our model. We give the evolutionary stable strategy （ESS） of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.展开更多
In this paper, the entanglement of two moving atoms induced by a single-mode field via a three-photon process is investigated. It is shown that the entanglement is dependent on the category of the field, the average p...In this paper, the entanglement of two moving atoms induced by a single-mode field via a three-photon process is investigated. It is shown that the entanglement is dependent on the category of the field, the average photon number N, the number p of half-wave lengths of the field mode and the atomic initial state. Also, the sudden death and the sudden birth of the entanglement are detected in this model and the results show that the existence of the sudden death and the sudden birth depends on the parameter and the category of the mode field. In addition, the three-photon process is a higher order nonlinear process.展开更多
formula of simulation proccss by In this paper, we employ monmnt generating function to obtain some exact transition probability of inlmigration-birth-death（IBD） model and discuss the of sample path and statistical ...formula of simulation proccss by In this paper, we employ monmnt generating function to obtain some exact transition probability of inlmigration-birth-death（IBD） model and discuss the of sample path and statistical inference with complete observations of the IBD the exact transition density formula.展开更多
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 absorb...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.展开更多
Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general pop...Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general population. These sub-groups have higher infectivity rates. We came up with a likelihood inference model of multi-type birth-death process that can be used to make inference for HIV epidemic in an African setting. We employ a likelihood inference that incorporates a probability of removal from infectious pool in the model. We have simulated trees and made parameter inference on the simulated trees as well as investigating whether the model distinguishes between heterogeneous and homogeneous dynamics. The model makes fairly good parameter inference. It distinguishes between heterogeneous and homogeneous dynamics well. Parameter estimation was also performed under sparse sampling scenario. We investigated whether trees obtained from a structured population are more balanced than those from a non-structured host population using tree statistics that measure tree balance and imbalance. Trees from non-structured population were more balanced basing on Colless and Sackin indices.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.12101186,11771046)the Scientific Research Foundation for Young Teachers in Capital University of Economics and Business(Grant No.XRZ2021036)。
文摘The explicit solution to the Poisson equation corresponding to the Q-matrix of a single birth process is obtained,thus the explicit inverse(if exists)is presented directly.As an application,inspired by the inverse power method,combining the explicit inverse with CollatzWielandt formula,a powerful approximation theorem for the maximal eigenpair corresponding to the Q-matrix of a single birth process is presented.Different from the classical acceleration method using some fixed shift in the iteration,the shift in each iteration step is varying and the sequence formed by these shifts is strictly monotone and increases to the eigenvalue needed,which effectively reduces the number of iterations.Some examples are studied to illustrate the power of these results.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20100003110005）, the ＂985＂ project from the Ministry of Education in China, the Fundamental Research Funds for the Central Universities, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems （including uniqueness, recurrence, ergodicity, exponential ergodicity, strong ergodicity, as well as extinction probability, etc.） for the processes are presented.
文摘The single birth process is a Markov chain, either time-continuous or time-discrete, valuedin the non-negative integers: the system jumps with positive rate from k to k + 1 but not tok +j for all j 2 (this explains the meaning of 'single birth') . However, there is no restrictionfor the jumps from k to k - j(1 j< k). This note mainly deals with the uniqueness problemfor the time-continuous processes with an extension: the jumps from k to k + 1 may also beforbidden for at most finite number of k. In both cases (time-continuous or -discrete), thehitting probability and the first moment of the hitting time are also studied
文摘We obtain sufficient criteria for central limit theorems （CLTs） for ergodic continuous-time Markov chains （CTMCs）. We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.
基金Supported by the National Natural Science Foundation of China(No.10471130,10371024)
文摘Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci is nondecreasing with respect to i and a（i） ＋ u（i） - ci ≥ 0 （i≥ 0）, then VarX（t）-EX（t）≥-X（0）e^-2ct,t≥0,or （ii） if there exists a constant u（i） - c≥ 0 such that b（i）-a（i）＋ci is non-increasing with respect to i and a（i）＋u（i）-ci≤0（i≥0）,then VarX（t） - EX（t） ≤ -X（0）e^-2c,t ≥ 0 Hereb（i） = qii＋1, a（0） = 0, a（i） = ∑j=^ijqii-j （i≥ 1）, u（0） = u（1） =0 and u（i） = 1/2∑j=^ij（j - 1）qii-j （i ≥ 2） . This result covers the results for birth-death processes obtained in [7].
基金Supported by National Defense Foundation of P. R. China (41319060206)
文摘This paper aims at two problems which exist in most of repairable spare part demand models at present: the exponential distribution as the basic assumption and one typical distribution corresponding to a model. A general repairable spare part demand model built on quasi birth-and-death process is developed. This model assumes that both the operational time of the unit and the maintenance time of the unit follow the continuous time phase type distributions. The first passage time distribution to be out of spares, the first mean time to be out of spares, and an algorithm to get the minimal amount of spares under certain restrictions are obtained. At the end of this paper, a numerical example is given.
基金Supported by the NNSF of China(10371092,10771185)the Foundation of Whuan University
文摘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 rateλ〈death rateμ,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.
文摘At first, the concept of bridge reliability is given, followed with its mathematic model. Then, based on the analysis about the mechanism of the damage and repair of bridges, and the state diversion of bridge network, the state diversion process is proved to be birth-and-death process. In the end, the state diversion balance equation of bridge network is built, and the evaluation model of wartime bridge reliability is got. The model is used in a certain example, and it is proved to be precise and credible.
基金Project supported by the National Natural Science Foundation of China （Grant Nos.10471088, 60572126）
文摘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.
文摘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.
基金the National Natural Science Foundation of China(10271091)
文摘In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type inequality and a criteria for a is also presented.
基金supported by the National Natural Science Foundation of China (Grant No. 71071119)the Fundamental Research Funds for the Central Universities
文摘By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when s is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the ＇Hawk-Dove＇ game and the ＇Coordinate＇ game are used to illustrate our model. We give the evolutionary stable strategy （ESS） of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.
基金supported by the National Natural Science Foundation of China (Grant No. 10374025)the Natural Science Foundation of Hunan Province (Grant Nos. 07JJ3013 and 07JJ5003)the Research Foundation of the Education Bureau of Hunan Province(Grant No. 06A038)
文摘In this paper, the entanglement of two moving atoms induced by a single-mode field via a three-photon process is investigated. It is shown that the entanglement is dependent on the category of the field, the average photon number N, the number p of half-wave lengths of the field mode and the atomic initial state. Also, the sudden death and the sudden birth of the entanglement are detected in this model and the results show that the existence of the sudden death and the sudden birth depends on the parameter and the category of the mode field. In addition, the three-photon process is a higher order nonlinear process.
基金Supported by the Fundamental Research Funds for the Central Universities(JBK120405)
文摘formula of simulation proccss by In this paper, we employ monmnt generating function to obtain some exact transition probability of inlmigration-birth-death（IBD） model and discuss the of sample path and statistical inference with complete observations of the IBD the exact transition density formula.
文摘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.
文摘Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general population. These sub-groups have higher infectivity rates. We came up with a likelihood inference model of multi-type birth-death process that can be used to make inference for HIV epidemic in an African setting. We employ a likelihood inference that incorporates a probability of removal from infectious pool in the model. We have simulated trees and made parameter inference on the simulated trees as well as investigating whether the model distinguishes between heterogeneous and homogeneous dynamics. The model makes fairly good parameter inference. It distinguishes between heterogeneous and homogeneous dynamics well. Parameter estimation was also performed under sparse sampling scenario. We investigated whether trees obtained from a structured population are more balanced than those from a non-structured host population using tree statistics that measure tree balance and imbalance. Trees from non-structured population were more balanced basing on Colless and Sackin indices.
