Quasi-stationary distributions for structured birth and death processes with mutations

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.     

Ramanujan theta functions and birth and death processes

This paper discusses birth and death processes that are related to the celebrated Ramanujan’s theta functions. The pertinent transient system size probabilities are calculated numerically via a truncation of continued fractions.     

A characterization of Wishart processes and Wishart distributions

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.     

Joint distributions of first hitting time and first hitting location after explosion for birth and death processes

For a birth and death processX=|X(t),t <σ| with explosion and lifespanu distributions and joint distributions of first hitting time and first hitting location after explosion of setB _n = |0,1,...,n| ,n have been found.     

Distributions of lifetime after explosion for birth and death processes

How long can a process live after explosion for a birth and death process with explosion and lifespan σ,X ={X(t),t<σ}?Distribution of lifetime from explosion to the end of life, expected lifetime and probability of explosion which means the end of life are found.     

A class of infinitely divisible distributions connected to branching processes and random walks

A class of infinitely divisible distributions on {0,1,2,…} is defined by requiring the (discrete) Lévy function to be equal to the probability function except for a very simple factor. These distributions turn out to be special cases of the total offspring distributions in (sub)critical branching processes and can also be interpreted as first passage times in certain random walks. There are connections with Lambert's W function and generalized negative binomial convolutions.     

On the exact distribution and mean value function of a geometric process with exponential interarrival times

The geometric process is considered when the distribution of the first interarrival time is assumed to be exponential. An analytical expression for the one dimensional probability distribution of this process is obtained as a solution to a system of recursive differential equations. A power series expansion is derived for the geometric renewal function by using an integral equation and evaluated in a computational perspective. Further, an extension is provided for the power series expansion of the geometric renewal function in the case of the Weibull distribution.     

Properties of birth and death processes with asymptotically periodic intensities

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 15–20, 1990.     

MULTIVARIATE SURVIVAL DISTRIBUTIONS OF AGE AND RESIDUAL LIFETIME PROCESSES IN NONHOMOGENEOUS POISSON PROCESS

Let{N(t),t≥0} be the nonhomogeneous Poisson process with cumulative intensity parameter Λ(t),{δt,t≥0} the age process,and {yi,t≥0) the residual lifetime process.In the present paper the expressions of n-dimensional survival distribution functions of the processes{δi} and {yi}, and their Lebesgue decompositions are derived.     

Quadratic geometric programming with application to machining economics

Geometric Programming is extended to include convex quadratic functions. Generalized Geometric Programming is applied to this class of programs to obtain a convex dual program. Machining economics problems fall into this class. Such problems are studied by applying this duality to a nested set of three problems. One problem is zero degree of difficulty and the solution is obtained by solving a simple system of equations. The inclusion of a constraint restricting the force on the tool to be less than or equal to the breaking force provides a more realistic solution. This model is solved as a program with one degree of difficulty. Finally the behavior of the machining cost per part is studied parametrically as a function of axial depth. This research was supported by the Air Force Office of Scientific Research Grant AFOSR-83-0234     

Extremes in autoregressive processes with uniform marginal distributions

Chernick (1981) derives a limit theorem for the maximum term for a class of first order autoregressive processes with uniform marginal distributions. The parameter for these processes is equal to 1/r where r is an integer, r 2. Based on this limit theorem, the asymptotic distribution of the minimum term and the joint asymptotic distribution of the maximum and minimum terms in the sequence are obtained. Since the condition D′(u_n) of Leadbetter (1974) fails, the condition of Davis (1979), D′(v_n, u_n), also fails. Negatively correlated uniform sequences are shown to exist. Asymptotic distributions for the maximum and minimum terms in the sequence are derived and it is shown that the maximum and minimum are not asymptotically independent.     

The birth and death processes with zero as their absorbing barrier

LetE=(0, 1,...), Q ^b=(q_ij), i, j=0, 1, ..., whereq _{i, i–1}=a_i, q_{i, i+1}=b_i, q_ii=–(a_i+b_i), q_ij=0, when|i–j|>1. a ₀=0, b₀=b>0, a_i, b_i>0 (i>0). Lettingb=0 inQ ^b, we get the matrixQ ⁰.The time homogeneous Markov processX ^b ={x ^b (t,w), 0t< ^b (w)} (X ⁰={x⁰(t,w), 0t<⁰(w)}), withQ ^b (Q ⁰, respectively) as its density matrix and withE as its state space, is calledQ ^b (Q ⁰, respectively) process in this paper.Q ^b andQ ⁰ processes are all called the birth and death processes, with zero being the reflecting barrier ofQ ^b processes, the absorbing barrier ofQ ⁰ processes.AllQ ^b processes have been constructed by both probability and analytical methods (Wang [2], Yang [1]). In this paper, theQ ⁰ processes are imbedded intoQ ^b processes and all theQ ⁰ processes are directly constructed from theQ ^b processes. The main results are:Letb>0 be arbitrarily fixed, then there is a one to one correspondence between theQ ⁰ processes and theQ ^b processes (in the sense of transition probability).TheQ ⁰ process is unique iffR ^*=. SupposingR<, then:IfX ⁰={x⁰(t,w), 0t<⁰(w)} is a non-minimalQ ⁰ process, then its eigensequence (p, q, r _n, n–1) satisfies § 4(7).Conversely, let a non-negative number sequence (p, q, r _n, n–1) satisfying § 4(7) be arbitrarily given, then there exists a unique non-minimalQ ⁰ processX ⁰ with eigensequence (p, q, r _n, n–1). The Laplace transform of the transition probability (p _ij ⁰ (t)) ofX ⁰ is determined by § 4(15). X ⁰ is honest iffr _–1=0.X ⁰ satisfies the forward equation iffp=0.     

Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called $(K\text{,}N)$ policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.     

A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations

This paper analyzes a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. The repairman may mean here the human beings who are used to repair the failed device. When the device functions smoothly, the repairman leaves the system for a vacation, the duration of which is an exponentially distributed random variable. In vacation period, the repairman can perform other secondary jobs to make some extra profits for the system. The lifetimes and the repair times of the device are governed by phase-type distributions (PH distributions), and the condition of device following repair is not “as good as new”. After a prefixed number of repairs, the device is replaced by a new and identical one. The spare device for replacement is available only by an order and the procurement lead time for delivering the spare device also follows a PH distribution. Under these assumptions, the vector-valued Markov process governing the system is constructed, and several important performance measures are studied in transient and stationary regimes. Furthermore, employing the standard results in renewal reward process, the explicit expression of the long-run average profit rate for the system is derived. Meanwhile, the optimal maintenance policy is also numerically determined.     

Exponential and Strong Ergodicity for Markov Processes with an Application to Queues

For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state.An application to the queue length process of M/G/1 queue with multiple vacations is given.     

Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

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.     

Some universal limits for nonhomogeneous birth and death processes

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP X_N. Finally we present some examples where these bounds are used in order to approximate the double mean. AMS Subject Classification: Primary: 60J27 Secondary: 60K25 34A30