scholarly journals Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1123
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Transition Probabilities ◽  
Growth Models ◽  
Death Process ◽  
Bell Polynomials ◽  
Bell Polynomial ◽  
Conditional Mean ◽  
Finite State ◽  
Potential Applications ◽  
Intensity Functions ◽  
Birth Death

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

On Separation for Birth-Death Processes

1994 ◽  
Vol 8 (1) ◽  
pp. 51-68
Author(s):  
Masaaki Kijima
Keyword(s):  
Infinitesimal Generator ◽  
Transition Probabilities ◽  
Upper Bounds ◽  
Death Process ◽  
Exponential Functions ◽  
Finite State ◽  
Variation Distance ◽  
Stationary Probabilities ◽  
Finite State Space ◽  
Birth Death

This article considers separation for a birth-death process on a finite state space S = [1,2,…, N]. Separation is defined by si(t) = 1 – minj∈sPij(t)/πj, as in Fill [5,6], where Pij(t) denotes the transition probabilities of the birth-death process and πj the stationary probabilities. Separation is a measure of nonstationarity of Markov chains and provides an upper bound of the variation distance. Easily computable upper bounds for si-(t) are given, which consist of simple exponential functions whose parameters are the eigenvalues of the infinitesimal generator or its submatrices of the birth-death process.

scholarly journals A Time-Inhomogeneous Prendiville Model with Failures and Repairs

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 251
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Markov Chain ◽  
State Space ◽  
The State ◽  
Random Time ◽  
Death Process ◽  
Inhomogeneous Markov Chain ◽  
Finite State ◽  
Intensity Functions ◽  
Finite State Space ◽  
Birth Death

We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.

scholarly journals Logistic Growth Described by Birth-Death and Diffusion Processes

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 489 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Paola Paraggio
Keyword(s):  
Diffusion Processes ◽  
Growth Models ◽  
First Passage Time ◽  
Death Process ◽  
Necessary Condition ◽  
Logistic Growth ◽  
Birth Process ◽  
Conditional Mean ◽  
Time Problem ◽  
Birth Death

We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

scholarly journals Approximating Quasistationary Distributions of Birth–Death Processes

2012 ◽  
Vol 49 (4) ◽  
pp. 1036-1051 ◽  
Author(s):  
Damian Clancy
Keyword(s):  
State Space ◽  
Population Biology ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
The State ◽  
Death Process ◽  
Sis Model ◽  
Finite State ◽  
Quasistationary Distributions ◽  
Birth Death

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (01) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

Population processes allowing emigration of families

1984 ◽  
Vol 21 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Abebe Tessera
Keyword(s):  
Population Growth ◽  
Growth Models ◽  
Death Process ◽  
Population Processes ◽  
The Individual ◽  
Population Growth Models ◽  
Birth Death

In the familiar immigration–birth–death process the events of immigration, birth and death relate to the individual. There are processes in which the whole family and not just an individual migrates. Such population growth models are studied in some detail.

Approximating Quasistationary Distributions of Birth–Death Processes

2012 ◽  
Vol 49 (04) ◽  
pp. 1036-1051
Author(s):  
Damian Clancy
Keyword(s):  
State Space ◽  
Population Biology ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
The State ◽  
Death Process ◽  
Sis Model ◽  
Finite State ◽  
Quasistationary Distributions ◽  
Birth Death

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

scholarly journals On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Symmetry ◽  
2009 ◽  
Vol 1 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Transition Probabilities ◽  
Death Process ◽  
Birth Death

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (1) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (1) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

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