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 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.

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.

HITTING TIME DISTRIBUTIONS FOR BIRTH–DEATH PROCESSES WITH BILATERAL ABSORBING BOUNDARIES

2016 ◽  
Vol 31 (3) ◽  
pp. 345-356
Author(s):  
Yong-Hua Mao ◽  
Chi Zhang
Keyword(s):  
State Space ◽  
Hitting Time ◽  
Death Process ◽  
Absorbing Boundaries ◽  
Finite State ◽  
Finite State Space ◽  
Birth Death

For the birth–death process on a finite state space with bilateral boundaries, we give a simpler derivation of the hitting time distributions by h-transform and φ-transform. These transforms can then be used to construct a quick derivation of the hitting time distributions of the minimal birth–death process on a denumerable state space with exit/regular boundaries.

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.

Strong consistency of a modified maximum likelihood estimator for controlled Markov chains

1980 ◽  
Vol 17 (03) ◽  
pp. 726-734 ◽  
Author(s):  
Bharat Doshi ◽  
Steven E. Shreve
Keyword(s):  
Likelihood Function ◽  
Transition Probabilities ◽  
Control Law ◽  
Likelihood Estimator ◽  
Modified Maximum Likelihood ◽  
Control Scheme ◽  
Finite State ◽  
Finite Set ◽  
Stationary Control ◽  
Finite State Space

A controlled Markov chain with finite state space has transition probabilities which depend on an unknown parameter α lying in a known finite set A. For each α, a stationary control law ϕ α is given. This paper develops a control scheme whereby at each stage t a parameter α t is chosen at random from among those parameters which nearly maximize the log likelihood function, and the control ut is chosen according to the control law ϕ αt. It is proved that this algorithm leads to identification of the true α under conditions weaker than any previously considered.

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.

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 Fluid queues driven by a birth and death process with alternating flow rates

2004 ◽  
Vol 2004 (5) ◽  
pp. 469-489
Author(s):  
P. R. Parthasarathy ◽  
K. V. Vijayashree ◽  
R. B. Lenin
Keyword(s):  
State Space ◽  
Death Process ◽  
Birth And Death Process ◽  
Fluid Queue ◽  
Flow Rates ◽  
Buffer Occupancy ◽  
Fluid Queues ◽  
Finite State ◽  
Service Rates ◽  
Finite State Space

Fluid queue driven by a birth and death process (BDP) with only one negative effective input rate has been considered in the literature. As an alternative, here we consider a fluid queue in which the input is characterized by a BDP with alternating positive and negative flow rates on a finite state space. Also, the BDP has two alternating arrival rates and two alternating service rates. Explicit expression for the distribution function of the buffer occupancy is obtained. The case where the state space is infinite is also discussed. Graphs are presented to visualize the buffer content distribution.

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.

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