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 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 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.

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 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.

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (03) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Letqn(t) be the conditioned probability of finding a birth-and-death process in statenat timet,given that absorption into state 0 has not occurred by then. A family {q1(t),q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (3) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

scholarly journals Why is Kemeny’s constant a constant?

2018 ◽  
Vol 55 (4) ◽  
pp. 1025-1036 ◽  
Author(s):  
Dario Bini ◽  
Jeffrey J. Hunter ◽  
Guy Latouche ◽  
Beatrice Meini ◽  
Peter Taylor
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Finite State ◽  
Finite Markov Chains ◽  
Infinite State ◽  
Special Case ◽  
Finite State Space

Abstract In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as Kemeny’s constant. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a finite state space. For Markov chains with denumerably infinite state space, the constant may be infinite and even if it is finite, there is no guarantee that the physical argument will hold. We show that the physical interpretation does go through for the special case of a birth-and-death process with a finite value of Kemeny’s constant.

scholarly journals Finite-State-Space Truncations for Infinite Quasi-Birth-Death Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-23
Author(s):  
Hendrik Baumann
Keyword(s):  
Markov Chain ◽  
Lower Bound ◽  
State Space ◽  
Upper Bound ◽  
Speed Of Convergence ◽  
Drift Condition ◽  
Finite State ◽  
Infinite State ◽  
Finite State Space ◽  
Birth Death

For dealing numerically with the infinite-state-space Markov chains, a truncation of the state space is inevitable, that is, an approximation by a finite-state-space Markov chain has to be performed. In this paper, we consider level-dependent quasi-birth-death processes, and we focus on the computation of stationary expectations. In previous literature, efficient methods for computing approximations to these characteristics have been suggested and established. These methods rely on truncating the process at some level N, and for N⟶∞, convergence of the approximation to the desired characteristic is guaranteed. This paper’s main goal is to quantify the speed of convergence. Under the assumption of an f-modulated drift condition, we derive terms for a lower bound and an upper bound on stationary expectations which converge quickly to the same value and which can be efficiently computed.

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.

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