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Questions tagged [birth-death-process]

This tag is for questions about birth and death processes. These processes are a special case of the continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one and they are used to model the size of a population, queuing systems, the evolution of bacteria, the number of people with a disease within a population etc.

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Determine $\lim_{t→∞} \mathbb{P}_i(X_t = 0) $ for $i = 0, 1, 2, 3$.

I have this problem, I figured out the part (a) but I'm having a little trouble with part (b) if anyone can help me with that. Two repairmen serve three machines (that is, at most two machines can ...
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I am having trouble figuring out how many lambda's (births) there are in a given birth-death Markov process problem.

These questions are not for assignment. I am just confused as to how to set up the problem. I also do not need help calculating the problems at hand. I understand that in a birth and death problem, $...
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Maximum size of a subcritical birth-death process

Beginning with a population of $n_0$ individuals, let each individual have a probability $p$ to survive until it replicates into two independent and identical individuals, where $p<\frac12$. It ...
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Expected Growth of General Pure Birth Process

Assume we have a non-explosive Markov Chain $(X_t)_t$ in continuous time with Q matrix $$ Q(k,l) = \begin{cases} \lambda_k &\text{if } l=k+1\\ -\lambda_k &\text{if } l=k\\ 0 &\text{...
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Difference between embedded chain and continuous-time Markov chain

I am using the book Understanding Markov Chain by Nicolas Privault I start having some confusions when it comes to Continuous-Time Markov Chain. As far as I understand, continuous-time Markov chain ...
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differential equations of a birth-death process

Given are the following differential equations from the paper by Thorne, Kishino and Felsenstein 1991 (reference): $ \frac{d f_n(t)}{dt} = \lambda(n-1)f_{n-1}(t)-(\lambda+\mu)nf_n(t)+\mu n f_{n+1}(t) ...
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Pure Birth Process With Random Time

I'm having a bit of trouble figuring out a problem presented in lecture. Given a pure birth process ${X_t}$ with $X_0 = 0$ and rates $\lambda_i, i\geq0$, we're supposed to figure out for some integer $...
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Birth and death process with probability of staying

I have a birth and death process with states $S=\{ 0,1 \}$. If I'm in state $0$ the process change to state $1$ in a time with distribution $\exp(\lambda)$. If I'm in state $1$ I wait a exponential ...
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Probability generating function of simple birth process

Question: Suppose that $(X_t)_{t \geq 0}$ is a $(1,(\lambda_n)_{n \geq 0})$ simple birth process with $\lambda_n = (n+1)\lambda$. Show that $$\phi(t) := \Bbb E[z^{X_t}]= ze^{-\lambda t}+ \int_0^t \...
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First passage time distribution in birth-death processes

I'm working on the following birth-death process $$X\rightleftharpoons A$$ in which the birth and death (transition) rates are $t^+=k_2a$ and $t^-=k_1n$ respectively. $n$ is the concentration of $X$...
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Birth-death process Expected waiting time in when in invariant distribution

Say I have a birth-death process, with birth having Poisson distribution with parameter $\lambda$ and death having poisson distribution $\mu$. Assuming that both stochastic processes, birth and death, ...
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Total time spent in state $i$ before reaching state $i+1$

Let $T_k$ be the total cumulative time spent in state $i$ before reaching state $i+1$ in a continuous time birth-death process, with birth and death rates $\lambda_k$ and $\mu_k$. What is the ...
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birth-death process in a two state system

I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying. Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can ...
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find expected time and variance for state by using birth and death process

Consider a birth and death process with birth rates $λ_n = (n + 1)λ$, $n \ge 0$, and death rates $μ_n = nμ$, $n \ge 0.$ (a) Determine the expected time to go from state 0 to state 3. (b) ...
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Can a birth death chain having a finite space contain any recurrent state?

I am studying Birth-Death Chains from Introduction to Stochastic Processes by Paul G. Hoel. From the definition of birth death chain it is immediate that it does not contain any absorbing state. We ...
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M/M/1 and M/M/2 in the same Queue based on number of customers

Suppose you have a queue that starts as M/M/1 with an initial birth rate and initial death rate but the birth rate is larger than the death rate. Once you have n customers in the Queue it switches to ...
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Stochastic simulation Gillespie algorithm for areas instead of volumes?

I am trying to find resources on the Gillespie stochastic simulation algorithm for my system which happens on a surface. The original algorithm was developed for a reactor of volume $V$, but my system ...
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How would I determine how large an M/M/c queue grows after a certain amount of time?

Most of the reading on queueing theory I've done focuses on when the arrival rate is less than the service rate so that the queue doesn't explode. However, if I have a queueing system (either M/M/1 or ...
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Expected time until next birth with different possibilities.

Consider a birth and death process $X(t)$ with birthrate $n(\lambda_1+\lambda_2)$ and death rate $n\mu$ for $n$ individuals at that time. We assume, that $\lambda_1$ and $\lambda_2$ are two ...
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Birth-death-process: probability for at least one birth up to $t$

Consider a Birth-and-Death Process with individual birth rates $\lambda(t)$ and individual death rates $\mu(t)$, starting at $n_0$. My question is if there is a formula for something like $$\mathbb ...
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Definition of non-homogeneous Birth–death process

I know that the usual definition of a birth-death processes found in books uses a homogeneous Markov processes, defines a transition function and uses the derived q-matrix to define the birth and ...
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Birth and death process, inequality between two times

I have a discrete times birth and death process $\{\Psi_n\}_{n\in \mathbb N}$ with birth probability $p$ and death probability $q$ defined as follows: \begin{align} \Psi_n=\sum_{i=1}^n\eta_i \end{...
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Modelling a path-dependent counting process as a Finite State Markov Chain

There are $C$ i.i.d processes running in parallel. Each process can be in any one of the $k$ states and is a birth-death process (with only one step transitions allowed). That is, for any process, the ...
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Birth/Death chain

How should I think about birth/death chains in terms of their generators? The transition probability is obviously only nonzero when going from state $i$ to $i+1$ or $i$ to $i-1$. But how should I ...
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Finding $E(\int_{0}^{T}X \big(t)dt \big)$ for a pure birth process

As the title suggests, I have a question on the find the expectation of a pure birth process. Let $\{X(t), t \geq0 \}$ be a pure birth process with $X(0)=1$ and birth rates $\lambda _{k}=k $ for $k=...
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Birth/Death Process: Each individual in a population gives birth to an offspring at the rate of 1.5 per unit time independently of other individuals.

Birth/Death Process: Each individual in a population gives birth to an offspring at the rate of 1.5 per unit time independently of other individuals. If the population starts with 4 individuals, find ...
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infinitesimal parameter $\lambda_n$, $\mu_n$

Consider a birth and death process with infinitesimal parameter $\lambda_n$, $\mu_n$. Then the expected length of time for reaching state $r + 1$ starting from state 0 is $$\sum_{n=0}^r \frac{1}{\...
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Using Laplace to solve pure death process

I have got all differential equations as shown below in the image, but I don't know how to proceed next to solve. In hint, it was given that you can use Laplace to solve it. Can someone show me how to ...
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Finding the PDE for a birth and death process

I have been having some trouble with the following problem. Suppose a general birth and death process has birth and death rates given by $\lambda_{i}=b_{o}+b_{1}i+b_{2}i^2 $ and $\mu_{i}=d_{1}i+d_{2}...
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Limiting proportion of time spent in each state of a cyclic birth and death chain

Let $\{X_n\}$ be a birth and death chain on $\mathbb{Z}$. Let $p_0, p_1, ..., p_m \in (0,1)$ and label state $i$ by $[i]_m = i \text{(mod $m$)}$ Suppose that $p_{i} = $ probability of going from ...
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A birth-death process sufficient condition for regularity

In Resnick's Adventures of Stochastic Processes, the author gives in page 411 a sufficient conditions for the birth-death process to be regular (a.s. that there are no infinite transitions in finite ...
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Exercise about Birth and Death process

I have some difficultes with an exercise that I must do. I try to post it there, maiby someone can help me to resolve it. Let ${X(t)}_{t>0}$ on $\{0,1,2,3\}$ a birth end death process, with $\...
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Departure rate of $k$ customers in a queue

$\textbf{Soft question:}$ If the probability of a customer leaving the queue (system) is $\frac{1}{4}$, am I right to say that the departure rate of a single customer is $\mu=\frac{1}{4}$, and that ...
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Mean number of customers in a queue at a car repair workshop

$\textbf{Q:}$ Suppose that a car repair workshop encounters on average $1$ customer per day, where the probability of encountering a customer is proportional to the length of a small time interval, ...
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Mean Value and Variance of a Birth and Death Process

Let $\{X(t)\}_{t>0}$ on $\{0,1,2,3\}$ a birth and death process, with $\lambda(s)=(3-s)^2$ and $\mu(s)=s^2+s$. Assume $P(X(0)=3)=1$ and determine: (a)$E[X(t)]$; (b)$Var[X(t)]$. I don't know how ...
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Probability of extinction in a simple birth and death process with carrying capacity?

I know that the probability of extinction for a simple birth and death process is given by $p_{0}(t) = \bigg(\frac{\mu-\mu e^{(\mu-\lambda)t}}{\lambda-\mu e^{(\mu-\lambda)t}}\bigg)^{N_0}$ where $\mu$...
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Poisson process in a disaster relief station

$\textbf{Q}: $Suppose you are helping out in a disaster relief, at a relief station. Assume that every morning, supplies will be delivered to the station, and that the amount of supplies delivered is ...
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190 views

Departure Process Of $M/M/S/K$ queue

Regarding to Burke Theorem it said that $M/M/S$ queue with Poisson arrival rate $\lambda$ will have departure process with rate $\lambda$. as far as i know it happen as the affect of time ...
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151 views

A Finite Capacity Queueing System with $N=3$

The following is a question from Pinksy and Karlin's An Introduction to Stochastic Modelling: The problem is to model a queueing system having finite capacity. We assume arrivals according to a ...
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Gridlock in Queueing Theory

The following is a question from Pinksy and Karlin's $\textit{An Introduction to Stochastic Modelling}$: Customers arrive at a service facility according to a Poisson process of rate λ. There is ...
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1answer
268 views

Extinction time of a simple birth-death process density

Here's what I have: Let X be a simple birth-death process where individuals have independent $\text{Exp}(\mu)$ lifetimes and, during their lifetime give birth at rate $λ$ independently of other ...
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Are death-birth stochastic process double stochastic?

I'm having an issue to justify a statement. We have seen in class that a double-stochastic discrete process is a process for which the sum of each column and the sum of each line in the transition ...
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152 views

Stationary distribution/expected return time of a birth and death process

A squirrel walks down a path. Every minute it either moves one yard forward, one yard backward, or stands still. At its beginning point there is a wall so that the squirrel either moves forward or ...
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How to use Inverse Laplace transform to obtain the Cumulative distribution function (CDF)?

Questions: 1. Given the Laplace transform of the CDF, which is $\mathcal{L}[G_T(t)]$, I want to use the inverse Laplace to obtain $G_T(t)$ evaluated at 0. But, by definition, inverse Laplace using ...
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Multivariable birth and death process

I read about birth and death process in a paper. The paper then generalized the birth and death process as multivariable. The paper part is like in the picture : . Can anyone explain to me about this ...
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Transition probability of a Markov Chain (Paper)

I've been checking out this paper and I got stuck. I want to know how to calculate the transition probability of a birth and death Markov process. To understand the logic of the paper and how they ...
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Expectation time - General birth and death process

I study continuous time Markov chain and more specifically birth and death processes. I am trying to understand how to calculate the expectation time it takes to start from a state $i$ to a state $i+1$...
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Calculating First Passage time in a Birth-Death Process

I am trying to find the mean first passage time in a birth-death process to get from state 2 to state 1. Here is the question in detail: We are in a birth-death system, so the state space ...
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Birth death process with searching for server

I have some difficulties with drawing up a birth death process for the following special case: Suppose that customers arrive at a system with a total of $n$ servers according to a Poisson process ...
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Mean hitting time formula

Suppose you have a random walk on $\{0,\ldots,N\}$, and let $v_k= E[T|x_o=k]$ be the mean hitting time starting in state $k$. Using fist step analysis we can easily see that $v_k$ satisfies the ...