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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.

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.

125 questions
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Likelihood for Births and Deaths?

I am trying to understand the general logic on how to write the Likelihood Function of a Stochastic Process. In some situations, I think I can understand how this is done. For example, suppose $X_t$ is a standard Wiener Process…
konofoso
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Birth processes, X-Bacterias in a petri dish abide to the following rules...

Question X-Bacterias in a petri dish abide to the following rules: each bacteria evolves identically and independently from the others. each bacteria is replaced by four new bacterias after a random time with $e(\beta)$-distribution, where $\beta…
George Cooper
  • 637
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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 P_{n_0} (\text{more than one birth happens up to…
user568706
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Queueing theory M/M/k - probability of number of busy servers seen by next arrival process

Consider a $n$ server parallel queueing system, need to calculate the probability of $1$ busy server as seen by next arrival process. $\lambda$$=$$arrival$ $rate$ $of$ $processes$ ; $\mu$$=$$service$ $rate$ $of$ $processes$ When there are $0$…
Phani
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Mean time to absorption in a birth and death process proof by Samuel Karlin

I am studying "A First Course in Stochastic Process" by Samuel Karlin. In this book, theorem 7.1 states that Consider a birth and death process with birth and death parameters $\lambda_n$ and $\mu_n$, where $\lambda_0 = 0$ so that $0$ is an…
Donguk Lim
  • 31
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Combinatorial identity: $\sum\limits_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1$

Let $i,j\in\mathbb Z_{\ge0}$ be nonnegative integers. How can we prove $$\sum_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1?$$ (Here, $i\land j=\min(i,j)=\min\{i,j\}=\min(\{i,j\})$ is the minimum of $i$ and $j.$ This problem comes from my study of…
xFioraMstr18
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Discrete time birth-death processes

Let $(X_{n})$ be a discrete time Markov chain with $E=\mathbb{N}$. Suppose that the transition matrix $P$ is given by: $$P(n,n+1)=p(n), \quad P(n,n)=q(n) \quad \mbox{and} \quad P(n,n-1)=r(n), \quad r(0)=0$$ Let…
MGF01
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birth-death processes orthogonal polynomials

Let $Z_t$ be a birth-death process with birth rates $\lambda_n$ and death rates $\mu_n$ defined on the non-negative integers. The family of orthogonal polynomials $\{P_n(x)\}_{n\ge0}$ associated with $Z_t$ is defined by the recurrence…
PabloSau
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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 follows that the population goes extinct in the…
Alex
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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 is quite similar to discrete-time Markov Chain,…
Raven Cheuk
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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 death rates. Clearly a thing as a non-homogeneous…
user565950
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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 $$\sum_{n=0}^r \frac{1}{\lambda_n\pi_n} \sum_{k=0}^n \pi_k$$ For the…
Rosa Maria Gtz.
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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 $\mu_{i}=d_{1}i+d_{2}i^2$ where $\lambda_{i} $ and $\mu_{i}$ are birth…
CeeTeaEn
  • 31
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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 I can start to resolve this exercise. At the beggin I…
user502400
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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 $i+1$. I am using the book "Introduction to…
user386721
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