## What does a Markov branching process is a continuous-time Markov chain?

A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix.

## Is queuing process a discrete-time Markov chain or a continuous-time Markov chain?

Then X = { X t : t ∈ [ 0 , ∞ ) } is a continuous-time Markov chain on , known as the M/M/ queuing chain . In terms of the basic structure of the chain, the important quantities are the exponential parameters for the states and the transition matrix for the embedded jump chain.

**What is the difference between discrete and continuous Markov chain?**

The Markov process and Markov chain are both “memoryless.” A Markov process or Markov chain contains either continuous-valued or finite discrete-valued states. A discrete-state Markov process contains a finite alphabet set (or finite state space), with each element representing a distinct discrete state.

**What is a holding time for a Markov chain?**

By time homogeneity, whenever the process enters state i, the way it evolves probabilistically from that point is the same as if the process started in state i at time 0. When the process enters state i, the time it spends there before it leaves state i is called the holding time in state i.

### What is the stationary distribution of a Markov chain?

The stationary distribution of a Markov chain describes the distribution of Xt after a sufficiently long time that the distribution of Xt does not change any longer. To put this notion in equation form, let π be a column vector of probabilities on the states that a Markov chain can visit.

### What do you understand by a Markov chain give suitable examples?

A Markov chain is a mathematical process that transitions from one state to another within a finite number of possible states. It is a collection of different states and probabilities of a variable, where its future condition or state is substantially dependent on its immediate previous state.

**What is time homogeneous Markov chain?**

The Markov chain X(t) is time-homogeneous if P(Xn+1 = j|Xn = i) = P(X1 = j|X0 = i), i.e. the transition probabilities do not depend on time n. If this is the case, we write pij = P(X1 = j|X0 = i) for the probability to go from i to j in one step, and P = (pij) for the transition matrix.

**Are all Markov processes stationary?**

A theorem that applies only for Markov processes: A Markov process is stationary if and only if i) P1(y, t) does not depend on t; and ii) P1|1(y2,t2 | y1,t1) depends only on the difference t2 − t1.

#### Does a Markov chain always have a stationary distribution?

Every finite state Markov chain has a stationary probability distribution.

#### What is state in Markov process?

A state is known as recurrent or transient depending upon whether or not the Markov chain will eventually return to it. A recurrent state is known as positive recurrent if it is expected to return within a finite number of steps, and null recurrent otherwise.

**What is meant by Markov process?**

Summary. A Markov process is a random process in which the future is independent of the past, given the present. Thus, Markov processes are the natural stochastic analogs of the deterministic processes described by differential and difference equations. They form one of the most important classes of random processes.

**What is steady state distribution of Markov chain?**

The idea of a steady state distribution is that we have reached (or converging to) a point in the process where the distributions will no longer change. The distributions for this step are equal to distributions for steps from hereon forward.

## What is the state space of a continuous time Markov chain?

Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.

## What is the transition rate in a continuous time Markov chain?

The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j. be the random variable describing the state of the process at time t, and assume the process is in a state i at time t .

**What is the identity matrix of the continuous time Markov chain?**

with initial condition P (0) being the identity matrix . The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j. be the random variable describing the state of the process at time t, and assume the process is in a state i at time t .

**Can a discrete-time process be derived from a continuous-time Markov chain?**

( S may be periodic, even if Q is not. Once π is found, it must be normalized to a unit vector .) Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X ( t) at intervals of δ units of time.