Markov chains steady state equation

These are called the steady-state equations. A Markov chain is said to be inedudble if any state can be reached from any other through a sequence of transitions whose probabdities are positive. For irreducible chains, there is precisely one steady-state distribution ir, and it is the only vector x that satisfies the steady-state equations xP = x and also satisfies the condition x, -f Xj H-. . . H- = 1. 

We summarize the discussion of steady-state behavior of Markov chains as follows. For well-behaved chains, such as irreducible, aperiodic, finite-state chains, there is a unique steady-state distribution that is also the limiting distribution. Furthermore, this distribution gives the long-run fraction of time spent in each state. It may be computed easily, by solving the steady-state equations (22). The steady-state behavior of the chain does not depend on the initial state. 

TMs relates it to the parameters of the cham—the entries of its generator Q. These are the steady-state equations in the continuous-time case. For finite-state irreducible chains, these equations have a unique solution whose components add to 1, and this solution is the steady-state distribution v. As in the discrete-time case, it is also the limiting distribution of the Markov chain and gives the long-run proportion of time spent in each state. These results extend to the infinite-state case, assuming positive recurrence, as in Subsection 3.3. 

In Section 5.1 we introduce the stochastic processes. In Section 5.2 we will introduce Markov chains and define some terms associated with them. In Section 5.3 we find the n-step transition probability matrix in terms of one-step transition probability matrix for time invariant Markov chains with a finite state space. Then we investigate when a Markov ehain has a long-run distribution and discover the relationship between the long-run distribution of the Markov chain and the steady state equation. In Section 5.4 we classify the states of a Markov chain with a discrete state space, and find that all states in an irreducible Markov chain are of the same type. In Section 5.5 we investigate sampling from a Markov chain. In Section 5.6 we look at time-reversible Markov chains and discover the detailed balance conditions, which are needed to find a Markov chain with a given steady state distribution. In Section 5.7 we look at Markov chains with a continuous state space to determine the features analogous to those for discrete space Markov chains. 

Equation 5.16 says that the Uj area solution of the steady state equation. Thus the theorem says that if a unique non-zero solution of the steady state equation exists the chain is ergodic, and vice-versa. A consequence of this theorem is that time averages from an ergodic Markov chain will approach the steady state probabilities of the chain. Note, however, for an aperiodic irreducible Markov chain that has all null recurrent states, the mean recurrence times are infinite, and hence Uj = 0 for such a chain. The only solution to the steady state equation for such a chain is Uj = 0 for all j. It is very important that we make sure all chains that we use are ergodic and contain only positive recurrent states Note also that the theorem does not say anything about the rate the time averages converges to the steady state probabilities. 

AH states in an eigodic (irreducible and aperiodic) Markov chain are positive recurrent if and only if there is a unique non-zero solution to the steady state Equation 5.18. 

If a Markov chain has a limiting distribution that is independent of the initial state, it is called the long-run distribution of the Markov chain and is a solution of the steady state equation... 

The Chapman-Kolmogorov equation and the steady state equation takes the form of infinite sums when the Markov chain has an infinite number of discrete states. 

An aperiodic irreducible Markov chain with positive recurrent states has a unique non-zero solution to the steady state equation, and vice-versa. These are known as ergodic Markov chains. 

For an irreducible Markov chain with null recurrent states, the only solution to the steady state equation is identically equal to zero. 

For a Markov chain with continuous state space, if the transition kernel is absolutely continuous, the Chapman-Kolmogorov and the steady state equations are written as integral equations involving the transition density function. 

We want to find a Markov chain that has the posterior distribution of the parameters given the data as its long-run distribution. Thus the parameter space will be the state space of the Markov chain. We investigate how to find a Markov chain that satisfies this requirement. We know that the long-run distribution of a ergodic Markov chain is a solution of the steady state equation. That means that the long-run distribution 7T of a finite ergodic Markov chain with one-step transition matrix P satisfies the equation... 

This says the long-run probability of a state equals the weighted sum of one-step probabilities of entering that state from all states each weighted by its long-run probability. The comparable steady state equation that ir 0), the long-run distribution of a Markov chain with a continuous state space, satisfies is given by... 

These equations are called the traffic equations. Invertibility of I - P implies that these equations have a unique solution in the Suppose now that p < s , > 0 and fjL > 0 for each m = 1, 2,. . . , Af. These conditions ensure that the network is an irreducible, positive recurrent Markov chain. Jackson proved the following result The steady-state distribution of the number of jobs at each station in the Jackson network is the same as that of M independent stations, the mth being an MIMIs queue with arrival rate and service rate... 

Using these approximations, we can model the demand for each type separately as a continuous-time Markov chain. Let us consider one type, with arrival rate X, service rate x and a production minimum x. In the Maikov chain two elements are pla3dng a role the number of ordo for the type and the state of the machine. The machine can be set for the production of the type or not set for the production. The states will be denoted by X or k, where k denotes the number of oidors for the type and indicates that the machine is ready to produce orders for the type. The steady-state probabilities for the states will be denoted by p or p respectively. We now have to solve the following set of equations ... 

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