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# 数学代写|运筹学代写Operations Research代考|Balance Equations

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## 数学代写|运筹学代写Operations Research代考|Balance Equations

In this subsection, we make the following assumption for the Markov chain.

Assumption 7.1. There is a state $r$ that can ultimately be reached from every initial state $i$ with probability 1 , and the expected value of the number of steps needed to return from state $r$ to itself is finite.

This assumption is satisfied in almost every practical application concerning the equilibrium of the Markov chain. For a Markov chain with a finite state space $I$, the assumption is automatically satisfied when the Markov chain has no two disjoint closed sets. Under Assumption 7.1, one can prove that for all $i, j \in I$,
$$\pi_j=\lim {n \rightarrow \infty} \frac{1}{n} \sum{t=1}^n p_{i j}^{(t)}$$
is independent of the initial state $i$, and moreover $\sum_{j \in I} \pi_j=1$. Thus, $\pi_j$ is the expected fraction of time $\lim {n \rightarrow \infty} \frac{1}{n} \sum{t=1}^n p_{i j}^{(t)}$ that the process is in state $j$. The fraction $\pi_j$ is related to the mean first passage time to reach state $j$ from state $j$, $m_{j j}$, as
$$\pi_j=\frac{1}{m_{j j}}$$

## 数学代写|运筹学代写Operations Research代考| Markov Chains with a Cost Structure

Many practical problems concern Markov chains with a cost or revenue structure. Suppose that every time the Markov chain makes a transition to state $j$, a cost of $c(j)$ is incurred. What is the long-run average cost per unit of time? Intuitively, it should be clear that this cost is equal to $\sum_{j \in I} c(j) \pi_j$ because $\pi_j$ can be interpreted as the average number of transitions to state $j$ per unit of time. To give a mathematical proof that this cost formula is correct, we need a technical assumption in addition to the earlier assumption that there is a state $r$ that is ultimately reached from the initial state $i$ with probability 1 . The additional assumption is that $\sum_{j \in I}|c(j)| \pi_j<$ $\infty$ and that for every initial state $i \in I$, the total cost made until the first return to state $r$ has a finite expected value. This assumption is automatically satisfied if the Markov chain has a finite state space and no two disjoint closed sets. The following main theorem holds.

Theorem 7.5. The actual long-run average cost per unit of time is given by
$$\lim {n \rightarrow \infty} \frac{1}{n} \sum{t=1}^n c\left(X_t\right)=\sum_{j \in I} c(j) \pi_j \quad \text { with probability } 1,$$
independently of the initial state $X_0=i$.
We do not give the proof of this so-called ergodic theorem. It relies on the famous renewal-reward theorem from the theory of regenerative stochastic processes. The proof moreover shows that the theorem also applies to the case of continuous costs between the state transitions rather than direct costs $c(j)$. If, in the continuous case, we define the cost function $c(j)$ by
$$c(j)=\mathbb{E}\left[\text { cost between the times } t=n-1 \text { and } t=n \mid X_{n-1}=j\right],$$

then it is also true that the actual long-run average cost per unit of time is equal to $\sum_{j \in I} c(j) \pi_j$ with probability 1. This is an extremely useful observation for real-world applications. In applications, it can also be helpful to interpret certain performance measures as average costs per unit of time.

## 数学代写|运筹学代写Operations Research代考|Balance Equations

$$\pi_j=\lim n \rightarrow \infty \frac{1}{n} \sum t=1^n p_{i j}^{(t)}$$

$$\pi_j=\frac{1}{m_{j j}}$$

## 数学代写|运筹学代写Operations Research代考| Markov Chains with a Cost Structure

$$\lim n \rightarrow \infty \frac{1}{n} \sum t=1^n c\left(X_t\right)=\sum_{j \in I} c(j) \pi_j \quad \text { with probability } 1$$

$$c(j)=\mathbb{E}\left[\text { cost between the times } t=n-1 \text { and } t=n \mid X_{n-1}=j\right],$$

## MATLAB代写

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