## 数学代写|运筹学作业代写operational research代考|The Uniformization Method

2023年4月7日

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## 数学代写|运筹学作业代写operational research代考|The Uniformization Method

The Markov jump chain with state space $I$ introduced in Definition 8.1 leaves state $i$ after an exponentially distributed time with mean $1 / \nu_i$ and then jumps to another state $j(j \neq i)$ with probability $p_{i j}$. Letting $X_n$ denote the state of the process just after the $n$th state transition, the discrete-time stochastic process $\left{X_n\right}$ is called the embedded Markov chain with one-step transition probabilities $p_{i j}$. If $\nu_i=\nu$ for all $i$, the transition epochs are generated by a Poisson process with rate $\nu$, and an expression for $p_{i j}(t)$ is directly obtained by conditioning on the number of Poisson events up to time $t$ and using the $n$-step transition probabilities of the embedded Markov chain $\left{X_n\right}$. However, the leaving rates $\nu_i$ are in general not identical. Fortunately, there is a simple trick for reducing the case of nonidentical leaving rates to the case of identical leaving rates. The uniformization method transforms the original continuous-time Markov chain with non-identical leaving rates into an equivalent stochastic process in which the transition epochs are generated by a Poisson process at a uniform rate. However, to achieve this, the discrete-time Markov chain describing the state transitions in the transformed process has to allow for self-transitions leaving the state of the process unchanged. The uniformization method is a powerful computational tool to solve Kolmogorov’s equations for the time-dependent state probabilities and to calculate first passage time probabilities.
To formulate the uniformization method, choose a finite number $\nu$ with
$$\nu \geq \nu_i, \quad i \in I$$
Define now $\left{\bar{X}n\right}$ as the discrete-time Markov chain whose one-step transition probabilities $\bar{p}{i j}$ are given by
$$\bar{p}{i j}= \begin{cases}\frac{\nu_i}{\nu} p{i j}, & j \neq i \ 1-\frac{\nu_i}{\nu}, & j=i\end{cases}$$
for any $i \in I$. Let ${N(t), t \geq 0}$ be a Poisson process with rate $\nu$ such that the process is independent of the discrete-time Markov chain $\left{\bar{X}n\right}$. Define now the continuous-time stochastic process ${\bar{X}(t), t \geq 0}$ by $$\bar{X}(t)=\bar{X}{N(t)}, \quad t \geq 0$$

## 数学代写|运筹学作业代写operational research代考|Equilibrium Probabilities

While studying discrete-time Markov chains, we already saw that an accessibility condition is needed to guarantee that the Markov chain’s equilibrium distribution does not depend on the initial state. For the continuous-time Markov chain ${X(t), t \geq 0}$, we make a similar assumption.

Assumption 8.1. There is a state $r$ such that for any initial state $i$, the probability is 1 that there will ultimately be a transition to state $r$, and the expected time needed to return from state $r$ to itself is finite.

This assumption is satisfied in almost all practical applications in operations research. Under this assumption, for every $j \in I$, the limiting probability
$$\pi_j=\lim {t \rightarrow \infty} p{i j}(t)$$
exists and is independent of the initial state $i$. The reason the ordinary limit always exists is that periodicity problems do not occur in continuous-time Markov chains because the times between state transitions have a continuous distribution. An irreducible continuous-time Markov chain satisfying Assumption 8.1 is called ergodic. The limiting probabilities for an ergodic Markov chain can be calculated by solving a system of linear equations.

Theorem 8.3. Let the Markov chain be ergodic, then the limiting probabilities $\pi_j$ for $j \in I$ are the unique solution to the linear equations
\begin{aligned} & \nu_j \pi_j=\sum_{k \neq j} \pi_k q_{k j} \quad \text { for } j \in I \ & \sum_{j \in I} \pi_j=1 \end{aligned}

The equations (8.8) are called the balance equations or global balance equations, while (8.9) is called the normalizing equation. 3 These equations characterize the stationary distribution that for an ergodic Markov chain coincides with the limiting distribution. Later, we will give a physical interpretation of the balance equations that makes them simple to remember. We will then also see why the limiting probabilities $\pi_j$ for $j \in I$ are also often called the equilibrium probabilities. We do not prove Theorem 8.3 in its full generality. For the case that $I$ is finite, we only show that it is plausible that the $\pi_j$ satisfy the equations. For this, we take $t \rightarrow \infty$ in the differential equations for the transient probabilities $p_{i j}(t)$ in Theorem 8.1. If we use that $p_{i j}(t) \rightarrow \pi_j$ and $p_{i j}^{\prime}(t) \rightarrow 0$ as $t \rightarrow \infty$, it follows that $0=\sum_{k \neq j} \pi_k q_{k j}-\pi_j \nu_j$ for all $j \in I$. The normalizing equation follows by taking $t \rightarrow \infty$ in $\sum_{j \in I} p_{i j}(t)=1$.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|The Uniformization Method

$$\nu \geq \nu_i, \quad i \in I$$

$$\bar{p} i j=\left{\frac{\nu_i}{\nu} p i j, \quad j \neq i 1-\frac{\nu_i}{\nu}, \quad j=i\right.$$

$$\bar{p} i j=\left{\frac{\nu_i}{\nu} p i j, \quad j \neq i 1-\frac{\nu_i}{\nu}, \quad j=i\right.$$

$$\bar{X}(t)=\bar{X} N(t), \quad t \geq 0$$

## 数学代写|运筹学作业代写operational research代考|Equilibrium Probabilities

$$\pi_j=\lim t \rightarrow \infty p i j(t)$$

$$\nu_j \pi_j=\sum_{k \neq j} \pi_k q_{k j} \quad \text { for } j \in I \quad \sum_{j \in I} \pi_j=1$$方程 (8.8) 称为平衡方程或全局平衡方程，而 (8.9) 称为归一化方程。 3 这些方程苗述了与极限分布一致的 遍历马尔可夫链的平稳分布。稍后，我们将对平衡方程 进行物理解释，使它们易于记忆。然后我们还将看到为 什么极限概率 $\pi_j$ 为了 $j \in I$ 通常也称为均衡概率。我们 不完全普遍地证明定理 8.3。对于这种情况 $I$ 是有限的， 我们只证明它是合理的 $\pi_j$ 满足方程。为此，我们取 $t \rightarrow \infty$ 在瞬态概率的微分方程中 $p_{i j}(t)$ 在定理 8.1 中。 如果我们用那个 $p_{i j}(t) \rightarrow \pi_j$ 和 $p_{i j}^{\prime}(t) \rightarrow 0$ 作为 $t \rightarrow \infty$ ， 它遵犑 $0=\sum_{k \neq j} \pi_k q_{k j}-\pi_j \nu_j$ 对全部 $j \in I$. 归一化方程如下 $t \rightarrow \infty$ 在 $\sum_{j \in I} p_{i j}(t)=1$.

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