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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 英国补考|随机控制代写Stochastic Control代考|Network Control Problems

Consider for each $r \in \mathbb{N}$ a stochastic processing network $\mathscr{N}^{r}$ with $J$ types of jobs and $I$ resources for processing them. Here $r$ is a scaling parameter and as $r \rightarrow \infty$, the sytem approaches criticality in a suitable sense. All the networks in the collection have a similar structure described through an $I \times J$ matrix $K$ with $K_{i j}=1$ if resource $i$ works on job type $j$ and $K_{i j}=0$ otherwise. We will assume that for each subset of resources, there is at most one job type with it as the associated set of resources, or equivalently that no two columns of $K$ are identical.

Let for $m \in \mathbb{N}, \mathbb{N}{m} \doteq{1, \ldots, m}$. In particular, $\mathbb{N}{I} \doteq{1, \ldots, I}$ and $\mathbb{N}_{J} \doteq{1, \ldots, J}$. We will assume the following local traffic condition:

Condition 1 Let $\mathscr{R}{j}=\left{i \in \mathbb{N}{I}: K_{i j}=1\right}$ be the set of resources that work on type $j$ jobs, and let $\mathscr{J}=\left{j \in \mathbb{N}{J}: \sum{i=1}^{I} K_{i j}=1\right}$ be the collection of all job types that use only one resource. Then, $\bigcup_{j \in \mathscr{J}} \mathscr{R}{j}=\mathbb{N}{I}$.

The above condition, which was first introduced in [12], says that for each resource there is a unique job-type that only requires service from that resource.

For job type $j \in \mathbb{N}{J}$, let $\left{u{j}^{r}(k)\right}_{k \in \mathbb{N}}$ be the i.i.d. inter-arrival times and $\left{v_{j}^{r}(k)\right}_{k \in \mathbb{N}}$ be the associated i.i.d. amounts of work. For each $r$, the random variables in the collection $\left{u_{j}^{r}(k), v_{j}^{r}(k), k \in \mathbb{N}, j \in \mathbb{N}_{J}\right}$ are taken to be mutually independent. We will assume that $$P\left(u_{j}^{r}(1)>0\right)=P\left(v_{j}^{r}(1)>0\right)=1 \quad \text { for all } r \text { and } j,$$
and that
$$\left{u_{j}^{r}(1)^{2}\right}_{r} \text { and }\left{v_{j}^{r}(1)^{2}\right}_{r} \text { are uniformly integrable for each } j .$$

## 英国补考|随机控制代写Stochastic Control代考|Equivalent workload formulations of Brownian control

The main results of this work will give a lower bound on the asymptotic discounted and ergodic control value functions in terms of value functions of certain control problems for Brownian motions $[9,11]$. We present below the Equivalent Workload Formulations (EWF) of these control problems. We refer the reader to for a discussion on equivalence between this formulation and the Brownian control problems as formulated in Harrison [8]. We begin by introducing the notion of an effective cost function. With our formulation of the workload process as in (5) in mind, let $G=K M$, where we recall that $M$ is the $J \times J$ diagonal matrix with entries $1 / \beta_{j}$, and let $\mathscr{W} \doteq \mathbb{R}{+}^{I}$. For each $w \in \mathscr{W}$, define the effective cost function as $$\hat{h}(w)=\min {h \cdot q: G q=w, q \geq 0}$$ Note that, from the local traffic condition (Condition 1), the set on the right side is nonempty for every $w \in \mathscr{W}$. It is known that we can select a continuous minimizer in the above linear program (cf. [2]), i.e. there is a continuous map $q^{}: \mathscr{W} \rightarrow \mathbb{R}{+}^{J}$ such that
$$q^{}(w) \in \underset{q}{\arg \min }{h \cdot q: G q=w, q \geq 0} .$$
Let $\theta=M^{-1} \eta$, and let $\Sigma$ denote the $J \times J$ matrix
$$\Sigma=\Sigma^{u}+\Sigma^{v} R,$$
where $\Sigma^{u}$ is the $J \times J$ diagonal matrix with entries $\alpha_{j}^{3}\left(\sigma_{j}^{u}\right)^{2}, \Sigma^{v}$ is the $J \times J$ diagonal matrix with entires $\beta_{j}^{3}\left(\sigma_{j}^{y}\right)^{2}$, and $R$ is the diagonal matrix with entries $\rho_{j}$. The EWF and the associated controls and state processes are defined as follows.

# 随机控制代写

## 英国补考|随机控制代写Stochastic Control代考|Network Control Problems

$P\left(u_{j}^{r}(1)>0\right)=P\left(v_{j}^{r}(1)>0\right)=1 \quad$ for all $r$ and $j$,

\left 的分隔符缺失或无法识别

## 英国补考|随机控制代写Stochastic Control代考|Equivalent workload formulations of Brownian control

$$\hat{h}(w)=\min h \cdot q: G q=w, q \geq 0$$

$$q(w) \in \underset{q}{\arg \min } h \cdot q: G q=w, q \geq 0 .$$

$$\Sigma=\Sigma^{u}+\Sigma^{v} R,$$

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