### 英国补考|随机控制代写Stochastic Control代考|MAST90059

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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|Multiclass G/M/n+ Mueues with heavy-tailed arrivals

As in [4, Subsection 4.1], consider $G / M / n+M$ queues with $m$ classes of customers and one server pool of $n$ parallel servers. Customers of each class form their own queue and are served in the first-come first-served (FCFS) service discipline. Customers of different classes are scheduled to receive service under the work conserving constraint, that is, non-idling whenever customers are in queue. We assume that the arrival process of each class is renewal with heavy-tailed interarrival times. The service and patience times are exponentially distributed with class-dependent rates. The arrival, service and abandonment processes of each class are mutually independent.

We consider a sequence of such queueing models indexed by $n$ and let $n \rightarrow \infty$. Let $A_{i}^{n}, i=1, \ldots, m$, be the arrival process of class-i customers with arrival rate $\lambda_{i}^{n}$. Assume that $A_{i}^{n}$ ‘s are mutually independent. Define the FCLT-scaled arrival processes $\widehat{A}^{n}=\left(\widehat{A}{1}^{n}, \ldots, \widehat{A}{m}^{n}\right)^{\prime}$ by $\widehat{A}{i}^{n}:=n^{-1 / \alpha}\left(A{i}^{n}-\lambda_{i}^{n} \varpi\right), i=1, \ldots, m$, where $\varpi(t) \equiv t$ for each $t \geq 0$, and $\alpha \in(1,2)$. We assume that
$$\lambda_{i}^{n} / n \rightarrow \lambda_{i}>0, \quad \text { and } \quad \ell_{i}^{n}:=n^{-1 / \alpha}\left(\lambda_{i}^{n}-n \lambda_{i}\right) \rightarrow \ell_{i} \in \mathbb{R},$$
for each $i=1, \ldots, m$, as $n \rightarrow \infty$, and that the arrival processes satisfy an FCLT
$$\widehat{A}^{n} \Rightarrow \widehat{A}=\left(\widehat{A}{1}, \ldots, \widehat{A}{m}\right)^{\prime} \quad \text { in }\left(D_{m}, M_{1}\right), \text { as } n \rightarrow \infty,$$
where the limit processes $\widehat{A}{i}, i=1, \ldots, m$, are mutually independent symmetric $\alpha$ stable processes with $\widehat{A}{i}(0) \equiv 0$, and $\Rightarrow$ denotes weak convergence and $\left(D_{m}, M_{1}\right)$ is the space of $\mathbb{R}^{m}$-valued càdlàg functions endowed with the product $M_{1}$ topology [15]. The processes $\widehat{A}{i}$ have the same stability parameter $\alpha$, with possibly different “scale” parameters $\xi{i}$. Note that if the arrival process of each class is renewal with regularly varying interarrival times of parameter $\alpha$, then we obtain the above limit process. Let $\mu_{i}$ and $\gamma_{i}$ be the service and abandonment rates for class-i customers, respectively.

## 英国补考|随机控制代写Stochastic Control代考|Homogenized Correlated Nonlinear Filtering

The theoretical aim of filtering is to derive representations and convergence results of the filter $\pi_{t}^{\varepsilon}$,
$$\pi_{t}^{\varepsilon}(\varphi) \equiv \mathbb{E}{\mathbb{Q}}\left[\varphi\left(X{t}^{\varepsilon}, Z_{t}^{\varepsilon}\right) \mid \mathscr{Y}{t}^{\varepsilon}\right],$$ a conditional measure, where $\mathscr{Y}{t}^{\varepsilon} \equiv \sigma\left(\left{Y_{s}^{\varepsilon} \mid s \in[0, t]\right}\right)$ is the $\sigma$-algebra generated by the observation process, $\varphi(x, z)$ is an integrable test function of interest, and the dynamics of $\left(X_{t}^{\varepsilon}, Z_{t}^{\varepsilon}, Y_{t}^{\varepsilon}\right)$ is given by (1). In the case where for each fixed $x, Z_{t}^{\varepsilon, x}$ is ergodic and converges rapidly to it’s stationary distribution; that is,
$$d Z_{t}^{\varepsilon, x}=\frac{1}{\varepsilon} f\left(x, Z_{t}^{\varepsilon, x}\right) d t+\frac{1}{\sqrt{\varepsilon}} g\left(x, Z_{t}^{\varepsilon, x}\right) d V_{t}$$
is ergodic, then the theory of stochastic averaging [28] tells us that $X_{t}^{\varepsilon} \Rightarrow X_{t}^{0}$ in distribution as $\varepsilon \rightarrow 0$, where $X_{t}^{0}$ satisfies the following averaged stochastic differential equation (SDE),
$$d X_{t}^{0}=\bar{b}\left(X_{t}^{0}\right) d t+\sqrt{\bar{a}\left(X_{t}^{0}\right)} d W_{t}$$
Equation (2) is also known as the effective dynamics. With the stationary distribution of $Z_{t}^{\varepsilon, x}$ denoted as $\mu_{\infty}(z ; x)$, the averaged coefficients $\bar{b}$ and $\bar{a}$ are defined as,
\begin{aligned} &\bar{b}(x) \equiv \int b(x, z) \mu_{\infty}(d z ; x) \ &\bar{a}(x) \equiv \int \sigma \sigma^{*}(x, z) \mu_{\infty}(d z ; x) \end{aligned}
Then $\sqrt{\bar{a}(x)}$ is the factor of the modified Cholesky decomposition of $\bar{a}(x)$.

## 英国补考|随机控制代写Stochastic Control代考|Multiclass G/M/n+ Mueues with heavy-tailed arrivals

$$\lambda_{i}^{n} / n \rightarrow \lambda_{i}>0, \quad \text { and } \quad \ell_{i}^{n}:=n^{-1 / \alpha}\left(\lambda_{i}^{n}-n \lambda_{i}\right) \rightarrow \ell_{i} \in \mathbb{R}$$

$$\widehat{A}^{n} \Rightarrow \widehat{A}=(\widehat{A} 1, \ldots, \widehat{A} m)^{\prime} \quad \text { in }\left(D_{m}, M_{1}\right), \text { as } n \rightarrow \infty$$

## 英国补考|随机控制代写Stochastic Control代考|Homogenized Correlated Nonlinear Filtering

$$\pi_{t}^{\varepsilon}(\varphi) \equiv \mathbb{E} \mathbb{Q}\left[\varphi\left(X t^{\varepsilon}, Z_{t}^{\varepsilon}\right) \mid \mathscr{Y} t^{\varepsilon}\right],$$

$$d Z_{t}^{\varepsilon, x}=\frac{1}{\varepsilon} f\left(x, Z_{t}^{\varepsilon, x}\right) d t+\frac{1}{\sqrt{\varepsilon}} g\left(x, Z_{t}^{\varepsilon, x}\right) d V_{t}$$

$$d X_{t}^{0}=\bar{b}\left(X_{t}^{0}\right) d t+\sqrt{\bar{a}\left(X_{t}^{0}\right)} d W_{t}$$

$$\bar{b}(x) \equiv \int b(x, z) \mu_{\infty}(d z ; x) \quad \bar{a}(x) \equiv \int \sigma \sigma^{*}(x, z) \mu_{\infty}(d z ; x)$$

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