﻿ 随机环境中的M/M/n排队系统 M/M/n Queuing System in Random Environment

Vol. 10  No. 02 ( 2021 ), Article ID: 40654 , 12 pages
10.12677/AAM.2021.102055

M/M/n Queuing System in Random Environment

Chaochao Shan, Xiaxia Chi

School of Science, Hangzhou Normal University, Hangzhou Zhejiang

Received: Jan. 23rd, 2021; accepted: Feb. 17th, 2021; published: Feb. 25th, 2021

ABSTRACT

In this paper, we give the definition of M/M/n queuing system in random environment, the equilibrium condition of M/M/n queuing system in random environment $P\left\{{\theta }^{\ast }\in {\Theta }^{\ast }|\rho \left({\theta }^{\ast }\right)<1\right\}=1$, the expressions of $L\left({\theta }^{\ast }\right)$,${L}_{q}\left({\theta }^{\ast }\right)$,$K\left({\theta }^{\ast }\right)$ and their relations, finally the random Kolmogorov forward and backward equations of $p\left({\theta }^{\ast },t;i,j\right)$ of ${X}^{\ast }$ are given.

Keywords:Random Environment, M/M/n Queuing System, Random Kolmogorov Forward and Backward Equations

1. 引言

2. 符号及定义

$X=N$ 为一状态空间， $\left(\Theta ,B\right)$ 是任一抽象可测空间， ${X}^{*}=\left\{X\left(t\right),t\in {R}_{+}\right\}$${\xi }^{*}=\left\{{\xi }_{t},t\in {R}_{+}\right\}$ 是概率空间 $\left(\Omega ,F,P\right)$ 上分别取值于X和 $\Theta$ 的两个随机过程， ${\Theta }^{\text{*}}$ 是从 ${R}_{+}$$\Theta$ 上的函数集，对每个 ${\theta }^{\text{*}}\in {\Theta }^{\text{*}}$$s\ge \text{0},t>0$${\theta }^{\text{*}}\left[s,s+t\right)$${\theta }^{\text{*}}$$\left[s,s+t\right)$ 上的局限， ${\Theta }^{\text{*}}\left[s,s+t\right)=\left\{{\theta }^{*}\left[s,s+t\right):{\theta }^{*}\in {\Theta }^{\text{*}}\right\}$，p为 ${\Theta }^{\text{*}}$ 上的时齐的随机转移函数，参见 [6]，即 $p\left({\theta }^{*}\left[s,s+t\right);i,j\right)$ 不依赖于s (对每个 ${\theta }^{\text{*}}\in {\Theta }^{\text{*}}$$t>0$$i,j\in X$ )。

1) 对 $\forall {\theta }^{\text{*}}\in {\Theta }^{\text{*}}$，记 ${N}^{*}=\left\{{N}^{*}\left(t\right),t\ge 0\right\}$ 是顾客进入服务系统的过程，且 ${N}^{*}$ 服从强度参数为 $\lambda \left({\theta }^{*}\right)$ 的随机环境中的泊松过程(参见 [9] )，其中 ${N}^{*}\left(t\right)$ 表示在 ${\xi }^{\text{*}}={\theta }^{\text{*}}$ 下，时刻t进入系统的人数。

2) 对任意固定的 ${\theta }^{\text{*}}\in {\Theta }^{\text{*}}$，若记 ${J}^{*}=\left\{{J}_{k}^{*},k\ge 0\right\}$ 为顾客相继到达时间间隔序列，那么随机变量序列 $\left\{{J}_{k}^{*},k\ge 0\right\}$ 为独立同分布随机变量序列，且 ${J}_{1}^{*}~\Gamma \left(1,\lambda \left({\theta }^{*}\right)\right)$

3) 对 $\forall {\theta }^{*}\in {\Theta }^{*}$，记 ${B}^{*}=\left\{{B}_{k}^{*},k\ge 0\right\}$ 为顾客的服务时间序列， ${B}^{*}=\left\{{B}_{k}^{*},k\ge 0\right\}$ 为独立同分布随机变量序列， ${B}_{1}^{*}~\Gamma \left(1,\mu \left({\theta }^{*}\right)\right)$，且满足 ${B}^{*}$${J}^{*}$ 相互独立。

4) 该系统中有 $n\left(n\ge 1\right)$ 个服务台。

$p\left(\theta ;0,x\right)=p\left(\theta ;1,x\right)={k}_{x}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\theta \in \Theta ,x\ge 1\right);$

$p\left(\theta ;0,0\right)=p\left(\theta ;1,0\right)=1-\underset{x=1}{\overset{\infty }{\sum }}{k}_{x}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\theta \in \Theta \right);$

$p\left(\theta ;n,x\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\theta \in \Theta ,n\ge 2,x

$p\left(\theta ;n,x\right)=p\left(\theta ;,1,x-n+1\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\theta \in \Theta ,n\ge 2,x\ge n-1\right),$

$P\left\{{B}^{*}>t+\Delta t|{B}^{*}>t,{\xi }^{*}\left[t,t+\Delta t\right)={\theta }^{*}\left[t,t+\Delta t\right)\right\}={\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}=1-\mu \left({\theta }^{*}\right)\Delta t+o\left( \Delta t \right)$

$P\left\{{B}^{*}>t+\Delta t|{B}^{*}>t,{\xi }^{*}\left[t,t+\Delta t\right)={\theta }^{*}\left[t,t+\Delta t\right)\right\}={\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}=1-\mu \left({\theta }^{*}\right)\Delta t+o\left( \Delta t \right)$

$\begin{array}{l}{p}_{i,i+1}\left(\Delta t,{\theta }^{*}\right)\equiv P\left\{{X}^{*}\left(t+\Delta t\right)=i+1|{X}^{*}\left(t\right)=i,{\xi }^{*}\left[t,t+\Delta t\right)={\theta }^{*}\left[t,t+\Delta t\right)\right\}\\ =\underset{k=0}{\overset{\mathrm{min}\left(i,n\right)}{\sum }}P\left\{t<{X}_{k}^{*}

$\begin{array}{l}{p}_{i,i-1}\left(\Delta t,{\theta }^{*}\right)\equiv P\left\{{X}^{*}\left(t+\Delta t\right)=i-1|{X}^{*}\left(t\right)=i,{\xi }^{*}\left[t,t+\Delta t\right)={\theta }^{*}\left[t,t+\Delta t\right)\right\}\\ =\underset{k=0}{\overset{\mathrm{min}\left(i,n\right)}{\sum }}P\left\{t<{X}_{k}^{*}

$\begin{array}{l}={C}_{\mathrm{min}\left(i,n\right)}^{1}\left(1-{\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}\right){\left({\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}\right)}^{\mathrm{min}\left(i,n\right)-1}{\text{e}}^{-\lambda \left({\theta }^{*}\right)\Delta t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{C}_{\mathrm{min}\left(i,n\right)}^{2}{\left(1-{\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}\right)}^{2}{\left({\text{e}}^{-\mu \left({\theta }^{*}\right)\Delta t}\right)}^{\mathrm{min}\left(i,n\right)-2}\lambda \left({\theta }^{*}\right)\Delta t{\text{e}}^{-\lambda \left({\theta }^{*}\right)\Delta t}+o\left(\Delta t\right)\\ =\mathrm{min}\left(i,n\right)\mu \left({\theta }^{*}\right)\Delta t+o\left(\Delta t\right)\\ =\left\{\begin{array}{l}i\mu \left({\theta }^{*}\right)\Delta t+o\left(\Delta t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n-1,\\ n\mu \left({\theta }^{*}\right)\Delta t+o\left(\Delta t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=n,n+1,n+2,\cdots .\end{array}\end{array}$

$\begin{array}{c}{p}_{i,j}\left(\Delta t,{\theta }^{*}\right)\equiv P\left\{{X}^{*}\left(t+\Delta t\right)=j|{X}^{*}\left(t\right)=i,{\xi }^{*}\left[t,t+\Delta t\right)={\theta }^{*}\left[t,t+\Delta t\right)\right\}\\ =o\left( \Delta t \right)\end{array}$

$\begin{array}{c}{p}_{i,i}\left(\Delta t,{\theta }^{*}\right)\equiv P\left\{{X}^{*}\left(t+\Delta t\right)=j|{X}^{*}\left(t\right)=i,{\xi }^{\text{*}}\left[t,t+\Delta t\right)={\theta }^{\text{*}}\left[t,t+\Delta t\right)\right\}\\ =1-\lambda \left({\theta }^{*}\right)\Delta t-{\mu }_{i}\left({\theta }^{*}\right)\Delta t+o\left( \Delta t \right)\end{array}$

${\mu }_{i}\left({\theta }^{*}\right)=\left\{\begin{array}{l}i\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n-1,\\ n\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=n,n+\text{1},n+2,\cdots .\end{array}$ (1)

${\lambda }_{i}\left({\theta }^{*}\right)=\lambda \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i,=0,1,2,\cdots .$

$\underset{k=1}{\overset{\infty }{\sum }}\frac{{\lambda }_{0}\left({\theta }^{*}\right){\lambda }_{1}\left({\theta }^{*}\right)\cdots {\lambda }_{k-1}\left({\theta }^{*}\right)}{{\mu }_{1}\left({\theta }^{*}\right){\mu }_{2}\left({\theta }^{*}\right)\cdots {\mu }_{k}\left({\theta }^{*}\right)}<\infty ,$

$\underset{k=1}{\overset{n-1}{\sum }}\frac{1}{k!}{\left(\frac{\lambda \left({\theta }^{\ast }\right)}{\mu \left({\theta }^{\ast }\right)}\right)}^{k}+\underset{k=n}{\overset{\infty }{\sum }}\frac{{n}^{n}}{n!}{\left(\frac{\lambda \left({\theta }^{\ast }\right)}{n\mu \left({\theta }^{\ast }\right)}\right)}^{k}<\infty ,$

$\rho \left({\theta }^{\ast }\right)\equiv \frac{\lambda \left({\theta }^{\ast }\right)}{n\mu \left({\theta }^{\ast }\right)},$

$P\left\{{\theta }^{\ast }\in {\Theta }^{\ast }|\rho \left({\theta }^{\ast }\right)<1\right\}=1$

${\pi }_{k}^{*}=\left\{\begin{array}{l}\frac{{\left(n\rho \left({\theta }^{*}\right)\right)}^{k}{\pi }_{0}^{*}}{k!},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,\cdots ,n-1\\ \frac{{n}^{n}{\rho }^{k}\left({\theta }^{*}\right){\pi }_{0}^{*}}{n!},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=n,n+1,\cdots \end{array}$ (2)

${\pi }_{\text{0}}^{\text{*}}={\left[\underset{k=0}{\overset{n-1}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{k!}+\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}}{n!\left(1-\rho \left({\theta }^{\ast }\right)\right)}\right]}^{-1}$ (3)

3. 平衡状态下的一些结果

$K\left({\theta }^{\ast }\right)=\frac{\lambda \left({\theta }^{\ast }\right)}{\mu \left({\theta }^{\ast }\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}E\left({L}_{q}\left({\theta }^{\ast }\right)\right)=\frac{\rho \left({\theta }^{\ast }\right){\pi }_{n}^{*}}{{\left(1-\rho \left({\theta }^{\ast }\right)\right)}^{2}},$

$E\left(L\left({\theta }^{\ast }\right)\right)=E\left({L}_{q}\left({\theta }^{\ast }\right)\right)+K\left({\theta }^{\ast }\right)=\frac{\rho \left({\theta }^{\ast }\right){\pi }_{n}^{*}}{{\left(1-\rho \left({\theta }^{\ast }\right)\right)}^{2}}+\frac{\lambda \left({\theta }^{\ast }\right)}{\mu \left({\theta }^{\ast }\right)}.$

$\left\{\begin{array}{l}P\left\{{L}_{q}\left({\theta }^{\ast }\right)=k\right\}={\pi }_{n+k}^{*}=\frac{{n}^{n}{\rho }^{n+k}\left({\theta }^{\ast }\right)}{n!}{\pi }_{0}^{*},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,3,\cdots \\ P\left\{{L}_{q}\left({\theta }^{\ast }\right)=0\right\}=\underset{j=0}{\overset{n}{\sum }}{\pi }_{j}^{*}=\underset{j=0}{\overset{n}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{j}}{j!}{\pi }_{0}^{*},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=0\end{array}$

$E\left({L}_{q}\left({\theta }^{*}\right)\right)=E\left[{L}_{q}\left({\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]=\underset{k=0}{\overset{\infty }{\sum }}k\frac{{\left(n\rho \left({\theta }^{*}\right)\right)}^{n}{\rho }^{k}\left({\theta }^{*}\right)}{n!}{\pi }_{0}^{*}=\frac{{\left(n\rho \left({\theta }^{*}\right)\right)}^{n}\rho \left({\theta }^{*}\right)}{n!{\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}{\pi }_{0}^{*}=\frac{\rho \left({\theta }^{*}\right){\pi }_{n}^{*}}{{\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}$

$\begin{array}{c}E\left(L\left({\theta }^{\ast }\right)\right)=E\left[L\left({\theta }^{\ast }\right)|{\xi }^{*}={\theta }^{*}\right]=\underset{k=0}{\overset{\infty }{\sum }}k{\pi }_{k}^{*}=\left[\underset{k=1}{\overset{n-1}{\sum }}\frac{k{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{k!}+\underset{k=n}{\overset{\infty }{\sum }}\frac{k{n}^{n}{\rho }^{k}\left({\theta }^{\ast }\right)}{n!}\right]{\pi }_{0}^{*}\\ =\left\{\underset{k=1}{\overset{n-1}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{\left(k-1\right)!}+\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}\left[\rho \left({\theta }^{\ast }\right)+n\left(1-\rho \left({\theta }^{\ast }\right)\right)\right]}{n!{\left(1-\rho \left({\theta }^{\ast }\right)\right)}^{2}}\right\}{\pi }_{0}^{*}\end{array}$

$\begin{array}{c}K\left({\theta }^{\ast }\right)=E\left(L\left({\theta }^{\ast }\right)\right)-E\left({L}_{q}\left({\theta }^{\ast }\right)\right)\\ =\left[\underset{k=1}{\overset{n-1}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{\left(k-1\right)!}+\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}\left[\rho \left({\theta }^{\ast }\right)+n\left(1-\rho \left({\theta }^{\ast }\right)\right)\right]}{n!{\left(1-\rho \left({\theta }^{\ast }\right)\right)}^{2}}\right]{\pi }_{0}^{*}-\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}\rho \left({\theta }^{\ast }\right){\pi }_{0}^{*}}{n!{\left(1-\rho \left({\theta }^{\ast }\right)\right)}^{2}}\\ =\left[\underset{k=1}{\overset{n-1}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{\left(k-1\right)!}+\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}}{\left(n-1\right)!\left(1-\rho \left({\theta }^{\ast }\right)\right)}\right]{\pi }_{0}^{*}\\ =n\rho \left({\theta }^{*}\right){\pi }_{0}^{*}\underset{k=0}{\overset{n-1}{\sum }}\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{k}}{k!}+\frac{n\rho \left({\theta }^{*}\right){\left(n\rho \left({\theta }^{*}\right)\right)}^{n}}{n!\left(1-\rho \left({\theta }^{\ast }\right)\right)}{\pi }_{0}^{*}\\ =n\rho \left({\theta }^{*}\right)=\frac{\lambda \left({\theta }^{*}\right)}{\mu \left( \theta * \right)}\end{array}$

$E\left(L\left({\theta }^{*}\right)\right)=E\left({L}_{q}\left({\theta }^{*}\right)\right)+K\left({\theta }^{*}\right)=\frac{\rho \left({\theta }^{*}\right){\pi }_{n}^{*}}{{\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}+\frac{\lambda \left({\theta }^{*}\right)}{\mu \left( \theta * \right)}$

${F}_{{W}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0,\\ 1-\frac{{\pi }_{n}^{*}}{1-\rho \left({\theta }^{*}\right)}{\text{e}}^{-n\mu \left({\theta }^{*}\right)\left(1-\rho \left({\theta }^{*}\right)\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0.\end{array}$

${p}^{*}=P\left\{L\left({\theta }^{*}\right)\ge n|{\xi }^{*}={\theta }^{*}\right\}=\underset{k=n}{\overset{\infty }{\sum }}{\pi }_{k}^{*}=\underset{k=n}{\overset{\infty }{\sum }}\frac{{n}^{n}{\rho }^{k}\left({\theta }^{*}\right)}{n!}{\pi }_{0}^{*}=\frac{{\left(n\rho \left({\theta }^{*}\right)\right)}^{n}}{n!\left(1-\rho \left({\theta }^{*}\right)\right)}{\pi }_{0}^{*}=\frac{{\pi }_{n}^{*}}{1-\rho \left( \theta * \right)}$

$1-{p}^{*}=1-\frac{{\left(n\rho \left({\theta }^{\ast }\right)\right)}^{n}}{n!\left(1-\rho \left({\theta }^{\ast }\right)\right)}{\pi }_{0}^{*}=\frac{1-\rho \left({\theta }^{\ast }\right)-{\pi }_{n}^{*}}{1-\rho \left( \theta \ast \right)}$

$P\left\{{W}^{*}=0\right\}=P\left\{L\left({\theta }^{*}\right)

$\forall t>0$

${F}_{{W}^{*}}\left(t,{\theta }^{*}\right)=P\left\{{W}^{*}

$\begin{array}{l}P\left\{0<{W}^{*}

${F}_{{W}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0,\\ 1-\frac{{\pi }_{n}^{*}}{1-\rho \left({\theta }^{\ast }\right)}{\text{e}}^{-n\mu \left({\theta }^{\ast }\right)\left(1-\rho \left({\theta }^{\ast }\right)\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0.\end{array}$

$f\left(x\right)=\left\{\begin{array}{l}\frac{{\alpha }^{n}{x}^{n-1}}{\Gamma \left(n\right)}{\text{e}}^{-\alpha x},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>0\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\le 0\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\alpha >0\right)$

${F}_{{W}^{*}}\left(t,{\theta }^{*}\right)$ 的表达式，我们可以得到 ${W}^{*}$ 的密度函数

${f}_{{W}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<0,\\ \left(1-\frac{{\pi }_{n}^{*}}{1-\rho \left({\theta }^{\ast }\right)}\right)\delta \left(t\right)+n\mu \left({\theta }^{\ast }\right){\pi }_{n}^{*}{\text{e}}^{-n\mu \left({\theta }^{\ast }\right)\left(1-\rho \left({\theta }^{\ast }\right)\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0,\end{array}$

1) 当 $t\ne 0$ 时， $\delta \left(t\right)=0$ ；当 $t=0$ 时， $\delta \left(t\right)=\infty$

2) $\delta \left(-t\right)=\delta \left(t\right)$

3) 对任意连续函数 $\phi \left(t\right)$$\underset{-\infty }{\overset{\infty }{\int }}\phi \left(t\right)\delta \left(t\right)\text{d}t=\phi \left(0\right)$$\delta \left(t\right)$ 可以看成单跳跃函数

$\mu \left(t\right)=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0\end{array}$

$\delta \left(t\right)=\frac{\text{d}\mu \left(t\right)}{\text{d}t}$

${F}_{{W}^{*}}\left(t,{\theta }^{*}\right)$，或 ${f}_{{W}^{*}}\left(t,{\theta }^{*}\right)$ 的表达式可以得到 ${W}^{*}$ 的数学期望：

$\begin{array}{c}E\left[{W}^{*}|{\xi }^{*}={\theta }^{*}\right]=\underset{{0}^{-}}{\overset{\infty }{\int }}t\text{d}{F}_{{W}^{*}}\left(t,{\theta }^{*}\right)\\ =\underset{{0}^{-}}{\overset{\infty }{\int }}t{f}_{{W}^{*}}\left(t,{\theta }^{*}\right)\text{d}t=\underset{{0}^{-}}{\overset{\infty }{\int }}\left(1-\frac{{\pi }_{n}^{*}}{1-\rho \left({\theta }^{*}\right)}\right)\delta \left(t\right)t\text{d}t+\underset{{0}^{-}}{\overset{\infty }{\int }}n\mu \left({\theta }^{*}\right){\pi }_{n}^{*}t{\text{e}}^{-n\mu \left({\theta }^{*}\right)\left(1-\rho \left({\theta }^{*}\right)\right)t}\text{d}t\\ =\frac{{\pi }_{n}^{*}}{n\mu \left({\theta }^{*}\right){\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}=\frac{\rho \left({\theta }^{*}\right){\pi }_{n}^{*}}{\lambda \left({\theta }^{*}\right){\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}=\frac{E\left({L}_{q}\left({\theta }^{*}\right)\right)}{\lambda \left( \theta * \right)}\end{array}$

${F}_{{T}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0,\\ 1-{\text{e}}^{-\mu \left({\theta }^{*}\right)t}-n\mu \left({\theta }^{*}\right)t{\pi }_{n}^{*}{\text{e}}^{-\mu \left({\theta }^{*}\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)=\lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\\ 1-{\text{e}}^{-\mu \left({\theta }^{*}\right)t}-\frac{\mu \left({\theta }^{*}\right){\pi }_{n}^{*}}{\left(1-\rho \left({\theta }^{*}\right)\right)\left(n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)-\mu \left({\theta }^{*}\right)\right)}\\ ×\left[{\text{e}}^{-\mu \left({\theta }^{*}\right)t}-{\text{e}}^{-\left(n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)\right)t}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)\ne \lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0.\end{array}$

${T}^{*}={W}^{*}+{B}^{*}$

$E\left({T}^{*}|{\xi }^{*}={\theta }^{*}\right)=E\left({W}^{*}|{\xi }^{*}={\theta }^{*}\right)+E\left({B}^{*}|{\xi }^{*}={\theta }^{*}\right)=\frac{\rho \left({\theta }^{*}\right){\pi }_{n}^{*}}{\lambda \left({\theta }^{*}\right){\left(1-\rho \left({\theta }^{*}\right)\right)}^{2}}+\frac{1}{\mu \left({\theta }^{*}\right)}=\frac{E\left(L\left({\theta }^{*}\right)\right)}{\lambda \left( \theta * \right)}$

2) 当 $t>0$ 时，我们有

$\begin{array}{l}P\left\{{T}^{*}

$=\left\{\begin{array}{l}1-{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-\frac{\mu \left({\theta }^{*}\right)t{\pi }_{n}^{*}}{1-\rho \left({\theta }^{\ast }\right)}{\text{e}}^{-\left(n\mu \left({\theta }^{\ast }\right)-\lambda \left({\theta }^{\ast }\right)\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{\ast }\right)=\lambda \left({\theta }^{\ast }\right)+\mu \left({\theta }^{\ast }\right)\\ 1-{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-\frac{\mu \left({\theta }^{*}\right)t{\pi }_{n}^{*}}{\left(1-\rho \left({\theta }^{*}\right)\right)\left(n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)-\mu \left({\theta }^{*}\right)\right)}\\ ×\left[{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-{\text{e}}^{-\left(n\mu \left({\theta }^{\ast }\right)-\lambda \left({\theta }^{\ast }\right)\right)t}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)=\lambda \left({\theta }^{*}\right)+\mu \left( \theta * \right)\end{array}$

${F}_{{T}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0,\\ 1-{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-n\mu \left({\theta }^{*}\right)t{\pi }_{n}^{*}{\text{e}}^{-\mu \left({\theta }^{*}\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)=\lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\\ 1-{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-\frac{\mu \left({\theta }^{*}\right){\pi }_{n}^{*}}{\left(1-\rho \left({\theta }^{*}\right)\right)\left(n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)-\mu \left({\theta }^{*}\right)\right)}\\ ×\left[{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}-{\text{e}}^{-\left(n\mu \left({\theta }^{\ast }\right)-\lambda \left({\theta }^{\ast }\right)\right)t}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)\ne \lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0.\end{array}$

${f}_{{T}^{*}}\left(t,{\theta }^{*}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 0,\\ \left(n{\mu }^{2}\left({\theta }^{*}\right)t{\pi }_{n}^{*}+\mu \left({\theta }^{*}\right)-n\mu \left({\theta }^{*}\right){\pi }_{n}^{*}\right){\text{e}}^{-\mu \left({\theta }^{\ast }\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)=\lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\\ \left[\mu \left({\theta }^{*}\right)+\frac{{\mu }^{2}\left({\theta }^{*}\right){\pi }_{n}^{*}}{\left(1-\rho \left({\theta }^{*}\right)\right)\left(n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)-\mu \left({\theta }^{*}\right)\right)}\right]{\text{e}}^{-\mu \left({\theta }^{\ast }\right)t}\\ -\frac{n{\mu }^{2}\left({\theta }^{*}\right){\pi }_{n}^{*}}{n\mu \left({\theta }^{*}\right)-\lambda \left({\theta }^{*}\right)-\mu \left({\theta }^{*}\right)}×{\text{e}}^{-\left(n\mu \left({\theta }^{\ast }\right)-\lambda \left({\theta }^{\ast }\right)\right)t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\mu \left({\theta }^{*}\right)\ne \lambda \left({\theta }^{*}\right)+\mu \left({\theta }^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0.\end{array}$

$E\left({W}^{*}|{\xi }^{*}={\theta }^{*}\right)=E\left({L}_{q}\left({\theta }^{*}\right)\right)/\stackrel{¯}{\lambda \left({\theta }^{*}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}E\left({T}^{*}|{\xi }^{*}={\theta }^{*}\right)=E\left(L\left({\theta }^{*}\right)\right)/\stackrel{¯}{\lambda \left({\theta }^{*}\right)}.$

$\stackrel{¯}{\lambda \left({\theta }^{*}\right)}=\frac{E\left[\alpha \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{t}.$

$\gamma \left(t,{\theta }^{*}\right)$ 表示 $\alpha \left(t,{\theta }^{*}\right)$ 个顾客到时刻t为止在系统中花费的总时间，那么在这段时间内每个顾客的平均逗留时间为：

$\stackrel{¯}{{T}_{t}\left({\theta }^{*}\right)}=\frac{E\left[\gamma \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{E\left[\alpha \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}$

$\stackrel{¯}{{L}_{t}\left({\theta }^{*}\right)}=\frac{E\left[\gamma \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{t}.$

$\stackrel{¯}{{L}_{t}\left({\theta }^{*}\right)}=\frac{E\left[\gamma \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{E\left[\alpha \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}.\frac{E\left[\alpha \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{t}.$ (4)

$\stackrel{¯}{\lambda \left({\theta }^{*}\right)}=\underset{t\to \infty }{\mathrm{lim}}\stackrel{¯}{{\lambda }_{t}\left({\theta }^{*}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}E\left({T}^{*}|{\xi }^{*}={\theta }^{*}\right)=\underset{t\to \infty }{\mathrm{lim}}\stackrel{¯}{{T}_{t}\left({\theta }^{*}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}E\left(L\left({\theta }^{*}\right)\right)=\underset{t\to \infty }{\mathrm{lim}}\stackrel{¯}{{L}_{t}\left({\theta }^{*}\right)}$

$E\left(L\left({\theta }^{*}\right)\right)=\stackrel{¯}{\lambda \left({\theta }^{*}\right)}E\left({T}^{*}|{\xi }^{\text{*}}={\theta }^{\text{*}}\right)$

$E\left({T}^{*}|{\xi }^{\text{*}}={\theta }^{\text{*}}\right)=\frac{E\left(L\left({\theta }^{*}\right)\right)}{\stackrel{¯}{\lambda \left({\theta }^{*}\right)}}$

2) 设 ${\gamma }^{\text{*}}\left(t,{\theta }^{*}\right)$ 表示在 $\left(0,t\right]$ 内进入系统的 $\alpha \left(t,{\theta }^{*}\right)$ 个顾客到时刻t为止等待时间之和。那么每个顾客的平均等待时间为：

$\stackrel{¯}{{W}_{t}\left({\theta }^{*}\right)}=\frac{E\left[{\gamma }^{*}\left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}{E\left[\alpha \left(t,{\theta }^{*}\right)|{\xi }^{*}={\theta }^{*}\right]}.$

$\stackrel{¯}{{L}_{qt}\left({\theta }^{*}\right)}=\frac{E\left[{\gamma }^{\text{*}}\left(t\right)|{\xi }^{*}={\theta }^{*}\right]}{t}=\frac{E\left[{\gamma }^{\text{*}}\left(t\right)|{\xi }^{*}={\theta }^{*}\right]}{E\left[{\alpha }^{\text{*}}\left(t\right)|{\xi }^{*}={\theta }^{*}\right]}\cdot \frac{E\left[{\alpha }^{\text{*}}\left(t\right)|{\xi }^{*}={\theta }^{*}\right]}{t}.$

$E\left[{L}_{q}\left({\theta }^{*}\right)\right]=\stackrel{¯}{\lambda \left({\theta }^{*}\right)}E\left({W}^{*}|{\xi }^{*}={\theta }^{*}\right),$

$E\left({W}^{*}|{\xi }^{*}={\theta }^{*}\right)=\frac{E\left({L}_{q}\left({\theta }^{*}\right)\right)}{\stackrel{¯}{\lambda \left({\theta }^{*}\right)}}$

4. 随机Kolmogorov向前与向后方程

1) 随机Kolmogorov向后方程

$\frac{\partial }{\partial t}p\left({\theta }^{*},t;i,j\right)=\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i+1,j\right)-\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j\right);$

2) 随机Kolmogorov向前方程

$\frac{\partial }{\partial t}p\left({\theta }^{*},t;i,j\right)=\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j-1\right)-\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j\right);$

$p\left({\theta }^{*},t+h;i,j\right)=\underset{k\in X}{\sum }p\left({\theta }^{*},h;i,k\right)p\left({\theta }^{*},t;k,j\right),$

$p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)=\underset{k\ne i}{\sum }p\left({\theta }^{*},h;i,k\right)p\left({\theta }^{*},t;k,j\right)-\left(1-p\left({\theta }^{*},h;i,i\right)\right)p\left({\theta }^{*},t;i,j\right),$

$\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)}{h}=\underset{h\to 0}{\mathrm{lim}}\underset{k\ne i}{\sum }\frac{p\left({\theta }^{*},h;i,k\right)}{h}p\left({\theta }^{*},t;k,j\right)-\underset{h\to 0}{\mathrm{lim}}\frac{1-p\left({\theta }^{*},h;i,i\right)}{h}p\left({\theta }^{*},t;i,j\right),$ (5)

$\underset{h\to 0}{\mathrm{lim}}\underset{k\ne i}{\sum }\frac{p\left({\theta }^{*},h;i,k\right)}{h}p\left({\theta }^{*},t;k,j\right)=\underset{k\ne i}{\sum }\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},h;i,k\right)}{h}p\left({\theta }^{*},t;k,j\right)$

$\begin{array}{l}\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)}{h}\\ =\underset{k\ne i}{\sum }\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},h;i,k\right)}{h}p\left({\theta }^{*},t;k,j\right)-\underset{h\to 0}{\mathrm{lim}}\frac{1-p\left({\theta }^{*},h;i,i\right)}{h}p\left({\theta }^{*},t;i,j\right),\end{array}$ (6)

$\begin{array}{l}\underset{h\to 0}{\mathrm{lim}}\frac{1-p\left({\theta }^{*},h;k,k\right)}{h}=\lambda \left({\theta }^{*}\right),\\ \underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},h;k,k+1\right)}{h}=\lambda \left({\theta }^{*}\right),\end{array}$ (7)

$\frac{\partial }{\partial t}p\left({\theta }^{*},t;i,j\right)=\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i+1,j\right)-\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j\right),$

2) 类似地，根据定义1的(3)有

$p\left({\theta }^{*},t+h;i,j\right)=\underset{k\in X}{\sum }p\left({\theta }^{*},t;i,k\right)p\left({\theta }^{*},h;k,j\right),$

$p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)=\underset{k\ne j}{\sum }p\left({\theta }^{*},t;i,k\right)p\left({\theta }^{*},h;k,j\right)-\left(1-p\left({\theta }^{*},h;j,j\right)\right)p\left({\theta }^{*},t;i,j\right),$

$\begin{array}{l}\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\underset{k\ne j}{\sum }p\left({\theta }^{*},t;i,k\right)\frac{p\left({\theta }^{*},h;k,j\right)}{h}-\underset{h\to 0}{\mathrm{lim}}\frac{1-p\left({\theta }^{*},h;j,j\right)}{h}p\left({\theta }^{*},t;i,j\right),\end{array}$

$\begin{array}{l}\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},t+h;i,j\right)-p\left({\theta }^{*},t;i,j\right)}{h}\\ =\underset{k\ne j}{\sum }p\left({\theta }^{*},t;i,k\right)\underset{h\to 0}{\mathrm{lim}}\frac{p\left({\theta }^{*},h;k,j\right)}{h}-\underset{h\to 0}{\mathrm{lim}}\frac{1-p\left({\theta }^{*},h;j,j\right)}{h}p\left({\theta }^{*},t;i,j\right),\end{array}$ (8)

$\frac{\partial }{\partial t}p\left({\theta }^{*},t;i,j\right)=\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j-1\right)-\lambda \left({\theta }^{*}\right)p\left({\theta }^{*},t;i,j\right).$

M/M/n Queuing System in Random Environment[J]. 应用数学进展, 2021, 10(02): 506-517. https://doi.org/10.12677/AAM.2021.102055

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