基于启动时间和完全故障的双阶段休假排队系统的流体模型性能分析

基于启动时间和完全故障的双阶段休假排队系统的流体模型性能分析

徐秀丽^,¹, 张怡通^,²^,^*, 王勋^,¹, 刘洺辛^,³

¹燕山大学理学院河北秦皇岛 066004

²燕山大学经济管理学院河北秦皇岛 066004

³广东海洋大学电子与信息工程学院广东湛江 524088

Performance Analysis of Fluid Model Based on Two-Stage Vacation Queue with Set-Up Time and Complete Failure

Xu Xiuli^,¹, Zhang Yitong^,²^,^*, Wang Xun^,¹, Liu Mingxin^,³

¹School of Sciences, Yanshan University, Hebei Qinhuangdao 066004

²School of Economics and Management, Yanshan University, Hebei Qinhuangdao 066004

³School of Electronics and Information Engineering, Guangdong Ocean University, Guangdong Zhanjiang 524088

通讯作者: *张怡通, E-mail: YitongZz@126.com

收稿日期: 2023-07-31 修回日期: 2024-01-25

基金资助:

国家自然科学基金(62171143)

Received: 2023-07-31 Revised: 2024-01-25

Fund supported:

NSFC(62171143)

作者简介 About authors

徐秀丽,E-mail:xxl-ysu@163.com;

王勋,E-mail:2635388468@qq.com;

刘洺辛,E-mail:mingxinliu001@sina.com.cn

摘要

基于风力发电路灯的能量转化机制, 该文构建并分析了具有启动时间和完全故障策略的 M/M/1 排队系统驱动的流体模型. 首先, 对驱动系统进行模型描述, 运用矩阵几何解方法得到驱动系统的稳态概率分布. 其次, 引入流体模型的净输入率结构, 并利用概率分析方法得到流体库存水平在稳态条件下的微分差分方程组, 进而运用 Laplace-Stieltjes transform(LST) 方法得到稳态条件下库存量的均值及空库概率. 根据性能指标构建系统的费用函数, 在数值分析中给出系统参数对性能指标和费用的影响.

关键词： 流体模型; 双阶段休假; 启动时间; 完全故障; 库存量

Abstract

Based on the energy conversion mechanism of wind power street lamps, this paper constructed and analyzed a fluid model driven by the M/M/1 queueing system with set-up time and complete failure. Firstly, the driving system is described and the stationary probability distribution of the driving system is obtained by using the matrix-geometry method. Secondly, the differential and difference equations of the fluid level in steady-state conditions are obtained based on the net input rate structure of the fluid model and using the probability analysis method. Then, the expected buffer content and the probability of the empty buffer under steady-state conditions are obtained by using the Laplace-Stieltjes transform(LST) method. The cost function of the system is constructed according to the performance index. Finally, the influence of parameters changing on the performance indicators and cost function are illustrated in numerical analysis.

Keywords： Fluid model; Two-stage vacation; Set-up time; Complete failure; Buffer content

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本文引用格式

徐秀丽, 张怡通, 王勋, 刘洺辛. 基于启动时间和完全故障的双阶段休假排队系统的流体模型性能分析[J]. 数学物理学报, 2024, 44(4): 1092-1009

Xu Xiuli, Zhang Yitong, Wang Xun, Liu Mingxin. Performance Analysis of Fluid Model Based on Two-Stage Vacation Queue with Set-Up Time and Complete Failure[J]. Acta Mathematica Scientia, 2024, 44(4): 1092-1009

1 引言

流体模型研究的是一个输入输出系统, 流体的流入和流出情况受外部驱动系统控制. 随着高新科技的迅速发展, 传统离散排队模型表现出很大的不便性, 这时关于流体模型的研究逐渐兴起, 并已在信息网络、数字化智慧工厂等方面有着广泛应用.

Virtamo 和 Norros^[1]研究了由 M/M/1 排队模型驱动的流体模型, 运用谱分析方法和第二类切比雪夫多项式得到无限状态环境中库存量的平稳分布. Anda 和 Resing^[2]使用嵌入点研究了由 M/M/1 队列驱动的流体模型, 并运用第一类修正 Bessel 函数给出库存量稳态分布的表达式. Kulkarni^[3]分析了具有单个缓冲器的流体模型, 但是由于计算量大, 不能得到较稳定的数值解. 除谱分析方法外, 出现了更加多样的研究方法. Sericola 和 Tuffin^[4]利用递归法研究并给出了一般分布的流体模型库存量的平稳分布函数. Lierde 等^[5] 使用迭代方法, 不仅解决了谱分析方法不能得到较稳定数值解的问题, 还可以用来研究更多更复杂的流体排队模型. Ammar 和 Sherif^[6]分析了由 M/M/1 灾难队列驱动的流体模型, 并利用生成函数技术和一阶修正 Bessel 函数得到缓冲器库存量平稳分布函数的显性表达式. Wang 和 Mao^[7] 将启动-关闭期引入到流体排队模型中, 运用矩阵几何解方法给出了稳态条件下库存量分布函数的 LST 及均值的表达式. 徐秀丽等^[8]将可选服务策略引入到流体模型的外部驱动系统中, 并利用 LT 得到了稳态时库存量分布函数的 LST 和平均库存量. 考虑服务台出现故障的情形, 孙红霜^[9]研究了具有工作故障策略 M/M/1 排队系统驱动的流体模型, 使用矩阵几何解、 LT 和 LST 得到了库存量平稳分布函数和均值的显性表达式.

现实生活中, 系统为减少运行成本或降低服务台发生故障的概率, 没有顾客时系统中的服务台会停止运行或以低服务速率运行, 此时将服务台的行为称为休假. 目前基于各类休假策略的流体模型的研究已有大量研究成果. Sherif 和 Ammar^[10]使用母函数法分析了多重休假 M/M/1 排队系统驱动的流体模型, 得到了缓冲器库存量平稳分布函数的显式解. Mao 等^[11]研究了基于多重休假和有限状态空间的排队系统驱动的流体模型, 得到了库存量均值的表达式. Vijayashree 和 Anjuka^[12]分析了具有多重工作休假策略的 M/M/1/ $N$ 排队系统驱动的流体模型. 随后, 徐秀丽等^[13,14]研究了基于可选服务策略和负顾客的 M/M/1 休假排队系统驱动的流体模型, 得到了缓冲器库存量的空库概率和均值的表达式. 刘煜飞和叶晴晴^[15]使用矩阵几何解方法得到了由双阶段休假排队系统驱动的流体模型中库存量的均值. 李子坤等^[16]将多服务台策略引入到工作休假排队系统驱动的流体模型, 使用矩阵几何解方法得到库存量平稳分布函数的 LST 表达式、空库概率和均值. 近年来, 王勋和徐秀丽^[17]基于店铺的外卖经营模式, 构建并分析了具有启动时间、可选服务和两种混合休假策略的 M/M/1/ $N$ 排队系统驱动的流体模型.

风力发电路灯作为将风能转化为电能的照明设备, 由于具有节能环保、低成本、方便安装维修等优势, 目前在海边城市及风力资源丰富的地区已有着广泛的应用. 考虑风力发电路灯的能量转化过程, 转化成的电流看作流体, 风力发电机看作流体缓冲器, 风能通过风力发电机转化成电流, 并贮存于蓄电池中. 风力变化影响着转化速率的高低. 当风力较大时, 转化速率较快, 类似于驱动系统处于忙期. 当风力较小时, 转化速率较慢, 类似于驱动系统处于工作休假期. 当风力大小无法带动风轮转动时, 此时无法进行能量转化, 类似于驱动系统处于休假期. 在此基础上, 本文研究了具有启动时间和完全故障的双阶段休假 M/M/1 排队系统驱动的流体模型, 通过建立稳态下系统的平衡方程, 得到了库存量平稳分布函数的 LST 及均值等性能指标, 并将其应用到风力发电路灯的能量转化过程中, 在数值分析中验证了参数变化对性能指标的影响. 根据实际情况构建了费用函数, 最终通过数值分析得到了系统参数对成本费用的影响.

2 驱动系统描述

考虑一个具有启动时间、多重休假、单重工作休假及工作中完全故障等策略的 M/M/1 排队系统, 模型假设如下

1) 顾客的到达率为 $\lambda$ , 且到达过程是一个 Possion 过程.

2) 正规忙期内, 服务台的服务速率为 ${\mu _b}$ . 当系统中没有顾客时, 服务台进入单重工作休假期. 服务台在工作休假期的服务速率为 ${\mu _w}$ $\left( {{\mu _w} < {\mu _b}} \right)$ , 工作休假时间服从参数为 ${\theta _w}$ 的指数分布. 当一次工作休假结束时, 若系统中有顾客, 则立即进入正规忙期以 ${\mu _b}$ 为顾客进行服务. 若系统中没有顾客, 则进入经典休假. 休假期内服务台不进行任何服务, 休假时间服从参数为 ${\theta _v}$ 的指数分布. 当一次休假期结束, 系统中有顾客时, 则服务台开始启动, 启动时间服从参数为 $\varepsilon$ 的指数分布. 服务台在启动结束后进入正规忙期. 否则, 服务台再进行一次新的休假, 直至某次休假期结束时系统中顾客数非空.

3) 服务台在工作休假期内会发生完全故障, 服务台发生故障的概率服从参数为 $\alpha$ 的指数分布, 维修时间服从参数为 $\beta$ 的指数分布. 顾客在故障发生后不会离开系统, 而是等待维修完成后继续接受服务, 那么为保证受到故障影响的顾客的服务完成效率, 在维修完成后, 服务台将立即进入正规忙期以高服务速率为顾客提供服务. 由于服务台是维修成功后才进入正规忙期, 因此假设服务台在忙期发生的概率忽略不计.

4) 假设顾客的到达间隔时间、服务台的启动时间、服务时间、维修时间、故障时间、休假时间和工作休假时间均相互独立. 假设该排队系统的服务规则为先到先服务 (FIFO).

设 $L\left( t \right)$ 为时刻 $t$ 系统中的顾客数, $J\left( t \right) = 0,1,2,3,4$ 代表服务台在时刻 $t$ 分别处于工作休假期、休假期、启动期、工作故障期、正规忙期, 则 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 是一个状态空间为 $\Omega = \left\{ {\left( {0,0} \right), \left( {0,1} \right)} \right\} \cup \left\{ {\left( {i,j} \right),i \ge 1,j = 0,1,2,3,4} \right\}$ 的马尔可夫过程. 将系统的所有状态按照字典排序, 则马尔可夫过程的无穷小生成元可记为

${Q} = \left( {\begin{array}{*{20}{c}} {{A_{00}}}&{{C_{01}}}&0&0&0&0\\ {{B_{10}}}&A&C&0&0&0\\ 0&B&A&C&0&0\\ 0&0&B&A&C&0\\ 0&0&0& \ddots & \ddots & \ddots \end{array}} \right),$

其中

${ {A}_{00}} = \left( {\begin{array}{*{20}{c}} { - \left( {\lambda + {\theta _w}} \right)}&{{\theta _w}}\\ 0&{ - \lambda } \end{array}} \right), { {C}_{01}} = \left( {\begin{array}{*{20}{c}} \lambda &0&0&0&0\\ 0&\lambda &0&0&0 \end{array}} \right), { {B}_{10}} = \left( {\begin{array}{*{20}{c}} {{\mu _w}}&0\\ 0&0\\ 0&0\\ 0&0\\ {{\mu _b}}&0 \end{array}} \right),$

${A} = \left( {\begin{array}{*{20}{c}} { - \left( {\lambda + {\theta _w} + {\mu _w} + \alpha } \right)}&{{\theta _w}}&0&\alpha &0\\ 0&{ - \left( {\lambda + {\theta _v}} \right)}&{{\theta _v}}&0&0\\ 0&0&{ - \left( {\lambda + \varepsilon } \right)}&0&\varepsilon \\ 0&0&0&{ - \left( {\lambda + \beta } \right)}&\beta \\ 0&0&0&0&{ - \left( {\lambda + {\mu _b}} \right)} \end{array}} \right),$

${B} = \left( {\begin{array}{*{20}{c}} {{\mu _w}}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&{{\mu _b}} \end{array}} \right), {C} = \left( {\begin{array}{*{20}{c}} \lambda &0&0&0&0\\ 0&\lambda &0&0&0\\ 0&0&\lambda &0&0\\ 0&0&0&\lambda &0\\ 0&0&0&0&\lambda \end{array}} \right).$

现在证明拟生灭过程 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 存在稳态分布, 给出下述结论.

引理 2.1 若系统负载 $\rho = \lambda /{\mu _b} < 1$ , 矩阵方程 ${ {R}^2} {B} + {RA} + {C} = {0}$ 的最小非负解 ${R}$ 为

${R} = \left( {\begin{array}{*{20}{c}} r& {\frac{{{\theta _w}r}}{{\lambda + {\theta _v}}}}& {\frac{{{\theta _v}{\theta _w}r}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}}} & {\frac{{\alpha r}}{{\lambda + \beta }}}& {\frac{{\left( {\alpha + {\theta _w}} \right)r}}{{{\mu _b}\left( {1 - r} \right)}}}\\[3mm] 0& {\frac{\lambda }{{\lambda + {\theta _v}}}}& {\frac{{\lambda {\theta _v}}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}}}&0&\rho \\[3mm] 0&0& {\frac{\lambda }{{\lambda + \varepsilon }}}&0&\rho \\[3mm] 0&0&0& {\frac{\lambda }{{\lambda + \beta }}}&\rho \\ 0&0&0&0&\rho \end{array}} \right),$

其中

$r = \frac{{\lambda + {\theta _w} + {\mu _w} + \alpha - \sqrt {{{\left( {\lambda + {\theta _w} + {\mu _w} + \alpha } \right)}^2} - 4\lambda {\mu _w}} }}{{2{\mu _w}}}.$

定理 2.1 拟生灭过程 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 的稳态分布存在当且仅当 $\rho = \lambda /{\mu _b} < 1$ .

证拟生灭过程 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 只有在矩阵 ${R}$ 的谱半径 $SP\left( {R} \right) < 1$ , 并且方程组 $\left( {{ {\pi } _0},{ {\pi } _1}} \right) {B}\left[ R \right] = {0}$ 有正解时稳态分布才存在, 其中

$\begin{equation} {B}\left[ R \right] = \left( {\begin{array}{*{20}{c}} { - \left( {\lambda + {\theta _w}} \right)}&{{\theta _w}}&\lambda &0&0&0&0\\ 0& { - \lambda } &0&\lambda &0&0&0\\ {{\mu _w}}&0&\xi &{{\theta _w}}&0&\alpha & {\frac{{\left( {\alpha + {\theta _w}} \right)r}}{{1 - r}}}\\ 0&0&0& { - \left( {\lambda + {\theta _v}} \right)} &{{\theta _v}}&0&\lambda \\ 0&0&0&0&{ - \left( {\lambda + \varepsilon } \right)}&0&{\lambda + \varepsilon }\\ 0&0&0&0&0& { - \left( {\lambda + \beta } \right)} &{\lambda + \beta }\\ {{\mu _b}}&0&0&0&0&0&{ - {\mu _b}} \end{array}} \right), \end{equation}$

(2.1)

这里 $\xi = {\mu _w}r - \left( {\lambda + {\theta _w} + {\mu _w} + \alpha } \right)$ .

由 (2.1) 式可知 ${B}\left[ R \right]$ 是有限维的不可约非周期矩阵, 则 $\left( { {\pi} _0},{ {\pi} _1} \right) {B}\left[ R \right] = 0$ 有正解. 此外, 谱半径 $SP\left( {R} \right) = \max \left\{ {r,\frac{\lambda }{{\lambda + {\theta _v}}},\frac{\lambda }{{\lambda + \varepsilon }},\frac{\lambda }{{\lambda + \beta }},\rho } \right\} < 1$ 当且仅当 $\rho < 1$ . 因此当 $\rho < 1$ 时, 拟生灭过程 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 的稳态分布存在. 证毕.

当 $\rho < 1$ 时, 记稳态分布为 ${{\pi} _{ij}} = \mathop {\lim }\limits_{t \to \infty } P\left\{ {L\left( t \right) = i,J\left( t \right) = j} \right\}$ , $\left( {i,j} \right) \in \Omega$ , 稳态分布向量为 ${ {\pi } _0} = \left( {{\pi _{00}},{\pi _{01}}} \right)$ , ${ {\pi } _i} = \left( {{\pi _{i0}},{\pi _{i1}},{\pi _{i2}},{\pi _{i3}},{\pi _{i4}}} \right)$ , $i \ge 1$ .

为方便给出稳态分布的表达式, 构建以下两个数列

$\begin{equation} \left\{ \begin{aligned} {\psi _0} =& 0,\quad {\psi _1} = \frac{{\left( {\alpha + {\theta _w}} \right)r}}{{{\mu _b}\left( {1 - r} \right)}},\\ {\psi _i} =& \frac{{\left( {\alpha + {\theta _w}} \right)r}}{{{\mu _b}\left( {1 - r} \right)}}\sum\limits_{k = 0}^{i - 1} {{r^k}{\rho ^{i - k - 1}}} + \frac{{{\theta _w}r}}{{\lambda + {\theta _v}}}\sum\limits_{j = 1}^{i - 1} {{\rho ^j}\sum\limits_{k = 0}^{i - j - 1} {{r^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - j - k - 1}}} } \\ &+ \frac{{\alpha r}}{{\lambda + \beta }}\sum\limits_{j = 1}^{i - 1} {{\rho ^j}\sum\limits_{k = 0}^{i - j - 1} {{r^k}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - j - k - 1}}} } + \frac{{{\theta _v}{\theta _w}r}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}}\sum\limits_{k = 1}^{i - 1} {{\rho ^k}{\eta _{i - k - 1}}},i \ge 2. \end{aligned} \right. \end{equation}$

(2.2)

$\begin{equation} \left\{ \begin{aligned} &{\eta _0} = 0,\quad {\eta _1} = 1,\\ &{\eta _i} = \left( {r + \frac{\lambda }{{\lambda + {\theta _v}}} + \frac{\lambda }{{\lambda + \varepsilon }}} \right){\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)^{i - 2}} + \sum\limits_{j = 1}^{i - 2} {{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{j - 1}}\sum\limits_{k = 0}^{i - j} {{r^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - j - k}}} },i \ge 2. \end{aligned} \right. \end{equation}$

(2.3)

利用矩阵几何解方法, 借助于式 (2.2) 和 (2.3), 导出拟生灭过程 $\left\{ {\left( {L\left( t \right),J\left( t \right)} \right),t \ge 0} \right\}$ 的稳态分布为

$\begin{eqnarray*} &&{\pi _{i0}} = \left\{ \begin{aligned} &{\pi _{00}},\ &i = 0,\\ &\frac{\lambda }{\xi }{r^{i - 1}}{\pi _{00}},\ &i \ge 1, \end{aligned} \right.\nonumber \\ &&{\pi _{i1}} = \left\{ \begin{aligned} &\frac{{{\theta _w}}}{\lambda }{\pi _{00}},\ &i = 0,\\ &\left[ {\frac{{\lambda {\theta _w}r}}{{\xi \left( {\lambda + {\theta _v}} \right)}}\sum\limits_{k = 0}^{i - 2} {{r^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - k - 2}}} + \frac{{{\theta _w}\left( {\lambda + \xi } \right)}}{{\left( {\lambda + {\theta _v}} \right)\xi }}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - 1}}} \right]{\pi _{00}},\ &i \ge 1, \end{aligned} \right.\nonumber\\ && {\pi _{i2}} = \left[ {\frac{\lambda }{\xi }{\eta _{i - 1}} + \frac{{{\theta _v}{\theta _w}\left( {\lambda + \xi } \right)}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)\xi }}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - 1}}} \right.\\ &&\left. {\ \quad \quad + \frac{{{\theta _w}\left( {\lambda + \xi } \right)}}{{\left( {\lambda + {\theta _v}} \right)\xi }}\frac{{\lambda {\theta _v}}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}}\sum\limits_{k = 0}^{i - 2} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - k - 2}}} } \right]{\pi _{00}}, i \ge 1, \nonumber \\ &&{\pi _{i3}} = \left[ {\frac{{\lambda \alpha r}}{{\xi \left( {\lambda + \beta } \right)}}\sum\limits_{k = 0}^{i - 2} {{r^k}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - k - 2}}} + \frac{{\lambda \alpha }}{{\left( {\lambda + \beta } \right)\xi }}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - 1}}} \right]{\pi _{00}}, i \ge 1,\nonumber\\ && {\pi _{i4}} = \left[ {\frac{\lambda }{\xi }{\psi _{i - 1}} + \sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - k - 1}}} + \frac{{\lambda \left( {\alpha + {\theta _w}} \right) + {\theta _w}\xi \left( {1 - r} \right)}}{{{\mu _b}\xi \left( {1 - r} \right)}}{\rho ^{i - 1}}} \right.\\ &&\quad\ \quad + \frac{{{\theta _v}{\theta _w}\left( {\lambda + \xi } \right)}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)\xi }}\sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - k - 1}}} + \frac{{\lambda \alpha }}{{\left( {\lambda + \beta } \right)\xi }}\sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - k - 1}}} \\ &&\quad\ \quad + \left. {\frac{{\lambda {\theta _v}{\theta _w}\left( {\lambda + \xi } \right)}}{{\xi \left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}\sum\limits_{j = 1}^{i - 2} {{\rho ^j}} \sum\limits_{k = 0}^{i - j - 2} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - j - k - 2}}} } \right]{\pi _{00}}, i \ge 2. \nonumber \end{eqnarray*}$

其中

$\begin{eqnarray*} {\pi _{00}} &=& \left\{ {1 + \frac{{{\theta _w}}}{\lambda } + \frac{{\lambda \left( {\alpha + \beta } \right)\left( {1 - r} \right)}}{{\left( {1 - r} \right)\xi \beta }} + \frac{{\left( {\lambda + \xi } \right)\left( {\varepsilon \left( {\lambda + {\theta _v}} \right) + {{\left( {{\theta _v}} \right)}^2}} \right)}}{{\varepsilon \xi {\theta _v}\left( {\lambda + {\theta _v}} \right)}} + \frac{{\lambda \left( {\alpha + {\theta _w}} \right) + {\theta _w}\xi \left( {1 - r} \right)}}{{{\mu _b}\xi \left( {1 - r} \right)\left( {1 - \rho } \right)}}} \right.\\ &&+ \sum\limits_{i = 2}^\infty {\left[ {\frac{\lambda }{\xi }\left( {\frac{{{\theta _w}r}}{{\lambda + {\theta _v}}}\sum\limits_{k = 0}^{i - 2} {{r^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - k - 2}}} + \frac{{\alpha r}}{{\lambda + \beta }}\sum\limits_{k = 0}^{i - 2} {{r^k}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - k - 2}}} + {\eta _{i - 1}} + {\psi _{i - 1}}} \right)} \right.} \\ && + \frac{{{\theta _w}\left( {\lambda + \xi } \right)\lambda {\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)\xi \left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}}\left( {\sum\limits_{k = 0}^{i - 2} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - k - 2}}} } \right.\\ && + \left. {\sum\limits_{j = 1}^{i - 2} {{\rho ^j}} \sum\limits_{k = 0}^{i - j - 2} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - j - k - 2}}} } \right) + \frac{{\lambda \alpha }}{{\left( {\lambda + \beta } \right)\xi }}\sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + \beta }}} \right)}^{i - k - 1}}} \\ && + {\left. {\left. { \sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v}}}} \right)}^{i - k - 1}}} + \frac{{{\theta _v}{\theta _w}\left( {\lambda + \xi } \right)}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)\xi }}\sum\limits_{k = 1}^{i - 1} {{\rho ^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon }}} \right)}^{i - k - 1}}} } \right]} \right\}^{ - 1}}. \end{eqnarray*}$

证毕.

3 流体模型稳态分析

设 $C\left( t \right)$ 表示 $t$ 时刻缓冲器内的库存量. 由于库存量是非负的, 则 $C\left( t \right)$ 是非负的随机变量. 设缓冲器的净流入率为随机过程 $\left\{ {\left( {L\left( t \right),J\left( t \right),C\left( t \right)} \right),t \ge 0} \right\}$ 的函数

$\begin{equation} \frac{{{\rm d}C\left( t \right)}}{{\rm{d}t}} = \left\{ \begin{array}{l} \sigma,\left( {L\left( t \right),J\left( t \right)} \right) = \left( {0,0} \right) \cup \left( {0,1} \right),C\left( t \right) > 0,\\ 0,\left( {L\left( t \right),J\left( t \right)} \right) = \left( {0,0} \right),C\left( t \right) = 0,\\ {\sigma _0},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,0} \right),k \ge 1,\\ {\sigma _1},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,1} \right),k \ge 1,\\ {\sigma _2},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,2} \right),k \ge 1,\\ {\sigma _3},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,3} \right),k \ge 1,\\ {\sigma _4},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,4} \right),k \ge 1, \end{array} \right. \end{equation}$

(3.1)

其中 $\sigma < 0$ , ${\sigma _4},{\sigma _3},{\sigma _2},{\sigma _1},{\sigma _0} > 0$ . 式 (3.1) 表明当驱动系统处于休假期或工作休假期, 且顾客数为空, 此时缓冲器内的库存量以速率 $-\sigma$ 减少, 直至库存量为空后保持不变. 当驱动系统处于工作休假期, 且顾客数非空, 此时库存量以速率 $\sigma _0$ 增加. 当驱动系统处于休假期, 且顾客数非空, 此时库存量以速率 $\sigma _1$ 增加. 当驱动系统处于启动期, 且顾客数非空, 此时库存量以速率 $\sigma _2$ 增加. 当驱动系统处于工作故障期, 且顾客数非空, 此时库存量以速率 $\sigma _3$ 增加. 当驱动系统处于正规忙期, 且顾客数非空, 此时库存量以速率 $\sigma _4$ 增加.

流体模型即为具有净输入率结构 (3.1) 的三维随机过程 $\left\{ {\left( {L\left( t \right),J\left( t \right),C\left( t \right)} \right),t \ge 0} \right\}$ , 其状态空间为 $\Omega ' = \left[ 0 \right.,\left. { + \infty } \right)\times \Omega$ . 定义模型的平均漂移为 $d$ , 记为

$d = \sigma \left( {{\pi _{00}} + {\pi _{01}}} \right) + {\sigma _0}\sum\limits_{i = 1}^\infty {{\pi _{i0}}} + {\sigma _1}\sum\limits_{i = 1}^\infty {{\pi _{i1}}} + {\sigma _2}\sum\limits_{i = 1}^\infty {{\pi _{i2}}} + {\sigma _3}\sum\limits_{i = 1}^\infty {{\pi _{i3}}} + {\sigma _4}\sum\limits_{i = 1}^\infty {{\pi _{i4}}},$

由文献 [16] 知, 当 $d < 0$ 且 $\rho < 1$ 时, 流体模型的稳态分布存在. 定义 $\left( {L,J,C} \right)$ 为流体模型的平稳向量, 且其稳态联合分布函数为

${F_{ij}}\left( u \right) = \mathop {\lim }\limits_{t \to \infty } P\left\{ {C\left( t \right) \le u,L\left( t \right) = i,J\left( t \right) = j} \right\} = P\left\{ {C \le u,L = i,J = j} \right\},\left( {i,j} \right) \in \Omega,$

则由全概率公式得到流体模型库存量的平稳分布函数

$F\left( u \right) = P\left\{ {C \le u} \right\} = {F_{00}}\left( u \right) + {F_{01}}\left( u \right) + \sum\limits_{i = 1}^\infty {\sum\limits_{j = 0}^4 {{F_{ij}}\left( u \right)} }.$

记流体模型稳态概率向量为

${ {F}_0}\left( u \right) = \left( {{F_{00}}\left( u \right),{F_{01}}\left( u \right)} \right), { {F}_i}\left( u \right) = \left( {{F_{i0}}\left( u \right),{F_{i1}}\left( u \right),{F_{i2}}\left( u \right),{F_{i3}}\left( u \right),{F_{i4}}\left( u \right)} \right), i \ge 1.$

运用概率分析方法, 得到 ${ {F}_i}\left( u \right)$ 满足以下矩阵微分方程形式

$\begin{equation} \frac{{\rm d}}{{{\rm d}u}}\left( {{ {F}_0}\left( u \right),{ {F}_1}\left( u \right), \cdots } \right) {H} = \left( {{ {F}_0}\left( u \right),{ {F}_1}\left( u \right), \cdots} \right) {Q}, \end{equation}$

(3.2)

且满足边界条件

$\begin{array}{l} {F_{00}}\left( 0 \right) = a,\ {F_{ij}}\left( 0 \right) = 0,\ \left( {i,j} \right) \in \Omega /\left( {0,0} \right),\ {F_{ij}}\left( \infty \right) = {\pi _{ij}}, \end{array}$

其中 ${H} = \rm{diag}\left( {\sigma,\sigma,{\sigma _0},{\sigma _1},{\sigma _2},{\sigma _3},{\sigma _4},{\sigma _0},{\sigma _1},{\sigma _2},{\sigma _3},{\sigma _4},\cdots } \right)$ , $a = {F_{00}}\left( 0 \right)$ 称为流体模型稳态时的空库概率.

记 ${F_{ij}}\left( u \right)$ 和 $F\left( u \right)$ 的 Lapalace 变换为

${{\hat F}_{ij}}\left( s \right) = \int_0^\infty {{{\rm e}^{ - su}}{F_{ij}}\left( u \right)}{\rm d}u,s \ge 0,\left( {i,j} \right) \in \Omega,\quad \hat F\left( s \right) = \int_0^\infty {{{\rm e}^{ - su}}F\left( u \right)}{\rm d}u,s \ge 0,$

对 (3.2) 式的两边取 Laplace 变换, 结合边界条件得到

$\begin{equation} \left( {{ {{\hat F}}_0}\left( s \right),{ {{\hat F}}_1}\left( s \right), \cdots } \right)\left( { {Q} - s {H}} \right) = \left( { - a\sigma,0,0, \cdots } \right), \end{equation}$

(3.3)

其中

${ {{\hat F}}_0}\left( s \right) = \left( {{{\hat F}_{00}}\left( s \right),{{\hat F}_{01}}\left( s \right)} \right),\ { {{\hat F}}_i}\left( s \right) = \left( {{{\hat F}_{i0}}\left( s \right),{{\hat F}_{i1}}\left( s \right),{{\hat F}_{i2}}\left( s \right),{{\hat F}_{i3}}\left( s \right),{{\hat F}_{i4}}\left( s \right)} \right), i\ge1.$

为给出稳态分布直观的数学表达式, 构建函数矩阵

${A}\left( s \right) = \left( {\begin{array}{*{20}{c}} {{a_{00}}\left( s \right)}&{{\theta _w}}&0&\alpha &0\\ 0&{ - \left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}&{{\theta _v}}&0&0\\ 0&0&{ - \left( {\lambda + \varepsilon + s{\sigma _2}} \right)}&0&\varepsilon \\ 0&0&0&{ - \left( {\lambda + \beta + s{\sigma _3}} \right)}&\beta \\ 0&0&0&0&{ - \left( {\lambda + {\mu _b} + s{\sigma _4}} \right)} \end{array}} \right),$

其中 ${a_{00}}\left( s \right) = - \left( {\lambda + {\theta _w} + {\mu _w} + \alpha + s{\sigma _0}} \right)$ .

引理 3.1 对任意 $s \ge 0$ , 二次矩阵方程 ${\left( { {R}\left( s \right)} \right)^2} {B} + {R}\left( s \right) {A}\left( s \right) + {C} = {0}$ 的最小非负解存在, 记为

${R}\left( s \right) = \left( {\begin{array}{*{20}{c}} {r\left( s \right)}& {\frac{{{\theta _w}r\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}} & {\frac{{{\theta _v}{\theta _w}r\left( s \right)}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}}& {\frac{{\alpha r\left( s \right)}}{{\lambda + \beta + s{\sigma _3}}}} &{m\left( s \right)}\\[3mm] 0& {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}}& {\frac{{\lambda {\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}}&0&{l\left( s \right)}\\[3mm] 0&0& {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}}&0&{n\left( s \right)}\\[3mm] 0&0&0& {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}}&{g\left( s \right)}\\ 0&0&0&0&{h\left( s \right)} \end{array}} \right),$

其中

$\begin{equation*}\begin{aligned} &r\left( s \right) = \frac{{\lambda + {\theta _w} + {\mu _w} + \alpha + s{\sigma _0} - \sqrt {{{\left( {\lambda + {\theta _w} + {\mu _w} + \alpha + s{\sigma _0}} \right)}^2} - 4\lambda {\mu _w}} }}{{2{\mu _w}}},\\ &h\left( s \right) = \frac{{\lambda + {\mu _b} + s{\sigma _4} - \sqrt {{{\left( {\lambda + {\mu _b} + s{\sigma _4}} \right)}^2} - 4\lambda {\mu _b}} }}{{2{\mu _b}}},\\ &g\left( s \right) = \frac{{\lambda \beta }}{{\left( {\lambda + \beta + s{\sigma _3}} \right)\left( {\lambda + s{\sigma _4} + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right) - \lambda {\mu _b}}},\\ &n\left( s \right) = \frac{{\lambda \varepsilon }}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + s{\sigma _4} + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right) - \lambda {\mu _b}}},\\ &l\left( s \right) = \frac{{\lambda {\theta _v}\left( {{\mu _b}n\left( s \right) + \varepsilon } \right)}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\frac{1}{{\lambda + s{\sigma _4} + {\mu _b} - {\mu _b}\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}} + h\left( s \right)} \right)}},\\ &m\left( s \right) = \left[ {\frac{{{\theta _v}{\theta _w}r\left( s \right)\left( {{\mu _b}n\left( s \right) + \varepsilon } \right)}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}} + \frac{{\alpha r\left( s \right)\left( {\beta + {\mu _b}g\left( s \right)} \right)}}{{\lambda + \beta + s{\sigma _3}}}} \right.\\ & \qquad\quad\left. { + \frac{{{\theta _w}r\left( s \right){\mu _b}l\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}} \right]\frac{1}{{\lambda + s{\sigma _4} + {\mu _b} - {\mu _b}\left( {r\left( s \right) + h\left( s \right)} \right)}}. \end{aligned}\end{equation*}$

特别的 $r\left( 0 \right) = r$ , $h\left( 0 \right) = g\left( 0 \right) = n\left( 0 \right) = l\left( 0 \right) = \rho$ , $m\left( 0 \right) = \frac{{\left( {\alpha + {\theta _w}} \right)r}}{{{\mu _b}\left( {1 - r} \right)}}$ .

为了方便, 构建以下函数序列

$\begin{equation} \left\{ \begin{array}{*{20}{c}}\begin{aligned} {\eta _0}\left( s \right) &= 0,{\eta _1}\left( s \right) = 1,\\ {\eta _i}\left( s \right) &= \left( {r\left( s \right) + \frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}} + \frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right){\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)^{i - 2}}\\ &+ \sum\limits_{j = 1}^{i - 2} {{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^{j - 1}}\sum\limits_{k = 0}^{i - j} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - j - k}}} },i \ge 2. \end{aligned}\end{array}\right. \end{equation}$

(3.4)

$\begin{equation} \left\{ \begin{array}{*{20}{c}}\begin{aligned} {\psi _1}\left( s \right) &= m\left( s \right),\\ {\psi _i}\left( s \right) &= m\left( s \right)\sum\limits_{k = 0}^{i - 1} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {h\left( s \right)} \right)}^{i - k - 1}}} + \frac{{{\theta _v}{\theta _w}r\left( s \right)n\left( s \right)}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\sum\limits_{k = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{\eta _{i - k - 2}}\left( s \right)} \\ & + \frac{{{\theta _w}r\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}l\left( s \right)\sum\limits_{j = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^j}\sum\limits_{k = 0}^{i - j - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - j - k - 2}}} } \\ & + \frac{{\alpha r\left( s \right)}}{{\lambda + \beta + s{\sigma _3}}}g\left( s \right)\sum\limits_{j = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^j}\sum\limits_{k = 0}^{i - j - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^{i - j - k - 2}}} },i \ge 2.\quad \end{aligned}\end{array}\right. \end{equation}$

(3.5)

利用式 (3.4) 和 (3.5) 得到下述结论.

定理 3.1 当 $d < 0,\rho < 1$ , ${{\hat F}_{kj}}\left( s \right)\left( {\left( {i,j} \right) \in \Omega } \right)$ 的表达式为

$\begin{eqnarray*} {{\hat F}_{i0}}\left( s \right)& =& \left\{ \begin{aligned} &{{\hat F}_{00}}\left( s \right),i = 0,\\ &\frac{\lambda }{{\gamma \left( s \right)}}{\left( {r\left( s \right)} \right)^{i - 1}}{{\hat F}_{00}}\left( s \right),i \ge 1, \end{aligned} \right. \\ {{\hat F}_{i1}}\left( s \right) &=& \left\{ \begin{aligned} &\frac{{{\theta _w}}}{{\lambda + s\sigma }}{{\hat F}_{00}}\left( s \right),i = 0\\ &\left[ {\frac{{{\theta _w}r\left( s \right)}}{{\gamma \left( s \right)}}\sum\limits_{k = 0}^{i - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - k - 1}}} } \right.\\ &\left. { + \left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right){{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^i}} \right]{{\hat F}_{00}}\left( s \right),i \ge 1, \end{aligned} \right.\\ {{\hat F}_{i2}}\left( s \right) &=& \left\{ {\frac{\lambda }{{\gamma \left( s \right)}}{\eta _{i - 1}}\left( s \right) + \left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right)\frac{{{\theta _v}}}{{\lambda + {\theta _v} + s{\sigma _1}}}\left[ {{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^i}} \right.} \right.\\ &&\left. {\left. { + \sum\limits_{k = 0}^{i - 2} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{k + 1}}{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^{i - k - 1}}} } \right]} \right\}{{\hat F}_{00}}\left( s \right),i \ge 1,\\ {{\hat F}_{i3}}\left( s \right) &=& \left[ {\frac{{\alpha r\left( s \right)}}{{\gamma \left( s \right)}}\sum\limits_{k = 0}^{i - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^{i - k - 1}}} + \frac{\alpha }{{\gamma \left( s \right)}}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^i}} \right]{{\hat F}_{00}}\left( s \right),i \ge 1,\\ {{\hat F}_{i4}}\left( s \right) &=& \Bigg\{ {\frac{\lambda }{{\gamma \left( s \right)}}{\psi _{i - 1}}\left( s \right) + \left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right)\left[ {l\left( s \right)\sum\limits_{k = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - k - 1}}} } \right.} \\ &&+ {\left. \frac{{n\left( s \right){\theta _v}}}{{\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\sum\limits_{j = 0}^{i - 3} {{{\left( {h\left( s \right)} \right)}^j}} \sum\limits_{k = 0}^{i - j - 3} {{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^{i - j - k - 2}}} \right]} \\ && + \frac{{{\theta _v}}}{{\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right)n\left( s \right)\sum\limits_{k = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^{i - k - 1}}} \\ && + \frac{\alpha }{{\gamma \left( s \right)}}g\left( s \right)\sum\limits_{k = 1}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^{i - k - 1}}} \\ && + \frac{1}{{\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)}}\left[ {\frac{{\lambda {\mu _b}m\left( s \right)}}{{\gamma \left( s \right)}} + \frac{{{\mu _b}l\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right)} \right.\\ &&+ \frac{\left( \varepsilon + \mu _bn\left( s \right) \right)\theta _v} {\left( \lambda + \varepsilon + s\sigma _2 \right)\left( \lambda + \theta _v + s\sigma _1 \right)} \left( \frac{\lambda \theta _w}{\lambda + s\sigma } + \frac{\lambda \theta _w} {\gamma \left( s \right)} \right)\\ &&+ \frac{\lambda \alpha \left( {\beta + {\mu _b}g\left( s \right)} \right)} {\gamma \left( s \right)\left( {\lambda + \beta + s{\sigma _3}} \right)}\bigg] \left( h\left( s \right) \right)^{i - 1} \Bigg\} {\hat F}_{00}\left( s \right),i \ge 1, \end{eqnarray*}$

其中

$\begin{eqnarray*} &&\gamma \left( s \right) = \lambda + {\theta _w} + \alpha + s{\sigma _0} - {\mu _w}\left( {1 - r\left( s \right)} \right),\nonumber \\ &&{{\hat F}_{00}}\left( s \right) = a\sigma \Bigg\{ {\lambda + {\theta _w} + s\sigma - \frac{{\lambda {\mu _w}}}{{\gamma \left( s \right)}} - \frac{{{\mu _b}}}{{\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)}}\left[ {\frac{{\lambda {\mu _b}m\left( s \right)}}{{\gamma \left( s \right)}}} \right.} \nonumber \\ &&\hspace{1.25cm} + \frac{{{\mu _b}l\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right) + \frac{{\lambda \alpha \left( {\beta + {\mu _b}g\left( s \right)} \right)}}{{\gamma \left( s \right)\left( {\lambda + \beta + s{\sigma _3}} \right)}}\nonumber\\ &&\hspace{1.2cm} { {\left. { + \frac{{\left( {\varepsilon + {\mu _b}n\left( s \right)} \right){\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right)} \right]} \Bigg\}^{ - 1}}.\nonumber \end{eqnarray*}$

证对矩阵方程 (3.3) 使用矩阵几何解法可以得到

$\begin{equation} \left( {{ {{\hat F}}_0}\left( s \right),{ {{\hat F}}_1}\left( s \right)} \right) {{B}}\left[ {R\left( s \right)} \right] = \left( { - a\sigma,0,0,0,0,0,0} \right), \end{equation}$

(3.6)

$\begin{equation} { {{\hat F}}_i}\left( s \right) = { {{\hat F}}_1}\left( s \right){\left( { {{R}}\left( s \right)} \right)^{i - 1}},i \ge 1, \end{equation}$

(3.7)

其中

${B}\left[ {R\left( s \right)} \right] = \left( {\begin{array}{*{20}{c}} {{ {A}_{00}}\left( s \right)}&{{ {C}_{01}}}\\ {{ {B}_{10}}}&{ {A}\left( s \right) + {R}\left( s \right) {B}} \end{array}} \right),\quad { {A}_{00}}\left( s \right) = \left( {\begin{array}{*{20}{c}} { - \left( {\lambda + {\theta _w} + s\sigma } \right)}&{{\theta _w}}\\ 0&{ - \left( {\lambda + s\sigma } \right)} \end{array}} \right).$

将 (3.6) 式展开, 整理得到

$\begin{equation} \left\{ \begin{array}{l} - \left( {\lambda + {\theta _w} + s\sigma } \right){{\hat F}_{00}}\left( s \right) + {\mu _w}{{\hat F}_{10}}\left( s \right) + {\mu _b}{{\hat F}_{14}}\left( s \right) = - a\sigma,\\ {\theta _w}{{\hat F}_{00}}\left( s \right) - \left( {\lambda + s\sigma } \right){{\hat F}_{01}}\left( s \right) = 0,\\ \lambda {{\hat F}_{00}}\left( s \right) - \left[ {\lambda + {\theta _w} + \alpha + s{\sigma _0} - {\mu _w}\left( {1 - r\left( s \right)} \right)} \right]{{\hat F}_{10}}\left( s \right) = 0,\\ \lambda {{\hat F}_{01}}\left( s \right) + {\theta _w}{{\hat F}_{10}}\left( s \right) - \left( {\lambda + {\theta _v} + s{\sigma _1}} \right){{\hat F}_{11}}\left( s \right) = 0,\\ {\theta _v}{{\hat F}_{11}}\left( s \right) - \left( {\lambda + \varepsilon + s{\sigma _2}} \right){{\hat F}_{12}}\left( s \right) = 0,\\ \alpha {{\hat F}_{10}}\left( s \right) - \left( {\lambda + \beta + s{\sigma _3}} \right){{\hat F}_{13}}\left( s \right) = 0,\\ {\mu _b}\left[ {m\left( s \right){{\hat F}_{10}}\left( s \right) + l\left( s \right){{\hat F}_{11}}\left( s \right) + n\left( s \right){{\hat F}_{12}}\left( s \right) + g\left( s \right){{\hat F}_{13}}\left( s \right)} \right]\\ + \varepsilon {{\hat F}_{12}}\left( s \right) + \beta {{\hat F}_{13}}\left( s \right) - \left( {\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right){{\hat F}_{14}}\left( s \right) = 0. \end{array} \right. \end{equation}$

(3.8)

求解方程组 (3.8)式, 得到以下结果

$\begin{equation} \left\{ \begin{aligned} &{{\hat F}_{01}}\left( s \right) = \frac{{{\theta _w}}}{{\lambda + s\sigma }}{{\hat F}_{00}}\left( s \right),\; {{\hat F}_{10}}\left( s \right) = \frac{\lambda }{{\gamma \left( s \right)}}{{\hat F}_{00}}\left( s \right),\\ &{{\hat F}_{11}}\left( s \right) = \frac{1}{{\lambda + {\theta _v} + s{\sigma _1}}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right){{\hat F}_{00}}\left( s \right),\\ &{{\hat F}_{12}}\left( s \right) = \frac{{{\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right){{\hat F}_{00}}\left( s \right),\\ &{{\hat F}_{13}}\left( s \right) = \frac{\alpha }{{\lambda + \beta + s{\sigma _3}}}{{\hat F}_{10}}\left( s \right) = \frac{\alpha }{{\lambda + \beta + s{\sigma _3}}}\frac{\lambda }{{\gamma \left( s \right)}}{{\hat F}_{00}}\left( s \right),\\ &{{\hat F}_{14}}\left( s \right) = \frac{1}{{\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)}}\left[ {\frac{{\lambda {\mu _b}m\left( s \right)}}{{\gamma \left( s \right)}} + \frac{{{\mu _b}l\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right)} \right.\\ &\left. {\quad \quad \;\;\;\;\;\; + \frac{{\left( {\varepsilon + {\mu _b}n\left( s \right)} \right){\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right) + \frac{{\alpha \left( {\beta + {\mu _b}g\left( s \right)} \right)}}{{\lambda + \beta + s{\sigma _3}}}\frac{\lambda }{{\gamma \left( s \right)}}} \right]{{\hat F}_{00}}\left( s \right). \end{aligned}\right. \end{equation}$

(3.9)

求解 (3.7) 式, 得到以下结果

$\begin{equation} \left\{ \begin{aligned} {{\hat F}_{i0}}\left( s \right) &= {\left( {r\left( s \right)} \right)^{i - 1}}{{\hat F}_{10}}\left( s \right),i \ge 1,\\ {{\hat F}_{i1}}\left( s \right) &=\frac{{{\theta _w}r\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}\sum\limits_{k = 0}^{i - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - k - 2}}} {{\hat F}_{10}}\left( s \right) \\ & + {\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)^{i - 1}}{{\hat F}_{11}}\left( s \right),i \ge 1,\\ {{\hat F}_{i2}}\left( s \right) &={\eta _{i - 1}}\left( s \right){{\hat F}_{10}}\left( s \right) + \frac{{\lambda {\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _3}} \right)}}\sum\limits_{k = 0}^{i - 2} \left( \frac{\lambda }{\lambda + \theta _v + s\sigma _1} \right)^k\\ & \times\left( \frac{\lambda }{\lambda + \varepsilon + s\sigma _2} \right)^{i - k - 2} {\hat F}_{11}\left( s \right) + {\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)^{i - 1}}{{\hat F}_{12}}\left( s \right),i \ge 1,\\ {{\hat F}_{i3}}\left( s \right) &=\frac{{\alpha r}}{{\lambda + \beta + s{\sigma _3}}}\sum\limits_{k = 0}^{i - 2} {{{\left( {r\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^{i - k - 2}}} {{\hat F}_{10}}\left( s \right) \\ & + {\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)^{i - 1}}{{\hat F}_{13}}\left( s \right),i \ge 1,\\ {{\hat F}_{i4}}\left( s \right) &= {\psi _{i - 1}}\left( s \right){{\hat F}_{10}}\left( s \right) + \left[ {l\left( s \right)\sum\limits_{k = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + {\theta _v} + s{\sigma _1}}}} \right)}^{i - k - 2}}} } \right.\\ & + \frac{n\left( s \right)\lambda \theta _v} {\left( \lambda + \varepsilon + s\sigma _2 \right)\left( \lambda + \theta _v + s\sigma _1 \right)} \sum\limits_{j = 0}^{i - 3} \left( h\left( s \right) \right)^j \sum\limits_{k = 0}^{i - j - 3} \left( \frac{\lambda }{\lambda + \theta _v + s\sigma _1} \right)^k \\ &\times \left( \frac{\lambda }{\lambda + \varepsilon + s\sigma _2} \right)^{i - j - k - 3} \Bigg] {\hat F}_{11}\left( s \right) + n\left( s \right)\sum\limits_{k = 0}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \varepsilon + s{\sigma _2}}}} \right)}^{i - k - 2}}} {{\hat F}_{12}}\left( s \right) \\ & + g\left( s \right)\sum\limits_{k = 1}^{i - 2} {{{\left( {h\left( s \right)} \right)}^k}{{\left( {\frac{\lambda }{{\lambda + \beta + s{\sigma _3}}}} \right)}^{i - k - 2}}} {{\hat F}_{13}}\left( s \right) + {\left( {h\left( s \right)} \right)^{i - 1}}{{\hat F}_{14}}\left( s \right),i \ge 1. \end{aligned}\right. \end{equation}$

(3.10)

将 (3.9) 式中的结果代入到 (3.10) 式, 定理 3.1 得证.

下面计算流体模型稳态时库存量的 LST, 由 (3.7) 式得到

$\begin{equation} \hat F\left( s \right) = \left( {{{\hat F}_{00}}\left( s \right),{{\hat F}_{01}}\left( s \right)} \right){ e} + \left( {{{\hat F}_{10}}\left( s \right),{{\hat F}_{11}}\left( s \right),{{\hat F}_{12}}\left( s \right),{{\hat F}_{13}}\left( s \right),{{\hat F}_{14}}\left( s \right)} \right){\left( { {I} - {R}\left( s \right)} \right)^{ - 1}} {e}, \end{equation}$

(3.11)

这里 ${I}$ 是五阶单位矩阵, ${e}$ 是相应维数的全1列向量.

由于矩阵 ${ {I} - {R}\left( s \right)}$ 的谱半径

$SP\left( { {I} - {R}\left( s \right)} \right) = \max \left\{ {1 - r\left( s \right),\frac{{{\theta _v} + s{\sigma _1}}}{{\lambda + {\theta _v} + s{\sigma _1}}},\frac{{\varepsilon + s{\sigma _2}}}{{\lambda + \varepsilon + s{\sigma _2}}},\frac{{\beta + s{\sigma _3}}}{{\lambda + \beta + s{\sigma _3}}},1 - h\left( s \right)} \right\} < 1,$

则矩阵 ${ {I} - {R}\left( s \right)}$ 是可逆的. 由式 (3.11) 得到 $\hat F\left( s \right)$ 的表达式为

$\hat F\left( s \right) = K\left( s \right){{\hat F}_{00}}\left( s \right),$

其中

$\begin{eqnarray*} K\left( s \right) &=& {1 + \frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{\lambda }{{\gamma \left( s \right)\left( {1 - r\left( s \right)} \right)}}\left[ {1 + \frac{{{\theta _w}r\left( s \right)}}{{\left( {{\theta _v} + s{\sigma _1}} \right)}} + \frac{{{\theta _v}{\theta _w}r\left( s \right)}}{{\left( {\varepsilon + s{\sigma _2}} \right)\left( {{\theta _v} + s{\sigma _1}} \right)}} + \frac{{\alpha r\left( s \right)}}{{\left( {\beta + s{\sigma _3}} \right)}}} \right.} \nonumber \\ && + \left. {\frac{1}{{1 - h\left( s \right)}}\left( {m\left( s \right) + \frac{{l\left( s \right){\theta _w}r\left( s \right)}}{{{\theta _v} + s{\sigma _1}}} + \frac{{{\theta _v}{\theta _w}r\left( s \right)n\left( s \right)}}{{\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)\left( {\varepsilon + s{\sigma _2}} \right)}} + \frac{{g\left( s \right)\alpha r\left( s \right)}}{{\beta + s{\sigma _3}}}} \right)} \right]\nonumber \\ && + \frac{\lambda }{{\left( {{\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right)\left[ {1 + \frac{{\lambda {\theta _v}}}{{\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)\left( {\varepsilon + s{\sigma _2}} \right)}}} \right.\nonumber \\ && + \left. { \frac{{l\left( s \right)}}{{\left( {1 - h\left( s \right)} \right)}} \!+\! \frac{{n\left( s \right)\lambda {\theta _v}}}{{\left( {1 - h\left( s \right)} \right)\left( {\varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}} \right]\!+\! \frac{{\lambda \alpha }}{{\left( {\beta + s{\sigma _3}} \right)\gamma \left( s \right)}}\left[ {1 \!+\! \frac{{g\left( s \right)}}{{1 - h\left( s \right)}}} \right]\nonumber \\ && + \frac{{\lambda {\theta _v}}}{{\left( {\varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{{\theta _w}}}{{\lambda + s\sigma }} + \frac{{{\theta _w}}}{{\gamma \left( s \right)}}} \right)\left[ {1 + \frac{{n\left( s \right)}}{{\left( {1 - h\left( s \right)} \right)}}} \right]\nonumber \\ && + \frac{1}{{\left( {\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right)\left( {1 - h\left( s \right)} \right)}}\left[ {\frac{{\lambda {\mu _b}m\left( s \right)}}{{\gamma \left( s \right)}} + \frac{{{\mu _b}l\left( s \right)}}{{\lambda + {\theta _v} + s{\sigma _1}}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right)} \right.\nonumber \\ &&+ {\left. { \frac{{\left( {\varepsilon + {\mu _b}n\left( s \right)} \right){\theta _v}}}{{\left( {\lambda + \varepsilon + s{\sigma _2}} \right)\left( {\lambda + {\theta _v} + s{\sigma _1}} \right)}}\left( {\frac{{\lambda {\theta _w}}}{{\lambda + s\sigma }} + \frac{{\lambda {\theta _w}}}{{\gamma \left( s \right)}}} \right) + \frac{{\alpha \left( {\beta + {\mu _b}g\left( s \right)} \right)}}{{\lambda + \beta + s{\sigma _3}}}\frac{\lambda }{{\gamma \left( s \right)}}} \right]}.\nonumber \end{eqnarray*}$

引入 $F\left( u \right)$ 的 LST, 记为 ${f^ * }\left( s \right) = \int_0^\infty {{{\rm e}^{ - su}}{\rm d}} F\left( u \right) = s\hat F\left( s \right)$ . 再由正规化条件 $\mathop {\lim }\limits_{s \to 0} {f^*}\left( s \right) = 1$ , 得到

$\begin{equation} \mathop {\lim }\limits_{s \to 0} \left( {sK\left( s \right){{\hat F}_{00}}\left( s \right)} \right) = 1. \end{equation}$

(3.12)

对 (3.12) 式运用洛必达法则, 得到库存量在稳态时的空库概率为

$\begin{eqnarray} a &=& \Bigg\{ {1 + \frac{1}{\sigma }\left[ {\frac{{\gamma '\left( 0 \right)\lambda {\mu _w}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} - \frac{{{{\left( {{\mu _b}} \right)}^2}h'\left( 0 \right)}}{{\gamma \left( 0 \right){{\left( {\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right)}^2}}}\left[ {\lambda {\mu _b}m\left( 0 \right) + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\left. { + \lambda \alpha } \right]} \right.} \right.} \nonumber \\ &&\hspace{0.4cm} - \frac{{\lambda {\mu _b}\left( {m'\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma '\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} + \frac{{{\mu _b}l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \lambda {\sigma _1}}}{{\gamma \left( 0 \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\nonumber\\ &&\hspace{0.4cm} - \frac{{\lambda {\theta _w}}}{{\lambda + {\theta _v}}}\left( {\frac{\sigma }{\lambda } + \frac{{\lambda \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) - \frac{{\lambda {\theta _w}{\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)}}\left( {\frac{\sigma }{{{\lambda ^2}}} + \frac{{\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right)\nonumber \\ &&\hspace{0.4cm} + \frac{{{\mu _b}n'\left( 0 \right){\theta _v}\left( {\lambda + {\theta _v}} \right) - {\theta _v}\left[ {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + \left( {\lambda + \varepsilon } \right){\sigma _1}} \right]}}{{\gamma \left( 0 \right)\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\nonumber \\ &&\hspace{0.4cm}+ {\left. {\frac{{\lambda \alpha {\mu _b}g'\left( 0 \right)\gamma \left( 0 \right) - \lambda \alpha \left( {\gamma '\left( s \right)\left( {\lambda + \beta } \right) + \gamma \left( 0 \right){\sigma _3}} \right)}}{{\gamma \left( 0 \right)\left( {\lambda + \beta } \right)}}} \right]} \Bigg\}{\left( {K\left( 0 \right)} \right)^{ - 1}},\nonumber \end{eqnarray}$

其中

$\begin{eqnarray*} &&\gamma '\left( 0 \right) =\frac{{\rm d}\gamma(s)}{{\rm d}s}\bigg|_{s= 0}={\sigma _0} + {\mu _w}r'\left( 0 \right),\quad r'\left( 0 \right) = \frac{{{\sigma _0}}}{{2{\mu _w}}}\left[ {1 - \frac{{\lambda + {\theta _w} + {\mu _w} + \alpha }}{{\sqrt {{{\left( {\lambda + {\theta _w} + {\mu _w} + \alpha } \right)}^2} - 4\lambda {\mu _w}} }}} \right],\nonumber \\ && h'\left( 0 \right) = \frac{{{\sigma _4}\rho }}{{\lambda - {\mu _b}}},\quad g'\left( 0 \right) = - \frac{{\lambda \beta }}{{{{\left( {\beta {\mu _b}} \right)}^2}}}\left[ {{\mu _b}{\sigma _3} + \left( {\lambda + \beta } \right)\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right)} \right],\nonumber \\ &&n'\left( 0 \right) = - \frac{{\lambda \varepsilon }}{{{{\left( {\varepsilon {\mu _b}} \right)}^2}}}\left[ {{\mu _b}{\sigma _2} + \left( {\lambda + \varepsilon } \right)\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right)} \right],\nonumber \\ &&l'\left( 0 \right) = \frac{{\left( {\lambda + {\theta _v}} \right)\left( {{\mu _b}n'\left( 0 \right) - {\sigma _2}} \right) - {\sigma _1}\left( {\lambda + \varepsilon } \right)}}{{\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)}} + \frac{{\lambda {\theta _v}}}{{\lambda + {\theta _v}}}\frac{{{{\left( {\lambda + {\theta _v}} \right)}^2}\left( {{\mu _b}h'\left( 0 \right) - {\sigma _4}} \right) - {\mu _b}\lambda {\sigma _1}}}{{{{\left( {{\theta _v}{\mu _b}} \right)}^2}}},\nonumber \\ &&m'\left( 0 \right) = \left\{ {\frac{{{\theta _v}{\theta _w}\left[ {\left( {r'\left( 0 \right)\left( {\lambda + \varepsilon } \right) + r\left( {{\mu _b}n'\left( 0 \right) + \varepsilon } \right)} \right)\left( {\lambda + {\theta _v}} \right) - r\left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right)} \right]}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}} \right.\nonumber \\ &&\hspace{1.3cm} + \frac{{\alpha \left[ {\left( {r'\left( 0 \right)\left( {\lambda + \beta } \right) + r\left( {{\mu _b}g'\left( 0 \right) + \beta } \right)} \right) - r{\sigma _3}} \right]}}{{\lambda + \beta }}\nonumber \\ &&\hspace{1.25cm} \left.+ { \frac{{{\theta _w}{\mu _b}\left[ {\left( {r'\left( 0 \right)\rho + rl'\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right) - r\rho {\sigma _1}} \right]}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}}}} \right\}\frac{1}{{{\mu _b}\left( {1 - r} \right)}}\nonumber \\ &&\hspace{1.3cm} + \frac{{\left( {{\theta _w} + \alpha } \right)r\left[ {{\mu _b}\left( {r'\left( 0 \right) + h'\left( 0 \right)} \right) - {\sigma _4}} \right]}}{{{{\left[ {{\mu _b}\left( {1 - r} \right)} \right]}^2}}},\nonumber \\ &&K\left( 0 \right) = {1 + \frac{{{\theta _w}}}{\lambda } + \frac{{{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)}}{{\gamma \left( 0 \right)\left( {1 - \rho } \right){\theta _v}}}\left( {1 + \frac{{\lambda {\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)\varepsilon }}} \right) + \frac{{\left( {\gamma \left( 0 \right) + \lambda } \right){\theta _v}}}{{\varepsilon \gamma \left( 0 \right)\left( {\lambda + {\theta _v}} \right)\left( {1 - \rho } \right)}}} \nonumber \\ &&\hspace{1.3cm} + \frac{\lambda }{{\gamma \left( 0 \right)\left( {1 - r} \right)}}\left[ {1 + \frac{{{\theta _w}r}}{{{\theta _v}}} \!+\! \frac{{{\theta _v}{\theta _w}r}}{{\varepsilon {\theta _v}}} \!+\! \frac{{\alpha r}}{\beta } \!+\! \frac{1}{{1 - \rho }}\left( {m\left( 0 \right) \!+\! \frac{{\rho {\theta _w}r}}{{{\theta _v}}} \!+\! \frac{{{\theta _v}{\theta _w}r\rho }}{{\left( {\lambda + {\theta _v}} \right)\varepsilon }} \!+\! \frac{{\rho \alpha r}}{\beta }} \right)} \right]\nonumber \\ &&\hspace{1.3cm} { + \frac{{\lambda \alpha }}{{\beta \left( {1 - \rho } \right)\gamma \left( 0 \right)}} + \frac{{\lambda {\mu _b}m\left( 0 \right) + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right) + \lambda \alpha }}{{\left( {\lambda + {\mu _b}\left( {1 - \rho } \right)} \right)\left( {1 - \rho } \right)\gamma \left( 0 \right)}}}.\nonumber \end{eqnarray*}$

进一步得到流体模型稳态时库存量均值的表达式

$E\left( C \right) = - \frac{\rm d}{{\rm d}s}\left( {f^ * }\left( s \right)\right){|_{s = 0}} = a\sigma K\left( 0 \right)\frac{{K''\left( 0 \right)}}{{2{{\left( {K'\left( 0 \right)} \right)}^2}}} + M\sum\limits_{i = 1}^5 {{K_i}},$

其中

$\begin{eqnarray*} &&{K_1} = \frac{{\lambda {\theta _v}{\theta _w}}}{{{{\left( {\lambda \varepsilon \left( {\lambda + {\theta _v}} \right)\gamma \left( s \right)0} \right)}^2}\left( {1 - \rho } \right)}}\left\{ {\lambda \varepsilon \left( {\sigma + \gamma '\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right)\gamma \left( 0 \right)} \right. \\ &&\hspace{0.8cm} - \left. { \left( {\lambda + \gamma \left( 0 \right)} \right)\left[ {\lambda \gamma \left( 0 \right)\left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + \varepsilon {\sigma _1}} \right) + \left( {\lambda + {\theta _v}} \right)\varepsilon \left( {\sigma \gamma \left( 0 \right) + \lambda \gamma '\left( 0 \right)} \right)} \right]} \right\} \\ &&\hspace{0.8cm} + \frac{{{\theta _v}{\theta _w}\left( {\lambda + \gamma \left( 0 \right)} \right)}}{{\varepsilon \left( {\lambda + {\theta _v}} \right)\gamma \left( 0 \right)}}\frac{{n'\left( 0 \right)\left( {1 - \rho } \right) + \rho h'\left( 0 \right)}}{{{{\left( {1 - \rho } \right)}^2}}}, \\ &&{K_2} = - \frac{{{\theta _w}\sigma }}{{{\lambda ^2}}}, \\ &&{K_3} = \frac{{ - \lambda \left( {\gamma '\left( 0 \right)\left( {1 - r} \right) - \gamma \left( 0 \right)r'\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)\left( {1 - r} \right)} \right)}^2}}}\left[ {1 + \frac{{{\theta _w}r}}{{{\theta _v}}} + \frac{{{\theta _w}r}}{\varepsilon } + \frac{{\alpha r}}{\beta }} \right. \\ &&\hspace{0.8cm} + \left. {\frac{1}{{1 - \rho }}\left( {m\left( 0 \right) + \frac{{\rho {\theta _w}r}}{{{\theta _v}}} + \frac{{\rho {\theta _v}{\theta _w}r}}{{\left( {\lambda + {\theta _v}} \right)\varepsilon }} + \frac{{\rho \alpha r}}{\beta }} \right)} \right] \\ &&\hspace{0.8cm} + \frac{\lambda }{\gamma \left( 0 \right)\left( 1 - r \right)} \Bigg[ \theta _w\left( \frac{\theta _vr'\left( 0 \right) - r\sigma _1} {\theta^2 _v} + \frac{\varepsilon \theta _vr'\left( 0 \right) - r\left( \sigma _2\theta _v + \varepsilon \sigma _1 \right)} {\varepsilon ^2\theta _v} \right) \\ &&\hspace{0.8cm}+ \frac{\alpha \beta r'\left( 0 \right) - \alpha r\sigma _3}{\beta } + \frac{{h'\left( 0 \right)}}{{{{\left( {1 - \rho } \right)}^2}}}\left( {m\left( 0 \right) + \frac{{\rho {\theta _w}r}}{{{\theta _v}}} + \frac{{\rho {\theta _v}{\theta _w}r}}{{\left( {\lambda + {\theta _v}} \right)\varepsilon }} + \frac{{\rho \alpha r}}{\beta }} \right) \Bigg], \\ &&{K_4} = \left[ {1 + \frac{{\lambda {\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)\varepsilon }}} \right.\left. { + \frac{\rho }{{1 - \rho }} + \frac{{\rho \lambda {\theta _v}}}{{\left( {1 - \rho } \right)\left( {\lambda + {\theta _v}} \right)\varepsilon }}} \right]{\left[ {{{\left( {\lambda {\theta _v}\gamma \left( 0 \right)} \right)}^2}} \right]^{ - 1}}\left[ {\lambda {\theta _w}\left( {\sigma + \gamma '\left( 0 \right)} \right)} \right) \\ &&\hspace{0.8cm} - \left. { \lambda {\theta _w}\left( {\lambda + \gamma \left( 0 \right)} \right)\left( {{\sigma _1}\lambda \gamma \left( 0 \right) + \sigma {\theta _v}\gamma \left( 0 \right) + {\sigma _1}\lambda \gamma '\left( 0 \right)} \right)} \right] \\ &&\hspace{0.8cm} + \frac{{{\theta _w}\left( {\lambda + \gamma \left( 0 \right)} \right)}}{{{\theta _v}\gamma \left( 0 \right)}}\left[ {\frac{{ - \lambda {\theta _v}\left( {\left( {\lambda + {\theta _v}} \right){\sigma _2} + \varepsilon {\sigma _1}} \right)}}{{{{\left( {\left( {\lambda + {\theta _v}} \right)\varepsilon } \right)}^2}}} + \frac{{l'\left( 0 \right)\left( {1 - \rho } \right) + \rho h'\left( 0 \right)}}{{{{\left( {1 - \rho } \right)}^2}}}} \right. \\ &&\hspace{0.8cm} + \left. { \frac{{\lambda {\theta _v}\left[ {n'\left( 0 \right)\left( {1 - \rho } \right)\varepsilon - \rho \left( { - h'\left( 0 \right)\varepsilon + \left( {1 - \rho } \right)\left( {{\sigma _2} + \varepsilon {\sigma _1}{{\left( {\lambda + {\theta _v}} \right)}^{ - 1}}} \right)} \right)} \right]}}{{{{\left( {\varepsilon \left( {1 - \rho } \right)\left( {\lambda + {\theta _v}} \right)} \right)}^2}}}} \right], \\ &&{K_5} = \frac{{\left[ {{\mu _b}h'\left( 0 \right)\left( {1 - \rho } \right) + \left( {\lambda + {\mu _b}\left( {1 - \rho } \right)} \right)h'\left( 0 \right)} \right]}}{{{{\left[ {\left( {\lambda + {\mu _b}\left( {1 - \rho } \right)} \right)\left( {1 - \rho } \right)} \right]}^2}}}\left[ {\frac{{\lambda {\mu _b}m\left( 0 \right)}}{{\gamma \left( 0 \right)}} + \frac{{{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)}}{{\gamma \left( 0 \right)}}} \right.\left. { + \frac{{\lambda \alpha }}{{\gamma \left( 0 \right)}}} \right] \\ &&\hspace{0.8cm} + \frac{1}{{\left( {\lambda + {\mu _b}\left( {1 - \rho } \right)} \right)\left( {1 - \rho } \right)}}\left[ {\lambda {\mu _b}\frac{{m'\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} } \right. \\ &&\hspace{0.8cm} + \frac{{\left( {{\mu _b}l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - {\mu _b}\rho {\sigma _1}} \right){\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}\gamma \left( 0 \right)}} + \frac{\lambda }{{\lambda + {\theta _v}}}\left( {\frac{{ - \lambda {\theta _w}\sigma }}{{{\lambda ^2}}} + \frac{{ - \lambda {\theta _w}\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) \\ &&\hspace{0.8cm} + \frac{{{\mu _b}n'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right)}}{{\left( {\lambda + \varepsilon } \right){{\left( {\left( {\lambda + {\theta _v}} \right)} \right)}^2}}}\frac{{{\theta _v}{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)}}{{\gamma \left( 0 \right)}} \\ &&\hspace{0.8cm} +\left. { \frac{{ - \lambda {\theta _w}{\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)}}\left( {\frac{\sigma }{{{\lambda ^2}}} + \frac{{\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + \frac{{\alpha {\mu _b}g'\left( 0 \right) - \alpha {\sigma _3}}}{{\lambda + \beta }}\frac{\lambda }{{\gamma \left( 0 \right)}} + \frac{{ - \lambda \alpha \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right], \\ &&M = a\sigma \left\{ { \frac{{{{\left( {{\mu _b}} \right)}^2}h'\left( 0 \right)}}{{\gamma \left( 0 \right){{\left( {\lambda + {\mu _b}\left( {1 - h\left( s \right)} \right)} \right)}^2}}}\left[ {\lambda {\mu _b}m\left( 0 \right) + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\left. { + \lambda \alpha } \right]} \right.} \right. \\ &&\hspace{0.7cm} + \frac{{\lambda {\mu _b}\left( {m'\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma '\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} + \sigma + \frac{{\gamma '\left( 0 \right)\lambda {\mu _w}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} \\ &&\hspace{0.7cm} + \frac{{\lambda {\theta _w}}}{{\lambda + {\theta _v}}}\left( {\frac{\sigma }{\lambda } + \frac{{\lambda \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + \frac{{{\mu _b}l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \lambda {\sigma _1}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}}}{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right) \\ &&\hspace{0.7cm} - \frac{{{\mu _b}n'\left( 0 \right){\theta _v}\left( {\lambda + {\theta _v}} \right) - {\theta _v}\left[ {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + \left( {\lambda + \varepsilon } \right){\sigma _1}} \right]}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}{\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right) \\ &&\hspace{0.69cm} + {\left. {\frac{{\lambda {\theta _w}{\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)}}\left( {\frac{\sigma }{{{\lambda ^2}}} + \frac{{\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + \frac{{\lambda \alpha {\mu _b}g'\left( 0 \right)\gamma \left( 0 \right) - \lambda \alpha \left( {\gamma '\left( s \right)\left( {\lambda + \beta } \right) + \gamma \left( 0 \right){\sigma _3}} \right)}}{{\gamma \left( 0 \right)\left( {\lambda + \beta } \right)}}}\right\}^{ - 1}}, \\ &&K'\left( 0 \right)=\frac{{\rm d}K(s)}{{\rm d}s}\bigg|_{s= 0} = \Bigg\{ {1 + \frac{1}{\sigma }\left[ {\frac{{\gamma '\left( 0 \right)\lambda {\mu _w}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} - \frac{{h'\left( 0 \right)}}{{\gamma \left( 0 \right)}}\left[ {\lambda {\mu _b}m\left( 0 \right) + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\left. { + \lambda \alpha } \right]} \right.} \right.} \\ &&\hspace{1.3cm} - \frac{{\lambda {\mu _b}\left( {m'\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma '\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\left( {\frac{{{\mu _b}l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \lambda {\sigma _1}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}\gamma \left( 0 \right)}}} \right. \\ &&\hspace{1.3cm} + \left. {\frac{{{\mu _b}n'\left( 0 \right){\theta _v}\left( {\lambda + {\theta _v}} \right) - {\theta _v}\left[ {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + \left( {\lambda + \varepsilon } \right){\sigma _1}} \right]}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}\gamma \left( 0 \right)}}} \right) - \frac{{\lambda {\theta _w}}}{{\lambda + {\theta _v}}}\left( {\frac{\sigma }{\lambda } + \frac{{\lambda \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) \\ &&\hspace{1.3cm} - {\left. {\frac{{\lambda {\theta _w}{\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)}}\left( {\frac{\sigma }{{{\lambda ^2}}} + \frac{{\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + \frac{{\lambda \alpha {\mu _b}g'\left( 0 \right)\gamma \left( 0 \right) - \lambda \alpha \left( {\gamma '\left( s \right)\left( {\lambda + \beta } \right) + \gamma \left( 0 \right){\sigma _3}} \right)}}{{\gamma \left( 0 \right)\left( {\lambda + \beta } \right)}}} \right]} \Bigg\}, \\ &&K''\left( 0 \right) = - \frac{{2\lambda {\mu _w}{{\left( {\gamma '\left( 0 \right)} \right)}^2}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^3}}} - \left( {h''\left( 0 \right) + 2{{\left( {h'\left( s \right)} \right)}^2}} \right)\left[ {\lambda {\mu _b}m\left( 0 \right) + {\theta _w}\left( {\gamma \left( 0 \right) + \lambda } \right)\left. { + \lambda \alpha } \right]} \right. \\ &&\hspace{1.4cm} - 2\frac{{h'\left( 0 \right)}}{{\gamma \left( 0 \right)}}\left\{ {\frac{{\lambda {\mu _b}\left( {m'\left( 0 \right)\gamma \left( 0 \right) \!-\! m\left( 0 \right)\gamma '\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}} \!+\! {\theta _w}\left( {\gamma \left( 0 \right) \!+\! \lambda } \right)\left( {\frac{{{\mu _b}l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right)\! -\! \lambda {\sigma _1}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}\gamma \left( 0 \right)}}} \right.} \right. \\ &&\hspace{1.4cm} + \left. {\frac{{{\mu _b}n'\left( 0 \right){\theta _v}\left( {\lambda + {\theta _v}} \right) - {\theta _v}\left[ {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + \left( {\lambda + \varepsilon } \right){\sigma _1}} \right]}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}\gamma \left( 0 \right)}}} \right) - \frac{{\lambda {\theta _w}}}{{\lambda + {\theta _v}}}\left( {\frac{\sigma }{\lambda } + \frac{{\lambda \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) \\ &&\hspace{1.4cm} - \left. {\frac{{\lambda {\theta _w}{\theta _v}}}{{\left( {\lambda + {\theta _v}} \right)}}\left( {\frac{\sigma }{{{\lambda ^2}}} + \frac{{\gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + \frac{{\lambda \alpha {\mu _b}g'\left( 0 \right)\gamma \left( 0 \right) - \lambda \alpha \left( {\gamma '\left( s \right)\left( {\lambda + \beta } \right) + \gamma \left( 0 \right){\sigma _3}} \right)}}{{\gamma \left( 0 \right)\left( {\lambda + \beta } \right)}}} \right\} \\ &&\hspace{1.4cm} - \lambda {\mu _b}\frac{{\left( {m''\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma ''\left( 0 \right)} \right)\gamma \left( 0 \right) - 2\gamma '\left( 0 \right)\left( {m'\left( 0 \right)\gamma \left( 0 \right) - m\left( 0 \right)\gamma '\left( 0 \right)} \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^3}}} \\ &&\hspace{1.4cm} - \frac{{\lambda \alpha }}{{{{\left( {\gamma \left( 0 \right)} \right)}^3}{{\left( {\lambda + \beta } \right)}^2}}}\left\{ {\left[ {{\mu _b}g''\left( 0 \right)\gamma \left( 0 \right) - \left( {2{\sigma _3}^\prime \gamma \left( 0 \right) + \gamma ''\left( 0 \right)\left( {\lambda + \beta } \right)} \right)} \right]} \right. \\ &&\hspace{1.4cm}- \left. {2\left( {{\sigma _3}\gamma \left( 0 \right) + \gamma '\left( 0 \right)\left( {\lambda + \beta } \right)} \right)\left[ {{\mu _b}g'\left( 0 \right)\gamma \left( 0 \right) - \left( {{\sigma _3}\gamma \left( 0 \right) + \gamma '\left( 0 \right)\left( {\lambda + \beta } \right)} \right)} \right]} \right\} \\ &&\hspace{1.4cm} - \lambda {\theta _w}{\theta _v}\left\{ {\frac{1}{{{{\left( {\lambda + \varepsilon } \right)}^2}{{\left( {\lambda + {\theta _v}} \right)}^3}}}\left[ {\left( {{\mu _b}n''\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - 2{\sigma _1}{\sigma _2}} \right)\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)} \right.} \right. \\ &&\hspace{1.4cm} - 2\left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right)\left( {{\mu _b}n'\left( 0 \right)\left( {\lambda + {\theta _v}} \right)} \right. \\ &&\hspace{1.4cm} - \left. {\left. { \left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right)} \right)} \right]\left( {\frac{1}{\lambda } + \frac{1}{{\gamma \left( 0 \right)}}} \right) + \frac{2}{{\lambda + {\theta _v}}}\left( {\frac{{{\sigma ^2}}}{{{\lambda ^3}}} + \frac{{{{\left( {\gamma '\left( 0 \right)} \right)}^2}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^3}}}} \right) \\ &&\hspace{1.4cm}+ \left. {2\left[ {\frac{{{\mu _b}n'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \left( {{\sigma _2}\left( {\lambda + {\theta _v}} \right) + {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right)}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}} \right]\left( {\frac{{ - \sigma }}{{{\lambda ^2}}} + \frac{{ - \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right)} \right\} \\ &&\hspace{1.4cm} - \lambda {\theta _w}{\mu _b}\left\{ {\frac{{l''\left( 0 \right){{\left( {\lambda + {\theta _v}} \right)}^2} - 2{\sigma _1}\left( {l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \rho {\sigma _1}} \right)}}{{{{\left( {\lambda + {\theta _v}} \right)}^3}}}\left( {\frac{1}{\lambda } + \frac{1}{{\gamma \left( 0 \right)}}} \right)} \right. \\ &&\hspace{1.4cm} + \left. {2\frac{{l'\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - \rho {\sigma _1}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}}}\left( {\frac{{ - \sigma }}{{{\lambda ^2}}} + \frac{{ - \gamma '\left( 0 \right)}}{{{{\left( {\gamma \left( 0 \right)} \right)}^2}}}} \right) + 2\frac{\rho }{{\lambda + {\theta _v}}}\left( {\frac{{{\sigma ^2}}}{{{\lambda ^3}}} + \frac{{{{\left( {\gamma '\left( 0 \right)} \right)}^2}}}{{{{\left( {\gamma \left( 0 \right)} \right)}^3}}}} \right)} \right\}, \\ &&r''\left( 0 \right) = 2\lambda {\left( {{\sigma _0}} \right)^2}{\left[ {{{\left( {\lambda + {\theta _w} + {\mu _w} + \alpha } \right)}^2} - 4\lambda {\mu _w}} \right]^{ - \frac{3}{2}}}, \\ &&h''\left( 0 \right) = 2\lambda {\left( {{\sigma _4}} \right)^2}{\left( {\lambda - {\mu _b}} \right)^{ - 3}},\quad \gamma ''\left( 0 \right) = {\mu _w}r''\left( 0 \right),\\ &&g''\left( 0 \right) = - \lambda \beta \left[ {2{\sigma _3}\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right) - \left( {\lambda + \beta } \right){\mu _b}h''\left( 0 \right)} \right]\beta {\mu _b} \\ &&\hspace{1.2cm} - 2\left[ {{\mu _b}{\sigma _3} + \left( {\lambda + \beta } \right)\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right)} \right]{\left( {\beta {\mu _b}} \right)^{ - 3}}, \\ &&n''\left( 0 \right) = - \lambda \varepsilon \left[ {2{\sigma _2}\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right) - \left( {\lambda + \varepsilon } \right){\mu _b}h''\left( 0 \right)} \right]\varepsilon {\mu _b} \\ &&\hspace{1.2cm} - 2\left[ {{\mu _b}{\sigma _2} + \left( {\lambda + \varepsilon } \right)\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right)} \right]{\left( {\varepsilon {\mu _b}} \right)^{ - 3}}, \\ &&l''\left( 0 \right) = \frac{\rho }{{{{\left[ {\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)} \right]}^2}}}\left\{ {\left[ {{\mu _b}n''\left( 0 \right)\left( {\lambda + {\theta _v}} \right) - 2{\mu _b}{\sigma _1}{\sigma _2}} \right]\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)} \right. \\ &&\hspace{1.2cm} - \left. { 2\left[ {\left( {\lambda + {\theta _v}} \right)\left( {{\mu _b}n'\left( 0 \right) - {\sigma _2}} \right) - {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right]\left( {{\sigma _1}\left( {\lambda + \varepsilon } \right) + {\sigma _2}\left( {\lambda + {\theta _v}} \right)} \right)} \right\} \\ &&\hspace{1.2cm} - \rho {\left( {\frac{{\lambda + {\theta _v}}}{{{\mu _b}{\theta _v}}}} \right)^2}\left\{ {\left[ {\frac{{ - \lambda {{\left( {{\sigma _1}} \right)}^2}}}{{{{\left( {\lambda + {\theta _v}} \right)}^3}}} - h''\left( 0 \right)} \right]\frac{{{{\left( {{\mu _b}} \right)}^2}{\theta _v}}}{{\lambda + {\theta _v}}}} \!-\! {2\left[ {{\sigma _4} + {\mu _b}\left( {\frac{{\lambda {\sigma _1}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}}} \!-\! h'\left( 0 \right)} \right)} \right]} \right\} \\ &&\hspace{1.4cm} + 2\lambda \frac{{\left[ {\left( {\lambda + {\theta _v}} \right)\left( {{\mu _b}n'\left( 0 \right) - {\sigma _2}} \right) - {\sigma _1}\left( {\lambda + \varepsilon } \right)} \right] + {{\left( {\lambda + {\theta _v}} \right)}^2}\left( {{\sigma _4} - {\mu _b}h'\left( 0 \right)} \right) + {\mu _b}\lambda {\sigma _1}}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}{{\left( {{\mu _b}} \right)}^2}{\theta _v}}}, \\ &&m''\left( 0 \right) = \Bigg\{ {\frac{{{\theta _v}{\theta _w}}}{{{{\left[ {\left( {\lambda + \varepsilon } \right)\left( {\lambda + {\theta _v}} \right)} \right]}^3}}}}\left[ {\left( {r''\left( 0 \right)\left( {\lambda + \varepsilon } \right) + 2{\mu _b}r'\left( 0 \right)n'\left( 0 \right) + r{\mu _b}n''\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right)\left( {\lambda + \varepsilon } \right)} \right. \\ &&\hspace{1.4cm} - \left. { 2r{\sigma _1}{\sigma _2}\left( {\lambda + \varepsilon } \right)} \right]\left( {\lambda + {\theta _v}} \right)\left( {\lambda + \varepsilon } \right) - 2\left[ {\left( {r'\left( 0 \right)\left( {\lambda + \varepsilon } \right) + {\mu _b}n'\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right)\left( {\lambda + \varepsilon } \right)} \right. \\ &&\hspace{1.4cm} - {\left. { r\left( {\lambda + \varepsilon } \right)\left( {{\sigma _1}\left( {\lambda + \varepsilon } \right) + {\sigma _2}\left( {\lambda + {\theta _v}} \right)} \right)} \right]} \Bigg\}\left( {{\sigma _1}\left( {\lambda + \varepsilon } \right) + {\sigma _2}\left( {\lambda + {\theta _v}} \right)} \right) \\ &&\hspace{1.4cm} + \frac{\alpha }{{{{\left( {\lambda + \beta } \right)}^3}}}\Bigg\{ {\left( {r''\left( 0 \right)\left( {\lambda + \beta } \right) + 2{\mu _b}r'\left( 0 \right)g'\left( 0 \right) + r{\mu _b}g''\left( 0 \right)} \right)\left( {\lambda + \beta } \right)} \\ &&\hspace{1.4cm} - {2{\sigma _3}\left( {\lambda + \beta } \right)\left[ {\left( {r'\left( 0 \right)\left( {\lambda + \beta } \right) + {\mu _b}g'\left( 0 \right)} \right) - r{\sigma _3}} \right]} \Bigg\} \\ &&\hspace{1.4cm} + \frac{{{\theta _w}}}{{{\left( {1 - r} \right){\left( {\lambda + {\theta _v}} \right)}^3}}}\Bigg\{ {\left( {r''\left( 0 \right)\rho + 2r'\left( 0 \right)l'\left( 0 \right) + rl''\left( 0 \right)} \right){{\left( {\lambda + {\theta _v}} \right)}^2}} \\ &&\hspace{1.4cm} - {2{\sigma _1}\left[ {\left( {r'\left( 0 \right)\rho + rl'\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right) - r\rho {\sigma _1}} \right]} \Bigg\} \\ &&\hspace{1.4cm} + 2\left\{ {\frac{{{\theta _v}{\theta _w}}}{{\left( {\lambda + \varepsilon } \right){{\left( {\lambda + {\theta _v}} \right)}^2}}}\left[ {\left( {r'\left( 0 \right)\left( {\lambda + \varepsilon } \right) + {\mu _b}n'\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right)} \right.} \right. \\ &&\hspace{1.4cm} - \left. { r\left( {{\sigma _1}\left( {\lambda + \varepsilon } \right) + {\sigma _2}\left( {\lambda + {\theta _v}} \right)} \right)} \right] + \frac{\alpha }{{\left( {\lambda + \beta } \right)}}\left[ {r'\left( 0 \right)\left( {\lambda + \beta } \right) + r\left( {\beta + {\mu _b}g'\left( 0 \right)} \right) - r{\sigma _3}} \right] \\ &&\hspace{1.4cm} + \left. {\frac{{{\mu _b}{\theta _w}}}{{{{\left( {\lambda + {\theta _v}} \right)}^2}}}\left[ {\left( {r'\left( 0 \right)\rho + rl'\left( 0 \right)} \right)\left( {\lambda + {\theta _v}} \right) - r\rho {\sigma _1}} \right]} \right\}\frac{{{\mu _b}\left( {r'\left( 0 \right) + h'\left( 0 \right)} \right) - {\sigma _4}}}{{{{\left( {{\mu _b}\left( {1 - r} \right)} \right)}^2}}} \\ &&\hspace{1.4cm} + \left( {{\theta _w} + \alpha } \right)r\frac{{{{\left( {{\mu _b}} \right)}^2}\left( {r''\left( 0 \right) + h''\left( 0 \right)} \right)\left( {1 - r} \right) + 2{{\left[ {{\mu _b}\left( {r'\left( 0 \right) + h'\left( 0 \right)} \right) - {\sigma _4}} \right]}^2}}}{{{{\left( {{\mu _b}\left( {1 - r} \right)} \right)}^3}}}. \end{eqnarray*}$

4 数值分析

风力发电路灯为在蓄电池中储存更多的电流, 则需要提高风能转化效率以及减少转化过程中的能量损失. 下面给出系统参数变化对性能指标和费用函数的影响. 本节考虑净流入率 $\sigma$ , 故障率和启动率对系统性能指标和费用函数的影响. 此时空库概率 $a$ 可看作蓄电池内没有电流的概率, 平均库存量 $E\left( C \right)$ 可看作系统的平均电流.

4.1 参数敏感性分析

为直观给出系统参数对性能指标的影响, 下面假设 $\lambda = 8$ , ${\mu _b} = 5$ , ${\mu _w} = 4$ , ${\theta _w} = 4$ , ${\theta _v} = 5$ , ${\alpha} = 2$ , $\beta = 1.5$ , $\varepsilon =1$ , $\sigma = -5$ , ${\sigma _0} = 1$ , ${\sigma _1} =2$ , ${\sigma _2} = 3$ , ${\sigma _3} = 4$ , ${\sigma _4} = 4.5$ .

通过图1可以看出, 当 $\sigma$ 一定, 随着 $\alpha$ 变大, $a$ 随之变大; 当 $\alpha$ 一定, 随着 $\sigma$ 变大, $a$ 随之变小. 这是因为当 $\sigma$ 一定时, $\alpha$ 增大说明系统在工作休假状态下发生故障的概率越大, 当系统故障时, 风力发电机不能及时地将风能转化为电流, 这会导致蓄电池内没有电流的概率变大, 即 $a$ 会随之变大. 通过图2可以看出, 当 $\varepsilon$ 一定时, 随着 $\sigma$ 变大, $a$ 随之变小; 当 $\sigma$ 一定时, 随着 $\varepsilon$ 变大, $a$ 随之变大. 这是因为当 $\sigma$ 一定时, $\varepsilon$ 增大代表当服务台的启动时间增加时, 也就是说风力发电机的启动时间变长, 导致单位时间内的转化量减少, 进而输送到蓄电池中的电流减少, 即 $a$ 会随之变大. 而当 $\varepsilon$ 和 $\alpha$ 一定时, 净流入率的增大表示会有更多的电流输送到蓄电池中, 即 $a$ 会随之变小.

图1

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图1 $a$ 随 $\alpha$ 和 $\sigma$ 的变化曲线

图2

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图2 $a$ 随 $\sigma$ 和 $\varepsilon$ 的变化曲线

通过图3可以看出, 当 $\alpha$ 一定时, 随着 $\sigma$ 变大, $E\left( C \right)$ 随之变大; 当 $\sigma$ 一定时, 随着 $\alpha$ 变大, $E\left( C \right)$ 随之变小. 这是因为当 $\sigma$ 一定时, $\alpha$ 变大代表服务台的故障率增大, 这会导致风力发电机因为故障原因频繁停止工作, 使得平均电流减少, 即 $E\left( C \right)$ 会随之变小. 通过图4可以看出, 当 $\varepsilon$ 一定时, 随着净流入率 $\sigma$ 变大, $E\left( C \right)$ 会随之变大; 当 $\sigma$ 一定时, 随着 $\varepsilon$ 变大, $E\left( C \right)$ 会随之变小. 这是因为当 $\sigma$ 一定时, $\varepsilon$ 变大代表随着服务台启动时间的增加, 风力发电机单位时间内的转化电流量减少, 这会导致平均电流减少, 即 $E\left( C \right)$ 随之变小. 此外, 当 $p$ 和 $\alpha$ 一定时, 净流入率的增大代表会有更多的电流及时地输送到蓄电池中, 导致电流的储存量增加, 即 $E\left( C \right)$ 随之变大, 这也符合现实情况.

图3

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图3 $E\left( C \right)$ 随 $\alpha$ 和 $\sigma$ 的变化曲线

图4

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图4 $E\left( C \right)$ 随 $\sigma$ 和 $\varepsilon$ 的变化曲线

4.2 费用分析

根据给出的性能指标, 记 $C_1$ 表示单位时间内的风能损失费用, $C_2$ 表示系统故障时单位时间内服务台的维修费用, $C_3$ 表示空库时单位时间系统的损耗费用, $C_4$ 表示单位时间内由于蓄电池储存量达到阈值时的损失费用, 则构造系统的费用函数为 $C = {C_1}L + {C_2}BF + {C_3}a + {C_4}E(C)$ , 其中驱动系统的平均等待队长 $L = \sum\limits_{i = 1}^\infty {\sum\limits_{j = 0}^4 {i{\pi _{ij}}} }$ , 系统的稳态故障频度 $BF = \alpha \sum\limits_{i = 0}^\infty {{\pi _{i3}}}$ .

假设 $C_1=1.5$ , $C_2=2$ , $C_3=4$ , $C_4=5$ , $\lambda = 8$ , ${\mu _b} = 5$ , ${\mu _w} = 4$ , ${\theta _w} = 4$ , ${\theta _v} = 5$ , ${\alpha} = 2$ , $\beta = 1.5$ , $\varepsilon =1$ , $\sigma = -5$ , ${\sigma _0} = 1$ , ${\sigma _1} =2$ , ${\sigma _2} = 3$ , ${\sigma _3} = 4$ , ${\sigma _4} = 4.5$ .

通过图5可以看出, 当 $\alpha$ 一定时, 随着 $\varepsilon$ 的增大, 系统所需费用随之增加. 这是因为 $\varepsilon$ 的增大导致系统设备启动时间增加, 使得风力发电机的启动时间较长, 导致单位时间风能的损失费用增加. 当 $\varepsilon$ 一定时, 随着 $\alpha$ 的增大, 所需费用也随之增加. 这是因为 $\alpha$ 的增大, 代表系统发生故障的概率变大, 当频繁发生故障时, 使得系统维修费用、风能损失费用和空库时损耗费用也会随之增加, 最终导致整体成本费用的增加. 在此参数水平下, 在参数水平为 $\varepsilon=1.5, \alpha=1$ 时取得最小的成本费用, 最小成本费用为 8.6.

图5

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图5 $C$ 随 $\alpha$ 和 $\sigma$ 的变化曲线

通过图6可以看出, 当 $\lambda$ 一定时, 随着 $\mu_w$ 的增大, 所需费用随之减小. 这是因为 $\mu_w$ 的增大代表完成风能转化成电流的效率变大, 这使得蓄电池内的电流增加, 从而使空库时的损失费用减少. 此外, 当 $\mu_w$ 一定时, 随着 $\lambda$ 的增大, 所需费用也随之增加. 这是因为随着 $\lambda$ 的增大, 系统将一直以高效率进行转化, 这会使系统发生故障的概率变得更大, 进而使维修费用会增加. 在此参数水平下, 在参数水平为 $\mu_w=1.5, \alpha=6$ 时取得最小的成本费用, 最小成本费用为 16.7.

图6

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图6 $C$ 随 $\sigma$ 和 $\varepsilon$ 的变化曲线

5 总结

本文研究了具有启动时间, 完全故障和双阶段休假 M/M/1 排队系统驱动的流体模型, 利用拟生灭过程和矩阵几何解方法给出流体模型在稳态条件下的空库概率及均值等性能指标. 并结合风力发电路灯的能量转化系统, 通过数值分析得到了系统参数变化对性能指标和费用函数的影响, 以减少能源损耗, 获得更多的电流, 提高风能的利用效率.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]

Virtamo

, Norros

Fluid queue driven by an M/M/1 queue

Queueing Systems, 1994, 16(3/4): 373-386