## 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

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

 基金资助: 国家自然科学基金(62171143)

 Fund supported: NSFC(62171143)

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

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

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.

## 2 驱动系统描述

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). 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} {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. 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 $${B}\left[ R \right] = \left( {\begin{array}{*{20}{c}} { - \left( {\lambda + {\theta _w}} \right)}&{{\theta _w}}&\lambda &0&0&0&0\\ 0& { - \lambda } &0&\lambda &0&0&0\\ {{\mu _w}}&0&\xi &{{\theta _w}}&0&\alpha & {\frac{{\left( {\alpha + {\theta _w}} \right)r}}{{1 - r}}}\\ 0&0&0& { - \left( {\lambda + {\theta _v}} \right)} &{{\theta _v}}&0&\lambda \\ 0&0&0&0&{ - \left( {\lambda + \varepsilon } \right)}&0&{\lambda + \varepsilon }\\ 0&0&0&0&0& { - \left( {\lambda + \beta } \right)} &{\lambda + \beta }\\ {{\mu _b}}&0&0&0&0&0&{ - {\mu _b}} \end{array}} \right),$$ 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. \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 \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. \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. 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 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 $$\frac{{{\rm d}C\left( t \right)}}{{\rm{d}t}} = \left\{ \begin{array}{l} \sigma,\left( {L\left( t \right),J\left( t \right)} \right) = \left( {0,0} \right) \cup \left( {0,1} \right),C\left( t \right) > 0,\\ 0,\left( {L\left( t \right),J\left( t \right)} \right) = \left( {0,0} \right),C\left( t \right) = 0,\\ {\sigma _0},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,0} \right),k \ge 1,\\ {\sigma _1},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,1} \right),k \ge 1,\\ {\sigma _2},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,2} \right),k \ge 1,\\ {\sigma _3},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,3} \right),k \ge 1,\\ {\sigma _4},\left( {L\left( t \right),J\left( t \right)} \right) = \left( {k,4} \right),k \ge 1, \end{array} \right.$$ 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\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

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

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

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

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

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

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