﻿ 蜂群无人机充电排队优化方法
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1. 中国科学院 空天信息创新研究院, 北京 100094;
2. 中国科学院大学, 北京 100049

Optimization of charging queuing of UAV swarming
WANG Yunzhe1,2, XU Guoning1,2, WANG Sheng1,2, LI Zhaojie1,2, CAI Rong1,2
1. Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100094, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China
Abstract: Aiming at the problem of fast charging queuing of UAV swarming, this paper, for the first time, proposes to use the average queueing length and the average waiting time as the evaluation indicators for UAV swarming charging queuing. Through theoretical analysis and numerical calculation, the charging queuing methods of distributed charging and concentrated charging are compared under the condition of multiple charging platforms. We discover that an intersection appears in the curves corresponding to the evaluation indicators with the increase of service intensity when UAV swarming meets the Poisson condition. Before the intersection of the curves, the service intensity is weak, and the method of concentrated charging is better than distributed charging. After the intersection, with the increase of service intensity, the method of distributed charging gradually becomes better than concentrated charging. This paper provides a fast charging queuing solution which helps UAV swarming complete tasks efficiently.
Keywords: UAV    charging queuing    M/M/1/m model    M/M/n/m model    concentrated charging    distributed charging

1 排队系统基本理论

 $A/B/N/S/C/Z$

 $A/B/N/S$

2 2种排队模型 2.1 M/M/1/m模型

M/M/1/m模型的状态转移图如图 1所示，图中：λμ为上述指数分布对应的参数。

 图 1 M/M/1/m模型状态转移图 Fig. 1 State transition diagram of M/M/1/m model

 $\left\{ \begin{array}{l} \lambda {P_0} = \mu {P_1}\\ \lambda {P_{k - 1}} + \mu {P_{k + 1}} = (\lambda + \mu ){P_k}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \cdots ,m - 1\\ \lambda {P_{m - 1}} = \mu {P_m} \end{array} \right.$ （1）

 $\rho = \frac{\lambda }{\mu }$ （2）

 $L = \left\{ {\begin{array}{*{20}{l}} {\frac{{\rho [(m - 1){\rho ^{m + 1}} - m{\rho ^m} + \rho ]}}{{(1 - \rho )(1 - {\rho ^{m + 1}})}}}&{\rho \ne 1}\\ {\frac{{{m^2} - m}}{{2(m + 1)}}}&{\rho = 1} \end{array}} \right.$ （3）

 $T = \left\{ {\begin{array}{*{20}{l}} {\frac{{\rho [(m - 1){\rho ^{m + 1}} - m{\rho ^m} + \rho ]}}{{\lambda (1 - \rho )(1 - {\rho ^{m + 1}})(1 - {P_m})}}}&{\rho \ne 1}\\ {\frac{{{m^2} - m}}{{2\lambda (m + 1)(1 - {P_m})}}}&{\rho = 1} \end{array}} \right.$ （4）
2.2 M/M/n/m模型

M/M/n/m模型也可采用如图 2所示的状态转移图求解概率分布。

 图 2 M/M/n/m模型状态转移图 Fig. 2 State transition diagram of M/M/n/m model

 $\left\{ {\begin{array}{*{20}{l}} {\lambda {{\hat P}_0} = \mu {{\hat P}_1}}\\ {\lambda {{\hat P}_{k - 1}} + (k + 1)\mu {{\hat P}_{k + 1}} = (\lambda + {k_\mu }){{\hat P}_k}\quad 1 \le k < n}\\ {\lambda {{\hat P}_{k - 1}} + {n_\mu }{{\hat P}_{k + 1}} = (\lambda + n\mu ){{\hat P}_k}\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n \le k < m}\\ {n\mu {{\hat P}_m} = \lambda {{\hat P}_{m - 1}}} \end{array}} \right.$ （5）

 $\hat \rho = \frac{\lambda }{{n\mu }}$ （6）

 $\hat L = \left\{ {\begin{array}{*{20}{l}} {\frac{{{n^n}{{\hat \rho }^{n + 1}}{{\hat P}_0}}}{{n!{{(1 - \hat \rho )}^2}}}[1 - (m - n + 1){{\hat \rho }^{m - n}} + }&{}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m - n){{\hat \rho }^{m - n + 1}}]}&{\hat \rho \ne 1}\\ {\frac{{{n^n}}}{{2n!}}(m - n)(m - n + 1){{\hat P}_0}}&{\hat \rho = 1} \end{array}} \right.$ （7）

 $\begin{array}{l} \hat T = \\ \left\{ {\begin{array}{*{20}{l}} {\frac{{{n^n}{{\hat \rho }^{n + 1}}{{\hat P}_0}}}{{n!{{(1 - \hat \rho )}^2}\lambda (1 - {{\hat P}_m})}}[1 - (m - n + 1) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\hat \rho }^{m - n}} + (m - n){{\hat \rho }^{m - n + 1}}]\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \rho \ne 1\quad }\\ {\frac{{{n^n}{{\hat P}_0}}}{{2n!\lambda (1 - {{\hat P}_m})}}(m - n)(m - n + 1)\ \ \ \ \ \quad \hat \rho = 1} \end{array}} \right. \end{array}$ （8）

 $\begin{array}{l} {{\hat P}_0} = \\ \left\{ {\begin{array}{*{20}{l}} {{{\left[ {\sum\limits_{k = 0}^{n - 1} {\frac{{{{(n\hat \rho )}^k}}}{{k!}}} + \frac{{{{(n\hat \rho )}^n}}}{{n!}} \cdot \frac{{1 - {{\hat \rho }^{m - n + 1}}}}{{1 - \hat \rho }}} \right]}^{ - 1}}}&{\hat \rho \ne 1}\\ {{{\left[ {\sum\limits_{k = 0}^{n - 1} {\frac{{{n^k}}}{{k!}}} + \frac{{{n^n}}}{{n!}}(m - n + 1)} \right]}^{ - 1}}}&{\hat \rho = 1} \end{array}} \right. \end{array}$ （9）
3 2种排队充电方式解析 3.1 分布式充电方式

 图 3 分布式充电示意图 Fig. 3 Schematic diagram of distributed charging

 ${\rho ^\prime } = \frac{\lambda }{{n\mu }}$ （10）
 图 4 分布式充电等效图 Fig. 4 Equivalent diagram of distributed charging

 ${L_{\rm{d}}} = \left\{ {\begin{array}{*{20}{l}} {\frac{{{\rho ^\prime }[(m - 1){\rho ^{\prime m + 1}} - m{\rho ^{\prime m}} + {\rho ^\prime }]}}{{(1 - {\rho ^\prime })(1 - {\rho ^{\prime m + 1}})}}}&{{\rho ^\prime } \ne 1}\\ {\frac{{{m^2} - m}}{{2(m + 1)}}}&{{\rho ^\prime } = 1} \end{array}} \right.$ （11）
 $\begin{array}{l} {T_{\rm{d}}} = \\ \left\{ {\begin{array}{*{20}{l}} {\frac{1}{\lambda } \cdot \frac{{n{\rho ^\prime }}}{{(1 - {\rho ^\prime })(1 - {\rho ^{\prime m}})}}[(m - 1){\rho ^{\prime m + 1}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} m{\rho ^{\prime m}} + {\rho ^\prime }]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rho ^\prime } \ne 1}\\ {\frac{1}{\lambda } \cdot \frac{{n(m - 1)}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ \ {\rho ^\prime } = 1} \end{array}} \right. \end{array}$ （12）
3.2 集中式充电方式

 图 5 集中式充电示意图 Fig. 5 Schematic diagram of concentrated charging

 ${L_{\rm{c}}} = \left\{ \begin{array}{l} \frac{{{n^n}{{\hat \rho }^{n + 1}}{{\hat P}^\prime }_0}}{{n!{{(1 - \hat \rho )}^2}}}\left[ {1 - (nm - n + 1){{\hat \rho }^{nm - n}} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {{\kern 1pt} {\kern 1pt} (nm - n){{\hat \rho }^{nm - n + 1}}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ \hat \rho \ne 1\\ \frac{{{n^n}}}{{2n!}}{{\hat P}^\prime }_0(nm - n)(nm - n + 1){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}\ \hat \rho = 1 \end{array} \right.$ （13）
 $\begin{array}{l} {T_{\rm{c}}} = \\ \left\{ \begin{array}{l} \frac{1}{\lambda } \cdot \frac{{{n^n}{{\hat \rho }^{n + 1}}{{\hat P}^\prime }_0}}{{n!{{(1 - \hat \rho )}^2}(1 - {{\hat P}_{nm}})}}[1 - (nm - n + 1) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\hat \rho }^{nm - n}} + (nm - n){{\hat \rho }^{nm - n + 1}}]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \rho \ne 1\\ \frac{1}{\lambda } \cdot \frac{{{n^n}{{\hat P}^\prime }_0}}{{2n!(1 - {{\hat P}_{nm}})}}(nm - n) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (nm - n + 1){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}\ \ \ \hat \rho = 1 \end{array} \right. \end{array}$ （14）

 $\begin{array}{l} {{\hat P}^\prime }_0 = \\ \left\{ {\begin{array}{*{20}{l}} {{{\left[ {\sum\limits_{k = 0}^{n - 1} {\frac{{{{(n\hat \rho )}^k}}}{{k!}}} + \frac{{{{(n\hat \rho )}^n}}}{{n!}} \cdot \frac{{1 - {{\hat \rho }^{nm - n + 1}}}}{{1 - \hat \rho }}} \right]}^{ - 1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}\ \ \ \ \ \ \hat \rho \ne 1}\\ {{{\left[ {\sum\limits_{k = 0}^{n - 1} {\frac{{{n^k}}}{{k!}}} + \frac{{{n^n}}}{{n!}}(nm - n + 1)} \right]}^{ - 1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \rho = 1} \end{array}} \right. \end{array}$ （15）
 ${\hat P_{nm}} = \frac{{{n^n}{{\hat \rho }^{nm}}{{\hat P}^\prime }_0}}{{n!}}$ （16）
4 数值计算及结果分析

 ${\rho ^\prime } = \frac{\lambda }{{n\mu }} = \hat \rho$ （17）

 服务强度 系统容量 充电平台数量 [0, 2] 6 4 [0, 2] 6 5 [0, 2] 7 6 [0, 2] 7 7 [0, 2] 8 8 [0, 2] 8 9 [0, 2] 9 10
4.1 平均队列长度

 图 6 系统容量为6时的平均队列长度对比 Fig. 6 Comparison of average queuing length when system capacity is 6

 图 7 系统容量为7时的平均队列长度对比 Fig. 7 Comparison of average queuing length when system capacity is 7
 图 8 系统容量为8时的平均队列长度对比 Fig. 8 Comparison of average queuing length when system capacity is 8
 图 9 系统容量为9时的平均队列长度对比 Fig. 9 Comparison of average queuing length when system capacity is 9

 平台数量 系统容量 m=6 m=7 m=8 m=9 n=4 0.70751953125 0.73193359375 0.75205078125 0.76884765625 n=5 0.72783203125 0.75048828125 0.76904296875 0.78447265625 n=6 0.74345703125 0.76455078125 0.78193359375 0.79619140625 n=7 0.75576171875 0.77568359375 0.79208984375 0.80556640625 n=8 0.76611328125 0.78486328125 0.80029296875 0.81318359375 n=9 0.77470703125 0.79267578125 0.80751953125 0.81982421875 n=10 0.78193359375 0.79931640625 0.81357421875 0.82529296875
4.2 平均等待时间

 ${t_{\rm{d}}} = \left\{ {\begin{array}{*{20}{l}} {\frac{{n_\rho ^\prime [(m - 1){\rho ^{\prime m + 1}} - m{\rho ^{\prime m}} + {\rho ^\prime }]}}{{(1 - {\rho ^\prime })(1 - {\rho ^{\prime m}})}}}&{{\rho ^\prime } \ne 1}\\ {\frac{{n(m - 1)}}{2}}&{{\rho ^\prime } = 1} \end{array}} \right.$ （18）
 ${t_{\rm{c}}} = \left\{ \begin{array}{l} \frac{{{n^n}{{\hat \rho }^{n + 1}}{{\hat P}^\prime }_0}}{{n!{{(1 - \hat \rho )}^2}(1 - {{\hat P}_{nm}})}}[1 - (nm - n + 1) \cdot \\ {{\hat \rho }^{nm - n}} + (nm - n){{\hat \rho }^{nm - n + 1}}]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \rho \ne 1\\ \frac{{{n^n}{{\hat P}^\prime }_0}}{{2n!(1 - {{\hat P}_{nm}})}}(nm - n)(nm - n + 1){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \rho = 1 \end{array} \right.$ （19）

 图 10 系统容量为6时的平均等待时间对比 Fig. 10 Comparison of average waiting time when system capacity is 6

 图 11 系统容量为7时的平均等待时间对比 Fig. 11 Comparison of average waiting time when system capacity is 7
 图 12 系统容量为8时的平均等待时间对比 Fig. 12 Comparison of average waiting time when system capacity is 8
 图 13 系统容量为9时的平均等待时间对比 Fig. 13 Comparison of average waiting time when system capacity is 9

 平台数量 系统容量 m=6 m=7 m=8 m=9 n=4 1.019188720703125 1.013821923828125 1.010406689453125 1.008089208984375 n=5 1.014553759765625 1.010406689453125 1.007845263671875 1.006137646484375 n=6 1.011504443359375 1.008211181640625 1.006259619140625 1.004795947265625 n=7 1.009430908203125 1.006747509765625 1.005039892578125 1.003942138671875 n=8 1.007967236328125 1.005649755859375 1.004308056640625 1.003332275390625 n=9 1.006869482421875 1.004917919921875 1.003698193359375 1.002844384765625 n=10 1.006015673828125 1.004308056640625 1.003210302734375 1.002478466796875
5 结论

1) 在服务强度的区间为[0, 2]时，随着服务强度的增强，评价指标对应的曲线存在交叉点。交叉点前，即服务强度较低的情形下，集中式充电的两项评价指标优于分布式充电。但在交叉点后，随着服务强度的增大，分布式充电的两项评价指标优于集中式充电。

2) 对于不同充电平台数量和系统容量给出了选择集中式充电和分布式充电的交叉参考点，对蜂群无人机充电排队提供重要的参考。

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http://dx.doi.org/10.7527/S1000-6893.2020.23928

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#### 文章信息

WANG Yunzhe, XU Guoning, WANG Sheng, LI Zhaojie, CAI Rong

Optimization of charging queuing of UAV swarming

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 323928.
http://dx.doi.org/10.7527/S1000-6893.2020.23928