图共 5个 表共 4
    • 图  FIG. 494.  FIG. 494.

      Figure FIG. 494..  FIG. 494.

    • 图  1  排列图A4,2、子图X4、节点24

      Figure 1.  Arrangement graph A4,2、subgraph X4、node 24

    • 图  2  $A_{{\rm{5,4}}}$的一组子图$A_{{\rm{5,4}}}^{{\rm{4,3}}}$

      Figure 2.  A subgraph $A_{{\rm{5,4}}}^{{\rm{4,3}}}$ of $A_{{\rm{5,4}}}$

    • 图  3  排列图A5,4子图可靠性的近似计算

      Figure 3.  Approximate calculations of subgraph reliability of A5,4

    • 图  4  排列图A3,2

      Figure 4.  Arrangement graph A3,2

    • Reliability
      One position Two positions Three positions Four positions
      ${R_1} = C_k^{\rm{1}}C_n^{\rm{4}}{p^{4A_{n - 1}^{k - 1}}}$ $\begin{aligned} & {R_2} = C_k^{\rm{2} }(2C_n^{\rm{1} }C_{n - 1}^{\rm{2} }{p^{4A_{n - 1}^{k - 1} - 2A_{n - 2}^{k - 2} } } + \\& 2C_n^{\rm{1} }C_{n - 1}^{\rm{3} }{p^{4A_{n - 1}^{k - 1} - 3A_{n - 2}^{k - 2} } } + 2C_n^{\rm{2} }C_{n - 2}^{\rm{1} }\\& {p^{4A_{n - 1}^{k - 1} - 3A_{n - 2}^{k - 2} } }+ \\& C_n^{\rm{2} }{p^{4A_{n - 1}^{k - 1} - {\rm{2} }A_{n - 2}^{k - 2} } } + \\& C_n^{\rm{2} }C_{n - 2}^{\rm{2} }{p^{4A_{n - 1}^{k - 1} - {\rm{4} }A_{n - 2}^{k - 2} } } \end{aligned}$ $\begin{aligned} & {R_{\rm{3} } } = C_k^{\rm{3} }({\rm{3} }C_n^{\rm{1} }C_{n - 1}^{\rm{1} }\\& {p^{4A_{n - 1}^{k - 1} - 2A_{n - 2}^{k - 2} } } + \\& {\rm{3} }C_n^{\rm{1} }C_{n - 1}^{\rm{2} }{p^{4A_{n - 1}^{k - 1} - {\rm{4} }A_{n - 2}^{k - 2} } } +\\& {\rm{6} }C_n^{\rm{1} }C_{n{\rm{ - 1} } }^{\rm{1} }C_{n - 2}^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{4} }A_{n - 2}^{k - 2} + A_{n - {\rm{3} } }^{k - {\rm{3} } } } }+\\& {\rm{3} }C_n^{\rm{1} }C_{n{\rm{ - 1} } }^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{3} }A_{n - 2}^{k - 2} } } \end{aligned}$ $\begin{aligned} & {R_{\rm{4} } } = C_k^{\rm{4} }(C_n^{\rm{1} }{p^{4A_{n - 1}^{k - 1} } } + \\& {\rm{4} }C_n^{\rm{1} }C_{n - 1}^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{3} }A_{n - 2}^{k - 2} } }+ \\& {\rm{3} }C_n^{\rm{1} }C_{n - 1}^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{4} }A_{n - 2}^{k - 2} } }+ \\& {\rm{6} }C_n^{\rm{1} }C_{n{\rm{ - 1} } }^{\rm{1} }C_{n - 2}^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{5} }A_{n - 2}^{k - 2} + {\rm{2} }A_{n - {\rm{3} } }^{k - {\rm{3} } } }}+\\& C_n^{\rm{1} }C_{n{\rm{ - 1} } }^{\rm{1} }C_{n - {\rm{2} } }^{\rm{1} }C_{n - {\rm{3} } }^{\rm{1} }{p^{4A_{n - 1}^{k - 1} - {\rm{6} }A_{n - 2}^{k - 2} + {\rm{4} }A_{n - {\rm{3} } }^{k - {\rm{3} } } - A_{n - {\rm{4} } }^{k - {\rm{4} } } }}) \end{aligned}$

      表 1  $\displaystyle\sum\limits_{i < j < l < q} {{r_{\left( {i,j,l,q} \right)}}\left( p \right)} $的计算

      Table 1.  Calculation of $\displaystyle\sum\limits_{i < j < l < q} {{r_{\left( {i,j,l,q} \right)}}\left( p \right)} $

    • Cycles Monte Carlo error/%
      100 0.218
      500 0.185
      1 000 0.084

      表 2  A3,2的子图可靠性蒙特卡罗近似计算

      Table 2.  Approximate calculation of the Monte Carlo Method for A3,2 subgraph

    • An,k Lower
      bound
      Algorithm1 Ref[1] Monte
      Carlo algorithm
      Upper
      bound
      A3,2 −4.048 0.144 0.999 5 0.988 4.335
      A4,2 −6.446 −0.766 0.999 5 0.994 4.913
      A5,4 0.281 0.297 0.336 0.301 0.313
      A6,3 0.344 0.425 0.509 0.434 0.506
      A7,3 0.139 0.140 0.149 0.141 0.142

      表 3  排列图An,k子图可靠性的界以及近似计算(n=3,4,5,6,7;k=2,3,4)

      Table 3.  Bounds and the approximate calculation of subgraph reliability for An,kn=3,4,5,6,7; k=2,3,4)

    • p Real value Ref [1] Monte Carlo
      0.5 0.718 8 0.822 0.718
      0.6 0.848 4 0.931 3 0.842
      0.7 0.934 8 0.982 4 0.933
      0.8 0.981 0.997 8 0.986
      0.9 0.997 8 0.999 95 0.997

      表 4  排列图A3,2子图可靠性的近似计算

      Table 4.  Approximate calculation of subgraph reliability for A3,2