Time-dependent reliability analysis for repairable consecutive-k-out-of-n:F system

Figures & data

Table 1. Time-dependent transition performance probabilities for the linear repairable $C(2,5:F)$ system when one component fails.

t	${\mathrm{\Psi }}_{-10}\left(t\right)$	${\mathrm{\Psi }}_{-1-1}\left(t\right)$	${\mathrm{\Psi }}_{-1-2}\left(t\right)$	${\mathrm{\Psi }}_{-1-3}\left(t\right)$	${\mathrm{\Psi }}_{-11}\left(t\right)$
0.5	0.2100	0.3310	0.1515	0.0095	0.2980
1.0	0.1675	0.2124	0.1156	0.0120	0.4925
1.5	0.1224	0.1517	0.0838	0.0103	0.6319
2.0	0.0887	0.1096	0.0608	0.0079	0.7330
2.5	0.0642	0.0794	0.0441	0.0059	0.8064
3.0	0.0465	0.0575	0.0320	0.0043	0.8596
3.5	0.0337	0.0417	0.0232	0.0031	0.8983
4.0	0.0244	0.0302	0.0168	0.0023	0.9263
4.5	0.0177	0.0219	0.0122	0.0016	0.9466
5.0	0.0128	0.0159	0.0088	0.0012	0.9613

Table 2. Time-dependent transition performance probabilities for the circular repairable C(2,6:F) system when two components fail.

t	${\mathrm{\Psi }}_{-20}\left(t\right)$	${\mathrm{\Psi }}_{-2-1}\left(t\right)$	${\mathrm{\Psi }}_{-2-2}\left(t\right)$	${\mathrm{\Psi }}_{-2-3}\left(t\right)$	${\mathrm{\Psi }}_{-21}\left(t\right)$
0.5	0.0569	0.1553	0.2387	0.0366	0.5125
1.0	0.0603	0.1033	0.0936	0.0197	0.7230
1.5	0.0441	0.0661	0.0485	0.0098	0.8316
2.0	0.0294	0.0422	0.0284	0.0052	0.8947
2.5	0.0191	0.0269	0.0175	0.0030	0.9335
3.0	0.0123	0.0172	0.0110	0.0018	0.9578
3.5	0.0078	0.0109	0.0070	0.0011	0.9731
4.0	0.0050	0.0069	0.0044	0.0007	0.9829
4.5	0.0032	0.0044	0.0028	0.0005	0.9891
5.0	0.0020	0.0028	0.0018	0.0003	0.9931

Table 3. The non-zero eigenvalues of Q for three parameter sets.

σ	$\lambda =0.5,\mu =1.5$	$\lambda =0.5,\mu =3.0$	$\lambda =0.25,\mu =1.5$
${\sigma }_1$	$-8.3126$	$-11.1812$	$-5.5906$
${\sigma }_2$	$-6.1101$	$-8.2729$	$-4.1365$
${\sigma }_3$	$-4.4007$	$-5.9962$	$-2.9981$
${\sigma }_4$	$-3.1176$	$-4.3836$	$-2.1918$
${\sigma }_5$	$-2.3398$	$-3.3967$	$-1.6938$
${\sigma }_6$	$-1.5893$	$-1.9523$	$-0.9761$
${\sigma }_7$	$-0.6299$	$-0.3171$	$-0.1586$

Figure 1. The plot of the reliability $R_0\left(t\right)$ of the linear repairable $C(3,8:F)$ system against time t.

Table 4. The constant coefficients in Equation (27(27) $\begin{array}{rl}{R}_0\left(t\right)& =1-{\mathrm{\Psi }}_{01}\left(t\right)\\ & =1-(A_0^{01}+A_1^{01}{\mathrm{e}}^{{\sigma }_{1}t}+A_2^{01}{\mathrm{e}}^{{\sigma }_{2}t}+\cdots \\ & +A_7^{01}{\mathrm{e}}^{{\sigma }_{7}t}),\end{array}$(27) ) for the three parameter sets.

$A_k^{01}$	$\lambda =0.5,\mu =1.5$	$\lambda =0.5,\mu =3.0$	$\lambda =0.25,\mu =1.5$
$A_0^{01}$	1.0000	1.0000	1.0000
$A_1^{01}$	$-0.0045$	$-0.0015$	$-0.0019$
$A_2^{01}$	0.0026	$-0.0015$	$-0.0015$
$A_3^{01}$	$-0.0251$	$-0.0069$	$-0.0069$
$A_4^{01}$	$-0.1525$	$-0.0162$	$-0.0162$
$A_5^{01}$	0.0116	$-0.0084$	$-0.0085$
$A_6^{01}$	1.1515	0.2937	0.2938
$A_7^{01}$	$-1.9836$	$-1.2589$	$-1.2587$

Table 5. The non-zero eigenvalues of Q for three parameter sets.

σ	$\lambda =0.5,\mu =1.5$	$\lambda =0.5,\mu =3.0$	$\lambda =0.25,\mu =1.5$
${\sigma }_1$	$-8.2922$	$-11.1417$	$-5.5709$
${\sigma }_2$	$-6.0051$	$-8.0973$	$-4.0486$
${\sigma }_3$	$-4.2440$	$-5.7354$	$-2.8677$
${\sigma }_4$	$-2.9306$	$-3.9541$	$-1.9770$
${\sigma }_5$	$-1.8225$	$-2.2057$	$-1.1028$
${\sigma }_6$	$-0.7057$	$-0.3659$	$-0.1830$

Figure 2. The plot of the reliability $R_0\left(t\right)$ of the circular repairable $C(3,8:F)$ system against time t.

Table 6. The constant coefficients in Equation (27(27) $\begin{array}{rl}{R}_0\left(t\right)& =1-{\mathrm{\Psi }}_{01}\left(t\right)\\ & =1-(A_0^{01}+A_1^{01}{\mathrm{e}}^{{\sigma }_{1}t}+A_2^{01}{\mathrm{e}}^{{\sigma }_{2}t}+\cdots \\ & +A_7^{01}{\mathrm{e}}^{{\sigma }_{7}t}),\end{array}$(27) ) for the three parameter sets.

$A_k^{01}$	$\lambda =0.5,\mu =1.5$	$\lambda =0.5,\mu =3.0$	$\lambda =0.25,\mu =1.5$
$A_0^{01}$	1.0000	1.0000	1.0000
$A_1^{01}$	$-0.0060$	$-0.0026$	$-0.0026$
$A_2^{01}$	0.0051	$-0.0025$	$-0.0025$
$A_3^{01}$	$-0.0890$	$-0.0165$	$-0.0165$
$A_4^{01}$	$-0.1391$	$-0.0212$	$-0.0212$
$A_5^{01}$	1.2074	0.3139	0.3140
$A_6^{01}$	$-1.9784$	$-1.2712$	$-1.2710$