Example 1

Suppose that we have a parallel-series system consisting of two subsystems connected in series. The first subsystem has two components connected in parallel and the

49 0 10 20 30 40 50 60 70 80 90 100 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 t h(t) α=0.029, β=−1.597×10−3, γ=2.608×10−5 ,θ=0.786

(a) Bath-tub shape.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 7 8 t h(t) α=8,β=−3,γ=0.3,θ=3

(b) Upside down bath-tub shape. Figure 2.3: Hazard function of the GQFRD(α, β, γ, θ) for different parameter values. second subsystem has three components connected in parallel. This means that

n = 2, m1 = 2, m2 = 3 and the total number of components is m = 5. All of the system’s components are assumed to be independent and identically distributed, with lifetimes that behave according to a generalized quadratic failure rate distribu- tion with parametersα = 0.029, β=₋1.597_×10−3_{, γ = 2.608×10}−5 _{and θ= 0.786.} The values of these parameters derive from real data as described in Aarset (1987) and Sarhan and Alghamdi (2009). The hazard function for each component in the system takes bath-tub shape, see Figure 2.3a. We define:

1. A(_ki,j),i= 0,1,2,j = 0,1,2,3 andk =i+j, to represent a reduction method that requires us to reduce the failure rate of i components from the first subsystem and j from the second subsystem.

2. B_k(i,j), i = 0,1,2, j = 0,1,2,3 and k = i +j, to represent hot duplication methods when i components are added to the first subsystem and j to the second subsystem.

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3. C_k(i,j), i = 0,1,2, j = 0,1,2,3 and k = i+j, to represent cold duplication methods with perfect switch whenicomponents are added to the first subsystem and j components are added to the second subsystem.

4. D_k(i,j),i= 0,1,2,j = 0,1,2,3 andk =i+j, to represent cold duplication meth- ods with imperfect switch when i components are added to the first subsystem and j components are added to the second subsystem.

Table 2.1: Hot survival reliability equivalence factors.

ω B(0₁,1) B(0₂,2) B₃(0,3) B(1₁,0) B(1₂,1) B₃(1,2) B(1₄,3) B(2₂,0) B₃(2,1) B(2₄,2) B(2₅,3) 0.1 0.6680 0.5083 0.4109 0.5514 0.3799 0.2846 0.2224 0.3799 0.2549 0.1824 0.1341 A(0₁,1) 0.5 0.4590 0.2541 0.1526 0.1961 0.0178 - - 0.0178 - - - 0.9 0.1995 0.0458 0.0107 - - - - 0.1 0.8022 0.6803 0.5953 0.7151 0.5663 0.4703 0.4011 0.5663 0.4380 0.3529 0.2901 A(0₂,2) 0.5 0.6621 0.4764 0.3581 0.4120 0.1115 - - 0.1115 - - - 0.9 0.4379 0.2056 0.0946 - - - - 0.1 0.8587 0.7618 0.6895 0.7903 0.6639 0.5756 0.5086 0.6639 0.5447 0.4601 0.3946 A(0₃,3) 0.5 0.7545 0.5985 0.4882 0.5397 0.2130 0.0259 0.0160 0.2130 0.0183 - - 0.9 0.5732 0.3430 0.0298 0.0846 0.0527 0.0337 0.0237 0.0527 0.0040 - - 0.1 0.7704 0.6403 0.5544 0.6767 0.5259 0.4346 0.3720 0.5259 0.4051 0.3301 0.2777 A(1₁,0) 0.5 0.6866 0.5325 0.4431 0.4828 0.2933 0.1962 0.1380 0.2933 0.1483 0.0751 0.0330 0.9 0.7320 0.6503 0.6239 0.2516 0.0903 0.0355 0.0142 0.0903 - - - 0.1 0.8538 0.7685 0.7116 0.7925 0.6927 0.6322 0.5908 0.6927 0.6127 0.5633 0.5291 A(1₂,1) 0.5 0.7870 0.6802 0.6178 0.6456 0.5120 0.4424 0.4000 0.5120 0.4076 0.3525 0.3185 0.9 0.7890 0.7251 0.7044 0.4146 0.2863 0.2389 0.2180 0.2863 0.1830 0.1442 0.1256 0.1 0.8897 0.8210 0.7732 0.8406 0.7571 0.7040 0.6666 0.7571 0.6865 0.6412 0.6090 A(1₃,2) 0.5 0.8335 0.7449 0.6912 0.7152 0.5965 0.5312 0.4901 0.5965 0.4976 0.4428 0.4079 0.9 0.8219 0.7663 0.7481 0.4816 0.3539 0.3045 0.2821 0.3539 0.2438 0.2000 0.1783 0.1 0.9107 0.8522 0.8103 0.8692 0.7958 0.7474 0.7126 0.7958 0.7312 0.6884 0.6575 A(1₄,3) 0.5 0.8621 0.7847 0.7363 0.7582 0.6483 0.5854 0.5448 0.6483 0.5522 0.4971 0.4612 0.9 0.8443 0.7939 0.7773 0.5216 0.3905 0.3381 0.3141 0.3905 0.2726 0.2244 0.2002 0.1 0.8702 0.7815 0.7158 0.8076 0.6927 0.6135 0.5544 0.6927 0.5862 0.5124 0.4570 A(2₂,0) 0.5 0.8231 0.7170 0.6473 0.6791 0.5120 0.4060 0.3315 0.5120 0.3455 0.2341 0.1482 0.9 0.8544 0.8044 0.7876 0.4911 0.2863 0.1756 0.1092 0.2863 - - - 0.1 0.9029 0.8392 0.7936 0.8576 0.7779 0.7255 0.6879 0.7779 0.7079 0.6619 0.6288 A(2₃,1) 0.5 0.8618 0.7832 0.7337 0.7561 0.6427 0.5771 0.5345 0.6427 0.5423 0.4843 0.4465 0.9 0.8739 0.8319 0.8179 0.5896 0.4603 0.4052 0.3792 0.4603 0.3329 0.2769 0.2479 0.1 0.9209 0.8686 0.8307 0.8838 0.8175 0.7733 0.7411 0.8175 0.7583 0.7187 0.6897 A(2₄,2) 0.5 0.8845 0.8186 0.7768 0.7957 0.6996 0.6433 0.6064 0.6996 0.6132 0.5622 0.5284 0.9 0.8876 0.8503 0.8378 0.6368 0.5234 0.4749 0.4518 0.5234 0.4106 0.3598 0.3331 0.1 0.9329 0.8875 0.8543 0.9008 0.8427 0.8032 0.7741 0.8427 0.7897 0.7536 0.7268 A(2₅,3) 0.5 0.9000 0.8420 0.8049 0.8217 0.7355 0.6840 0.6499 0.7355 0.6561 0.6085 0.5765 0.9 0.8979 0.8638 0.8525 0.6669 0.5603 0.5141 0.4920 0.5603 0.4521 0.4026 0.3761

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Table 2.2: Cold survival reliability equivalence factors with perfect switch.

ω C(0₁,1) C(0₂,2) C₃(0,3) C(1₁,0) C(1₂,1) C₃(1,2) C(1₄,3) C(2₂,0) C₃(2,1) C(2₄,2) C(2₅,3) 0.1 0.2577 0.1064 0.0503 0.1105 - - - 0.0005 - - - A(0₁,1) 0.5 0.2776 0.0966 0.0361 0.0084 - - - - 0.9 0.1391 0.0236 0.0042 - - - - 0.1 0.4411 0.2508 0.1588 0.2567 - - - 0.0136 - - - A(0₂,2) 0.5 0.5006 0.2777 0.1625 0.0754 - - - - 0.9 0.3638 0.1466 0.0627 - - - - 0.1 0.5477 0.3521 0.2472 0.3587 - - - 0.0428 - - - A(0₃,3) 0.5 0.6198 0.4071 0.2776 0.1615 - - - 0.0064 - - - 0.9 0.5054 0.2726 0.1557 0.0784 - - - - 0.1 0.4079 0.2465 0.1795 0.2511 - - - 0.1056 - - - A(1₁,0) 0.5 0.5516 0.3877 0.3185 0.2783 - - - 0.1014 - - - 0.9 0.7031 0.6346 0.6174 0.2077 0.0544 0.0017 - 0.0707 - - - 0.1 0.6145 0.5089 0.4658 0.5119 0.1832 0.1180 0.0983 0.4186 0.1180 0.0851 0.0743 A(1₂,1) 0.5 0.6935 0.5788 0.5300 0.5014 0.2751 0.1924 0.1559 0.3726 0.1748 0.1062 0.0773 0.9 0.7664 0.7127 0.6993 0.3802 0.2558 0.2027 0.1793 0.2699 0.1405 0.0898 0.0686 0.1 0.6881 0.5896 0.5474 0.5925 0.2336 0.1512 0.1260 0.4998 0.1512 0.1091 0.0952 A(1₃,2) 0.5 0.7562 0.6569 0.6129 0.5867 0.3620 0.2691 0.2252 0.4630 0.2482 0.1612 0.1209 0.9 0.8023 0.7555 0.7436 0.4482 0.3223 0.2656 0.2397 0.3370 0.1957 0.1348 0.1075 0.1 0.7327 0.6386 0.5968 0.6414 0.2629 0.1704 0.1419 0.5487 0.1703 0.1229 0.1073 A(1₄,3) 0.5 0.7947 0.7049 0.6638 0.6390 0.4131 0.3125 0.2634 0.5176 0.2892 0.1903 0.1433 0.9 0.8267 0.7841 0.7732 0.4879 0.3571 0.2962 0.2680 0.3726 0.2196 0.1515 0.1208 0.1 0.5888 0.4220 0.3419 0.4273 - - - 0.2422 - - - A(2₂,0) 0.5 0.7310 0.6004 0.5368 0.4969 - - - 0.2777 - - - 0.9 0.8371 0.7944 0.7833 0.4440 0.2194 0.0376 - 0.2517 - - - 0.1 0.7096 0.6087 0.5645 0.6117 0.2343 0.1512 0.1260 0.5141 0.1512 0.1091 0.0952 A(2₃,1) 0.5 0.7934 0.7013 0.6588 0.6329 0.3958 0.2907 0.2404 0.5059 0.2667 0.1677 0.1234 0.9 0.8593 0.8236 0.8144 0.5574 0.4254 0.3595 0.3278 0.4417 0.2713 0.1871 0.1474 0.1 0.7598 0.6719 0.6322 0.6746 0.2911 0.1892 0.1577 0.5857 0.1892 0.1366 0.1192 A(2₄,2) 0.5 0.8271 0.7494 0.7134 0.6913 0.4822 0.3814 0.3295 0.5813 0.3571 0.2474 0.1908 0.9 0.8746 0.8429 0.8347 0.6086 0.4927 0.4343 0.4059 0.5071 0.3547 0.2751 0.2350 0.1 0.7910 0.7102 0.6728 0.7127 0.3237 0.2110 0.1759 0.6283 0.2110 0.1523 0.1329 A(2₅,3) 0.5 0.8496 0.7804 0.7479 0.7279 0.5321 0.4323 0.3789 0.6265 0.4074 0.2912 0.2278 0.9 0.8860 0.8571 0.8496 0.6405 0.5311 0.4751 0.4476 0.5448 0.3975 0.3179 0.2768

For this scenario, in Tables 2.1, 2.2 and 2.3 the SREFs for hot and cold (perfect and imperfect) duplication are calculated using Matlab according to the above formulae where ω is chosen to be 0.1,0.5,0.9 and the imperfect switch has a constant failure rate λ= 0.01. For more discussions based on the results presented in the Tables 2.1, 2.2 and 2.3, it may be observed that:

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denote as A(0₁,1)) by setting ρ = 0.6680 improves the reliability of the system like adding one component to the second subsystem (which we denote asB₁(0,1)) according to a hot duplication method where the reliability function of the system is chosen to beω = 0.1, see Table 2.1.

• Reducing the failure rate of each component belonging to the set A(2₅,3) of the system components by settingρ= 0.8496 improves the reliability of the system like adding a set C₁(0,1) of components to the system according to a cold dupli- cation method with perfect switch where the reliability function of the system is chosen to be ω = 0.5, see Table 2.2.

• Reducing the failure rate of each component belonging to the set A(2₅,3) of the system components by setting factor ρ= 0.3661 improves the reliability of the system like adding a setD₅(2,3) of components to the system according to a cold duplication method with perfect switch where the reliability function of the system is chosen to beω = 0.9, see Table 2.3.

• Missing values of the SREFs mean that it is not possible to reduce the failure rate for the set A of components in order to improve the system reliability to be equivalent with the system reliability that can be obtained by improving the sets B, C, D of components according to duplication methods.

• In the same manner, one can interpret the other results presented in Tables 2.1, 2.2 and 2.3.

Tables 2.4, 2.5 and 2.6 present the MREFs for hot and cold (perfect and imperfect) duplication. Based on the results presented in those tables, we see that:

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• The modified system that can be obtained by improving the set H₁(0,1), where

H =B, C, Dof the system components, according to hot and cold (perfect and imperfect) duplication has the same mean time to failure of that system which can be obtained be reducing the failure rate of each component belonging to the set A(0₁,1) by factors ρ= 0.465,0.257,0.367 respectively.

• Empty cells of MREFs mean that it is not possible to reduce the failure rate of the set A components in order to improve the mean time to failure of the system to be equivalent with the mean time to failure of the system that can be obtained by improving the sets B, C, D of components according to the duplication methods.

• In the same manner, one can interpret the other results presented in Tables 2.4, 2.5 and 2.6.

Table 2.7 presents the mean time to failure of the modified systems assuming hot and cold duplication methods, the latter with perfect and imperfect switch, assuming two constant failure rates λ = 0.01 and λ = 0.05. The mean time to failure of the original system is 53.063. From this table, one can conclude that

• If the failure rate of the imperfect switch is λ= 0.01, then

M T T F < M T T F(B)< M T T F(D) < M T T F(C)

• If the failure rate of the imperfect switch is λ= 0.05, then

M T T F < M T T F(D) < M T T F(B) < M T T F(C)

• This implies that the improvement due to hot duplication is better than using cold duplication with low reliability switch.

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Figure 2.4 explains the improvement strategies to calculate the SREFs. Figure 2.5 presents reliability functions of the original and some modified systems. From this figure, one may observe that, for this scenario:

• Improving the reliability of all components according to cold duplication with perfect switch gives the best system.

• For the same number of components

R(t)< R(B)(t)< R(D)(t)< R(C)(t) where λ= 0.01.

Figures 2.6 and 2.7 present the behaviour of MTTF against the appropriate re- duction factor ρ. It seems from these two figures that:

• MTTFs are non-decreasing with decreasing ρ for all possible setsA.

• Reducing the failure rate of one or two components from the first subsystem gives a better system than that obtained by reducing the failure rate of one or two components in the second subsystem, see Figure 2.6. This means that improving a component from the subsystem with the smaller number of compo- nents is better than improving a component from the subsystem with the larger number of components.

• Reducing the failure rates of all components in the system gives the best system, see Figure 2.7.

• It is not possible to reduce the failure rate of the sets A(0₁,1) or A(0₂,2) of the system components to reach the mean time to failure which we can achieve by

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improving the sets B₂(1,1) or C₃(1,2) of the system components according to hot duplication and cold duplication with perfect switch respectively, see Figure 2.6.

• Improving a number of components selected from two subsystems, with equal numbers if they are even, gives a better system than that obtained by improving the number of components selected from the same subsystem or selected from the two subsystems with unequal numbers, see Figure 2.7.