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$ \bf{M/G/1} $ fault-tolerant machining system with imperfection

  • * Corresponding author: Chandra Shekhar

    * Corresponding author: Chandra Shekhar 

The third author is supported by CSIR, New Delhi (India) grant: 09/719(0068)/2015-EMR-1. Also, First three authors are supported by DST FIST (India) grant SR/FST/MSI-090/2013(C)

Abstract / Introduction Full Text(HTML) Figure(17) / Table(18) Related Papers Cited by
  • The internet of things (IoT) is an emerging archetype of technology for the guaranteed quality of services (QoS). The availability of the uninterrupted power supply (UPS) is one of the most challenging criteria in the successful implementation of the service system of IoT. In this paper, we consider a fault-tolerant power generation system of finite operating machines along with warm standby machine provisioning. The time-to-failure for each of the operating and standby machines are assumed to be exponentially distributed. The time-to-repair by the single service facility for the failed machine follows the arbitrary distribution. For modeling purpose, we have also incorporated realistic machining behaviors like imperfect coverage of the failure of machines, switching failure of standby machine, reboot delay, switch over delay, etc. For the evaluation of the explicit expression for steady-state probabilities of the system, the only required input is the Laplace-Stieltjes transform (LST) of the repair time distribution. The step-wise recursive procedure, illustrative examples, and numerical results have been presented for the following different type of repair time distribution: exponential ($ M $), $ n $-stage Erlang ($ Er_{n} $), deterministic ($ D $), uniform ($ U(a, b) $), $ n $-stage generalized Erlang ($ GE_n $) and hyperexponential ($ HE_n $). Concluding remarks and future scopes have also been included.

    Mathematics Subject Classification: Primary: 68, 90, 94, 60, 62; Secondary: 68M15, 90B25, 94C12, 60K10, 62N05.

    Citation:

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  • Figure 1.  The state transition diagram

    Figure 2.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\mu$ for different repair time distribution

    Figure 3.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\lambda$ for different repair time distribution

    Figure 4.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\nu$ for different repair time distribution

    Figure 5.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\beta$ for different repair time distribution

    Figure 6.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\sigma$ for different repair time distribution

    Figure 7.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $p$ for different repair time distribution

    Figure 8.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $c$ for different repair time distribution

    Figure 9.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $q$ for different repair time distribution

    Figure 10.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\mu$ for different repair time distribution

    Figure 11.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\lambda$ for different repair time distribution

    Figure 12.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\nu$ for different repair time distribution

    Figure 13.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\beta$ for different repair time distribution

    Figure 14.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $\sigma$ for different repair time distribution

    Figure 15.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $p$ for different repair time distribution

    Figure 16.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $c$ for different repair time distribution

    Figure 17.  State probability $P_{2, 1}$ and availability of the system ($Av$) wrt $q$ for different repair time distribution

    Table 1.  State probabilities and availability of the system

    Distribution $ M $ $ Er_3 $ $ D $ $ U(a, b) $ $ GE_4 $ $ HE_2 $
    Parameter(s) $ \mu=25 $ $ \mu=25 $ $ \mu=25 $ $ a=0.02 $ $ \mu_1=60 $ $ \alpha_1=0.2 $
    $ b=0.06 $ $ \mu_2=100 $ $ \alpha_2=0.8 $
    $ \mu_3=120 $ $ \mu_1=15 $
    $ \mu_4=200 $ $ \mu_2=30 $
    $ P_{21} $ 0.9123644 0.9123430 0.9123320 0.9123347 0.9123417 0.9123711
    $ P_{20} $ 0.0437935 0.0443790 0.0446796 0.0446037 0.0444134 0.0436102
    $ P_{10} $ 0.0371515 0.0365876 0.0362980 0.0363711 0.0365544 0.0373280
    $ Q_{11} $ 0.0054742 0.0054741 0.0054740 0.0054740 0.0054741 0.0054742
    $ R_O $ 0.0007299 0.0007299 0.0007299 0.0007299 0.0007299 0.0007299
    $ R_S $ 0.0004866 0.0004866 0.0004866 0.0004866 0.0004866 0.0004866
    $ Av $ 0.9561579 0.9567219 0.9570115 0.9569385 0.9567551 0.9559813
     | Show Table
    DownLoad: CSV

    Table 2.  Performance indices corresponding to Fig. 2

    Indices Distribution $ \mu $
    24 26 28 30 32 34 36 38 40
    $ P_{2, 1} $ $ M $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9125
    $ Er_3 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ D $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ Av $ $ M $ 0.9556 0.9557 0.9559 0.9560 0.9561 0.9562 0.9562 0.9563 0.9564
    $ Er_3 $ 0.9561 0.9563 0.9564 0.9565 0.9566 0.9567 0.9568 0.9569 0.9569
    $ D $ 0.9564 0.9566 0.9567 0.9568 0.9569 0.9570 0.9571 0.9571 0.9572
     | Show Table
    DownLoad: CSV

    Table 3.  Performance indices corresponding to Fig. 3

    Indices Distribution $ \lambda $
    0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
    $ P_{2, 1} $ $ M $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9351 0.9274 0.9198 0.9124
    $ Er_3 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ D $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ U(a, b) $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ GE_4 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ HE_2 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9351 0.9274 0.9198 0.9124
    $ Av $ $ M $ 0.9905 0.9860 0.9816 0.9773 0.9730 0.9687 0.9645 0.9603 0.9562
    $ Er_3 $ 0.9905 0.9861 0.9818 0.9775 0.9732 0.9690 0.9649 0.9608 0.9567
    $ D $ 0.9905 0.9861 0.9818 0.9775 0.9733 0.9692 0.9651 0.9610 0.9570
    $ U(a, b) $ 0.9905 0.9861 0.9818 0.9775 0.9733 0.9691 0.9650 0.9610 0.9569
    $ GE_4 $ 0.9905 0.9861 0.9818 0.9775 0.9732 0.9690 0.9649 0.9608 0.9568
    $ HE_2 $ 0.9905 0.9860 0.9816 0.9772 0.9729 0.9686 0.9644 0.9602 0.9560
     | Show Table
    DownLoad: CSV

    Table 4.  Performance indices corresponding to Fig. 4

    Indices Distribution $ \nu $
    0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250
    $ P_{2, 1} $ $ M $ 0.9179 0.9170 0.9161 0.9151 0.9142 0.9133 0.9124 0.9114 0.9105
    $ Er_3 $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ D $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ U(a, b) $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ GE_4 $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ HE_2 $ 0.9179 0.9170 0.9161 0.9151 0.9142 0.9133 0.9124 0.9115 0.9105
    $ Av $ $ M $ 0.9565 0.9564 0.9564 0.9563 0.9563 0.9562 0.9562 0.9561 0.9561
    $ Er_3 $ 0.9570 0.9569 0.9569 0.9568 0.9568 0.9568 0.9567 0.9567 0.9566
    $ D $ 0.9572 0.9572 0.9572 0.9571 0.9571 0.9570 0.9570 0.9570 0.9569
    $ U(a, b) $ 0.9572 0.9571 0.9571 0.9571 0.9570 0.9570 0.9569 0.9569 0.9569
    $ GE_4 $ 0.9570 0.9570 0.9569 0.9569 0.9568 0.9568 0.9568 0.9567 0.9567
    $ HE_2 $ 0.9563 0.9563 0.9562 0.9562 0.9561 0.9560 0.9560 0.9559 0.9559
     | Show Table
    DownLoad: CSV

    Table 5.  Performance indices corresponding to Fig. 5

    Indices Distribution $ \beta $
    50 55 60 65 70 75 80 85 90
    $ P_{2, 1} $ $ M $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9125
    $ Er_3 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ D $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ U(a, b) $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ GE_4 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ HE_2 $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9126
    $ Av $ $ M $ 0.9556 0.9557 0.9559 0.9560 0.9561 0.9562 0.9562 0.9563 0.9564
    $ Er_3 $ 0.9561 0.9563 0.9564 0.9565 0.9566 0.9567 0.9568 0.9569 0.9569
    $ D $ 0.9564 0.9566 0.9567 0.9568 0.9569 0.9570 0.9571 0.9571 0.9572
    $ U(a, b) $ 0.9564 0.9565 0.9566 0.9568 0.9569 0.9569 0.9570 0.9571 0.9571
    $ GE_4 $ 0.9562 0.9563 0.9565 0.9566 0.9567 0.9568 0.9568 0.9569 0.9569
    $ HE_2 $ 0.9554 0.9556 0.9557 0.9558 0.9559 0.9560 0.9561 0.9561 0.9562
     | Show Table
    DownLoad: CSV

    Table 6.  Performance indices corresponding to Fig. 6

    Indices Distribution $ \sigma $
    30 35 40 45 50 55 60 65 70
    $ P_{2, 1} $ $ M $ 0.9090 0.9102 0.9111 0.9118 0.9124 0.9128 0.9132 0.9135 0.9138
    $ Er_3 $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ D $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ U(a, b) $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ GE_4 $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ HE_2 $ 0.9091 0.9102 0.9111 0.9118 0.9124 0.9128 0.9132 0.9135 0.9138
    $ Av $ $ M $ 0.9527 0.9539 0.9549 0.9556 0.9562 0.9566 0.9570 0.9574 0.9577
    $ Er_3 $ 0.9532 0.9545 0.9554 0.9561 0.9567 0.9572 0.9576 0.9579 0.9582
    $ D $ 0.9535 0.9548 0.9557 0.9564 0.9570 0.9575 0.9579 0.9582 0.9585
    $ U(a, b) $ 0.9535 0.9547 0.9556 0.9564 0.9569 0.9574 0.9578 0.9581 0.9584
    $ GE_4 $ 0.9533 0.9545 0.9554 0.9562 0.9568 0.9572 0.9576 0.9580 0.9583
    $ HE_2 $ 0.9525 0.9537 0.9547 0.9554 0.9560 0.9565 0.9569 0.9572 0.9575
     | Show Table
    DownLoad: CSV

    Table 7.  Performance indices corresponding to Fig. 7

    Indices Distribution $ p $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9154 0.9144 0.9134 0.9124
    $ Er_3 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9153 0.9143 0.9133 0.9123
    $ D $ 0.9204 0.9194 0.9184 0.9174 0.9163 0.9153 0.9143 0.9133 0.9123
    $ U(a, b) $ 0.9204 0.9194 0.9184 0.9174 0.9163 0.9153 0.9143 0.9133 0.9123
    $ GE_4 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9153 0.9143 0.9133 0.9123
    $ HE_2 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9154 0.9144 0.9134 0.9124
    $ Av $ $ M $ 0.9646 0.9635 0.9625 0.9614 0.9604 0.9593 0.9583 0.9572 0.9562
    $ Er_3 $ 0.9652 0.9641 0.9630 0.9620 0.9609 0.9599 0.9588 0.9578 0.9567
    $ D $ 0.9655 0.9644 0.9633 0.9623 0.9612 0.9602 0.9591 0.9581 0.9570
    $ U(a, b) $ 0.9654 0.9643 0.9633 0.9622 0.9611 0.9601 0.9590 0.9580 0.9569
    $ GE_4 $ 0.9652 0.9641 0.9631 0.9620 0.9610 0.9599 0.9589 0.9578 0.9568
    $ HE_2 $ 0.9644 0.9634 0.9623 0.9612 0.9602 0.9591 0.9581 0.9570 0.9560
     | Show Table
    DownLoad: CSV

    Table 8.  Performance indices corresponding to Fig. 8

    Indices Distribution $ c $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9113 0.9118 0.9124 0.9129
    $ Er_3 $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ D $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ U(a, b) $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ GE_4 $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ HE_2 $ 0.9085 0.9091 0.9096 0.9102 0.9107 0.9113 0.9118 0.9124 0.9129
    $ Av $ $ M $ 0.9521 0.9527 0.9533 0.9538 0.9544 0.9550 0.9556 0.9562 0.9567
    $ Er_3 $ 0.9527 0.9532 0.9538 0.9544 0.9550 0.9556 0.9561 0.9567 0.9573
    $ D $ 0.9530 0.9535 0.9541 0.9547 0.9553 0.9558 0.9564 0.9570 0.9576
    $ U(a, b) $ 0.9529 0.9535 0.9540 0.9546 0.9552 0.9558 0.9564 0.9569 0.9575
    $ GE_4 $ 0.9527 0.9533 0.9539 0.9544 0.9550 0.9556 0.9562 0.9568 0.9573
    $ HE_2 $ 0.9519 0.9525 0.9531 0.9537 0.9542 0.9548 0.9554 0.9560 0.9566
     | Show Table
    DownLoad: CSV

    Table 9.  Performance indices corresponding to Fig. 9

    Indices Distribution $ q $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9031 0.9047 0.9062 0.9077 0.9093 0.9108 0.9124 0.9139 0.9155
    $ Er_3 $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ D $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ U(a, b) $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ GE_4 $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ HE_2 $ 0.9031 0.9047 0.9062 0.9077 0.9093 0.9108 0.9124 0.9139 0.9155
    $ Av $ $ M $ 0.9465 0.9481 0.9497 0.9513 0.9529 0.9545 0.9562 0.9578 0.9594
    $ Er_3 $ 0.9470 0.9486 0.9502 0.9519 0.9535 0.9551 0.9567 0.9584 0.9600
    $ D $ 0.9473 0.9489 0.9505 0.9521 0.9538 0.9554 0.9570 0.9586 0.9603
    $ U(a, b) $ 0.9473 0.9489 0.9505 0.9521 0.9537 0.9553 0.9569 0.9586 0.9602
    $ GE_4 $ 0.9471 0.9487 0.9503 0.9519 0.9535 0.9551 0.9568 0.9584 0.9600
    $ HE_2 $ 0.9463 0.9479 0.9495 0.9511 0.9527 0.9544 0.9560 0.9576 0.9592
     | Show Table
    DownLoad: CSV

    Table 10.  State probabilities and availability of the system

    Distribution $ M $ $ Er_3 $ $ D $ $ U(a, b) $ $ GE_4 $ $ HE_2 $
    Parameter(s) $ \mu=25 $ $ \mu=25 $ $ \mu=25 $ $ a=0.02 $ $ \mu_1=60 $ $ \alpha_1=0.2 $
    $ b=0.06 $ $ \mu_2=100 $ $ \alpha_2=0.8 $
    $ \mu_3=120 $ $ \mu_1=15 $
    $ \mu_4=200 $ $ \mu_2=30 $
    $ P_{21} $ 0.9123644 0.9123430 0.9123320 0.9123347 0.9123417 0.9123711
    $ P_{20} $ 0.0437935 0.0443790 0.0446796 0.0446037 0.0444134 0.0436102
    $ P_{10} $ 0.0371515 0.0365876 0.0362980 0.0363711 0.0365544 0.0373280
    $ Q_{11} $ 0.0054742 0.0054741 0.0054740 0.0054740 0.0054741 0.0054742
    $ R_O $ 0.0007299 0.0007299 0.0007299 0.0007299 0.0007299 0.0007299
    $ R_S $ 0.0004866 0.0004866 0.0004866 0.0004866 0.0004866 0.0004866
    $ Av $ 0.9561579 0.9567219 0.9570115 0.9569385 0.9567551 0.9559813
     | Show Table
    DownLoad: CSV

    Table 11.  Performance indices corresponding to Fig. 2

    Indices Distribution $ \mu $
    24 26 28 30 32 34 36 38 40
    $ P_{2, 1} $ $ M $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9125
    $ Er_3 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ D $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ Av $ $ M $ 0.9556 0.9557 0.9559 0.9560 0.9561 0.9562 0.9562 0.9563 0.9564
    $ Er_3 $ 0.9561 0.9563 0.9564 0.9565 0.9566 0.9567 0.9568 0.9569 0.9569
    $ D $ 0.9564 0.9566 0.9567 0.9568 0.9569 0.9570 0.9571 0.9571 0.9572
     | Show Table
    DownLoad: CSV

    Table 12.  Performance indices corresponding to Fig. 3

    Indices Distribution $ \lambda $
    0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
    $ P_{2, 1} $ $ M $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9351 0.9274 0.9198 0.9124
    $ Er_3 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ D $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ U(a, b) $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ GE_4 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9350 0.9274 0.9198 0.9123
    $ HE_2 $ 0.9749 0.9667 0.9586 0.9507 0.9428 0.9351 0.9274 0.9198 0.9124
    $ Av $ $ M $ 0.9905 0.9860 0.9816 0.9773 0.9730 0.9687 0.9645 0.9603 0.9562
    $ Er_3 $ 0.9905 0.9861 0.9818 0.9775 0.9732 0.9690 0.9649 0.9608 0.9567
    $ D $ 0.9905 0.9861 0.9818 0.9775 0.9733 0.9692 0.9651 0.9610 0.9570
    $ U(a, b) $ 0.9905 0.9861 0.9818 0.9775 0.9733 0.9691 0.9650 0.9610 0.9569
    $ GE_4 $ 0.9905 0.9861 0.9818 0.9775 0.9732 0.9690 0.9649 0.9608 0.9568
    $ HE_2 $ 0.9905 0.9860 0.9816 0.9772 0.9729 0.9686 0.9644 0.9602 0.9560
     | Show Table
    DownLoad: CSV

    Table 13.  Performance indices corresponding to Fig. 4

    Indices Distribution $ \nu $
    0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250
    $ P_{2, 1} $ $ M $ 0.9179 0.9170 0.9161 0.9151 0.9142 0.9133 0.9124 0.9114 0.9105
    $ Er_3 $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ D $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ U(a, b) $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ GE_4 $ 0.9179 0.9170 0.9160 0.9151 0.9142 0.9133 0.9123 0.9114 0.9105
    $ HE_2 $ 0.9179 0.9170 0.9161 0.9151 0.9142 0.9133 0.9124 0.9115 0.9105
    $ Av $ $ M $ 0.9565 0.9564 0.9564 0.9563 0.9563 0.9562 0.9562 0.9561 0.9561
    $ Er_3 $ 0.9570 0.9569 0.9569 0.9568 0.9568 0.9568 0.9567 0.9567 0.9566
    $ D $ 0.9572 0.9572 0.9572 0.9571 0.9571 0.9570 0.9570 0.9570 0.9569
    $ U(a, b) $ 0.9572 0.9571 0.9571 0.9571 0.9570 0.9570 0.9569 0.9569 0.9569
    $ GE_4 $ 0.9570 0.9570 0.9569 0.9569 0.9568 0.9568 0.9568 0.9567 0.9567
    $ HE_2 $ 0.9563 0.9563 0.9562 0.9562 0.9561 0.9560 0.9560 0.9559 0.9559
     | Show Table
    DownLoad: CSV

    Table 14.  Performance indices corresponding to Fig. 5

    Indices Distribution $ \beta $
    50 55 60 65 70 75 80 85 90
    $ P_{2, 1} $ $ M $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9125
    $ Er_3 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ D $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ U(a, b) $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ GE_4 $ 0.9118 0.9119 0.9121 0.9122 0.9123 0.9123 0.9124 0.9125 0.9125
    $ HE_2 $ 0.9118 0.9120 0.9121 0.9122 0.9123 0.9124 0.9124 0.9125 0.9126
    $ Av $ $ M $ 0.9556 0.9557 0.9559 0.9560 0.9561 0.9562 0.9562 0.9563 0.9564
    $ Er_3 $ 0.9561 0.9563 0.9564 0.9565 0.9566 0.9567 0.9568 0.9569 0.9569
    $ D $ 0.9564 0.9566 0.9567 0.9568 0.9569 0.9570 0.9571 0.9571 0.9572
    $ U(a, b) $ 0.9564 0.9565 0.9566 0.9568 0.9569 0.9569 0.9570 0.9571 0.9571
    $ GE_4 $ 0.9562 0.9563 0.9565 0.9566 0.9567 0.9568 0.9568 0.9569 0.9569
    $ HE_2 $ 0.9554 0.9556 0.9557 0.9558 0.9559 0.9560 0.9561 0.9561 0.9562
     | Show Table
    DownLoad: CSV

    Table 15.  Performance indices corresponding to Fig. 6

    Indices Distribution $ \sigma $
    30 35 40 45 50 55 60 65 70
    $ P_{2, 1} $ $ M $ 0.9090 0.9102 0.9111 0.9118 0.9124 0.9128 0.9132 0.9135 0.9138
    $ Er_3 $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ D $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ U(a, b) $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ GE_4 $ 0.9090 0.9102 0.9111 0.9118 0.9123 0.9128 0.9132 0.9135 0.9138
    $ HE_2 $ 0.9091 0.9102 0.9111 0.9118 0.9124 0.9128 0.9132 0.9135 0.9138
    $ Av $ $ M $ 0.9527 0.9539 0.9549 0.9556 0.9562 0.9566 0.9570 0.9574 0.9577
    $ Er_3 $ 0.9532 0.9545 0.9554 0.9561 0.9567 0.9572 0.9576 0.9579 0.9582
    $ D $ 0.9535 0.9548 0.9557 0.9564 0.9570 0.9575 0.9579 0.9582 0.9585
    $ U(a, b) $ 0.9535 0.9547 0.9556 0.9564 0.9569 0.9574 0.9578 0.9581 0.9584
    $ GE_4 $ 0.9533 0.9545 0.9554 0.9562 0.9568 0.9572 0.9576 0.9580 0.9583
    $ HE_2 $ 0.9525 0.9537 0.9547 0.9554 0.9560 0.9565 0.9569 0.9572 0.9575
     | Show Table
    DownLoad: CSV

    Table 16.  Performance indices corresponding to Fig. 7

    Indices Distribution $ p $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9154 0.9144 0.9134 0.9124
    $ Er_3 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9153 0.9143 0.9133 0.9123
    $ D $ 0.9204 0.9194 0.9184 0.9174 0.9163 0.9153 0.9143 0.9133 0.9123
    $ U(a, b) $ 0.9204 0.9194 0.9184 0.9174 0.9163 0.9153 0.9143 0.9133 0.9123
    $ GE_4 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9153 0.9143 0.9133 0.9123
    $ HE_2 $ 0.9204 0.9194 0.9184 0.9174 0.9164 0.9154 0.9144 0.9134 0.9124
    $ Av $ $ M $ 0.9646 0.9635 0.9625 0.9614 0.9604 0.9593 0.9583 0.9572 0.9562
    $ Er_3 $ 0.9652 0.9641 0.9630 0.9620 0.9609 0.9599 0.9588 0.9578 0.9567
    $ D $ 0.9655 0.9644 0.9633 0.9623 0.9612 0.9602 0.9591 0.9581 0.9570
    $ U(a, b) $ 0.9654 0.9643 0.9633 0.9622 0.9611 0.9601 0.9590 0.9580 0.9569
    $ GE_4 $ 0.9652 0.9641 0.9631 0.9620 0.9610 0.9599 0.9589 0.9578 0.9568
    $ HE_2 $ 0.9644 0.9634 0.9623 0.9612 0.9602 0.9591 0.9581 0.9570 0.9560
     | Show Table
    DownLoad: CSV

    Table 17.  Performance indices corresponding to Fig. 8

    Indices Distribution $ c $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9113 0.9118 0.9124 0.9129
    $ Er_3 $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ D $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ U(a, b) $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ GE_4 $ 0.9085 0.9090 0.9096 0.9101 0.9107 0.9112 0.9118 0.9123 0.9129
    $ HE_2 $ 0.9085 0.9091 0.9096 0.9102 0.9107 0.9113 0.9118 0.9124 0.9129
    $ Av $ $ M $ 0.9521 0.9527 0.9533 0.9538 0.9544 0.9550 0.9556 0.9562 0.9567
    $ Er_3 $ 0.9527 0.9532 0.9538 0.9544 0.9550 0.9556 0.9561 0.9567 0.9573
    $ D $ 0.9530 0.9535 0.9541 0.9547 0.9553 0.9558 0.9564 0.9570 0.9576
    $ U(a, b) $ 0.9529 0.9535 0.9540 0.9546 0.9552 0.9558 0.9564 0.9569 0.9575
    $ GE_4 $ 0.9527 0.9533 0.9539 0.9544 0.9550 0.9556 0.9562 0.9568 0.9573
    $ HE_2 $ 0.9519 0.9525 0.9531 0.9537 0.9542 0.9548 0.9554 0.9560 0.9566
     | Show Table
    DownLoad: CSV

    Table 18.  Performance indices corresponding to Fig. 9

    Indices Distribution $ q $
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
    $ P_{2, 1} $ $ M $ 0.9031 0.9047 0.9062 0.9077 0.9093 0.9108 0.9124 0.9139 0.9155
    $ Er_3 $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ D $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ U(a, b) $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ GE_4 $ 0.9031 0.9046 0.9062 0.9077 0.9092 0.9108 0.9123 0.9139 0.9155
    $ HE_2 $ 0.9031 0.9047 0.9062 0.9077 0.9093 0.9108 0.9124 0.9139 0.9155
    $ Av $ $ M $ 0.9465 0.9481 0.9497 0.9513 0.9529 0.9545 0.9562 0.9578 0.9594
    $ Er_3 $ 0.9470 0.9486 0.9502 0.9519 0.9535 0.9551 0.9567 0.9584 0.9600
    $ D $ 0.9473 0.9489 0.9505 0.9521 0.9538 0.9554 0.9570 0.9586 0.9603
    $ U(a, b) $ 0.9473 0.9489 0.9505 0.9521 0.9537 0.9553 0.9569 0.9586 0.9602
    $ GE_4 $ 0.9471 0.9487 0.9503 0.9519 0.9535 0.9551 0.9568 0.9584 0.9600
    $ HE_2 $ 0.9463 0.9479 0.9495 0.9511 0.9527 0.9544 0.9560 0.9576 0.9592
     | Show Table
    DownLoad: CSV
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