January  2021, 17(1): 1-28. doi: 10.3934/jimo.2019096

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

1. 

Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani, Rajasthan, 333 031, India

2. 

Mathematics Department, Faculty of Science, Menofia University, Menofia, 32511, Egypt

* Corresponding author: Chandra Shekhar

Received  December 2018 Revised  February 2019 Published  January 2021 Early access  July 2019

Fund Project: 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)

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.

Citation: Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial and Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096
References:
[1]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Math. Proceedings of the Cambridge Phil. Society, 51 (1955), 433-441.  doi: 10.1017/S0305004100030437.

[2]

U. C. Gupta and T. S. S. S. Rao, A recursive method to compute the steady state probabilities of the machine interference model: $(M/G/1)/K$, Comp. and Ops. Research, 21 (1994), 597-605. 

[3]

U. C. Gupta and T. S. S. S. Rao, On the $M/G/1$ machine interference model with spares, European J. of Oper. Research, 89 (1996), 164-171. 

[4]

L. Haque and M. J. Armstrong, A survey of the machine interference problem, European J. of Oper. Research, 179 (2007), 469-482.  doi: 10.1016/j.ejor.2006.02.036.

[5]

P. Hokstad, A supplementary variable technique applied to the $M/G/1$ queue, Scandinavian J. of Stats., 2 (1975), 95-98. 

[6]

H. I. Huang and J. C. Ke, Comparative analysis on a redundant repairable system with different configurations, Engineering Computations, 26 (2009), 422-439.  doi: 10.1108/02644400910959197.

[7]

M. Jain, Availability prediction of imperfect fault coverage system with reboot and common cause failure, Int. J. of Oper. Research, 17 (2013), 374-397.  doi: 10.1504/IJOR.2013.054441.

[8]

M. JainC. Shekhar and S. Shukla, Markov model for switching failure of warm spares in machine repair system, J. of Reliability and Stat. Studies, 7 (2014), 57-68. 

[9]

T. Jiang, L. Liu and J. Li, Analysis of the $M/G/1$ queue in multi-phase random environment with disasters, J. of Math. Analysis and Applications, 430, 857–873. doi: 10.1016/j.jmaa.2015.05.028.

[10]

J. C. KeT. H. Liu and D. Y. Yang, Machine repairing systems with standby switching failure, Comp. and Indust. Engineering, 99 (2016), 223-228.  doi: 10.1016/j.cie.2016.07.016.

[11]

A. Kumar and M. Agarwal, A review of standby systems, IEEE Transaction on Reliability, 29 (1980), 290-294. 

[12]

C. C. Kuo and J. C. Ke, Comparative analysis of standby systems with unreliable server and switching failure, Reliability Engineering and System Safety, 145 (2016), 74-82.  doi: 10.1016/j.ress.2015.09.001.

[13]

E. E. Lewis, Introduction to Reliability Engineering, 2$^{nd}$ ed., John Wiley & Sons, New York, 1996.

[14]

C. D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, J. of Indust. and Mgmt. Optimization, 11 (2015), 83-104.  doi: 10.3934/jimo.2015.11.83.

[15]

M. Manglik and M. Ram, Multistate multifailures system analysis with reworking strategy and imperfect fault coverage, in Advances in System Reliability Engineering, Academic Press, 2019, 243-265. doi: 10.1016/B978-0-12-815906-4.00010-5.

[16]

M. S. Moustafa, Reliability analysis of $K$-out-of-$N$:$G$ systems with dependent failures and imperfect coverage, Reliability Engineering and System Safety, 58 (1997), 15-17. 

[17]

H. Pham, Reliability analysis of a high voltage system with dependent failures and imperfect coverage, Reliability Engineering and System Safety, 37 (1992), 25-28.  doi: 10.1016/0951-8320(92)90054-O.

[18]

C. ShekharM. Jain and S. Bhatia, Fuzzy analysis of machine repair problem with switching failure and reboot, J. of Reliability and Stat. Studies, 7 (2014), 41-55. 

[19]

C. ShekharM. JainA. A. Raina and J. Iqbal, Reliability prediction of fault tolerant machining system with reboot and recovery delay, Int. J. of System Assurance Engineering and Mgmt., 9 (2018), 377-400.  doi: 10.1007/s13198-017-0680-y.

[20]

C. ShekharM. JainA. A. Raina and R. P. Mishra, Sensitivity analysis of repairable redundant system with switching failure and geometric reneging, Decision Science Letters, 6 (2017), 337-350.  doi: 10.5267/j.dsl.2017.2.004.

[21]

C. ShekharA. A. RainaA. Kumar and J. Iqbal, A survey on queues in machining system: Progress from 2010 to 2017, Yugoslav J. of Ops. Research, 27 (2017), 391-413.  doi: 10.2298/YJOR161117006R.

[22]

K. H. Wang and Y. J. Chen, Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures, Appl. Math. and Computation, 215 (2009), 384-394.  doi: 10.1016/j.amc.2009.05.023.

[23]

K. H. WangJ. H. Su and D. Y. Yang, Analysis and optimization of an $M/G/1$ machine repair problem with multiple imperfect coverage, Appl. Math. and Computation, 242 (2014), 590-600.  doi: 10.1016/j.amc.2014.05.132.

[24]

D. Y. Yang and C. L. Tsao, Reliability and availability analysis of standby systems with working vacations and retrial of failed components, Reliability Engineering and System Safety, 182 (2019), 46-55.  doi: 10.1016/j.ress.2018.09.020.

show all references

References:
[1]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Math. Proceedings of the Cambridge Phil. Society, 51 (1955), 433-441.  doi: 10.1017/S0305004100030437.

[2]

U. C. Gupta and T. S. S. S. Rao, A recursive method to compute the steady state probabilities of the machine interference model: $(M/G/1)/K$, Comp. and Ops. Research, 21 (1994), 597-605. 

[3]

U. C. Gupta and T. S. S. S. Rao, On the $M/G/1$ machine interference model with spares, European J. of Oper. Research, 89 (1996), 164-171. 

[4]

L. Haque and M. J. Armstrong, A survey of the machine interference problem, European J. of Oper. Research, 179 (2007), 469-482.  doi: 10.1016/j.ejor.2006.02.036.

[5]

P. Hokstad, A supplementary variable technique applied to the $M/G/1$ queue, Scandinavian J. of Stats., 2 (1975), 95-98. 

[6]

H. I. Huang and J. C. Ke, Comparative analysis on a redundant repairable system with different configurations, Engineering Computations, 26 (2009), 422-439.  doi: 10.1108/02644400910959197.

[7]

M. Jain, Availability prediction of imperfect fault coverage system with reboot and common cause failure, Int. J. of Oper. Research, 17 (2013), 374-397.  doi: 10.1504/IJOR.2013.054441.

[8]

M. JainC. Shekhar and S. Shukla, Markov model for switching failure of warm spares in machine repair system, J. of Reliability and Stat. Studies, 7 (2014), 57-68. 

[9]

T. Jiang, L. Liu and J. Li, Analysis of the $M/G/1$ queue in multi-phase random environment with disasters, J. of Math. Analysis and Applications, 430, 857–873. doi: 10.1016/j.jmaa.2015.05.028.

[10]

J. C. KeT. H. Liu and D. Y. Yang, Machine repairing systems with standby switching failure, Comp. and Indust. Engineering, 99 (2016), 223-228.  doi: 10.1016/j.cie.2016.07.016.

[11]

A. Kumar and M. Agarwal, A review of standby systems, IEEE Transaction on Reliability, 29 (1980), 290-294. 

[12]

C. C. Kuo and J. C. Ke, Comparative analysis of standby systems with unreliable server and switching failure, Reliability Engineering and System Safety, 145 (2016), 74-82.  doi: 10.1016/j.ress.2015.09.001.

[13]

E. E. Lewis, Introduction to Reliability Engineering, 2$^{nd}$ ed., John Wiley & Sons, New York, 1996.

[14]

C. D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, J. of Indust. and Mgmt. Optimization, 11 (2015), 83-104.  doi: 10.3934/jimo.2015.11.83.

[15]

M. Manglik and M. Ram, Multistate multifailures system analysis with reworking strategy and imperfect fault coverage, in Advances in System Reliability Engineering, Academic Press, 2019, 243-265. doi: 10.1016/B978-0-12-815906-4.00010-5.

[16]

M. S. Moustafa, Reliability analysis of $K$-out-of-$N$:$G$ systems with dependent failures and imperfect coverage, Reliability Engineering and System Safety, 58 (1997), 15-17. 

[17]

H. Pham, Reliability analysis of a high voltage system with dependent failures and imperfect coverage, Reliability Engineering and System Safety, 37 (1992), 25-28.  doi: 10.1016/0951-8320(92)90054-O.

[18]

C. ShekharM. Jain and S. Bhatia, Fuzzy analysis of machine repair problem with switching failure and reboot, J. of Reliability and Stat. Studies, 7 (2014), 41-55. 

[19]

C. ShekharM. JainA. A. Raina and J. Iqbal, Reliability prediction of fault tolerant machining system with reboot and recovery delay, Int. J. of System Assurance Engineering and Mgmt., 9 (2018), 377-400.  doi: 10.1007/s13198-017-0680-y.

[20]

C. ShekharM. JainA. A. Raina and R. P. Mishra, Sensitivity analysis of repairable redundant system with switching failure and geometric reneging, Decision Science Letters, 6 (2017), 337-350.  doi: 10.5267/j.dsl.2017.2.004.

[21]

C. ShekharA. A. RainaA. Kumar and J. Iqbal, A survey on queues in machining system: Progress from 2010 to 2017, Yugoslav J. of Ops. Research, 27 (2017), 391-413.  doi: 10.2298/YJOR161117006R.

[22]

K. H. Wang and Y. J. Chen, Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures, Appl. Math. and Computation, 215 (2009), 384-394.  doi: 10.1016/j.amc.2009.05.023.

[23]

K. H. WangJ. H. Su and D. Y. Yang, Analysis and optimization of an $M/G/1$ machine repair problem with multiple imperfect coverage, Appl. Math. and Computation, 242 (2014), 590-600.  doi: 10.1016/j.amc.2014.05.132.

[24]

D. Y. Yang and C. L. Tsao, Reliability and availability analysis of standby systems with working vacations and retrial of failed components, Reliability Engineering and System Safety, 182 (2019), 46-55.  doi: 10.1016/j.ress.2018.09.020.

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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[20]

Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1145-1185. doi: 10.3934/dcdss.2020068

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