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April  2017, 13(2): 901-916. doi: 10.3934/jimo.2016052

Throughput of flow lines with unreliable parallel-machine workstations and blocking

1. 

Department of Statistics, Changwon National University, Changwon, Gyeongnam 641-773, Korea

2. 

School of Industrial Engineering and Naval Architecture, Changwon National University, Changwon, Gyeongnam 641-773, Korea

Received  May 2015 Revised  April 2016 Published  August 2016

Flow lines in which workstations and buffers are linked along a single flow path one after another are widely used for modeling manufacturing systems. In this paper we consider the flow lines with multiple independent unreliable machines at each workstation and blocking. The processing times, time to failure and time to repair of each machine are assumed to exponentially distributed and blocking after service blocking protocol is also assumed. An approximate analysis for throughput in the flow lines is presented. The method developed here is based on the decomposition method using the subsystems with three workstations including virtual station and two buffers between workstations. Some numerical examples are presented for accuracy of approximation.

Citation: Yang Woo Shin, Dug Hee Moon. Throughput of flow lines with unreliable parallel-machine workstations and blocking. Journal of Industrial & Management Optimization, 2017, 13 (2) : 901-916. doi: 10.3934/jimo.2016052
References:
[1] T. Altiok, Performance Analysis of Manufacturing Systems, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-1924-8.  Google Scholar
[2] J. A. Buzzacott and J. G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice-Hall, 1993.   Google Scholar
[3]

Y. Dallery and B. Gershwin, Manufacturing flow line systems: A review of models and analytical results, Queueing Systems, 12 (1992), 3-94.  doi: 10.1007/BF01158636.  Google Scholar

[4]

A. C. DiamantidisC. T. Papadopoulos and C. Heavey, Approximate analysis of serial flow lines with multiple parallel-machine stations, IIE Transactions, 39 (2007), 361-375.  doi: 10.1080/07408170600838423.  Google Scholar

[5]

A. C. Diamantidis and C. T. Papadopoulos, Exact analysis of a two-workstation one-buffer flow line with parallel unreliable machines, European Journal of Operational Research, 197 (2009), 592-580.   Google Scholar

[6] S. B. Gershwin, Manufacturing Systems Engineering, Prentice-Hall, 1994.   Google Scholar
[7] W. D. KeltonR. P. Sadowski and D. A. Sadowski, Simulation with ARENA, 2edition, McGraw-Hill, New York, 1998.   Google Scholar
[8]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadephia, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[9]

J. Li, Overlapping decomposition: A system-theoretic method for modeling and analysis of complex manufacturing systems, IEEE Transactions on Automation Science and Engineering, 2 (2005), 40-53.  doi: 10.1109/TASE.2004.835576.  Google Scholar

[10]

J. LiD. E. BlumenfeldN. Huang and J. M. Alden, Throughput analysis of production systems: Recent advances and future topics, International Journal of Production Research, 47 (2009), 3823-3851.  doi: 10.1080/00207540701829752.  Google Scholar

[11]

J. Li and S. M. Meerkov, Production Systems Engineering, Springer, 2009. doi: 10.1007/978-0-387-75579-3.  Google Scholar

[12]

J. LiuS. YangA. Wu and S. J. Hu, Multi-stage throughput analysis of a two-stage manufacturing system with parallel unreliable machines and a finite buffer, European Journal of Operational Research, 219 (2012), 296-304.  doi: 10.1016/j.ejor.2011.12.025.  Google Scholar

[13]

A. Patchong and D. Willaeys, Modeling and analysis of an unreliable flow line composed of parallel-machine stages, IIE Transactions, 33 (2001), 559-568.  doi: 10.1080/07408170108936854.  Google Scholar

[14] H. G. Perros, Queueing Networks with Blocking, Oxford University Press, 1994.   Google Scholar
[15]

Y. W. Shin and D. H. Moon, Approximation of throughput in tandem queues with multiple servers and blocking, Applied Mathematical Modelling, 38 (2014), 6122-6132.  doi: 10.1016/j.apm.2014.05.015.  Google Scholar

[16]

M. van VuurenI. J. B. F. Adan and S. A. E. Resing-Sassen, Performance analysis of multi-server tandem queues with finite buffers and blocking, OR Spectrum, 27 (2005), 315-338.  doi: 10.1007/s00291-004-0189-z.  Google Scholar

show all references

References:
[1] T. Altiok, Performance Analysis of Manufacturing Systems, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-1924-8.  Google Scholar
[2] J. A. Buzzacott and J. G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice-Hall, 1993.   Google Scholar
[3]

Y. Dallery and B. Gershwin, Manufacturing flow line systems: A review of models and analytical results, Queueing Systems, 12 (1992), 3-94.  doi: 10.1007/BF01158636.  Google Scholar

[4]

A. C. DiamantidisC. T. Papadopoulos and C. Heavey, Approximate analysis of serial flow lines with multiple parallel-machine stations, IIE Transactions, 39 (2007), 361-375.  doi: 10.1080/07408170600838423.  Google Scholar

[5]

A. C. Diamantidis and C. T. Papadopoulos, Exact analysis of a two-workstation one-buffer flow line with parallel unreliable machines, European Journal of Operational Research, 197 (2009), 592-580.   Google Scholar

[6] S. B. Gershwin, Manufacturing Systems Engineering, Prentice-Hall, 1994.   Google Scholar
[7] W. D. KeltonR. P. Sadowski and D. A. Sadowski, Simulation with ARENA, 2edition, McGraw-Hill, New York, 1998.   Google Scholar
[8]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadephia, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[9]

J. Li, Overlapping decomposition: A system-theoretic method for modeling and analysis of complex manufacturing systems, IEEE Transactions on Automation Science and Engineering, 2 (2005), 40-53.  doi: 10.1109/TASE.2004.835576.  Google Scholar

[10]

J. LiD. E. BlumenfeldN. Huang and J. M. Alden, Throughput analysis of production systems: Recent advances and future topics, International Journal of Production Research, 47 (2009), 3823-3851.  doi: 10.1080/00207540701829752.  Google Scholar

[11]

J. Li and S. M. Meerkov, Production Systems Engineering, Springer, 2009. doi: 10.1007/978-0-387-75579-3.  Google Scholar

[12]

J. LiuS. YangA. Wu and S. J. Hu, Multi-stage throughput analysis of a two-stage manufacturing system with parallel unreliable machines and a finite buffer, European Journal of Operational Research, 219 (2012), 296-304.  doi: 10.1016/j.ejor.2011.12.025.  Google Scholar

[13]

A. Patchong and D. Willaeys, Modeling and analysis of an unreliable flow line composed of parallel-machine stages, IIE Transactions, 33 (2001), 559-568.  doi: 10.1080/07408170108936854.  Google Scholar

[14] H. G. Perros, Queueing Networks with Blocking, Oxford University Press, 1994.   Google Scholar
[15]

Y. W. Shin and D. H. Moon, Approximation of throughput in tandem queues with multiple servers and blocking, Applied Mathematical Modelling, 38 (2014), 6122-6132.  doi: 10.1016/j.apm.2014.05.015.  Google Scholar

[16]

M. van VuurenI. J. B. F. Adan and S. A. E. Resing-Sassen, Performance analysis of multi-server tandem queues with finite buffers and blocking, OR Spectrum, 27 (2005), 315-338.  doi: 10.1007/s00291-004-0189-z.  Google Scholar

Figure 1.  Flow line
Figure 2.  Subsystems Li
Figure 3.  Two-stage line ${\hat L}_i$
Table 1.  Throughput for flow lines with $m_i=1$ and $\mu_i=1.0$
$(\nu_i,\gamma_i)$$N$$b_i$Sim(c.i.)AppErr(%)DDXErr(%)
(0.04, 0.2)600.3325 (±0.0006)0.34463.60.296210.9
30.4962 (±0.0013)0.50311.40.48043.2
50.5485 (±0.0013)0.54970.20.54081.4
1000.2870 (±0.0007)0.30385.80.243015.3
30.4583 (±0.0016)0.47183.00.44333.3
50.5157 (±0.0016)0.51910.70.51120.9
(0.1, 0.5)600.3550 (±0.0006)0.35730.60.314911.3
30.5443 (±0.0008)0.54320.20.52723.1
50.6003 (±0.0009)0.59770.40.59171.4
1000.3182 (±0.0003)0.31750.20.269115.4
30.5174 (±0.0006)0.51820.20.50103.2
50.5777 (±0.0008)0.57550.40.57171.0
$(\nu_i,\gamma_i)$$N$$b_i$Sim(c.i.)AppErr(%)DDXErr(%)
(0.04, 0.2)600.3325 (±0.0006)0.34463.60.296210.9
30.4962 (±0.0013)0.50311.40.48043.2
50.5485 (±0.0013)0.54970.20.54081.4
1000.2870 (±0.0007)0.30385.80.243015.3
30.4583 (±0.0016)0.47183.00.44333.3
50.5157 (±0.0016)0.51910.70.51120.9
(0.1, 0.5)600.3550 (±0.0006)0.35730.60.314911.3
30.5443 (±0.0008)0.54320.20.52723.1
50.6003 (±0.0009)0.59770.40.59171.4
1000.3182 (±0.0003)0.31750.20.269115.4
30.5174 (±0.0006)0.51820.20.50103.2
50.5777 (±0.0008)0.57550.40.57171.0
Table 2.  Throughput of flow lines with $N=6$ and $m_i\mu_i=1$
${\pmb m}$$b_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
SimApp (Err(%))SimApp (Err(%)
(2, 2, 2, 2, 2, 2)00.4584 (±0.0007)0.4604 (0.4)0.4725 (±0.0004)0.4678 (1.0)
30.5843 (±0.0010)0.5872 (0.5)0.6077 (±0.0009)0.6067 (0.2)
50.6254 (±0.0010)0.6258 (0.1)0.6504 (±0.0010)0.6488 (0.2)
(1, 1, 2, 2, 3, 3)00.4260 (±0.0012)0.4319 (1.4)0.4421 (±0.0007)0.4407 (0.3)
30.5613 (±0.0010)0.5665 (0.9)0.5930 (±0.0013)0.5918 (0.2)
50.6056 (±0.0011)0.6067 (0.2)0.6388 (±0.0008)0.6369 (0.3)
(3, 3, 2, 2, 1, 1)00.4241 (±0.0013)0.4284 (1.0)0.4398 (±0.0006)0.4382 (0.4)
30.5608 (±0.0016)0.5649 (0.7)0.5926 (±0.0007)0.5909 (0.3)
50.6058 (±0.0009)0.6065 (0.1)0.6385 (±0.0008)0.6367 (0.3)
(1, 2, 3, 3, 2, 1)00.4507 (±0.0009)0.4562 (1.2)0.4707 (±0.0007)0.4671 (0.8)
30.5754 (±0.0010)0.5828 (1.3)0.6069 (±0.0007)0.6070 (0.0)
50.6177 (±0.0013)0.6211 (0.5)0.6486 (±0.0009)0.6488 (0.0)
(3, 2, 1, 1, 2, 3)00.4131 (±0.0011)0.4173 (1.0)0.4277 (±0.0006)0.4264 (0.3)
30.5532 (±0.0010)0.5548 (0.3)0.5839 (±0.0009)0.5811 (0.5)
50.5977 (±0.0018)0.5968 (0.1)0.6320 (±0.0007)0.6283 (0.6)
${\pmb m}$$b_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
SimApp (Err(%))SimApp (Err(%)
(2, 2, 2, 2, 2, 2)00.4584 (±0.0007)0.4604 (0.4)0.4725 (±0.0004)0.4678 (1.0)
30.5843 (±0.0010)0.5872 (0.5)0.6077 (±0.0009)0.6067 (0.2)
50.6254 (±0.0010)0.6258 (0.1)0.6504 (±0.0010)0.6488 (0.2)
(1, 1, 2, 2, 3, 3)00.4260 (±0.0012)0.4319 (1.4)0.4421 (±0.0007)0.4407 (0.3)
30.5613 (±0.0010)0.5665 (0.9)0.5930 (±0.0013)0.5918 (0.2)
50.6056 (±0.0011)0.6067 (0.2)0.6388 (±0.0008)0.6369 (0.3)
(3, 3, 2, 2, 1, 1)00.4241 (±0.0013)0.4284 (1.0)0.4398 (±0.0006)0.4382 (0.4)
30.5608 (±0.0016)0.5649 (0.7)0.5926 (±0.0007)0.5909 (0.3)
50.6058 (±0.0009)0.6065 (0.1)0.6385 (±0.0008)0.6367 (0.3)
(1, 2, 3, 3, 2, 1)00.4507 (±0.0009)0.4562 (1.2)0.4707 (±0.0007)0.4671 (0.8)
30.5754 (±0.0010)0.5828 (1.3)0.6069 (±0.0007)0.6070 (0.0)
50.6177 (±0.0013)0.6211 (0.5)0.6486 (±0.0009)0.6488 (0.0)
(3, 2, 1, 1, 2, 3)00.4131 (±0.0011)0.4173 (1.0)0.4277 (±0.0006)0.4264 (0.3)
30.5532 (±0.0010)0.5548 (0.3)0.5839 (±0.0009)0.5811 (0.5)
50.5977 (±0.0018)0.5968 (0.1)0.6320 (±0.0007)0.6283 (0.6)
Table 3.  Throughput of flow lines with $N=6$ and $\mu_i=1$
${\pmb m}$$b_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
SimApp (Err(%))SimApp (Err(%)
(2, 2, 2, 2, 2, 2)00.8986 (±0.0011)0.9110 (1.4)0.9257 (±0.0008)0.9243 (0.1)
31.1255 (±0.0011)1.1415 (1.4)1.1842 (±0.0012)1.1848 (0.1)
51.2020 (±0.0026)1.2077 (0.5)1.2663 (±0.0012)1.2646 (0.1)
(1, 1, 2, 2, 3, 3)00.5179 (±0.0017)0.5176 (0.1)0.5254 (±0.0011)0.5245 (0.2)
30.6454 (±0.0009)0.6434 (0.3)0.6645 (±0.0012)0.6612 (0.5)
50.6783 (±0.0022)0.6778 (0.1)0.7010 (±0.0014)0.6983 (0.4)
(3, 3, 2, 2, 1, 1)00.5165 (±0.0018)0.5178 (0.3)0.5241 (±0.0010)0.5242 (0.0)
30.6448 (±0.0013)0.6439 (0.1)0.6639 (±0.0010)0.6613 (0.4)
50.6806 (±0.0017)0.6780 (0.4)0.7002 (±0.0013)0.6983 (0.3)
(1, 2, 3, 3, 2, 1)00.6405 (±0.0018)0.6811 (6.3)0.6637 (±0.0008)0.6898 (3.9)
30.7483 (±0.0026)0.7748 (3.5)0.7697 (±0.0013)0.7878 (2.4)
50.7739 (±0.0020)0.7854 (1.5)0.7905 (±0.0013)0.7994 (1.1)
(3, 2, 1, 1, 2, 3)00.5099 (±0.0022)0.5108 (0.2)0.5191 (±0.0010)0.5178 (0.2)
30.6442 (±0.0023)0.6411 (0.5)0.6646 (±0.0009)0.6596 (0.7)
50.6782 (±0.0022)0.6764 (0.3)0.7006 (±0.0010)0.6977 (0.4)
${\pmb m}$$b_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
SimApp (Err(%))SimApp (Err(%)
(2, 2, 2, 2, 2, 2)00.8986 (±0.0011)0.9110 (1.4)0.9257 (±0.0008)0.9243 (0.1)
31.1255 (±0.0011)1.1415 (1.4)1.1842 (±0.0012)1.1848 (0.1)
51.2020 (±0.0026)1.2077 (0.5)1.2663 (±0.0012)1.2646 (0.1)
(1, 1, 2, 2, 3, 3)00.5179 (±0.0017)0.5176 (0.1)0.5254 (±0.0011)0.5245 (0.2)
30.6454 (±0.0009)0.6434 (0.3)0.6645 (±0.0012)0.6612 (0.5)
50.6783 (±0.0022)0.6778 (0.1)0.7010 (±0.0014)0.6983 (0.4)
(3, 3, 2, 2, 1, 1)00.5165 (±0.0018)0.5178 (0.3)0.5241 (±0.0010)0.5242 (0.0)
30.6448 (±0.0013)0.6439 (0.1)0.6639 (±0.0010)0.6613 (0.4)
50.6806 (±0.0017)0.6780 (0.4)0.7002 (±0.0013)0.6983 (0.3)
(1, 2, 3, 3, 2, 1)00.6405 (±0.0018)0.6811 (6.3)0.6637 (±0.0008)0.6898 (3.9)
30.7483 (±0.0026)0.7748 (3.5)0.7697 (±0.0013)0.7878 (2.4)
50.7739 (±0.0020)0.7854 (1.5)0.7905 (±0.0013)0.7994 (1.1)
(3, 2, 1, 1, 2, 3)00.5099 (±0.0022)0.5108 (0.2)0.5191 (±0.0010)0.5178 (0.2)
30.6442 (±0.0023)0.6411 (0.5)0.6646 (±0.0009)0.6596 (0.7)
50.6782 (±0.0022)0.6764 (0.3)0.7006 (±0.0010)0.6977 (0.4)
Table 4.  The number of machines of the lines in Tables 5-6
$N=10$${\pmb m}=(m_1,\cdots,m_{10})$ $N=15$${\pmb m}=(m_1,\cdots,m_{15})$
Line 1$(2,2,2,2,2,2,2,2,2,2)$Line 5$(2,2,2,2,2,2,2,2,2,2,2,2,2,2,2)$
Line 2$(1,1,2,2,3,3,2,2,1,1)$Line 6$(1,1,1,2,2,2,3,3,3,2,2,2,1,1,1)$
Line 3$(3,3,2,2,1,1,2,2,3,3)$Line 7$(3,3,3,2,2,2,1,1,1,2,2,2,3,3,3)$
Line 4$(3,2,2,1,2,2,3,2,2,1)$Line 8$(3,3,2,2,1,1,2,2,3,3,2,2,1,1,3)$
$N=10$${\pmb m}=(m_1,\cdots,m_{10})$ $N=15$${\pmb m}=(m_1,\cdots,m_{15})$
Line 1$(2,2,2,2,2,2,2,2,2,2)$Line 5$(2,2,2,2,2,2,2,2,2,2,2,2,2,2,2)$
Line 2$(1,1,2,2,3,3,2,2,1,1)$Line 6$(1,1,1,2,2,2,3,3,3,2,2,2,1,1,1)$
Line 3$(3,3,2,2,1,1,2,2,3,3)$Line 7$(3,3,3,2,2,2,1,1,1,2,2,2,3,3,3)$
Line 4$(3,2,2,1,2,2,3,2,2,1)$Line 8$(3,3,2,2,1,1,2,2,3,3,2,2,1,1,3)$
Table 5.  Throughput for the lines with $N=10, 15$, $\mu_i=\frac{1}{m_i}$ and $l_i=m_i+b_i=3$
Lines$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
Sim(c.i.)App (Err(%))Sim(c.i.)App (Err(%)
10.4874 (±0.0003)0.4949 (1.5)0.5104 (±0.0004)0.5082 (0.4)
20.4597 (±0.0007)0.4729 (2.9)0.4934 (±0.0009)0.4914 (0.4)
30.4698 (±0.0011)0.4772 (1.6)0.4982 (±0.0008)0.4954 (0.6)
40.4742 (±0.0009)0.4836 (2.0)0.5023 (±0.0006)0.5004 (0.4)
50.4717 (±0.0007)0.4790 (1.6)0.4973 (±0.0004)0.4930 (0.9)
60.4430 (±0.0009)0.4567 (3.1)0.4808 (±0.0006)0.4767 (0.9)
70.4497 (±0.0007)0.4581 (1.9)0.4839 (±0.0005)0.4781 (1.2)
80.4474 (±0.0008)0.4596 (2.7)0.4828 (±0.0006)0.4790 (0.8)
Lines$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
Sim(c.i.)App (Err(%))Sim(c.i.)App (Err(%)
10.4874 (±0.0003)0.4949 (1.5)0.5104 (±0.0004)0.5082 (0.4)
20.4597 (±0.0007)0.4729 (2.9)0.4934 (±0.0009)0.4914 (0.4)
30.4698 (±0.0011)0.4772 (1.6)0.4982 (±0.0008)0.4954 (0.6)
40.4742 (±0.0009)0.4836 (2.0)0.5023 (±0.0006)0.5004 (0.4)
50.4717 (±0.0007)0.4790 (1.6)0.4973 (±0.0004)0.4930 (0.9)
60.4430 (±0.0009)0.4567 (3.1)0.4808 (±0.0006)0.4767 (0.9)
70.4497 (±0.0007)0.4581 (1.9)0.4839 (±0.0005)0.4781 (1.2)
80.4474 (±0.0008)0.4596 (2.7)0.4828 (±0.0006)0.4790 (0.8)
Table 6.  Throughput for the lines with $N=10, 15$, $\mu_i=1.0$ and $l_i=m_i+b_i=3,5$
Lines$l_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
Sim(c.i.)App (Err(%))Sim(c.i.)App (Err(%)
130.9367 (±0.0008)0.9706 (3.6)0.9908 (±0.0015)0.9965 (0.6)
51.0663 (±0.0014)1.0948 (2.7)1.1382 (±0.0013)1.1427 (0.4)
230.5671 (±0.0010)0.5932 (4.6)0.6018 (±0.0011)0.6116 (1.6)
50.6296 (±0.0016)0.6421 (2.0)0.6634 (±0.0010)0.6663 (0.4)
330.6085 (±0.0010)0.6075 (0.1)0.6279 (±0.0014)0.6244 (0.6)
50.6600 (±0.0017)0.6586 (0.2)0.6820 (±0.0019)0.6797 (0.3)
430.6840 (±0.0013)0.7218 (5.5)0.7162 (±0.0011)0.7362 (2.8)
50.7452 (±0.0014)0.7719 (3.6)0.7722 (±0.0012)0.7878 (2.0)
530.8994 (±0.0012)0.9376 (4.3)0.9606 (±0.0013)0.9651 (0.5)
51.0334 (±0.0016)1.0688 (3.4)1.1134 (±0.0015)1.1197 (0.6)
630.5167 (±0.0008)0.5257 (1.7)0.5539 (±0.0007)0.5515 (0.4)
50.5808 (±0.0015)0.5811 (0.1)0.6215 (±0.0010)0.6156 (0.9)
730.5414 (±0.0009)0.5420 (0.1)0.5689 (±0.0011)0.5658 (0.5)
50.6000 (±0.0013)0.5975 (0.4)0.6324 (±0.0016)0.6285 (0.6)
830.5657 (±0.0009)0.5899 (4.3)0.6006 (±0.0011)0.6091 (1.4)
50.6287 (±0.0012)0.6406 (1.9)0.6620 (±0.0010)0.6656 (0.5)
Lines$l_i$$(\nu_i,\gamma_i)=(0.04, 0.2)$$(\nu_i,\gamma_i)=(0.1, 0.5)$
Sim(c.i.)App (Err(%))Sim(c.i.)App (Err(%)
130.9367 (±0.0008)0.9706 (3.6)0.9908 (±0.0015)0.9965 (0.6)
51.0663 (±0.0014)1.0948 (2.7)1.1382 (±0.0013)1.1427 (0.4)
230.5671 (±0.0010)0.5932 (4.6)0.6018 (±0.0011)0.6116 (1.6)
50.6296 (±0.0016)0.6421 (2.0)0.6634 (±0.0010)0.6663 (0.4)
330.6085 (±0.0010)0.6075 (0.1)0.6279 (±0.0014)0.6244 (0.6)
50.6600 (±0.0017)0.6586 (0.2)0.6820 (±0.0019)0.6797 (0.3)
430.6840 (±0.0013)0.7218 (5.5)0.7162 (±0.0011)0.7362 (2.8)
50.7452 (±0.0014)0.7719 (3.6)0.7722 (±0.0012)0.7878 (2.0)
530.8994 (±0.0012)0.9376 (4.3)0.9606 (±0.0013)0.9651 (0.5)
51.0334 (±0.0016)1.0688 (3.4)1.1134 (±0.0015)1.1197 (0.6)
630.5167 (±0.0008)0.5257 (1.7)0.5539 (±0.0007)0.5515 (0.4)
50.5808 (±0.0015)0.5811 (0.1)0.6215 (±0.0010)0.6156 (0.9)
730.5414 (±0.0009)0.5420 (0.1)0.5689 (±0.0011)0.5658 (0.5)
50.6000 (±0.0013)0.5975 (0.4)0.6324 (±0.0016)0.6285 (0.6)
830.5657 (±0.0009)0.5899 (4.3)0.6006 (±0.0011)0.6091 (1.4)
50.6287 (±0.0012)0.6406 (1.9)0.6620 (±0.0010)0.6656 (0.5)
Table 7.  The number of iterations for $(\mu_i,\nu_i,\gamma_i)=(1.0/m_i,0.1,0.5)$ and $\epsilon=10^{-5}$
$m_i$123
$l_i$13523535
$N$555445545
1077668868
1589881011811
$m_i$123
$l_i$13523535
$N$555445545
1077668868
1589881011811
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