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

Yang Woo Shin; Dug Hee Moon

doi:10.3934/jimo.2016052

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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

Yang Woo Shin ^1, and Dug Hee Moon ^2,

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.

Keywords: Tandem queue, flow line, unreliable parallel-machine workstation, decomposition method.

Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22, 90B30.

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. Diamantidis, C. 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. Kelton, R. 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. Li, D. E. Blumenfeld, N. 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. Liu, S. Yang, A. 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 Vuuren, I. 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. Diamantidis, C. 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. Kelton, R. 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. Li, D. E. Blumenfeld, N. 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. Liu, S. Yang, A. 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 Vuuren, I. 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

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Figure 2. Subsystems L_i

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Figure 3. Two-stage line ${\hat L}_i$

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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.)	App	Err(%)	DDX	Err(%)
(0.04, 0.2)	6	0	0.3325 (±0.0006)	0.3446	3.6	0.2962	10.9
		3	0.4962 (±0.0013)	0.5031	1.4	0.4804	3.2
		5	0.5485 (±0.0013)	0.5497	0.2	0.5408	1.4
	10	0	0.2870 (±0.0007)	0.3038	5.8	0.2430	15.3
		3	0.4583 (±0.0016)	0.4718	3.0	0.4433	3.3
		5	0.5157 (±0.0016)	0.5191	0.7	0.5112	0.9
(0.1, 0.5)	6	0	0.3550 (±0.0006)	0.3573	0.6	0.3149	11.3
		3	0.5443 (±0.0008)	0.5432	0.2	0.5272	3.1
		5	0.6003 (±0.0009)	0.5977	0.4	0.5917	1.4
	10	0	0.3182 (±0.0003)	0.3175	0.2	0.2691	15.4
		3	0.5174 (±0.0006)	0.5182	0.2	0.5010	3.2
		5	0.5777 (±0.0008)	0.5755	0.4	0.5717	1.0

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$(\nu_i,\gamma_i)$	$N$	$b_i$	Sim(c.i.)	App	Err(%)	DDX	Err(%)
(0.04, 0.2)	6	0	0.3325 (±0.0006)	0.3446	3.6	0.2962	10.9
		3	0.4962 (±0.0013)	0.5031	1.4	0.4804	3.2
		5	0.5485 (±0.0013)	0.5497	0.2	0.5408	1.4
	10	0	0.2870 (±0.0007)	0.3038	5.8	0.2430	15.3
		3	0.4583 (±0.0016)	0.4718	3.0	0.4433	3.3
		5	0.5157 (±0.0016)	0.5191	0.7	0.5112	0.9
(0.1, 0.5)	6	0	0.3550 (±0.0006)	0.3573	0.6	0.3149	11.3
		3	0.5443 (±0.0008)	0.5432	0.2	0.5272	3.1
		5	0.6003 (±0.0009)	0.5977	0.4	0.5917	1.4
	10	0	0.3182 (±0.0003)	0.3175	0.2	0.2691	15.4
		3	0.5174 (±0.0006)	0.5182	0.2	0.5010	3.2
		5	0.5777 (±0.0008)	0.5755	0.4	0.5717	1.0

Table 2. Throughput of flow lines with $N=6$ and $m_i\mu_i=1$

${\pmb m}$	$b_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
${\pmb m}$	$b_i$	Sim	App (Err(%))	Sim	App (Err(%)
(2, 2, 2, 2, 2, 2)	0	0.4584 (±0.0007)	0.4604 (0.4)	0.4725 (±0.0004)	0.4678 (1.0)
	3	0.5843 (±0.0010)	0.5872 (0.5)	0.6077 (±0.0009)	0.6067 (0.2)
	5	0.6254 (±0.0010)	0.6258 (0.1)	0.6504 (±0.0010)	0.6488 (0.2)
(1, 1, 2, 2, 3, 3)	0	0.4260 (±0.0012)	0.4319 (1.4)	0.4421 (±0.0007)	0.4407 (0.3)
	3	0.5613 (±0.0010)	0.5665 (0.9)	0.5930 (±0.0013)	0.5918 (0.2)
	5	0.6056 (±0.0011)	0.6067 (0.2)	0.6388 (±0.0008)	0.6369 (0.3)
(3, 3, 2, 2, 1, 1)	0	0.4241 (±0.0013)	0.4284 (1.0)	0.4398 (±0.0006)	0.4382 (0.4)
	3	0.5608 (±0.0016)	0.5649 (0.7)	0.5926 (±0.0007)	0.5909 (0.3)
	5	0.6058 (±0.0009)	0.6065 (0.1)	0.6385 (±0.0008)	0.6367 (0.3)
(1, 2, 3, 3, 2, 1)	0	0.4507 (±0.0009)	0.4562 (1.2)	0.4707 (±0.0007)	0.4671 (0.8)
	3	0.5754 (±0.0010)	0.5828 (1.3)	0.6069 (±0.0007)	0.6070 (0.0)
	5	0.6177 (±0.0013)	0.6211 (0.5)	0.6486 (±0.0009)	0.6488 (0.0)
(3, 2, 1, 1, 2, 3)	0	0.4131 (±0.0011)	0.4173 (1.0)	0.4277 (±0.0006)	0.4264 (0.3)
	3	0.5532 (±0.0010)	0.5548 (0.3)	0.5839 (±0.0009)	0.5811 (0.5)
	5	0.5977 (±0.0018)	0.5968 (0.1)	0.6320 (±0.0007)	0.6283 (0.6)

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${\pmb m}$	$b_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
${\pmb m}$	$b_i$	Sim	App (Err(%))	Sim	App (Err(%)
(2, 2, 2, 2, 2, 2)	0	0.4584 (±0.0007)	0.4604 (0.4)	0.4725 (±0.0004)	0.4678 (1.0)
	3	0.5843 (±0.0010)	0.5872 (0.5)	0.6077 (±0.0009)	0.6067 (0.2)
	5	0.6254 (±0.0010)	0.6258 (0.1)	0.6504 (±0.0010)	0.6488 (0.2)
(1, 1, 2, 2, 3, 3)	0	0.4260 (±0.0012)	0.4319 (1.4)	0.4421 (±0.0007)	0.4407 (0.3)
	3	0.5613 (±0.0010)	0.5665 (0.9)	0.5930 (±0.0013)	0.5918 (0.2)
	5	0.6056 (±0.0011)	0.6067 (0.2)	0.6388 (±0.0008)	0.6369 (0.3)
(3, 3, 2, 2, 1, 1)	0	0.4241 (±0.0013)	0.4284 (1.0)	0.4398 (±0.0006)	0.4382 (0.4)
	3	0.5608 (±0.0016)	0.5649 (0.7)	0.5926 (±0.0007)	0.5909 (0.3)
	5	0.6058 (±0.0009)	0.6065 (0.1)	0.6385 (±0.0008)	0.6367 (0.3)
(1, 2, 3, 3, 2, 1)	0	0.4507 (±0.0009)	0.4562 (1.2)	0.4707 (±0.0007)	0.4671 (0.8)
	3	0.5754 (±0.0010)	0.5828 (1.3)	0.6069 (±0.0007)	0.6070 (0.0)
	5	0.6177 (±0.0013)	0.6211 (0.5)	0.6486 (±0.0009)	0.6488 (0.0)
(3, 2, 1, 1, 2, 3)	0	0.4131 (±0.0011)	0.4173 (1.0)	0.4277 (±0.0006)	0.4264 (0.3)
	3	0.5532 (±0.0010)	0.5548 (0.3)	0.5839 (±0.0009)	0.5811 (0.5)
	5	0.5977 (±0.0018)	0.5968 (0.1)	0.6320 (±0.0007)	0.6283 (0.6)

Table 3. Throughput of flow lines with $N=6$ and $\mu_i=1$

${\pmb m}$	$b_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
${\pmb m}$	$b_i$	Sim	App (Err(%))	Sim	App (Err(%)
(2, 2, 2, 2, 2, 2)	0	0.8986 (±0.0011)	0.9110 (1.4)	0.9257 (±0.0008)	0.9243 (0.1)
	3	1.1255 (±0.0011)	1.1415 (1.4)	1.1842 (±0.0012)	1.1848 (0.1)
	5	1.2020 (±0.0026)	1.2077 (0.5)	1.2663 (±0.0012)	1.2646 (0.1)
(1, 1, 2, 2, 3, 3)	0	0.5179 (±0.0017)	0.5176 (0.1)	0.5254 (±0.0011)	0.5245 (0.2)
	3	0.6454 (±0.0009)	0.6434 (0.3)	0.6645 (±0.0012)	0.6612 (0.5)
	5	0.6783 (±0.0022)	0.6778 (0.1)	0.7010 (±0.0014)	0.6983 (0.4)
(3, 3, 2, 2, 1, 1)	0	0.5165 (±0.0018)	0.5178 (0.3)	0.5241 (±0.0010)	0.5242 (0.0)
	3	0.6448 (±0.0013)	0.6439 (0.1)	0.6639 (±0.0010)	0.6613 (0.4)
	5	0.6806 (±0.0017)	0.6780 (0.4)	0.7002 (±0.0013)	0.6983 (0.3)
(1, 2, 3, 3, 2, 1)	0	0.6405 (±0.0018)	0.6811 (6.3)	0.6637 (±0.0008)	0.6898 (3.9)
	3	0.7483 (±0.0026)	0.7748 (3.5)	0.7697 (±0.0013)	0.7878 (2.4)
	5	0.7739 (±0.0020)	0.7854 (1.5)	0.7905 (±0.0013)	0.7994 (1.1)
(3, 2, 1, 1, 2, 3)	0	0.5099 (±0.0022)	0.5108 (0.2)	0.5191 (±0.0010)	0.5178 (0.2)
	3	0.6442 (±0.0023)	0.6411 (0.5)	0.6646 (±0.0009)	0.6596 (0.7)
	5	0.6782 (±0.0022)	0.6764 (0.3)	0.7006 (±0.0010)	0.6977 (0.4)

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${\pmb m}$	$b_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
${\pmb m}$	$b_i$	Sim	App (Err(%))	Sim	App (Err(%)
(2, 2, 2, 2, 2, 2)	0	0.8986 (±0.0011)	0.9110 (1.4)	0.9257 (±0.0008)	0.9243 (0.1)
	3	1.1255 (±0.0011)	1.1415 (1.4)	1.1842 (±0.0012)	1.1848 (0.1)
	5	1.2020 (±0.0026)	1.2077 (0.5)	1.2663 (±0.0012)	1.2646 (0.1)
(1, 1, 2, 2, 3, 3)	0	0.5179 (±0.0017)	0.5176 (0.1)	0.5254 (±0.0011)	0.5245 (0.2)
	3	0.6454 (±0.0009)	0.6434 (0.3)	0.6645 (±0.0012)	0.6612 (0.5)
	5	0.6783 (±0.0022)	0.6778 (0.1)	0.7010 (±0.0014)	0.6983 (0.4)
(3, 3, 2, 2, 1, 1)	0	0.5165 (±0.0018)	0.5178 (0.3)	0.5241 (±0.0010)	0.5242 (0.0)
	3	0.6448 (±0.0013)	0.6439 (0.1)	0.6639 (±0.0010)	0.6613 (0.4)
	5	0.6806 (±0.0017)	0.6780 (0.4)	0.7002 (±0.0013)	0.6983 (0.3)
(1, 2, 3, 3, 2, 1)	0	0.6405 (±0.0018)	0.6811 (6.3)	0.6637 (±0.0008)	0.6898 (3.9)
	3	0.7483 (±0.0026)	0.7748 (3.5)	0.7697 (±0.0013)	0.7878 (2.4)
	5	0.7739 (±0.0020)	0.7854 (1.5)	0.7905 (±0.0013)	0.7994 (1.1)
(3, 2, 1, 1, 2, 3)	0	0.5099 (±0.0022)	0.5108 (0.2)	0.5191 (±0.0010)	0.5178 (0.2)
	3	0.6442 (±0.0023)	0.6411 (0.5)	0.6646 (±0.0009)	0.6596 (0.7)
	5	0.6782 (±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	$(3,3,2,2,1,1,2,2,3,3,2,2,1,1,3)$

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$N=10$	${\pmb m}=(m_1,\cdots,m_{10})$	$N=15$	${\pmb m}=(m_1,\cdots,m_{15})$
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	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
Lines	Sim(c.i.)	App (Err(%))	Sim(c.i.)	App (Err(%)
1	0.4874 (±0.0003)	0.4949 (1.5)	0.5104 (±0.0004)	0.5082 (0.4)
2	0.4597 (±0.0007)	0.4729 (2.9)	0.4934 (±0.0009)	0.4914 (0.4)
3	0.4698 (±0.0011)	0.4772 (1.6)	0.4982 (±0.0008)	0.4954 (0.6)
4	0.4742 (±0.0009)	0.4836 (2.0)	0.5023 (±0.0006)	0.5004 (0.4)
5	0.4717 (±0.0007)	0.4790 (1.6)	0.4973 (±0.0004)	0.4930 (0.9)
6	0.4430 (±0.0009)	0.4567 (3.1)	0.4808 (±0.0006)	0.4767 (0.9)
7	0.4497 (±0.0007)	0.4581 (1.9)	0.4839 (±0.0005)	0.4781 (1.2)
8	0.4474 (±0.0008)	0.4596 (2.7)	0.4828 (±0.0006)	0.4790 (0.8)

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Lines	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
Lines	Sim(c.i.)	App (Err(%))	Sim(c.i.)	App (Err(%)
1	0.4874 (±0.0003)	0.4949 (1.5)	0.5104 (±0.0004)	0.5082 (0.4)
2	0.4597 (±0.0007)	0.4729 (2.9)	0.4934 (±0.0009)	0.4914 (0.4)
3	0.4698 (±0.0011)	0.4772 (1.6)	0.4982 (±0.0008)	0.4954 (0.6)
4	0.4742 (±0.0009)	0.4836 (2.0)	0.5023 (±0.0006)	0.5004 (0.4)
5	0.4717 (±0.0007)	0.4790 (1.6)	0.4973 (±0.0004)	0.4930 (0.9)
6	0.4430 (±0.0009)	0.4567 (3.1)	0.4808 (±0.0006)	0.4767 (0.9)
7	0.4497 (±0.0007)	0.4581 (1.9)	0.4839 (±0.0005)	0.4781 (1.2)
8	0.4474 (±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	$l_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
Lines	$l_i$	Sim(c.i.)	App (Err(%))	Sim(c.i.)	App (Err(%)
1	3	0.9367 (±0.0008)	0.9706 (3.6)	0.9908 (±0.0015)	0.9965 (0.6)
	5	1.0663 (±0.0014)	1.0948 (2.7)	1.1382 (±0.0013)	1.1427 (0.4)
2	3	0.5671 (±0.0010)	0.5932 (4.6)	0.6018 (±0.0011)	0.6116 (1.6)
	5	0.6296 (±0.0016)	0.6421 (2.0)	0.6634 (±0.0010)	0.6663 (0.4)
3	3	0.6085 (±0.0010)	0.6075 (0.1)	0.6279 (±0.0014)	0.6244 (0.6)
	5	0.6600 (±0.0017)	0.6586 (0.2)	0.6820 (±0.0019)	0.6797 (0.3)
4	3	0.6840 (±0.0013)	0.7218 (5.5)	0.7162 (±0.0011)	0.7362 (2.8)
	5	0.7452 (±0.0014)	0.7719 (3.6)	0.7722 (±0.0012)	0.7878 (2.0)
5	3	0.8994 (±0.0012)	0.9376 (4.3)	0.9606 (±0.0013)	0.9651 (0.5)
	5	1.0334 (±0.0016)	1.0688 (3.4)	1.1134 (±0.0015)	1.1197 (0.6)
6	3	0.5167 (±0.0008)	0.5257 (1.7)	0.5539 (±0.0007)	0.5515 (0.4)
	5	0.5808 (±0.0015)	0.5811 (0.1)	0.6215 (±0.0010)	0.6156 (0.9)
7	3	0.5414 (±0.0009)	0.5420 (0.1)	0.5689 (±0.0011)	0.5658 (0.5)
	5	0.6000 (±0.0013)	0.5975 (0.4)	0.6324 (±0.0016)	0.6285 (0.6)
8	3	0.5657 (±0.0009)	0.5899 (4.3)	0.6006 (±0.0011)	0.6091 (1.4)
	5	0.6287 (±0.0012)	0.6406 (1.9)	0.6620 (±0.0010)	0.6656 (0.5)

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Lines	$l_i$	$(\nu_i,\gamma_i)=(0.04, 0.2)$		$(\nu_i,\gamma_i)=(0.1, 0.5)$
Lines	$l_i$	Sim(c.i.)	App (Err(%))	Sim(c.i.)	App (Err(%)
1	3	0.9367 (±0.0008)	0.9706 (3.6)	0.9908 (±0.0015)	0.9965 (0.6)
	5	1.0663 (±0.0014)	1.0948 (2.7)	1.1382 (±0.0013)	1.1427 (0.4)
2	3	0.5671 (±0.0010)	0.5932 (4.6)	0.6018 (±0.0011)	0.6116 (1.6)
	5	0.6296 (±0.0016)	0.6421 (2.0)	0.6634 (±0.0010)	0.6663 (0.4)
3	3	0.6085 (±0.0010)	0.6075 (0.1)	0.6279 (±0.0014)	0.6244 (0.6)
	5	0.6600 (±0.0017)	0.6586 (0.2)	0.6820 (±0.0019)	0.6797 (0.3)
4	3	0.6840 (±0.0013)	0.7218 (5.5)	0.7162 (±0.0011)	0.7362 (2.8)
	5	0.7452 (±0.0014)	0.7719 (3.6)	0.7722 (±0.0012)	0.7878 (2.0)
5	3	0.8994 (±0.0012)	0.9376 (4.3)	0.9606 (±0.0013)	0.9651 (0.5)
	5	1.0334 (±0.0016)	1.0688 (3.4)	1.1134 (±0.0015)	1.1197 (0.6)
6	3	0.5167 (±0.0008)	0.5257 (1.7)	0.5539 (±0.0007)	0.5515 (0.4)
	5	0.5808 (±0.0015)	0.5811 (0.1)	0.6215 (±0.0010)	0.6156 (0.9)
7	3	0.5414 (±0.0009)	0.5420 (0.1)	0.5689 (±0.0011)	0.5658 (0.5)
	5	0.6000 (±0.0013)	0.5975 (0.4)	0.6324 (±0.0016)	0.6285 (0.6)
8	3	0.5657 (±0.0009)	0.5899 (4.3)	0.6006 (±0.0011)	0.6091 (1.4)
	5	0.6287 (±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 $\epsilon=10^{-5}$

	$m_i$	1			2			3
	$l_i$	1	3	5	2	3	5	3	5
$N$	5	5	5	4	4	5	5	4	5
	10	7	7	6	6	8	8	6	8
	15	8	9	8	8	10	11	8	11

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	$m_i$	1			2			3
	$l_i$	1	3	5	2	3	5	3	5
$N$	5	5	5	4	4	5	5	4	5
	10	7	7	6	6	8	8	6	8
	15	8	9	8	8	10	11	8	11

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American Institute of Mathematical Sciences

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

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