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

Yang Woo Shin; Dug Hee Moon

doi:10.3934/jimo.2016052

Article Contents

2017, Volume 13, Issue 2: 901-916. Doi: 10.3934/jimo.2016052

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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 2017

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Abstract

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:

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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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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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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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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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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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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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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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