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

• 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.

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

 Citation:

• 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.) 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 $m_i\mu_i=1$

 ${\pmb m}$ $b_i$ $(\nu_i,\gamma_i)=(0.04, 0.2)$ $(\nu_i,\gamma_i)=(0.1, 0.5)$ 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 $\mu_i=1$

 ${\pmb m}$ $b_i$ $(\nu_i,\gamma_i)=(0.04, 0.2)$ $(\nu_i,\gamma_i)=(0.1, 0.5)$ 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 $(3,3,2,2,1,1,2,2,3,3,2,2,1,1,3)$

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(%) 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 $l_i$ $(\nu_i,\gamma_i)=(0.04, 0.2)$ $(\nu_i,\gamma_i)=(0.1, 0.5)$ 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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Tables(7)