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

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  • 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:

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  • 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
     | Show Table
    DownLoad: CSV

    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)
     | Show Table
    DownLoad: CSV

    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)
     | Show Table
    DownLoad: CSV

    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$(3,3,2,2,1,1,2,2,3,3,2,2,1,1,3)$
     | Show Table
    DownLoad: CSV

    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)
     | Show Table
    DownLoad: CSV

    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)
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [1] T. AltiokPerformance Analysis of Manufacturing Systems, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-1924-8.
    [2] J. A. Buzzacott and  J. G. ShanthikumarStochastic Models of Manufacturing Systems, Prentice-Hall, 1993. 
    [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.
    [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.
    [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. 
    [6] S. B. GershwinManufacturing Systems Engineering, Prentice-Hall, 1994. 
    [7] W. D. KeltonR. P. Sadowski and  D. A. SadowskiSimulation with ARENA, 2edition, McGraw-Hill, New York, 1998. 
    [8] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadephia, 1999. doi: 10.1137/1.9780898719734.
    [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.
    [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.
    [11] J. Li and S. M. Meerkov, Production Systems Engineering, Springer, 2009. doi: 10.1007/978-0-387-75579-3.
    [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.
    [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.
    [14] H. G. PerrosQueueing Networks with Blocking, Oxford University Press, 1994. 
    [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.
    [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.
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