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doi: 10.3934/jimo.2018085

Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model

College of information engineering, Shenzhen University, Shenzhen 518060, China

* Corresponding author: Jianyu Cao

Received  April 2016 Revised  March 2018 Published  July 2018

Fund Project: This work is supported by the National Natural Science Foundations of China (Grant Nos. 61401286, 61702341 and 61771319) and the Research Project of Shenzhen Technology University (Grant No. 201727)

A stockless production system is considered, in which the products are not produced until the orders are accepted. Due to this character, the duration of the preventive maintenance has an influence on the lead time. In addition, in this stockless production system, the cost of the preventive maintenance depends on its duration; if the lead time exceeds to the quoted lead time, some penalty cost should be considered; and the non-conforming products can still be sold by a discount. A condition-based duration-varying preventive maintenance policy is designed for the stockless production system, by making a tradeoff among the duration of the preventive maintenance, the time for the machine continuously producing, and the lead time of the order. According to the characters of the stockless production system with the designed preventive maintenance policy, it can be modeled by a BMAP/G/1 infinite-buffer queueing model with gated service and queue-length dependent vacation. Based on this queueing model, the stationary probability distributions of four performance measures for the stockless production system are analyzed, including, the number of the products produced in a production cycle, the number of the unfulfilled orders at arbitrary time, the time required to fulfill the tasks present at arbitrary time, and the lead time of the order accepted at arbitrary time. Moreover, based on some information of these performance measures, a profit function, which represents the average profit of the manufacture in a production cycle, is constructed to optimize the designed preventive maintenance policy according to specific conditions. Finally, given an example with the purchasers having different sensitivities to the lead time, some numerical experiments are carried out; and from the numerical experiments, some general results can be inferred for the stockless production system with the designed preventive maintenance policy.

Citation: Jianyu Cao, Weixin Xie. Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018085
References:
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E.-H. AghezzafA. Khatab and P. L. Tam, Optimizing production and imperfect preventive maintenance planning's integration in failure-prone manufacturing systems, Reliability Engineering & System Safety, 145 (2016), 190-198. doi: 10.1016/j.ress.2015.09.017.

[2]

A. Arreola-Risa and M. F. Keblis, Design of stockless production systems, Production and Operations Management, 22 (2013), 203-215. doi: 10.1111/j.1937-5956.2012.01343.x.

[3]

J.-B. Bacot and J. H. Dshalalow, A bulk input queueing system with batch gated service and multiple vacation policy, Mathematical and Computer Modelling, 34 (2001), 873-886. doi: 10.1016/S0895-7177(01)00106-6.

[4]

A. D. Banik, The infinite-buffer single server queue with a variant of multiple vacation policy and batch {M}arkovian arrival process, Applied Mathematical Modelling, 33 (2009), 3025-3039. doi: 10.1016/j.apm.2008.10.021.

[5]

A. D. Banik and S. K. Samanta, Controlling packet loss of bursty and correlated traffics in a variant of multiple vacation policy, in 2013 International Conference on Distributed Computing and Internet Technology (ICDCIT), Vol. 7753 of Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2013,208-219. doi: 10.1007/978-3-642-36071-8_16.

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R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960.

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F. BerthautA. GharbiJ.-P. Kenné and J.-F. Boulet, Improved joint preventive maintenance and hedging point policy, International Journal of Production Economics, 127 (2010), 60-72. doi: 10.1016/j.ijpe.2010.04.030.

[8]

B. BouslahA. Gharbi and R. Pellerin, Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint, Omega, 61 (2016), 110-126. doi: 10.1016/j.omega.2015.07.012.

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B. BouslahA. Gharbi and R. Pellerin, Joint economic design of production, continuous sampling inspection and preventive maintenance of a deteriorating production system, International Journal of Production Economics, 173 (2016), 184-198. doi: 10.1016/j.ijpe.2015.12.016.

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L. Breuer and D. Baum, An Introduction to Queueing Theory and Matrix-Analytic Methods, Springer, Dordrecht, 2005.

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J. Cao and W. Xie, Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines, Queueing Systems, 85 (2017), 117-147. doi: 10.1007/s11134-016-9504-z.

[12]

A. Chelbi and N. Rezg, Analysis of a production/inventory system with randomly failing production unit subjected to a minimum required availability level, International Journal of Production Economics, 99 (2006), 131-143. doi: 10.1016/j.ijpe.2004.12.012.

[13]

A. ChelbiN. Rezg and M. Radhoui, Simultaneous determination of production lot size and preventive maintenance schedule for unreliable production system, Journal of Quality in Maintenance Engineering, 14 (2008), 161-176. doi: 10.1108/13552510810877665.

[14]

E. Çinlar, Markov renewal theory, Advances in Applied Probability, 1 (1969), 123-187. doi: 10.2307/1426216.

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E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1975.

[16]

D. DasA. Roy and S. Kar, A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment, Computers & Mathematics with Applications, 61 (2011), 2388-2400. doi: 10.1016/j.camwa.2011.02.015.

[17]

K. DhouibA. Gharbi and M. N. Ben Aziza, Joint optimal production control/preventive maintenance policy for imperfect process manufacturing cell, International Journal of Production Economics, 137 (2012), 126-136. doi: 10.1016/j.ijpe.2012.01.023.

[18]

L. Doyen, Semi-parametric estimation of Brown-Proschan preventive maintenance effects and intrinsic wear-out, Computational Statistics & Data Analysis, 77 (2014), 206-222. doi: 10.1016/j.csda.2014.02.022.

[19]

D. Gibson and E. Seneta, Monotone infinite stochastic matrices and their augmented truncations, Stochastic Processes and their Applications, 24 (1987), 287-292. doi: 10.1016/0304-4149(87)90019-6.

[20]

W. K. GrassmannM. I. Taksar and D. P. Heyman, Regenerative analysis and steady state distributions for Markov chains, Operations Research, 33 (1985), 1107-1116. doi: 10.1287/opre.33.5.1107.

[21]

D. Gupta and S. Benjaafar, Make-to-order, make-to-stock, or delay product differentiation? A common framework for modeling and analysis, IIE Transactions, 36 (2004), 529-546. doi: 10.1080/07408170490438519.

[22]

E. M. Jewkes and A. S. Alfa, A queueing model of delayed product differentiation, European Journal of Operational Research, 199 (2009), 734-743. doi: 10.1016/j.ejor.2008.08.001.

[23]

Y. -L. Jin, Multi-objective optimization of flexible period preventive maintenance on a single machine, in 2012 International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (ICQR2MSE), Chengdu, China, 2012,475-479. doi: 10.1109/ICQR2MSE.2012.6246277.

[24]

V. V. Krivtsov, Recent advances in theory and applications of stochastic point process models in reliability engineering, Reliability Engineering & System Safety, 92 (2007), 549-551. doi: 10.1016/j.ress.2006.05.001.

[25]

G.-L. Liao, Production and maintenance policies for an {EPQ} model with perfect repair, rework, free-repair warranty, and preventive maintenance, IEEE Transactions on Systems, Man, Cybernetics: Systems, 46 (2016), 1129-1139. doi: 10.1109/TSMC.2015.2465961.

[26]

J. D. C. Little, A proof for the queuing formula: L = λW, Operations Research, 9 (1961), 383-387. doi: 10.1287/opre.9.3.383.

[27]

C. LiuY. FanC. Zhao and J. Wang, Multiple common due-dates assignment and optimal maintenance activity scheduling with linear deteriorating jobs, Journal of Industrial & Management Optimization, 13 (2017), 713-720. doi: 10.3934/jimo.2016042.

[28]

D. M. Lucantoni, New results on the single server queue with a bath Markovian arrival process, Stochastic Models, 7 (1991), 1-46. doi: 10.1080/15326349108807174.

[29]

D. M. LucantoniK. S. Meier-Hellstern and M. F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Advances in Applied Probability, 22 (1990), 676-705. doi: 10.2307/1427464.

[30]

L. MannA. Saxena and G. M. Knapp, Statistical-based or condition-based preventive maintenance?, Journal of Quality in Maintenance Engineering, 1 (1995), 46-59. doi: 10.1108/13552519510083156.

[31]

R. MehdiR. Nidhal and C. Anis, Integrated maintenance and control policy based on quality control, Computers & Industrial Engineering, 58 (2010), 443-451. doi: 10.1016/j.cie.2009.11.002.

[32]

T. Nakagawa, Periodic and sequential preventive maintenance policies, Journal of Applied Probability, 23 (1986), 536-542. doi: 10.2307/3214197.

[33]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models. An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.

[34]

M. RadhouiN. Rezg and A. Chelbi, Integrated model of preventive maintenance, quality control and buffer sizing for unreliable and imperfect production systems, International Journal of Production Research, 47 (2009), 389-402. doi: 10.1080/00207540802426201.

[35]

V. Ramaswami, The N/G/1 queue and its detailed analysis, Advances in Applied Probability, 12 (1980), 222-261. doi: 10.2307/1426503.

[36]

J. SchutzN. Rezg and J.-B. Léger, Periodic and sequential preventive maintenance policies over a finite planning horizon with a dynamic failure law, Journal of Intelligent Manufacturing, 22 (2011), 523-532. doi: 10.1007/s10845-009-0313-7.

[37]

M. ShahriariN. ShojaA. E. ZadeS. Barak and M. Sharifi, JIT single machine scheduling problem with periodic preventive maintenance, Journal of Industrial Engineering International, 12 (2016), 299-310. doi: 10.1007/s40092-016-0147-9.

[38]

O. TangR. W. Grubbström and S. Zanoni, Planned lead time determination in a make-to-order remanufacturing system, International Journal of Production Economics, 108 (2007), 426-435. doi: 10.1016/j.ijpe.2006.12.034.

[39]

V. M. VishnevskyA. N. DudinO. V. Semenova and V. I. Klimenok, Performance analysis of the BMAP/G/1 queue with gated servicing and adaptive vacations, Performance Evaluation, 68 (2011), 446-462. doi: 10.1016/j.peva.2011.02.003.

[40]

H.-M. WeeW.-T. Wang and P.-C. Yang, A production quantity model for imperfect quality items with shortage and screening constraint, International Journal of Production Research, 51 (2013), 1869-1884. doi: 10.1080/00207543.2012.718453.

[41]

L. XiaoS. SongX. Chen and D. W. Coit, Joint optimization of production scheduling and machine group preventive maintenance, Reliability Engineering & System Safety, 146 (2016), 68-78. doi: 10.1016/j.ress.2015.10.013.

[42]

K.-C. YingC.-C. Lu and J.-C. Chen, Exact algorithms for single-machine scheduling problems with a variable maintenance, Computers & Industrial Engineering, 98 (2016), 427-433. doi: 10.1016/j.cie.2016.05.037.

[43]

W. ZhouR. Zhang and Y. Zhou, A queuing model on supply chain with the form postponement strategy, Computers & Industrial Engineering, 66 (2013), 643-652. doi: 10.1016/j.cie.2013.09.022.

show all references

References:
[1]

E.-H. AghezzafA. Khatab and P. L. Tam, Optimizing production and imperfect preventive maintenance planning's integration in failure-prone manufacturing systems, Reliability Engineering & System Safety, 145 (2016), 190-198. doi: 10.1016/j.ress.2015.09.017.

[2]

A. Arreola-Risa and M. F. Keblis, Design of stockless production systems, Production and Operations Management, 22 (2013), 203-215. doi: 10.1111/j.1937-5956.2012.01343.x.

[3]

J.-B. Bacot and J. H. Dshalalow, A bulk input queueing system with batch gated service and multiple vacation policy, Mathematical and Computer Modelling, 34 (2001), 873-886. doi: 10.1016/S0895-7177(01)00106-6.

[4]

A. D. Banik, The infinite-buffer single server queue with a variant of multiple vacation policy and batch {M}arkovian arrival process, Applied Mathematical Modelling, 33 (2009), 3025-3039. doi: 10.1016/j.apm.2008.10.021.

[5]

A. D. Banik and S. K. Samanta, Controlling packet loss of bursty and correlated traffics in a variant of multiple vacation policy, in 2013 International Conference on Distributed Computing and Internet Technology (ICDCIT), Vol. 7753 of Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2013,208-219. doi: 10.1007/978-3-642-36071-8_16.

[6]

R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960.

[7]

F. BerthautA. GharbiJ.-P. Kenné and J.-F. Boulet, Improved joint preventive maintenance and hedging point policy, International Journal of Production Economics, 127 (2010), 60-72. doi: 10.1016/j.ijpe.2010.04.030.

[8]

B. BouslahA. Gharbi and R. Pellerin, Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint, Omega, 61 (2016), 110-126. doi: 10.1016/j.omega.2015.07.012.

[9]

B. BouslahA. Gharbi and R. Pellerin, Joint economic design of production, continuous sampling inspection and preventive maintenance of a deteriorating production system, International Journal of Production Economics, 173 (2016), 184-198. doi: 10.1016/j.ijpe.2015.12.016.

[10]

L. Breuer and D. Baum, An Introduction to Queueing Theory and Matrix-Analytic Methods, Springer, Dordrecht, 2005.

[11]

J. Cao and W. Xie, Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines, Queueing Systems, 85 (2017), 117-147. doi: 10.1007/s11134-016-9504-z.

[12]

A. Chelbi and N. Rezg, Analysis of a production/inventory system with randomly failing production unit subjected to a minimum required availability level, International Journal of Production Economics, 99 (2006), 131-143. doi: 10.1016/j.ijpe.2004.12.012.

[13]

A. ChelbiN. Rezg and M. Radhoui, Simultaneous determination of production lot size and preventive maintenance schedule for unreliable production system, Journal of Quality in Maintenance Engineering, 14 (2008), 161-176. doi: 10.1108/13552510810877665.

[14]

E. Çinlar, Markov renewal theory, Advances in Applied Probability, 1 (1969), 123-187. doi: 10.2307/1426216.

[15]

E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1975.

[16]

D. DasA. Roy and S. Kar, A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment, Computers & Mathematics with Applications, 61 (2011), 2388-2400. doi: 10.1016/j.camwa.2011.02.015.

[17]

K. DhouibA. Gharbi and M. N. Ben Aziza, Joint optimal production control/preventive maintenance policy for imperfect process manufacturing cell, International Journal of Production Economics, 137 (2012), 126-136. doi: 10.1016/j.ijpe.2012.01.023.

[18]

L. Doyen, Semi-parametric estimation of Brown-Proschan preventive maintenance effects and intrinsic wear-out, Computational Statistics & Data Analysis, 77 (2014), 206-222. doi: 10.1016/j.csda.2014.02.022.

[19]

D. Gibson and E. Seneta, Monotone infinite stochastic matrices and their augmented truncations, Stochastic Processes and their Applications, 24 (1987), 287-292. doi: 10.1016/0304-4149(87)90019-6.

[20]

W. K. GrassmannM. I. Taksar and D. P. Heyman, Regenerative analysis and steady state distributions for Markov chains, Operations Research, 33 (1985), 1107-1116. doi: 10.1287/opre.33.5.1107.

[21]

D. Gupta and S. Benjaafar, Make-to-order, make-to-stock, or delay product differentiation? A common framework for modeling and analysis, IIE Transactions, 36 (2004), 529-546. doi: 10.1080/07408170490438519.

[22]

E. M. Jewkes and A. S. Alfa, A queueing model of delayed product differentiation, European Journal of Operational Research, 199 (2009), 734-743. doi: 10.1016/j.ejor.2008.08.001.

[23]

Y. -L. Jin, Multi-objective optimization of flexible period preventive maintenance on a single machine, in 2012 International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (ICQR2MSE), Chengdu, China, 2012,475-479. doi: 10.1109/ICQR2MSE.2012.6246277.

[24]

V. V. Krivtsov, Recent advances in theory and applications of stochastic point process models in reliability engineering, Reliability Engineering & System Safety, 92 (2007), 549-551. doi: 10.1016/j.ress.2006.05.001.

[25]

G.-L. Liao, Production and maintenance policies for an {EPQ} model with perfect repair, rework, free-repair warranty, and preventive maintenance, IEEE Transactions on Systems, Man, Cybernetics: Systems, 46 (2016), 1129-1139. doi: 10.1109/TSMC.2015.2465961.

[26]

J. D. C. Little, A proof for the queuing formula: L = λW, Operations Research, 9 (1961), 383-387. doi: 10.1287/opre.9.3.383.

[27]

C. LiuY. FanC. Zhao and J. Wang, Multiple common due-dates assignment and optimal maintenance activity scheduling with linear deteriorating jobs, Journal of Industrial & Management Optimization, 13 (2017), 713-720. doi: 10.3934/jimo.2016042.

[28]

D. M. Lucantoni, New results on the single server queue with a bath Markovian arrival process, Stochastic Models, 7 (1991), 1-46. doi: 10.1080/15326349108807174.

[29]

D. M. LucantoniK. S. Meier-Hellstern and M. F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Advances in Applied Probability, 22 (1990), 676-705. doi: 10.2307/1427464.

[30]

L. MannA. Saxena and G. M. Knapp, Statistical-based or condition-based preventive maintenance?, Journal of Quality in Maintenance Engineering, 1 (1995), 46-59. doi: 10.1108/13552519510083156.

[31]

R. MehdiR. Nidhal and C. Anis, Integrated maintenance and control policy based on quality control, Computers & Industrial Engineering, 58 (2010), 443-451. doi: 10.1016/j.cie.2009.11.002.

[32]

T. Nakagawa, Periodic and sequential preventive maintenance policies, Journal of Applied Probability, 23 (1986), 536-542. doi: 10.2307/3214197.

[33]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models. An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.

[34]

M. RadhouiN. Rezg and A. Chelbi, Integrated model of preventive maintenance, quality control and buffer sizing for unreliable and imperfect production systems, International Journal of Production Research, 47 (2009), 389-402. doi: 10.1080/00207540802426201.

[35]

V. Ramaswami, The N/G/1 queue and its detailed analysis, Advances in Applied Probability, 12 (1980), 222-261. doi: 10.2307/1426503.

[36]

J. SchutzN. Rezg and J.-B. Léger, Periodic and sequential preventive maintenance policies over a finite planning horizon with a dynamic failure law, Journal of Intelligent Manufacturing, 22 (2011), 523-532. doi: 10.1007/s10845-009-0313-7.

[37]

M. ShahriariN. ShojaA. E. ZadeS. Barak and M. Sharifi, JIT single machine scheduling problem with periodic preventive maintenance, Journal of Industrial Engineering International, 12 (2016), 299-310. doi: 10.1007/s40092-016-0147-9.

[38]

O. TangR. W. Grubbström and S. Zanoni, Planned lead time determination in a make-to-order remanufacturing system, International Journal of Production Economics, 108 (2007), 426-435. doi: 10.1016/j.ijpe.2006.12.034.

[39]

V. M. VishnevskyA. N. DudinO. V. Semenova and V. I. Klimenok, Performance analysis of the BMAP/G/1 queue with gated servicing and adaptive vacations, Performance Evaluation, 68 (2011), 446-462. doi: 10.1016/j.peva.2011.02.003.

[40]

H.-M. WeeW.-T. Wang and P.-C. Yang, A production quantity model for imperfect quality items with shortage and screening constraint, International Journal of Production Research, 51 (2013), 1869-1884. doi: 10.1080/00207543.2012.718453.

[41]

L. XiaoS. SongX. Chen and D. W. Coit, Joint optimization of production scheduling and machine group preventive maintenance, Reliability Engineering & System Safety, 146 (2016), 68-78. doi: 10.1016/j.ress.2015.10.013.

[42]

K.-C. YingC.-C. Lu and J.-C. Chen, Exact algorithms for single-machine scheduling problems with a variable maintenance, Computers & Industrial Engineering, 98 (2016), 427-433. doi: 10.1016/j.cie.2016.05.037.

[43]

W. ZhouR. Zhang and Y. Zhou, A queuing model on supply chain with the form postponement strategy, Computers & Industrial Engineering, 66 (2013), 643-652. doi: 10.1016/j.cie.2013.09.022.

Figure 1.  ${{K''}_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}} = 0$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server does not attend to the queue.
Figure 2.  ${{K''}_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}}\ge 1$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server does not attend to the queue.
Figure 3.  ${K''_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for $1\le {{i}_{0}}\le i$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server is attending to the queue.
Figure 4.  ${K''_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}}\ge 2$ and ${{i}_{0}}>i>0$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server is attending to the queue.
Figure 5.  Schematic diagrams of ${\varphi }(n)$, ${M_{c}}(x)$ and ${W_{c}}(x)$.
Figure 6.  $\eta(q)$ vs. $q$ for the purchasers with different sensitivities to the lead time.
Figure 7.  ${{\sigma }_{m}}$ vs. $q$.
Table 1.  The correspondence between performance measures of the stockless production system and the queueing model
The stockless production system The queueing model
The number of the products produced in a production cycle The queue length just after the server travels to the queue
The number of the unfulfilled orders at arbitrary time The queue length (including the customer in service) at arbitrary time
The time required to fulfill the tasks 1 present at arbitrary time The virtual waiting time2 at arbitrary time
The lead time of the order accepted at arbitrary time The actual waiting time3 at arbitrary time
1 The tasks contain the future or remaining machine set-up, machine close-down as well as PM (or the idle period) in the current production cycle, and the present unfulfilled orders.
2 The definition of the virtual waiting time is given in Section 5.
3 The definition of the actual waiting time is given in Section 6.
The stockless production system The queueing model
The number of the products produced in a production cycle The queue length just after the server travels to the queue
The number of the unfulfilled orders at arbitrary time The queue length (including the customer in service) at arbitrary time
The time required to fulfill the tasks 1 present at arbitrary time The virtual waiting time2 at arbitrary time
The lead time of the order accepted at arbitrary time The actual waiting time3 at arbitrary time
1 The tasks contain the future or remaining machine set-up, machine close-down as well as PM (or the idle period) in the current production cycle, and the present unfulfilled orders.
2 The definition of the virtual waiting time is given in Section 5.
3 The definition of the actual waiting time is given in Section 6.
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