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January  2011, 7(1): 19-30. doi: 10.3934/jimo.2011.7.19

## A queueing analysis of multi-purpose production facility's operations

 1 Saint Mary's University, Sobey School of Business, Department of Finance, Information Systems, and Management Science, Halifax, Nova Scotia, B3H 3C3, Canada 2 Western Washington University, College of Business and Economics, Department of Decision Sciences, Bellingham, WA 98225, United States 3 École de Technologie Supérieure, Département de Génie Électrique, Montréal, Québec, H3C 1K3, Canada

Received  July 2009 Revised  September 2010 Published  January 2011

In this paper, we study the production time allocation issue for a multi-purpose manufacturing facility. This production facility can produce different types of make-to-order and make-to-stock products. Using a vacation queueing model, we develop a set of quantitative performance measures for a two-parameter time allocation policy. Based on the renewal cycle analysis, we derive an average cost expression and propose a search algorithm to find the optimal time allocation policy that minimizes the average cost. Some numerical examples are presented to demonstrate the effectiveness of the search algorithm. The vacation model used in this paper is also a generalization of some previous vacation queueing models in the literature. The results obtained in this study are useful for production managers to design the operating policy in practice.
Citation: Lotfi Tadj, Zhe George Zhang, Chakib Tadj. A queueing analysis of multi-purpose production facility's operations. Journal of Industrial & Management Optimization, 2011, 7 (1) : 19-30. doi: 10.3934/jimo.2011.7.19
##### References:
 [1] L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications,, Journal of Applied Mathematics and Stochastic Analysis, 4 (1991), 335.  doi: 10.1155/S1048953391000254.  Google Scholar [2] J. R. Artalejo and G. Choudhury, Steady state analysis of an M/G/1 queue with repeated attempts and two phase service,, Quality Technology and Quantitative Management, 1 (2004), 189.   Google Scholar [3] D. Bertsimas and X. Papaconstantinou, On the steady-state solution of the M/C$_2(a,b)$/$S$ queueing system,, Transportation Sciences, 22 (1988), 125.  doi: 10.1287/trsc.22.2.125.  Google Scholar [4] D. Bertsimas and X. Papaconstantinou, Analysis of the stationary $E_k$/$C_2$/S queueing system,, European Journal of Operational Research, 37 (1988), 272.  doi: 10.1016/0377-2217(88)90336-0.  Google Scholar [5] G. Choudhury, Some aspects of an M/G/1 queueing system with optional second service,, TOP, 11 (2003), 141.  doi: 10.1007/BF02578955.  Google Scholar [6] G. Choudhury and K. C. Madan, A two-phase batch arrival queueing system with a vacation time under Bernoulli schedule,, Applied Mathematics and Computation, 149 (2004), 337.  doi: 10.1016/S0096-3003(03)00138-3.  Google Scholar [7] G. Choudhury and K. C. Madan, A two-stage arrival queueing system with a modified Bernoulli schedule vacation under N-policy,, Mathematical and Computer Modelling, 42 (2005), 71.  doi: 10.1016/j.mcm.2005.04.003.  Google Scholar [8] G. Choudhury and M. Paul, A batch arrival queue with an additional service channel under N-policy,, Applied Mathematics and Computation, 156 (2004), 115.  doi: 10.1016/j.amc.2003.07.006.  Google Scholar [9] G. Choudhury and M. Paul, Analysis of a two phases batch arrival queueing model with Bernoulli vacation schedule,, Revista Investigatión Operacional, 25 (2004), 217.   Google Scholar [10] B. T. Doshi, Queueing systems with vacations: A survey,, Queueing Systems, 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [11] B. T. Doshi, Single-server queues with vacations,, in, (1990), 217.   Google Scholar [12] B. T. Doshi, Analysis of a two-phase queueing system with general service times,, Operations Research Letters, 10 (1991), 265.  doi: 10.1016/0167-6377(91)90012-E.  Google Scholar [13] A. Federgruen and K. C. So, Optimality of threshold policies in single server queueing system with vacations,, Advances in Applied Probability, 23 (1991), 388.  doi: 10.2307/1427755.  Google Scholar [14] O. Kella, The threshold policy in the M/G/1 queue with server vacations,, Naval Research Logistics, 36 (1989), 111.  doi: 10.1002/1520-6750(198902)36:1<111::AID-NAV3220360109>3.0.CO;2-3.  Google Scholar [15] T. S. Kim and K. C. Chae, Two-phase queueing system with generalized vacation,, Journal of the Korean Institute of Industrial Engineers, 22 (1996), 95.   Google Scholar [16] T. S. Kim and A. Q. Park, Cycle analysis of a two-phase queueing model with threshold,, European Journal of Operational Research, 144 (2003), 157.   Google Scholar [17] C. M. Krishna and Y. H. Lee, A study of two-phase service,, Operations Research Letters, 9 (1990), 91.  doi: 10.1016/0167-6377(90)90047-9.  Google Scholar [18] H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, Analysis of M$^x$/G/1 queue with N policy and multiple vacations,, Journal of Applied Probability, 31 (1994), 467.  doi: 10.2307/3215040.  Google Scholar [19] K. C. Madan, A cyclic queueing system with three servers and optional two-way feedback,, Microelectron. Rel., 28 (1988), 873.  doi: 10.1016/0026-2714(88)90285-5.  Google Scholar [20] K. C. Madan, An M/G/1 queue with second optional service,, Queueing Systems, 34 (2000), 37.  doi: 10.1023/A:1019144716929.  Google Scholar [21] K. C. Madan, On a single server queue with two stage general heterogeneous service and binomial schedule server vacations,, The Egyptian Statistical Journal, 44 (2000), 39.   Google Scholar [22] K. C. Madan, On a single server queue with two stage general heterogeneous service and deterministic schedule server vacations,, International Journal of System Science, 32 (2001), 837.  doi: 10.1080/00207720121488.  Google Scholar [23] K. C. Madan and M. Al-Rawwash, On the M$^x$/G/1 queue with feedback and optional server vacations based on a single vacation policy,, Applied Mathematics and Computation, 160 (2005), 909.   Google Scholar [24] K. C. Madan and A. Z. Abu Al-Rub, On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy,, Applied Mathematics and Computation, 149 (2004), 723.  doi: 10.1016/S0096-3003(03)00174-7.  Google Scholar [25] K. C. Madan, A. D. Al-Nasser and A. Q. Al-Masri, On M$^x$/(G1,G2)/1 queue with optional re-service,, Applied Mathematics and Computation, 152 (2004), 71.  doi: 10.1016/S0096-3003(03)00545-9.  Google Scholar [26] J. Medhi, A single server Poisson input queue with a second optional channel,, Queueing Systems, 42 (2002), 239.  doi: 10.1023/A:1020519830116.  Google Scholar [27] D. D. Selvam and V. Sivasankaran, A two-phase queueing system with server vacations,, Operations Research Letters, 15 (1994), 163.  doi: 10.1016/0167-6377(94)90052-3.  Google Scholar [28] L. Tadj and G. Choudhury, Optimal design and control of queues,, TOP, 13 (2005), 359.  doi: 10.1007/BF02579061.  Google Scholar [29] L. Tadj and J-.C. Ke, Control policy of a hysteretic queueing system,, Mathematical Methods of Operations Research, 57 (2003), 367.   Google Scholar [30] L. Tadj and J-.C. Ke, Control policy of a hysteretic bulk queueing system,, Mathematical and Computer Modelling, 5 (2004), 571.   Google Scholar [31] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation,", Vol. 1, (1991).   Google Scholar [32] N. Tian and Z. G. Zhang, "Vacation Queueing Models - Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [33] Z. G. Zhang, R. G. Vickson and M. J. A. van Eenige, Optimal two threshold policies in an M/G/1 queue with two vacation types,, Performance Evaluation, 29 (1997), 63.  doi: 10.1016/S0166-5316(96)00005-3.  Google Scholar [34] J. Wang, An M/G/1 queue with second optional service and server breakdowns,, Computers and Mathematics with Applications, 47 (2004), 1713.  doi: 10.1016/j.camwa.2004.06.024.  Google Scholar

show all references

##### References:
 [1] L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications,, Journal of Applied Mathematics and Stochastic Analysis, 4 (1991), 335.  doi: 10.1155/S1048953391000254.  Google Scholar [2] J. R. Artalejo and G. Choudhury, Steady state analysis of an M/G/1 queue with repeated attempts and two phase service,, Quality Technology and Quantitative Management, 1 (2004), 189.   Google Scholar [3] D. Bertsimas and X. Papaconstantinou, On the steady-state solution of the M/C$_2(a,b)$/$S$ queueing system,, Transportation Sciences, 22 (1988), 125.  doi: 10.1287/trsc.22.2.125.  Google Scholar [4] D. Bertsimas and X. Papaconstantinou, Analysis of the stationary $E_k$/$C_2$/S queueing system,, European Journal of Operational Research, 37 (1988), 272.  doi: 10.1016/0377-2217(88)90336-0.  Google Scholar [5] G. Choudhury, Some aspects of an M/G/1 queueing system with optional second service,, TOP, 11 (2003), 141.  doi: 10.1007/BF02578955.  Google Scholar [6] G. Choudhury and K. C. Madan, A two-phase batch arrival queueing system with a vacation time under Bernoulli schedule,, Applied Mathematics and Computation, 149 (2004), 337.  doi: 10.1016/S0096-3003(03)00138-3.  Google Scholar [7] G. Choudhury and K. C. Madan, A two-stage arrival queueing system with a modified Bernoulli schedule vacation under N-policy,, Mathematical and Computer Modelling, 42 (2005), 71.  doi: 10.1016/j.mcm.2005.04.003.  Google Scholar [8] G. Choudhury and M. Paul, A batch arrival queue with an additional service channel under N-policy,, Applied Mathematics and Computation, 156 (2004), 115.  doi: 10.1016/j.amc.2003.07.006.  Google Scholar [9] G. Choudhury and M. Paul, Analysis of a two phases batch arrival queueing model with Bernoulli vacation schedule,, Revista Investigatión Operacional, 25 (2004), 217.   Google Scholar [10] B. T. Doshi, Queueing systems with vacations: A survey,, Queueing Systems, 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [11] B. T. Doshi, Single-server queues with vacations,, in, (1990), 217.   Google Scholar [12] B. T. Doshi, Analysis of a two-phase queueing system with general service times,, Operations Research Letters, 10 (1991), 265.  doi: 10.1016/0167-6377(91)90012-E.  Google Scholar [13] A. Federgruen and K. C. So, Optimality of threshold policies in single server queueing system with vacations,, Advances in Applied Probability, 23 (1991), 388.  doi: 10.2307/1427755.  Google Scholar [14] O. Kella, The threshold policy in the M/G/1 queue with server vacations,, Naval Research Logistics, 36 (1989), 111.  doi: 10.1002/1520-6750(198902)36:1<111::AID-NAV3220360109>3.0.CO;2-3.  Google Scholar [15] T. S. Kim and K. C. Chae, Two-phase queueing system with generalized vacation,, Journal of the Korean Institute of Industrial Engineers, 22 (1996), 95.   Google Scholar [16] T. S. Kim and A. Q. Park, Cycle analysis of a two-phase queueing model with threshold,, European Journal of Operational Research, 144 (2003), 157.   Google Scholar [17] C. M. Krishna and Y. H. Lee, A study of two-phase service,, Operations Research Letters, 9 (1990), 91.  doi: 10.1016/0167-6377(90)90047-9.  Google Scholar [18] H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, Analysis of M$^x$/G/1 queue with N policy and multiple vacations,, Journal of Applied Probability, 31 (1994), 467.  doi: 10.2307/3215040.  Google Scholar [19] K. C. Madan, A cyclic queueing system with three servers and optional two-way feedback,, Microelectron. Rel., 28 (1988), 873.  doi: 10.1016/0026-2714(88)90285-5.  Google Scholar [20] K. C. Madan, An M/G/1 queue with second optional service,, Queueing Systems, 34 (2000), 37.  doi: 10.1023/A:1019144716929.  Google Scholar [21] K. C. Madan, On a single server queue with two stage general heterogeneous service and binomial schedule server vacations,, The Egyptian Statistical Journal, 44 (2000), 39.   Google Scholar [22] K. C. Madan, On a single server queue with two stage general heterogeneous service and deterministic schedule server vacations,, International Journal of System Science, 32 (2001), 837.  doi: 10.1080/00207720121488.  Google Scholar [23] K. C. Madan and M. Al-Rawwash, On the M$^x$/G/1 queue with feedback and optional server vacations based on a single vacation policy,, Applied Mathematics and Computation, 160 (2005), 909.   Google Scholar [24] K. C. Madan and A. Z. Abu Al-Rub, On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy,, Applied Mathematics and Computation, 149 (2004), 723.  doi: 10.1016/S0096-3003(03)00174-7.  Google Scholar [25] K. C. Madan, A. D. Al-Nasser and A. Q. Al-Masri, On M$^x$/(G1,G2)/1 queue with optional re-service,, Applied Mathematics and Computation, 152 (2004), 71.  doi: 10.1016/S0096-3003(03)00545-9.  Google Scholar [26] J. Medhi, A single server Poisson input queue with a second optional channel,, Queueing Systems, 42 (2002), 239.  doi: 10.1023/A:1020519830116.  Google Scholar [27] D. D. Selvam and V. Sivasankaran, A two-phase queueing system with server vacations,, Operations Research Letters, 15 (1994), 163.  doi: 10.1016/0167-6377(94)90052-3.  Google Scholar [28] L. Tadj and G. Choudhury, Optimal design and control of queues,, TOP, 13 (2005), 359.  doi: 10.1007/BF02579061.  Google Scholar [29] L. Tadj and J-.C. Ke, Control policy of a hysteretic queueing system,, Mathematical Methods of Operations Research, 57 (2003), 367.   Google Scholar [30] L. Tadj and J-.C. Ke, Control policy of a hysteretic bulk queueing system,, Mathematical and Computer Modelling, 5 (2004), 571.   Google Scholar [31] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation,", Vol. 1, (1991).   Google Scholar [32] N. Tian and Z. G. Zhang, "Vacation Queueing Models - Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [33] Z. G. Zhang, R. G. Vickson and M. J. A. van Eenige, Optimal two threshold policies in an M/G/1 queue with two vacation types,, Performance Evaluation, 29 (1997), 63.  doi: 10.1016/S0166-5316(96)00005-3.  Google Scholar [34] J. Wang, An M/G/1 queue with second optional service and server breakdowns,, Computers and Mathematics with Applications, 47 (2004), 1713.  doi: 10.1016/j.camwa.2004.06.024.  Google Scholar
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