On the strong approximation for a simple reentrant line in light traffic under first-buffer first-served service discipline. (English) Zbl 1542.90075
Summary: For a 2-station and 3-class reentrant line under first-buffer first-served (FBFS) service discipline in light traffic, we firstly construct the strong approximations for performance measures including the queue length, workload, busy time and idle time processes. Based on the obtained strong approximations, we use a strong approximation method to find all the law of the iterated logarithms (LILs) for the above four performance measures, which are expressed as some functions of system parameters: means and variances of interarrival and service times, and characterize the fluctuations around their fluid approximations.
MSC:
| 90B22 | Queues and service in operations research |
| 68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |
| 60K25 | Queueing theory (aspects of probability theory) |
Keywords:
reentrant line queueing network; FBFS service discipline; strong approximation; LIL; Brownian motionReferences:
| [1] | Armbruster, D.; Marthaler, DE; Ringhofer, C.; Kempf, K.; Jo, TC, A continuum model for a re-entrant factory, Operations Research, 54, 5, 933-950, 2006 · Zbl 1167.90477 · doi:10.1287/opre.1060.0321 |
| [2] | Banks, J.; Dai, JG, Simulation studies of multiclass queueing networks, IIE Transactions, 29, 213-219, 1997 · doi:10.1080/07408179708966328 |
| [3] | Bramson, M.; D’Auria, B.; Walton, N., Stability and instability of the MaxWeight policy, Mathematics of Operations Research, 46, 4, 1611-1638, 2021 · Zbl 1532.90027 · doi:10.1287/moor.2020.1106 |
| [4] | Cao, J.; Guo, Y.; Yang, K., Variability analysis for a two-station queueing network in heavy traffic with arrival processes driven by queues, Acta Mathematicae Applicatae Sinica, English Series, 40, 2, 445-466, 2024 · Zbl 1536.90046 · doi:10.1007/s10255-024-1089-4 |
| [5] | Chen, H.; Harrison, JM; Mandelbaum, A.; Van Ackere, A.; Wein, LM, Empirical evaluation of a queueing network model for semi-conductor wafer fabrication, Operations Research, 36, 202-215, 1988 · doi:10.1287/opre.36.2.202 |
| [6] | Chen, H.; Shen, X., Strong approximations for multiclass feedforward queueing networks, Annals of Applied Probability, 10, 828-876, 2000 · Zbl 1083.60511 · doi:10.1214/aoap/1019487511 |
| [7] | Chen, H.; Yao, DD, Fundamentals of Queueing Networks, 2001, New York: Springer-Verlag, New York · Zbl 0992.60003 · doi:10.1007/978-1-4757-5301-1 |
| [8] | Chen, H.; Zhang, H., Diffusion approximation for re-entrant lines with a first-buffer-first-served priority discipline, Queueing Systems, 23, 177-195, 1997 · Zbl 0879.60088 · doi:10.1007/BF01206556 |
| [9] | Chen, H.; Zhang, H., Stability of multiclass queueing networks under priority service disciplines, Operations Research, 48, 1, 26-37, 2000 · Zbl 1106.60312 · doi:10.1287/opre.48.1.26.12456 |
| [10] | Csörgö, M.; Révész, P., Strong Approximations in Probability and Statistics, 1981, New York: Academic Press, New York · Zbl 0539.60029 |
| [11] | Csörgö, M.; Horváth, L., Weighted Approximations in Probability and Statistics, 1993, New York: Wiley, New York · Zbl 0770.60038 |
| [12] | Csörgö, M.; Horváth, L.; Steinebach, J., Invariance principlies for renewal processes, The Annals of Probability, 15, 4, 1441-1660, 1987 · Zbl 0635.60032 · doi:10.1214/aop/1176991986 |
| [13] | Dai, JG, On the positive Harris recurrence for multiclass queueing networks, Annals of Applied Probability, 5, 49-77, 1995 · Zbl 0822.60083 · doi:10.1214/aoap/1177004828 |
| [14] | Dai, JG; VandeVate, J., The stability of two-station multi-type fluid networks, Operations Research, 48, 721-744, 2000 · Zbl 1106.90311 · doi:10.1287/opre.48.5.721.12408 |
| [15] | Dai, JG; Weiss, G., Stability and instability of fluid models for reentrant lines, Mathematics of Operations Research, 21, 1, 115-134, 1996 · Zbl 0848.60086 · doi:10.1287/moor.21.1.115 |
| [16] | DeJong, CD; Wu, SP; Snowdon, JL; Charnes, JM, Simulating the transport and scheduling of priority lots in semiconductor factories, Proc. 34th Winter Simulation Conf, 1387-1391, 2002, San Diego, CA: ACM, San Diego, CA · doi:10.1109/WSC.2002.1166407 |
| [17] | Guo, Y., Rate of convergence of fluid approximation for re-entrant lines under FBFS discipline, Asia-Pacific Journal of Operational Research, 28, 3, 401-417, 2011 · Zbl 1216.90030 · doi:10.1142/S0217595911003296 |
| [18] | Guo, Y.; Hou, X., Strong approximation method and the (functional) law of iterated logarithm for GI/G/1 queue, Journal of Systems Science and Complexity, 30, 5, 1097-1106, 2017 · Zbl 1394.60091 · doi:10.1007/s11424-017-5226-5 |
| [19] | Guo, Y., Hou, X. Functional law of the iterated logarithm for multiclass queues with preemptive priority service discipline: the overloaded case. Stochastic Models in Reliability, Network Security and System Safety Essays Dedicated to Professor Jinhua Cao on the Occasion of His 80th Birthday, edited by Li Q., Wang J. and Yu H., Springer, 2019, 315-343 · Zbl 1427.68014 |
| [20] | Guo, Y., Hou, X. Functional law of the iterated logarithm for multiclass queues with preemptive priority service discipline: the underloaded and critically loaded case. Stochastic Models in Reliability, Network Security and System Safety Essays Dedicated to Professor Jinhua Cao on the Occasion of His 80th Birthday, edited by Li Q., Wang J. and Yu H., Springer, 2019, 344-360 · Zbl 1536.90052 |
| [21] | Guo, Y.; Hou, X.; Liu, Y., A functional law of the iterated logarithm for multi-class queues with batch arrivals, Annals of Operations Research, 300, 51-77, 2021 · Zbl 1480.60277 · doi:10.1007/s10479-020-03864-6 |
| [22] | Guo, Y.; Lefeber, E.; Nazarathy, Y.; Weiss, G.; Zhang, H., Stability and performance for multi-class queueing networks with infinite virtual queues, Queueing Systems, 76, 3, 309-342, 2014 · Zbl 1312.60110 · doi:10.1007/s11134-013-9362-x |
| [23] | Guo, Y.; Li, Z., Asymptotic variability analysis for a two-stage tandem queue, part I: The functional law of the iterated logarithm, Journal of Mathematical Analysis and Applications, 450, 2, 1479-1509, 2017 · Zbl 1383.60079 · doi:10.1016/j.jmaa.2017.01.062 |
| [24] | Guo, Y.; Li, Z., Asymptotic variability analysis for a two-stage tandem queue, part II: The law of the iterated logarithm, Journal of Mathematical Analysis and Applications, 450, 2, 1510-1534, 2017 · Zbl 1383.60080 · doi:10.1016/j.jmaa.2016.10.054 |
| [25] | Guo, Y.; Liu, Y., A law of iterated logarithm for multiclass queues with preemptive priority service discipline, Queueing Systems, 79, 3, 251-291, 2015 · Zbl 1310.60131 · doi:10.1007/s11134-014-9419-5 |
| [26] | Guo, Y.; Liu, Y.; Pei, R., Functional law of iterated logarithm for multi-server queues with batch arrivals and customer feedback, Annals of Operations Research, 264, 157-191, 2018 · Zbl 1391.90186 · doi:10.1007/s10479-017-2529-9 |
| [27] | Guo, Y.; Song, Y., The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue, Journal of Industrial and Management Optimization, 16, 3, 1049-1076, 2020 · Zbl 1449.60129 · doi:10.3934/jimo.2018192 |
| [28] | Guo, Y.; Yang, J.; Wang, X., Stability of a 2-station-5-class re-entrant line with infinite supply of work, Asia-Pacific Journal of Operational Research, 25, 4, 477-493, 2008 · Zbl 1151.90341 · doi:10.1142/S0217595908001821 |
| [29] | Guo, Y.; Zhang, H., On the stability of a simple re-entrant line with infinite supply, OR Transactions, 10, 2, 75-85, 2006 |
| [30] | Horváth, L., Strong approximations of renewal processes and their applications, Acta Mathematica Hungarica, 47, 1-2, 13-27, 1986 · Zbl 0606.60037 · doi:10.1007/BF01949120 |
| [31] | Horváath, L., Strong approximations of open queueing network, Mathematics of Operations Research, 17, 487-508, 1990 · Zbl 0758.60099 · doi:10.1287/moor.17.2.487 |
| [32] | Kumar, PR, Re-entrant lines, Queueing Systems, 13, 87-110, 1993 · Zbl 0772.90049 · doi:10.1007/BF01158930 |
| [33] | Leahu, H.; Mandjes, M., Stochastic monotonicity of markovian multiclass queueing networks, Stochastic Systems, 9, 2, 141-154, 2019 · Zbl 1446.60056 · doi:10.1287/stsy.2018.0022 |
| [34] | Lu, SCH; Ramaswamy, D.; Kumar, PR, Efficient scheduling policies to reduce mean and variance of cycle-time in semiconductor manufacturing plants, IEEE Trans. Semiconductor Manufacturing, 7, 3, 374-385, 1994 · doi:10.1109/66.311341 |
| [35] | Pai, HM, A differential game formulation of a controlled network, Queueing Systems, 64, 325-358, 2010 · Zbl 1188.90069 · doi:10.1007/s11134-009-9161-6 |
| [36] | Ross, M.S. Stochastic Processes. John Wiley & Sons Inc., 1996 · Zbl 0888.60002 |
| [37] | Sakalauskas, LL; Minkevičius, S., On the law of the iterated logarithm in open queueing networks, European Journal of Operational Research, 120, 3, 632-640, 2000 · Zbl 0981.60093 · doi:10.1016/S0377-2217(99)00003-X |
| [38] | Shikalgar, ST; Fronckowiak, D.; MacNair, EA; Snowdon, JL; Charnes, JM, 300 mm wafer fabrication line simulation model, Proc. 34th Winter Simulation Conf, 1365-1368, 2002, San Diego, CA: ACM, San Diego, CA · doi:10.1109/WSC.2002.1166403 |
| [39] | Veatch, MH; Senning, JR, Fluid analysis of an input control problem, Queueing Systems, 61, 87-112, 2009 · Zbl 1166.90334 · doi:10.1007/s11134-008-9101-x |
| [40] | Weiss, G., Stability of a simple re-entrant line with infinite supply of work C the case of exponential processing times, Journal of the Operations Research Society of Japan, 47, 304-313, 2004 · Zbl 1106.90323 · doi:10.15807/jorsj.47.304 |
| [41] | Zhang, H.; Hsu, G., Strong approximations for priority queues: Head-of-the-line-first discipline, Queueing Systems, 10, 213-233, 1992 · Zbl 0743.60106 · doi:10.1007/BF01159207 |
| [42] | Zhang, H.; Hsu, G.; Wang, R., Strong approximitions for multiple channel queues in heavy traffic, Journal of Applied Probability, 28, 658-670, 1990 · Zbl 0715.60113 |
| [43] | Zhang, H., Strong approximations for irreducible closed queueing networks, Advances in Applied Probability, 29, 498-522, 1997 · Zbl 0905.60069 · doi:10.2307/1428014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
