Modelling and Computation for a Queueing Manufacturing Plant System with Server Working Vacation under Control Policies

Abstract

This study investigates a queueing manufacturing plant system operating under sustainability principles, including green logistics and carbon tax policies. The focus is on the queueing dynamics of servicing failed machines, with server unreliability incorporated. The system includes advanced features such as N policy, reneging, multiple working vacations, Bernoulli phase repairs, server recovery Q policy, and multiple standby machines. A novel approach is introduced by combining two threshold policies, (Q, N) policies, where Q < N, to model complex operational and repair dynamics. The system begins in a working vacation state, transitioning to a busy state when the number of failed machines reaches or exceeds the threshold N. During the busy period, the server breaks down and can’t service failed machines until it is repaired. Repairs of the server begin when the number of failed machines reaches Q and follow a Bernoulli phase process, with up to p repair steps. Reneging behavior is incorporated in the ongoing service states of the server, namely the working vacation and busy states. Transient state probabilities are computed using the Runge-Kutta’s method to analyze the system’s performance. A comprehensive set of queueing and reliability metrics is developed, with numerical analyses examining the effects of varying system parameters. Sensitivity and relative sensitivity analyses of reliability, throughput, and cost function are presented. Results from the Runge-Kutta’s method are compared with adaptive neuro fuzzy inference system to verify accuracy. Managerial insights and conclusions are provided to enhance understanding of the system.

Keywords

Threshold control policy / working vacation / threshold recovery policy / Bernoulli phase repair / Runge-Kutta’s method / adaptive neuro fuzzy inference system (ANFIS)

References

[1] AbbasiS, MazaheriS, TalaieH R, GhasemiP. Designing green logistics networks under carbon tax policy: Post-COVID condition. Results in Engineering, 2024, 23: 102830

[2] AhujaA, JainA. Fuzzy analysis of a queueing system featuring an unreliable service provider and geometric arrivals by incorporating constant retrial policy and delayed threshold recovery. Journal of Ambient Intelligence and Humanized Computing, 2022, 14(6): 7499-7518

[3] AhujaA, JainA, JainM. Transient analysis and ANFIS computing of unreliable single server queueing model with multiple stage service and functioning vacation. Mathematics and Computers in Simulation, 2022, 192: 464-490

[4] BegumA, ChoudhuryG. Analysis of a bulk arrival N-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy. RAIRO-Operation Research, 2022, 56(2): 979-1012

[5] BegumA, ChoudhuryG. Analysis of an unreliable N-policy queue with set-up period under Bernoulli schedule and re-service approach. Communications in Statistics-Theory and Methods, 2024, 53(1): 1-33

[6] ChahalP K, KumarK, SoodanB S. Grey wolf algorithm for cost optimization of cloud computing repairable system with N-policy, discouragement and two-level Bernoulli feedback. Mathematics and Computers in Simulation, 2024, 225: 545-569

[7] ChakravarthyS R, KulshresthaR. A queueing model with server breakdowns, repairs, vacations, and backup server. Operations Research Perspectives, 2020, 7: 100131

[8] ChangC J, KeJ C, ChangaF M. Unreliable retrial queue with loss and feedback under threshold-based policy. International Journal of Industrial and Systems Engineering, 2018, 30(1): 1-20

[9] ChenW L. System reliability analysis of retrial machine repair systems with warm standbys and a single server of working breakdown and recovery policy. Systems Engineering, 2018, 21(1): 59-69

[10] ChenW L. Reliability and sensitivity analysis of the controllable repair system with warm standbys under the recovery threshold policy. Journal of Testing and Evaluation, 2019, 47(2): 1311-1331

[11] ChenW L, WangK H. Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy. Reliability Engineering & System Safety, 2018, 180: 476-486

[12] HaoY, WangJ, WangZ, YangM. Equilibrium joining strategies in the M/M/1 queues with setup times under N-policy. Journal of Systems Science and Systems Engineering, 2019, 28: 141-153

[13] JainM. Priority queue with batch arrival, balking, threshold recovery, unreliable server and optimal service. RAIRO Operations Research, 2017, 51(2): 417-432

[14] JainM, BhargavaC. N-Policy machine repair system with mixed standbys and unreliable server. Quality Technology & Quantitative Management, 2009, 6(2): 171-184

[15] JainM, MeenaR K. Vacation model for Markov machine repair problem with two heterogeneous unreliable servers and threshold recovery. Journal of Industrial Engineering International, 2018, 14: 143-152

[16] JainM, MittalR, KumariR. (m, M) Machining system with two unreliable servers, mixed spares and common-cause failure. Journal of Industrial Engineering International, 2015, 11: 171-178

[17] JainM, ShekharC, ShuklaS. Vacation queueing model for a machining system with two unreliable repairmen. International Journal of Operational Research, 2014, 20(4): 469-491

[18] JainM, KumarP, SinghM, GuptaR. Cost optimization and reliability analysis of fault tolerant system with service interruption and reboot. Reliability Engineering & System Safety, 2024, 249: 110229

[19] JayamaniV N, MadheswariS P, SuganthiP, JosephineS A. An N-policy M/M/1 queueing model for energy saving mechanism in Networks. Sustainable Computing: Informatics & Systems, 2024, 44: 101016

[20] KeJ C. The optimal control of an M/G/1 queueing system with server vacations, start up and breakdowns. Computers & Industrial Engineering, 2003, 44(4): 567-579

[21] KumarK, JainM. Threshold N-policy for (M, m) degraded machining system with K-heterogeneous servers, standby switching failure and multiple vacations. International Journal of Mathematics in Operational Research, 2013, 5(4): 423-445

[22] KumarK, JainM, ShekharC. Machine repair system with threshold recovery policy, unreliable servers and phase repair. Quality Technology & Quantitative Management, 2024, 21(5): 587-610

[23] KumarP, JainM, MeenaR K. Transient analysis and reliability modeling of fault-tolerant system operating under admission control policy with double retrial features and working vacation. ISA Transactions, 2023, 134: 183-99

[24] LiJ, TianN. The M/M/1 queue with working vacations and vacation interruptions. Journal of Systems Science and Systems Engineering, 2007, 16(1): 121-127

[25] MaQ, ZhangX. Analysis and comparison of queue with N-Policy and unreliable server. Hindawi Discrete Dynamics in Nature and Society, 2020, 1: 6195080

[26] MeenaR K, JainM, AssadA, SethiR, GargD. Performance and cost comparative analysis for M/G/1 repairable machining system with N-policy vacation. Mathematics and Computers in Simulation, 2022, 200: 315-328

[27] NiranjanS P, LathaS D, MahdalM, KarthikK. Multiple control policy in unreliable Two-Phase Bulk queueing system with active Bernoulli feedback and vacation. Mathematics, 2024, 12(1): 75

[28] RaniS, JainM, MeenaR K. Queueing modeling and optimization of a fault-tolerant system with reboot, recovery, and vacationing server operating under admission control policy. Mathematics and Computers in Simulation, 2023, 209: 408-425

[29] Raychaudhuri C, Jain A (2024). Efficiency of retrial queueing system under N threshold during vacation. International Journal of Information Technology: 1–12.

[30] SangaSS, AntalaK S. Performance analysis and cost investigations for state-dependent single unreliable server finite queue under F-policy using GA and QNM. Journal of Computational and Applied Mathematics, 2024, 441: 115679

[31] SelvarajuN, GoswamiC. Impatient customers in an M/M/1 queue with single and multiple working vacations. Computers & Industrial Engineering, 2013, 65(2): 207-215

[32] ServiL D, FinnS G. M/M/1 queues with working vacations (M/M/1/WV). Performance Evaluation, 2002, 50(1): 41-52

[33] ShaoQ, HuL, XuF. Reliability and optimization for k-out-of-n: G Mixed standby retrial system with dependency and J-vacation. Methodology and Computing in Applied Probability, 2024, 26(1): 8

[34] SharmaS, KumarK. Cost optimal analysis for a differentiated vacation machining system with discouragement under the two threshold control policies. Results in Control and Optimization, 2024, 15: 100409

[35] SharmaS, KumarK, GargS. Cost analysis for machine repair problem under triadic policy with discouragement and multiple working vacations. International Journal of Management Science and Engineering Management, 2024, 20(1): 122-140

[36] SharmaS, KumarK, GargS, JainM. Cost optimization of a single server machine repair system with hybrid vacation under a bi-level (N, M) policy and reneging. Quality Technology & Quantitative Management, 20241-29

[37] SunW, LiS, WangY, TianN. Comparisons of exhaustive and nonexhaustive M/M/1/N queues with working vacation and threshold policy. Journal of Systems Science and Systems Engineering, 2019, 28: 154-167

[38] Sun W, Zhang Z, Xie X, Li S (2024). Strategic behavior and social welfare optimization in a batch-service queueing system with threshold policy. Journal of Systems Science and Systems Engineering: 1–31.

[39] TadjL, YoonK P. Binomial schedule for an M/G/1 type queueing system with an unreliable server under N-policy. Advances in Decision Sciences, 20141-6

[40] Varshney S, Kaswan S, Devanda M, Shekhar C (2024). Finite capacity service system with partial server breakdown and recovery policy: An economic perspective. Journal of Systems Science and Systems Engineering: 1–31.

[41] WangK H. Optimal control of an M/E_K/1 queueing system with removable service station subject to breakdowns. Journal of the Operational Research Society, 1997, 48(9): 936-942

[42] WangK H, KaoH T, ChenG. Optimal management of a removable and non-reliable server in an Infinite and a finite M/H_k. Quality Technology & Quantitative Management, 2004, 1(2): 325-339

[43] WangK H, KeJ B, LeeW C. Reliability and sensitivity analysis of a repairable system with warm standbys and R unreliable service stations. International Journal of Advanced Manufacturing and Technology, 2007, 31: 1223-1232

[44] WangK H, YenT C, ChenJ Y. Optimization analysis of retrial machine repair problem with server breakdown and threshold recovery policy. Journal of Testing and Evaluation, 2018, 46(6): 2630-2640

[45] WangY, HuL, ZhaoB, TianR. Stochastic modeling and cost-benefit evaluation of consecutive k/n: F repairable retrial systems with two-phase repair and vacation. Computers & Industrial Engineering, 2023, 175: 108851

[46] YangD Y, ChiangY C. An evolutionary algorithm for optimizing the machine repair problem under a threshold recovery policy. Journal of the Chinese Institute of Engineers, 2014, 37(2): 224-231

[47] YangD Y, ChiangY C, TsouC S. Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy. Journal of Manufacturing Systems, 2013, 32(1): 174-179

[48] YangD Y, WuC H. Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Computers & Industrial Engineering, 2015, 82: 151-158

[49] YangD Y, WuZ R, TsouC S. Reliability analysis of a repairable system with geometric reneging and threshold-based recovery policy. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2015, 229(11): 2047-2062

[50] YangD Y, ChungC H, WuC H. Sojourn times in a Markovian queue with working breakdowns and delayed working vacations. Computers & Industrial Engineering, 2021, 156: 107239

[51] YenT C, ChenW L, ChenJ Y. Reliability and sensitivity analysis of the controllable repair system with warm standbys and working breakdown. Computers & Industrial Engineering, 2016, 97: 84-92

RIGHTS & PERMISSIONS

Systems Engineering Society of China and Springer-Verlag GmbH Germany