在 $(m,N)$-策略下不中断多重休假 Bernoulli 反馈排队的性能分析与最优控制策略

摘要/Abstract

摘要：

该文研究一个在$(m,N)$-策略下不中断多重休假 Bernoulli 反馈的$M/G/1$排队系统, 其中当服务员休假归来时, 若系统中的顾客数达到或超过事先设定的较低阈值$m(\geq1)$时服务员立即启动系统. 当系统的启动完成后, 如果系统中的顾客数不小于另一个事先设定的较高阈值$N(\geq m)$时服务员立即开始服务直到系统再次变空. 利用随机分解定理得到了系统稳态队长的概率母函数和平均队长的显式表达式, 并探讨了稳态下服务员处于忙期和非忙期的概率、服务员连续休假次数等其他重要排队系统性能指标. 通过数值实例进一步进行了平均附加队长和系统空闲率对参数$m$, $N$、系统启动时间参数和服务员休假时间参数的敏感性分析. 最后, 应用更新报酬定理推导出系统长期单位时间内期望费用的目标函数, 并通过数值实例获得了使得系统期望费用最小的二维最优控制策略$(m^*,N^*)$, 同时分析了系统启动时间参数、服务员休假时间参数和顾客反馈率对系统期望费用和最优控制策略的影响.

关键词: $M/G/1$ 排队, 双阈值 $(m,N)$-策略, 不中断多重休假, Bernoulli 反馈, 最优控制策略

Abstract:

This paper studies an $M/G/1$ queueing system with Bernoulli feedback and uninterrupted multiple vacations under $(m,N)$-policy, in which the server immediately activates the system upon returning from vacation if the number of customers reaches or exceeds a predefined lower threshold $m(\geq 1)$. After the system startup is completed, the server begins service only if the number of customers is not less than a predefined upper threshold $N(\geq m)$, continuing until the system becomes empty again. Utilizing the stochastic decomposition theorem for the steady-state queue length distribution, explicit expressions for the probability generating function (PGF) and the average queue size are derived. Furthermore, other important queueing performance indicators of the system are also discussed, such as the probabilities of the server busy and non-busy period in the steady state, as well as the number of consecutive vacations taken by the server. Through numerical examples, sensitivity analyses are conducted to examine the effects of the parameters $m$, $N$, the startup time, and the server vacation time on the mean additional queue length and the system idle probability. Finally, the objective function of the long-run expected cost per unit time of the system is constructed by applying the renewal reward theorem. Numerical examples are presented to discuss the bi-level optimal control strategy $(m^*,N^*)$ that minimizes the long-run expected cost per unit time of the system, and the impact of system activation time, server vacation time and customer feedback rate on the expected cost and optimal control strategy are also analyzed.

Key words: $M/G/1$ queue, bi-level threshold $(m,N)$-policy, uninterrupted multiple vacation, Bernoulli feedback, optimal control policy

中图分类号:

O226

朱芸, 唐应辉, 余玅妙, 刘琼琳. 在 $(m,N)$-策略下不中断多重休假 Bernoulli 反馈排队的性能分析与最优控制策略[J]. 数学物理学报, 2026, 46(6): 2433-2450.

Yun Zhu, Yinghui Tang, Miaomiao Yu, Qionglin Liu. Performance Analysis and Optimal Control Policy for a Uninterrupted Multiple Vacations and Bernoulli Feedback Queue Under the $(m,N)$-Policy[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2433-2450.

图/表 21

图1

图2

图3

图4

图5

图6

图7

图8

图9

图10

图11

表1

图12

图13

表2

表3

图14

图15

表4

图16

图17

参考文献 30

[1]	Doshi B T. Queueing systems with vacations-a survey. Queueing Syst, 1986, 1(1): 29-66
[2]	田乃硕. 休假随机服务系统. 北京: 科学出版社, 2001
	Tian N S. Stochastic Service System with Vacations. Beijing: Science Press, 2001
[3]	唐应辉, 唐小我. 排队论-基础与分析技术. 北京: 科学出版社, 2006
	Tang Y H, Tang X W. Queueing Theory-Foundations and Analysis Techniques. Beijing: Science Press, 2006
[4]	Lan S J, Tang Y H. The structure of departure process and optimal control strategy $N^*$ for $Geo/G/1$ discrete-time queue with multiple server vacations and $Min(N,V)$-policy. J Syst Sci Complex, 2017, 30(6): 1382-1402 doi: 10.1007/s11424-017-5279-5
[5]	Lee H W, Seo W J. The performance of the $M/G/1$ queue under the dyadic $Min(N,D)$-policy and its cost optimization. Perform Eval, 2008, 65(10): 742-758 doi: 10.1016/j.peva.2008.04.006
[6]	魏瑛源, 唐应辉, 余玅妙. 基于 $Min(N,D)$-策略的 $M/G/1$ 排队系统的队长分布及最优策略. 系统科学与数学, 2015, 35(6): 729-744 doi: 10.12341/jssms12594
	Wei Y Y, Tang Y H, Yu M M. Queue length distribution and optimum policy for $M/G/1$ queueing system under $Min(N,D)$-policy. J Syst Sci & Math Sci, 2015, 35(6): 729-744
[7]	Lou L, Tang Y H, Yu M M, et al. Optimal control policy of $M/G/1$ queueing system with delayed randomized multiple vacations under the modified $Min(N,D)$-policy control. J Oper Res Soc China, 2023, 11(4): 857-874 doi: 10.1007/s40305-022-00413-9
[8]	Liu Q L, Tang Y H, Yu M M. A new discrete-time queueing model to optimize cargo dispatch for a warehouse. Oper Res Int J, 2025, 25(1): Art 7
[9]	袁雨梅, 唐应辉, 刘雨欣, 等. 在随机启动 $(p,N)$-策略控制下不中断多重休假排队系统的性能分析. 应用数学, 2023, 36(4): 1069-1085
	Yuan Y M, Tang Y H, Liu Y X, et al. Performance analysis for a queueing system with uninterrupted multiple vacations under the control of randomized setup $(p,N)$-policy. Appl Math, 2023, 36(4): 1069-1085
[10]	Kuang X Y, Tang Y H, Yu M M, et al. Performance analysis of an $M/G/1$ queue with bi-level randomized $(p,N_1,N_2)$-policy. RAIRO Oper Res, 2022, 56(1): 395-413 doi: 10.1051/ro/2022008
[11]	陈镰元, 唐应辉, 袁雨梅, 等. 在双水平 $(p,N_1,N_2)$- 策略控制下随机启动时间的 $M/G/1$ 排队的性能与约束优化. 系统科学与数学, 2024, 44(7): 2088-2111 doi: 10.12341/jssms23155
	Chen L Y, Tang Y H, Yuan Y M, et al. Performance and constrain optimization of $M/G/1$ queue with randomized setup time and the bi-level $(p,N_1,N_2)$-policy. J Syst Sci & Math Sci, 2024, 44(7): 2088-2111
[12]	柯淇淋, 唐应辉, 余玅妙. 具有检修策略和顾客进入控制策略的排队系统分析. 系统科学与数学, 2023, 43(8): 2164-2181 doi: 10.12341/jssms22157
	Ke Q L, Tang Y H, Yu M M. Analysis of queueing system with admission control and overhaul policy. J Syst Sci & Math Sci, 2023, 43(8): 2164-2181
[13]	刘雨欣, 唐应辉, 陈镰元, 等. 有 Bernoulli 检修与顾客进入控制策略的 $M/G/1$ 可修排队的队长分布与费用优化. 应用数学学报, 2024, 47(4): 567-591 doi: 10.20142/j.cnki.amas.202401038
	Liu Y X, Tang Y H, Chen L Y, et al. Queue-length distribution and cost optimization for an $M/G/1$ repairable queue with Bernoulli maintenance and admission control strategy. Acta Math Appl Sin, 2024, 47(4): 567-591
[14]	He Y X, Tang Y H, Yu M M, et al. Performance and optimization analysis of a queue with delayed uninterrupted multiple vacations and $N$-policy. Methodol Comput Appl Probab, 2024, 26(2): Art 21
[15]	Ke J C. Bi-level control for batch arrival queues with an early startup and unreliable server. Appl Math Model, 2004, 28(5): 469-485 doi: 10.1016/j.apm.2003.10.007
[16]	Begum M I A, Fijy P J, Kalpana K. A batch arrival queue with a second optional service channel under $(m,N)$-policy and single Bernoulli vacation. J Eng Comput Appl Sci, 2014, 3(10): 10-15
[17]	Luo C Y, Tang Y H, Yu K Z, et al. Optimal $(r,N)$-policy for discrete-time $Geo/G/1$ queue with different input rate and setup time. Appl Stoch Models Bus Ind, 2015, 31(4): 405-423 doi: 10.1002/asmb.v31.4
[18]	旷欣宇, 唐应辉. 带随机启动时间与双阈值 $(m,N)$-策略的 $M/G/1$ 可修排队系统的最优控制策略. 运筹与管理, 2021, 30(10): 64-70 doi: 10.12005/orms.2021.0315
	Kuang X Y, Tang Y H. Optimal control policy for the $M/G/1$ repairable queueing system with random start-up time and bi-level threshold $(m,N)$-policy. Oper Res & Manage Sci, 2021, 30(10): 64-70
[19]	高文萍, 唐应辉. 双阈值 $(m,N)$-策略不中断单重休假 $M/G/1$ 排队分析. 数学学报, 2023, 66(2): 363-388 doi: 10.12386/A20200026
	Gao W P, Tang Y H. Analysis of $M/G/1$ queue with bi-level threshold $(m,N)$-policy and uninterrupted single vacation. Acta Math Sin, 2023, 66(2): 363-388
[20]	王金亭. 排队博弈论基础. 北京: 科学出版社, 2016
	Wang J T. Fundamentals of Queueing Game Theory. Beijing: Science Press, 2016
[21]	高珊, 王金亭,Do T V. 具有 Bernoulli 休假的不可见 $M/M/1$ 重试排队模型的进队策略分析. 应用数学学报, 2017, 40(1): 106-120 doi: 10.12387/C2017010
	Gao S, Wang J T, Do T V. Analysis of the entrance strategies for an unobservable equilibrium joining strategies in unobservable $M/M/1$ retrial queue with Bernoulli vacation. Acta Math Appl Sin, 2017, 40(1): 106-120
[22]	王硕, 徐秀丽. 带启动时间的休假流体模型的止步策略分析. 应用数学学报, 2018, 41(6): 846-857 doi: 10.12387/C2018066
	Wang S, Xu X L. The balking strategies of fluid vacation with setup time. Acta Math Appl Sin, 2018, 41(6): 846-857
[23]	何柳青, 田瑞玲. 带有负顾客和启动时间的排队系统最优策略分析. 应用数学, 2024, 37(1): 226-237
	He L Q, Tian R L. Analysis of optimal strategies in a queueing system with negative customers and setup time. Appl Math, 2024, 37(1): 226-237
[24]	Choi B D, Kim B, Choi S H. On the $M/G/1$ Bernoulli feedback queue with multi-class customers. Comput Oper Res, 2000, 27(3): 269-286 doi: 10.1016/S0305-0548(99)00036-2
[25]	余玅妙, 唐应辉. 反馈次数服从几何分布的 $M/G/1$ 排队系统的队长分布. 电子学报, 2007, 35(2): 275-278
	Yu M M, Tang Y H. The queue length distribution of $M/G/1$ queueing system with feedback follows geometric distribution. Acta Electron Sin, 2007, 35(2): 275-278
[26]	魏瑛源, 唐应辉, 顾建雄. 带有 Bernoulli 反馈的多级适应性休假的 $Geo/G/1$ 排队系统分析. 高校应用数学学报, 2010, 25(1): 27-37
	Wei Y Y, Tang Y H, Gu J X. Analysis of $Geo/G/1$ queueing system with multiple adaptive vacations and Bernoulli feedback. Appl Math J Chin Univ, 2010, 25(1): 27-37
[27]	Lan S J, Tang Y H. Performance and reliability analysis of a repairable discrete-time $Geo/G/1$ queue with Bernoulli feedback and randomized policy. Appl Stoch Models Bus Ind, 2017, 33(5): 522-543 doi: 10.1002/asmb.v33.5
[28]	Poonam G, Naveen K. Cost optimization of single server retrial queueing model with Bernoulli schedule working vacation, vacation interruption and balking. J Math Comput Sci, 2021, 11(3): 2508-2523
[29]	Fuhrmann S W, Cooper R B. Stochastic decomposition in the $M/G/1$ queue with generalized vacations. Oper Res, 1985, 33(5): 1117-1129 doi: 10.1287/opre.33.5.1117
[30]	Ross S M. Stochastic Processes. New York: John Wiley and Sons, 1996