In this paper, we consider the M/G/1 queueing system with multiple server vacations and Min(N,D,V) -policy. By using the total probability decomposition technique and the Laplace transformation tool, the transient queue-length distribution and the steady queue-length distribution are discussed. Both the expressions of the Laplace transformation of the transient queue-length distribution and the recursive expressions of the steady queue-length distribution are obtained. Meanwhile, we present the stochastic decomposition result of the steady queue length and the explicit expression of the additional queue length distribution. Furthermore, some special cases are discussed when N→1, D→1, p{V=∞}=1 or p{V=0}=1. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. And by through numerical calculation, we determine the optimal control policy (N*; D*) for minimizing the long-run expected cost per unit time as well as compare with the single optimal N*-policy and the single optimal D*-policy.
[1] Yadin M, Naor P. Queueing systems with a removable service station[J]. Operational Research Quarterly, 1963, 14:393-405.
[2] Balachandran K R. Control policies for a single server system[J]. Management Science, 1973, 19(9):1013-1018.
[3] Heyman D P. T-policy for the M/G/1 queue[J]. Management Science, 1977, 23(7):775-778.
[4] 唐应辉, 黄蜀娟, 余玅妙, 等. 推广的(t,T)策略下M/G/1排队系统队长分布的递推解及最优策略[J]. 工程数学学报}, 2009, 26(2):251-259.
[5] 吴锦标, 尹小玲, 刘再明. 具有N-策略和负顾客的反馈抢占型的M/G/1重试可修排队系统[J]. 应用数学学报}, 2009, 32(2):323-335.
[6] 唐应辉, 蒲会, 余玅妙. 带启动时间的N-策略M/G/1排队系统的队长[J]. 系统工程理论与实践, 2011, 31(1):131-137.
[7] Tadj L, Choudhury G, Rekab K. A two-phase quorum queueing system with Bernoulli vacation schedule, setup, and N-policy for an unreliable server with delaying repair[J]. International Journal Services and Operations Management, 2012, 12(2):139-164.
[8] Lee H W, Beak J W, Jeon J. Analysis of the Mx/G/1 queue under D-policy[J]. Stochastic Analysis and Applications, 2005, 23(4):785-808.
[9] Wang K H, Kuo C C, Ke J C. Optical control of the D-Policy M/G/1 queueing system with server breakdowns[J]. American Journal of Applied Science, 2008, 5(5):565-573.
[10] Gakis G K, Rhee H K, Sivazlian B D. Distributions and first moments of the busy and idle periods in controllable M/G/1 queueing models with simple and dyadic policies[J]. Stochastic Analysis and Applications, 1995, 13(1):47-81.
[11] 井彩霞, 崔颖, 田乃硕. Min(N,V)-策略休假的M/G/1排队系统分析[J]. 运筹与管理}, 2006, 15(3):53-58.
[12] 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[J]. Performance Evaluation, 2008, 65(10):742-758.
[13] Lee H W, Seo W J, Lee S W, et al. Analysis of the MAP/G/1 queue under the Min(N, D)-policy[J]. Stochastic Models, 2010, 26(1):98-123.
[14] 唐应辉, 吴文青, 刘云颇, 等. 基于多重休假的Min(N,V)-策略M/G/1排队系统的队长分布[J]. 系统工程理论与实践}, 2014, 34(6):1533-1546.
[15] 唐应辉, 吴文青, 刘云颇. 基于单重休假的Min(N,V)-策略M/G/1排队系统分析[J]. 应用数学学报}, 2014, 37(6):976-996.
[16] 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[J]. 系统科学与数学}, 2015, 35(6):729-744.
[17] 朱莎, 朱翼隽, 王逢佳. 带负顾客启动期N-策略Geom/Geom/1工作休假排队[J]. 系统科学与数学}, 2012, 32(1):36-44.
[18] 兰绍军, 唐应辉. 具有多重休假和Min(N,V)-策略控制的Geo/G/1离散时间排队[J]. 系统工程理论与实践}, 2015, 35(3):799-810.
[19] 兰绍军, 唐应辉. 具有单重休假和Min(N,V)-策略控制的Geo/G/1离散时间排队的离去过程分析[J]. 数学物理学报}, 2016, 36A(2):380-392.
[20] 兰绍军, 唐应辉. 具有Bernoulli反馈和Min(N,D)-策略控制的Geoλ1,λ2/G/1离散时间可修排队的可靠性分析[J]. 系统科学与数学}, 2016, 36(11):2070-2086.
[21] Lan S J, Tang Y H, Yu M M. System capacity optimization design and optimal threshold N* for a Geo/G/1 discrete-time queue with single server vacation and under the control of Min(N, V)-policy[J]. Journal of Industrial & Management Optimization, 2016, 12(4):1435-1464.
[22] Lan S J, Tang Y H. Analysis of D-policy discrete-time Geo/G/1 queue with second J-optional service and unreliable server[J]. RAIRO-Operations Research, 2017, 51(1):101-122.
[23] Lan S J, Tang Y H. Analysis of a discrete-time Geoλ1,λ2[24] 唐应辉, 唐小我. 排队论——基础与分析技术}[M]. 北京:科学出版社, 2006.
