延迟多重休假离散时间的Geomx/G/1可修排队系统---一些排队指标

数学物理学报 ›› 2011, Vol. 31 ›› Issue (2): 546-558.

延迟多重休假离散时间的Geom^x/G/1可修排队系统---一些排队指标

唐应辉¹|余玅妙^1,2|李才良³|黄蜀娟4|云曦⁴

1.四川师范大学数学与软件科学学院成都 610066|2. 四川理工学院理学院四川自贡 643000; 成都电子机械高等专科学校信息与计算科学系成都 610032|4. 电子科技大学数学学院成都 610054

收稿日期:2008-09-25 修回日期:2010-12-13 出版日期:2011-04-25 发布日期:2011-04-25
基金资助:
国家自然科学基金(70871084)、教育部高校博士点专项研究基金(200806360001)和四川省教育厅自然科学基金(08ZC028)资助

The Discrete Time Geom^x/G/1 Repairable Queueing System with Delay Multiple Vacations---Some Queueing Indices

TANG Ying-Hui¹, YU Miao-Miao^1,2, LI Cai-Liang³, HUANG Shu-Juan⁴, YUN Xi⁴

1.School of Mathematics &|Software Science, Sichuan Normal University, Chengdu 610066|2.School of Science, Sichuan University of Science and Engineering, Sichuan |Zigong 643000|3.Department of Information and Computer Science, Chengdu Electro-mechanical College, Chengdu 610032|4.School of Mathematics, University of Electronic Science &|Technology of China, Chengdu 610054

Received:2008-09-25 Revised:2010-12-13 Online:2011-04-25 Published:2011-04-25
Supported by:
国家自然科学基金(70871084)、教育部高校博士点专项研究基金(200806360001)和四川省教育厅自然科学基金(08ZC028)资助

摘要/Abstract

摘要：

该文首次考虑延迟多重休假离散时间成批到达的Geom^x/G/1可修排队系统, 在假定到达间隔时间和服务台的寿命服从几何分布, 而服务时间, 延迟休假时间, 休假时间和服务台失效后的修理时间均服从一般离散分布下, 通过引进``服务员忙期”和使用全概率分解技术, 从任意初始状态i~(i=0,1,…)出发, 研究了系统在任意时刻队长的瞬态性质, 导出了队长瞬态分布的Z -变换, 其次研究了系统在任意时刻队长的稳态分布, 获得了稳态分布的递推表达式, 进一步也得出了系统稳态队长的随机分解结果. 特别地, 通过该文可直接获得一系列特殊的离散时间排队系统相应的结果.

关键词: 离散时间排队, Geom^x/G/1, 可修, 瞬态分布, 稳态分布, 随机分解

Abstract:

This paper firstly considers the discrete time Geom^x/G/1 repairable queue system with delay multiple vacations. It's assumed that both the inter-arrival times and the life of the service station are independent random variables with geometric distribution, while the service time, the delayed vacation time, the vacation time and the repair time have general discrete distribution. By introducing the server busy period and using the total probability decomposition technique we discuss the transient properties of the queue length from the beginning of any initial state i~(i=0,1,…), and obtain the z-transformation of the transient distribution of the queue length. Furthermore, we obtain the recursion expressions of the steady distribution and the stochastic decomposition of the queue length at a random point in equilibrium. Especially we can obtain some corresponding results of the discrete queue system under some special cases.

Key words: Discrete time queue, Geom^x/G/1, Repairable, Transient distribution, Steady distribution, Stochastic
decomposition

中图分类号:

60K25

唐应辉, 余玅妙, 李才良, 黄蜀娟, 云曦. 延迟多重休假离散时间的Geom^x/G/1可修排队系统---一些排队指标[J]. 数学物理学报, 2011, 31(2): 546-558.

TANG Ying-Hui, YU Miao-Miao, LI Cai-Liang, HUANG Shu-Juan, YUN Xi. The Discrete Time Geom^x/G/1 Repairable Queueing System with Delay Multiple Vacations---Some Queueing Indices[J]. Acta mathematica scientia,Series A, 2011, 31(2): 546-558.

参考文献

[1] Meisling T. Discrete time queue theory. Open Res, 1958, 6: 96--105

[2] 朱翼隽, 胥秀珍. 空竭服务多级适应性休假Geom^x/G/1排队系统分析. 江苏大学学报(自然科学版), 2005, 26(3): 133--136

[3] 朱翼隽, 胥秀珍. 空竭服务多级适应性休假Geom^x/G/1可修排队系统分析. 江苏大学学报(自然科学版), 2005, 26(7): 316--319

[4] 田乃硕. GI/Geometric/1离散时间排队. 运筹与决策, 1992, 2: 1671--1677

[5] 田乃硕. Geometric/G/1休假随机服务系统. 应用数学与计算数学学报, 1993, 2(1): 71--78

[6] 马占友, 徐秀丽, 田乃硕. 多重休假的带启动期-关闭期的Geom/G/1离散排队及其在ATM网络中的应用. 运筹与管理, 2004, 13(5): 21--25

[7] 马占友, 田乃硕. 多重休假的带启动期Geom/G/1排队. 运筹与管理, 2002, 11(4): 5--10

[8] 侯玉梅. 服务台可修的Geomtric/G/1离散时间排队. 数学的实践与认识, 1996, 26(4): 328--333

[9] Yu Haibo. The MAP/PH(PH/PH)/1 discrete time queuing system with repairable server. Chinese Quarterly Journal of Mathematics,
2001, 6: 59--63

[10] Takagi H. Analysis of a discrete time queueing system with time-limited service. Questa, 1994, 18(2): 183--197

[11] Tian N, Zhang Z G. The discrete time GI/Geo/1 queue with multiple vacations. Queueing Systems, 2002, 40: 283--294

[12] 田乃硕, 徐秀丽, 马占友. 离散时间排队论. 北京: 科学出版社, 2008

[13] 曹晋华, 程侃. 服务台可修的M/G/1排队系统分析. 应用数学学报, 1982, 5(2): 113--127

[14] 唐应辉, 唐小我. 排队论---基础与分析技术. 北京: 科学出版社, 2006

[15] 唐应辉, 唐小我. 推广的多重休假M^x/G/1排队系统. 系统科学与数学, 2005, 25(1): 39--49

[16] Tang Y H, Tang X W. The queue length distribution for M^x/G/1 queue with single server vacation. Acta Math Scientia, 2001, 21(3): 397--408

[17] Hunter J J. Mathematical Techniques of Applied Probability, Vol II. Discrete Time Models: Techniques and Applications. New York: Academic Press, 1983