基于PH服务的工作休假排队的流模型
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摘要
研究了带有单重工作休假的M/PH/1排队系统驱动的流体模型.首先,通过拟生灭过程和矩阵几何解法分别得到无穷小生成元和驱动过程的稳态队长分布.其次,建立并分析流体模型,根据平衡方程给出流体模型的稳态联合分布函数满足的矩阵微分方程组,利用Laplace变换(LT)和Laplace-Stieltjes变换(LST)的方法,推导出平稳缓冲器(库)容量的空库概率表达式和稳态条件下的缓冲器(库)容量的均值表达式.最后,给出模型在移动自组织网络(Ad Hoc)中的应用,并通过数值例子讨论系统参数对系统性能指标的影响.
This paper is concerned with the fluid model which is driven by a PHservice time and single-server queue with server working vacation. To analyze the fluid model, we first establish the stationary distribution of the queue length process. Based on the steady-state distribution, then the matrix-type ordinary differential equation is obtained for the joint distribution characterizing the fluid model dynamics. With the help of the Laplace transform(LT) and the Laplace-Stieltjes transform(LST), as usual,the probability for the system empty and the average fluid level are given. An application of these obtained results to the mobile Ad Hoc networks is provided. The sensitivities about the system primitive parameters to the performance measures such as the average fluid level are discussed by some numerical experiments.
This paper is concerned with the fluid model which is driven by a PHservice time and single-server queue with server working vacation. To analyze the fluid model, we first establish the stationary distribution of the queue length process. Based on the steady-state distribution, then the matrix-type ordinary differential equation is obtained for the joint distribution characterizing the fluid model dynamics. With the help of the Laplace transform(LT) and the Laplace-Stieltjes transform(LST), as usual,the probability for the system empty and the average fluid level are given. An application of these obtained results to the mobile Ad Hoc networks is provided. The sensitivities about the system primitive parameters to the performance measures such as the average fluid level are discussed by some numerical experiments.
引文
[1] Virtamo J,Norros I. Fluid queue driven by an M/M/1 queue[J].Queueing Systems, 1994,16(3):1261-1273.
[2] Kulkarni V G. Fluid models for single buffer systems[M]//Fronties in Queueing:Models and Applications in Science and Engineering, Florida:CRC Press, 1997:321-338.
[3] Barbot N, Sericola B. Stationary solution to the fluid queue fed by an M/M/1 queue[J]. Journal of Applied Probability,2002, 39(2):359-369.
[4] Malhotra R, Mandjes M, Scheinhardt W, et al. A feedback fluid queue with two congestion control thresholds[J]. Mathematical Methods of Operations Research, 2009, 70(1):149-169.
[5] Mao B W, Wang F W, Tian N S. Fluid model driven by an M/M/1/N queue with multiple exponential vacations[J]. Journal of Computational Information Systems,2010, 6(6):1809-1816.
[6]刘晓艳,梅宁,徐秀丽.带负顾客的M/M/1多重休假排队系统驱动的流模型[J].西北师范大学学报(自然科学版),2013, 49(5):25-28.
[7] Xu X L, Geng J, Liu M X, et al. Stationary analysis for the fluid model driven by the M/M/c working vacation queue[J]. Journal of Mathematical Analysis and Applications, 2013, 403(2):423-433.
[8] Xu X L, Zhao Y Z, Geng J, et al. Analysis of the fluid model driven by an M/PH/1 queue[J].Journal of Information and Computational Sciences,2013, 10(11):3489-3496.
[9] Mao B W, Wang F W, Zhao H. Fluid model fed by an M/PH/1 queue with multiple vacations[J]. ICIC Express Letters, 2015, 9(10):2795-2800.
[10]徐秀丽,宋晓凤,靖欣,等.PH/M/1排队系统驱动的流体模型[J].系统科学与数学,2017, 37(3):838-845.
[11]李云,隆克平,赵为粮,等.无线多跳Ad hoc网络中MAC机制的公平性与网络容量利用率[J].电子与信息学报,2007, 29(2):333-339.
[12]马琳,张军,刘凯.无线Ad Hoc网络的MAC层拥塞控制算法[J].华中科技大学学报(自然科学版),2012, 40(8):112-117.
[13]杨顺利.N-策略M/M/1和M/PH/1工作休假排队[D].秦皇岛:燕山大学,2008.
[2] Kulkarni V G. Fluid models for single buffer systems[M]//Fronties in Queueing:Models and Applications in Science and Engineering, Florida:CRC Press, 1997:321-338.
[3] Barbot N, Sericola B. Stationary solution to the fluid queue fed by an M/M/1 queue[J]. Journal of Applied Probability,2002, 39(2):359-369.
[4] Malhotra R, Mandjes M, Scheinhardt W, et al. A feedback fluid queue with two congestion control thresholds[J]. Mathematical Methods of Operations Research, 2009, 70(1):149-169.
[5] Mao B W, Wang F W, Tian N S. Fluid model driven by an M/M/1/N queue with multiple exponential vacations[J]. Journal of Computational Information Systems,2010, 6(6):1809-1816.
[6]刘晓艳,梅宁,徐秀丽.带负顾客的M/M/1多重休假排队系统驱动的流模型[J].西北师范大学学报(自然科学版),2013, 49(5):25-28.
[7] Xu X L, Geng J, Liu M X, et al. Stationary analysis for the fluid model driven by the M/M/c working vacation queue[J]. Journal of Mathematical Analysis and Applications, 2013, 403(2):423-433.
[8] Xu X L, Zhao Y Z, Geng J, et al. Analysis of the fluid model driven by an M/PH/1 queue[J].Journal of Information and Computational Sciences,2013, 10(11):3489-3496.
[9] Mao B W, Wang F W, Zhao H. Fluid model fed by an M/PH/1 queue with multiple vacations[J]. ICIC Express Letters, 2015, 9(10):2795-2800.
[10]徐秀丽,宋晓凤,靖欣,等.PH/M/1排队系统驱动的流体模型[J].系统科学与数学,2017, 37(3):838-845.
[11]李云,隆克平,赵为粮,等.无线多跳Ad hoc网络中MAC机制的公平性与网络容量利用率[J].电子与信息学报,2007, 29(2):333-339.
[12]马琳,张军,刘凯.无线Ad Hoc网络的MAC层拥塞控制算法[J].华中科技大学学报(自然科学版),2012, 40(8):112-117.
[13]杨顺利.N-策略M/M/1和M/PH/1工作休假排队[D].秦皇岛:燕山大学,2008.