有限容量M/M/1/N多重工作休假排队系统的性能分析
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摘要
随着通讯与计算机技术的迅猛发展,各种各样复杂的排队系统也随之不断地出现,尤其是带有止步、中途退出和工作休假等类型的排队系统模型,在制造系统、计算机系统与通信网络等领域中有着广泛的应用,具有重要的实际意义。
论文研究了等待空间有限的带有止步、中途退出和多重工作休假的两个排队模型。这些模型是已有文献中相关模型的推广。
首先,研究了等待空间有限的M/M/1/N多重工作休假排队系统。根据马尔可夫过程理论导出了稳态概率所满足的方程组,并通过将转移率矩阵写成分块矩阵的形式,求出了系统稳态概率的矩阵形式解,并给出了具体的算法。此外,还针对N=3的特殊情况,利用Matlab数学计算软件的符号计算功能,得到了稳态概率的明显表达式。最后通过数值方法分析了系统各参数对系统性能指标的影响。
其次,研究了系统等待空间有限的带有止步和中途退出的M/M/1/N多重工作休假排队系统。通过将转移率矩阵写成分块矩阵的形式,给出了系统稳态概率的矩阵形式解。然后以N=3为例建立了一个以服务员正规忙期的服务率μb为控制变量的单位时间的稳态费用模型。由于费用函数的表达式非常复杂,难以求出最优服务率的明显表达式,所以采用数值方法计算最优服务率和单位时间最优费用。最后,通过费用模型的几个数值例子,分析了系统各参数对最优服务率、最优费用以及系统各个性能指标的影响。
With the quick development of the communication systems and computer techniques, there are many kinds of complicated queuing systems. The queuing systems are widely applied in the manufacture system, computer system and communication network, especially those with balking, reneging and multiple working vacations, so they have more practical significance.
In this paper, two finite buffer queuing systems with balking, reneging, and multiple working vacations are analyzed. They are expansions of other models in the documents. The main result of this paper has two parts.
Firstly, we investigate a finite buffer M/M/1/N queuing system with multiple working vacations. First, we derive the steady-state equations by the Markov process method .By writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities and present a algorithm for calculating the steady-state probabilities. In addition, using the symbol function of Matlab, we get the explicit expression of the steady-state probabilities for the special case of N=3. Finally, we analyze the influence of the parameters of the system to the system performance measures .
Secondly, we investigate a finite buffer M/M/1/N queuing system with balking, reneging and multiple working vacations. First, by writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities . In addition, we develop a steady-state cost model where the busy service rateμb is the control variable. However, the expression of the cost function is too complex to get the explicit expression of the optimal service rate, so we use the numerical method to get the optimal service rate and the optimal cost. Finally, we investigate the effect of the parameters of the system on the optimal service rate, the optimal cost and the system performance measures by several numerical examples.
论文研究了等待空间有限的带有止步、中途退出和多重工作休假的两个排队模型。这些模型是已有文献中相关模型的推广。
首先,研究了等待空间有限的M/M/1/N多重工作休假排队系统。根据马尔可夫过程理论导出了稳态概率所满足的方程组,并通过将转移率矩阵写成分块矩阵的形式,求出了系统稳态概率的矩阵形式解,并给出了具体的算法。此外,还针对N=3的特殊情况,利用Matlab数学计算软件的符号计算功能,得到了稳态概率的明显表达式。最后通过数值方法分析了系统各参数对系统性能指标的影响。
其次,研究了系统等待空间有限的带有止步和中途退出的M/M/1/N多重工作休假排队系统。通过将转移率矩阵写成分块矩阵的形式,给出了系统稳态概率的矩阵形式解。然后以N=3为例建立了一个以服务员正规忙期的服务率μb为控制变量的单位时间的稳态费用模型。由于费用函数的表达式非常复杂,难以求出最优服务率的明显表达式,所以采用数值方法计算最优服务率和单位时间最优费用。最后,通过费用模型的几个数值例子,分析了系统各参数对最优服务率、最优费用以及系统各个性能指标的影响。
With the quick development of the communication systems and computer techniques, there are many kinds of complicated queuing systems. The queuing systems are widely applied in the manufacture system, computer system and communication network, especially those with balking, reneging and multiple working vacations, so they have more practical significance.
In this paper, two finite buffer queuing systems with balking, reneging, and multiple working vacations are analyzed. They are expansions of other models in the documents. The main result of this paper has two parts.
Firstly, we investigate a finite buffer M/M/1/N queuing system with multiple working vacations. First, we derive the steady-state equations by the Markov process method .By writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities and present a algorithm for calculating the steady-state probabilities. In addition, using the symbol function of Matlab, we get the explicit expression of the steady-state probabilities for the special case of N=3. Finally, we analyze the influence of the parameters of the system to the system performance measures .
Secondly, we investigate a finite buffer M/M/1/N queuing system with balking, reneging and multiple working vacations. First, by writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities . In addition, we develop a steady-state cost model where the busy service rateμb is the control variable. However, the expression of the cost function is too complex to get the explicit expression of the optimal service rate, so we use the numerical method to get the optimal service rate and the optimal cost. Finally, we investigate the effect of the parameters of the system on the optimal service rate, the optimal cost and the system performance measures by several numerical examples.
引文
1.徐光辉.随机服务系统.北京:科学出版社,1988
2.唐应辉,唐小我.排队论一基础与应用.成都:电子科技大学出版社,2000
3. D.R. Cox. The Analysis of Non-Markovian Stochastic Processes by the Inclusion of Supplementary Variable. Proc. Camb. Phil. Soc.1995,51:433-441
4.孙荣恒,李建平.排队论基础.北京:科学出版社,2002:1-4
5. A.K. Erlang. the Theory of Probabilities and Telephone Conversations. Nyt Tidsskrift Matematik,1909,(B20):33-39
6. A.K. Erlang. Solution of Some Problems in the Theory of Probability of Singuificance in Automatic Telephone Exchanges. Electroteknikeren,1917,(13):5-13
7. Pollaczek. FLosung Eines Geometrischen Wahrsceinlichkeits Problems. Math Z,1932, (35):230-278
8. A.Ya. Khinchin. Mathematical Theory of A Stationary Queu (in Russian). Mat, Sbornik, 1932,(39):73-84
9. C. Palm. Analysis of the Erlang Traffic Formulae for Busy Signal Arrangements. Ericsson Tech,1938,(6):39-58
10. D.G. Kendall. Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of Imbedded Markov Chains. Ann Math Statist,1953,(24): 338-354
11. P. Moran. A Probability Theory of Dams and Storage Systems: Modifications of the Release Rule. Australia Journal of Applied Society,1955,(7):130-137
12. K.W. Ross, D.K. Tsang, J. Wang. Monte Carlo Summation and Integration Applied to Multichain Queueing Networks. Journal of Association Computers Mathematics, 1994,(41):1110-1135
13.盛招友.排队论及其在计算机通讯中的应用.北京:北京邮电大学出版社,1998:3-4
14.陈祖安,王凯.利用排队模型优化零件机械加工时间.现代制造工程, 2002, (09):27-30
15.陈庆宏,温渤.排队论在生产过程时间组织中的应用.北方经贸,2003,(11):7-11
16.杜智华.排队论在局域网规划设计中的应用.新疆师范大学学报,2002,(01):4-6
17.周玉元,何清泉,周铁军.一类休假排队模型及其在红绿灯问题中的应用.湘潭大学自然科学学报,2002,(02):35-39
18.张蕊.服务行业排队论问题分析.齐齐哈尔大学学报,2002,(06):26-27
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20. Y. Levy, U. Yechiali. Utilization of Idle Time in An M/G/1 Queueing System. Management Science,1975,35:708-721
21. O. Kella. the Threshold Policy in the M/G/1 Queue with Server Vacations. Naval Research Logistics,1989,(36):111-123
22. H.S. Lee, M.M. Srinivasan. Control Policies for the Mx/G/1 Queueing System. Management Science,1989,35:708-721
23. T.T. Lee. M/G/1 Queue with Vacation Time and Limited Service Discipline. Performance Evaluation,1989,9(3):181-190
24. K.K. Leung. On the Additional Delay in An M/G/1 Queue with Generalized Vacations and Exhaustive Service. Operations Research ,1992,40(supple):5272-5283
25. H.W. Lee, S.S. Lee. Batch Arrival Queue with Different Vacations. Computers & Operations Research,1991,18(1):51-58
26. B. Doshi. Queueing Systems with Vacations and Survey. Queueing System,1986,(1): 29-66
27. J. Teghem. Control of the Service Process in A Queueing System. European Journal of Operational Research.1986,(23):141-158
28.田乃硕.休假随机服务系统.运筹学杂志,1990,9(1):17-31
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30.田乃硕.单重指数休假的GI/M/1排队系统.系统科学与数学,2000,13:1-9
31.田乃硕.异步休假M/M/C排队系统的稳态理论.应用数学学报,2000,(23):2-2
32. N. Tian, Q. Li. the M/M/C Queue with Phase Type Synchronous Vacations. J.Sys. Sci. & Math.Scis,2001,(1):7-16
33.马占友,田乃硕.多重休假带启动期的Geom/G/1排队.运筹与管理,2002,(4):5-10
34.唐应辉,毛勇.服务员假期中以概率p进入的M/G/1排队系统的随机分解.数学物理学报,2004,24(6):683-688
35.田乃硕,岳德权.拟生灭过程与矩阵几何解.北京:科学出版社,2002:16-26
36. Servi L D, Finn S G. M/M/1 queue with working vacations (M/M/1/WV). Perform. Evaluation, 2002, 50: 41-52
37. Kim J D, Choi D W, Chae K C. Analysis of queue-length distribution of the M/G/1 queue with working vacations . In: Hawaii International Conference on statistics and Related Fields. 2003, (6): 5-8
38. Wu D, Takagi H. M/G/1 queue with multiple working vacations . Perform. Evaluation, 2006, 63: 654-681
39. Baba Y, Analysis of GI/M/1 queue with multiple working vacations. Operat. Res. Letters, 2005, 33: 201-209.
40. W. Liu, X. Xu, N. Tian. Stochastic decompositions in the M/M/1queue with working vacations, Operation Research Letters, 2007, 35(5):595-600
41. J. Li, N. Tian. The M/M/1 queue with working vacations and vacation interruptions. Syst Sci Syst Eng., 2007, (16):121-127
42. J. Li, N. Tian, W. Liu. Discrete-time GI/Geo/1 queue with multiple working vacations. Queueing systems, 2007, 56(1):53-63
43. J. Li, N. Tian. Analysis of the discrete time Geo/Geo/1 queue with single working vacation. Quality Technology and Quantity Management(Special issue on Queueing Models with vacations), 2008, 5(1):77-89
44. J. Li, N. Tian. The discrete-time GI/Geo/1 queue with working vacations and vacation interruption. Applied Mathematics and Computation, 2007, 185(1):1-10
45. C. Palm. Etude Des Delais D’attente. Ericsson technics,1937,5:37-56
46. K. Udagawa, G. Nakamura. On A Queue in which Joining Customers Give Up Their Services Halfway. Jour of Operations Research Society of Japan,1957:186-197
47. F.A. Haight. Queueing with Reneging. Metrika,1959,2:186-197
48. F.A. Haight. Queueing with balking. Biometrika,1957,44:360-369
49. J.F. Reynolds. The Stationary Solution of A Multiserver Queueing Model with Discouragement. Operations Research,1968,16:64-71
50. B. Natvig. On the Transient-state Probability of A Queueing Model where Potential Customers are Discouraged by Queueing Length. Scotland Journal Statistics,1975,(2): 34-42
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2.唐应辉,唐小我.排队论一基础与应用.成都:电子科技大学出版社,2000
3. D.R. Cox. The Analysis of Non-Markovian Stochastic Processes by the Inclusion of Supplementary Variable. Proc. Camb. Phil. Soc.1995,51:433-441
4.孙荣恒,李建平.排队论基础.北京:科学出版社,2002:1-4
5. A.K. Erlang. the Theory of Probabilities and Telephone Conversations. Nyt Tidsskrift Matematik,1909,(B20):33-39
6. A.K. Erlang. Solution of Some Problems in the Theory of Probability of Singuificance in Automatic Telephone Exchanges. Electroteknikeren,1917,(13):5-13
7. Pollaczek. FLosung Eines Geometrischen Wahrsceinlichkeits Problems. Math Z,1932, (35):230-278
8. A.Ya. Khinchin. Mathematical Theory of A Stationary Queu (in Russian). Mat, Sbornik, 1932,(39):73-84
9. C. Palm. Analysis of the Erlang Traffic Formulae for Busy Signal Arrangements. Ericsson Tech,1938,(6):39-58
10. D.G. Kendall. Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of Imbedded Markov Chains. Ann Math Statist,1953,(24): 338-354
11. P. Moran. A Probability Theory of Dams and Storage Systems: Modifications of the Release Rule. Australia Journal of Applied Society,1955,(7):130-137
12. K.W. Ross, D.K. Tsang, J. Wang. Monte Carlo Summation and Integration Applied to Multichain Queueing Networks. Journal of Association Computers Mathematics, 1994,(41):1110-1135
13.盛招友.排队论及其在计算机通讯中的应用.北京:北京邮电大学出版社,1998:3-4
14.陈祖安,王凯.利用排队模型优化零件机械加工时间.现代制造工程, 2002, (09):27-30
15.陈庆宏,温渤.排队论在生产过程时间组织中的应用.北方经贸,2003,(11):7-11
16.杜智华.排队论在局域网规划设计中的应用.新疆师范大学学报,2002,(01):4-6
17.周玉元,何清泉,周铁军.一类休假排队模型及其在红绿灯问题中的应用.湘潭大学自然科学学报,2002,(02):35-39
18.张蕊.服务行业排队论问题分析.齐齐哈尔大学学报,2002,(06):26-27
19.田乃硕.休假随机服务系统.北京:北京大学出版社,2000:1-3
20. Y. Levy, U. Yechiali. Utilization of Idle Time in An M/G/1 Queueing System. Management Science,1975,35:708-721
21. O. Kella. the Threshold Policy in the M/G/1 Queue with Server Vacations. Naval Research Logistics,1989,(36):111-123
22. H.S. Lee, M.M. Srinivasan. Control Policies for the Mx/G/1 Queueing System. Management Science,1989,35:708-721
23. T.T. Lee. M/G/1 Queue with Vacation Time and Limited Service Discipline. Performance Evaluation,1989,9(3):181-190
24. K.K. Leung. On the Additional Delay in An M/G/1 Queue with Generalized Vacations and Exhaustive Service. Operations Research ,1992,40(supple):5272-5283
25. H.W. Lee, S.S. Lee. Batch Arrival Queue with Different Vacations. Computers & Operations Research,1991,18(1):51-58
26. B. Doshi. Queueing Systems with Vacations and Survey. Queueing System,1986,(1): 29-66
27. J. Teghem. Control of the Service Process in A Queueing System. European Journal of Operational Research.1986,(23):141-158
28.田乃硕.休假随机服务系统.运筹学杂志,1990,9(1):17-31
29. N. Tian, D. Zhang, C. Cao.(田乃硕,张大庆,曹成弦) the GI/M/1 Queue with Exponential Vacations. Queueing Systems,1989,(5):331-344
30.田乃硕.单重指数休假的GI/M/1排队系统.系统科学与数学,2000,13:1-9
31.田乃硕.异步休假M/M/C排队系统的稳态理论.应用数学学报,2000,(23):2-2
32. N. Tian, Q. Li. the M/M/C Queue with Phase Type Synchronous Vacations. J.Sys. Sci. & Math.Scis,2001,(1):7-16
33.马占友,田乃硕.多重休假带启动期的Geom/G/1排队.运筹与管理,2002,(4):5-10
34.唐应辉,毛勇.服务员假期中以概率p进入的M/G/1排队系统的随机分解.数学物理学报,2004,24(6):683-688
35.田乃硕,岳德权.拟生灭过程与矩阵几何解.北京:科学出版社,2002:16-26
36. Servi L D, Finn S G. M/M/1 queue with working vacations (M/M/1/WV). Perform. Evaluation, 2002, 50: 41-52
37. Kim J D, Choi D W, Chae K C. Analysis of queue-length distribution of the M/G/1 queue with working vacations . In: Hawaii International Conference on statistics and Related Fields. 2003, (6): 5-8
38. Wu D, Takagi H. M/G/1 queue with multiple working vacations . Perform. Evaluation, 2006, 63: 654-681
39. Baba Y, Analysis of GI/M/1 queue with multiple working vacations. Operat. Res. Letters, 2005, 33: 201-209.
40. W. Liu, X. Xu, N. Tian. Stochastic decompositions in the M/M/1queue with working vacations, Operation Research Letters, 2007, 35(5):595-600
41. J. Li, N. Tian. The M/M/1 queue with working vacations and vacation interruptions. Syst Sci Syst Eng., 2007, (16):121-127
42. J. Li, N. Tian, W. Liu. Discrete-time GI/Geo/1 queue with multiple working vacations. Queueing systems, 2007, 56(1):53-63
43. J. Li, N. Tian. Analysis of the discrete time Geo/Geo/1 queue with single working vacation. Quality Technology and Quantity Management(Special issue on Queueing Models with vacations), 2008, 5(1):77-89
44. J. Li, N. Tian. The discrete-time GI/Geo/1 queue with working vacations and vacation interruption. Applied Mathematics and Computation, 2007, 185(1):1-10
45. C. Palm. Etude Des Delais D’attente. Ericsson technics,1937,5:37-56
46. K. Udagawa, G. Nakamura. On A Queue in which Joining Customers Give Up Their Services Halfway. Jour of Operations Research Society of Japan,1957:186-197
47. F.A. Haight. Queueing with Reneging. Metrika,1959,2:186-197
48. F.A. Haight. Queueing with balking. Biometrika,1957,44:360-369
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