带有止步和中途退出的可修和休假排队
摘要
可修排队和休假排队是经典排队理论的延伸和发展,在通信系统、管理系统以及运输系统等方面有广泛的应用。在现实生活中我们经常遇到顾客的止步和中途退出现象。因此,研究带有止步和中途退出现象的可修排队系统和休假排队系统具有重要的理论意义和实际应用价值。
论文考虑了带有止步和中途退出的可修排队模型和休假排队模型。首先,研究了等待空间有限的带有止步和中途退出的部分服务员不可靠可修排队系统,其中系统中有两种类型的服务员,一种完全可靠,另一种不可靠。利用马尔可夫过程方法建立了稳态概率满足的方程组,采用了分块矩阵的解法求出了稳态概率向量简洁明了的迭代计算公式,进而得到了系统的一些性能指标,并使用Matlab软件进行了数值分析。
其次,研究了等待空间有限带有止步和中途退出的可修排队系统,其中两个服务员有不同的服务率,故障率和修复率。利用马尔可夫过程的方法,建立稳态概率满足的方程组,求出了系统稳态概率向量的迭代计算公式,并给出了系统的一些性能指标和两个不可靠服务员的可靠性指标。
最后,研究了等待空间有限带有止步和中途退出的休假排队系统,其中两个服务员有不同的服务率,并进行同步多重休假。运用马尔可夫过程的方法,建立了系统的稳态概率方程组,利用矩阵分块的方法,得到了系统稳态概率向量的矩阵解,并且通过计算一些分块矩阵的逆矩阵,给出了系统稳态概率精确的解析表达式,推导出了系统的一些性能指标和两个不同服务员全忙时进入系统并最终接受服务的顾客的条件等待时间分布及条件平均等待时间的精确解析表达式。
The repairable queue and the vacation queue are extension and development of classical queuing theory and have widespread application in communications system ,management system and transportation system. the phenomena of the customers’balking and reneging offen happens in our real life. Thus, the study on the repairable queuing system and the vacation queuing system with balking and reneging has important theoretical significance and actual application value.
In this paper, we consider the queuing models with server breakdowns and the phenomena of balking and reneging, as well as the model with vacation.
Firstly, we investigate a finite waiting room repairable queuing system with balking, reneging and partial server breakdowns, in which there are two types servers ,one is completely reliable and another is unreliable. By Markov process method, we develop the steady-state probability equations, and obtain the simple and obvious iterative formulas of the steady-State probability vectors by using a method of blocking matrix. Then we obtain some performance measures of the system and make numerical analysis by using Matlab software.
Secondly, we study a finite waiting room repairable queuing system with balking and reneging, in which two servers have different service rate, failure rate and repair rate. Using Markov process method, we develop the steady-state probability equations, and derive the iterative formulas of the steady-State probability vectors .we obtain some performance measures of the system and reliability indices of two unreliable servers.
Finally, we study a finite waiting room vacation queuing system with balking and reneging, in which two servers have different service rate and carry on synchronous multiple vacations. By Markov process method, we develop the steady-state probability equations and derive matric solutions of the steady-state probability vectors. Meantime, by computing inverse matrices of some blocking matrices, we obtain the close-form precise expression of steady-state probability and some performance measures of the system. Furthermore, we give the precise close-form expressions of condition waiting time distribution and condition mean time of the arrival customers who finally accept service when two heterogeneous servers are busy.
论文考虑了带有止步和中途退出的可修排队模型和休假排队模型。首先,研究了等待空间有限的带有止步和中途退出的部分服务员不可靠可修排队系统,其中系统中有两种类型的服务员,一种完全可靠,另一种不可靠。利用马尔可夫过程方法建立了稳态概率满足的方程组,采用了分块矩阵的解法求出了稳态概率向量简洁明了的迭代计算公式,进而得到了系统的一些性能指标,并使用Matlab软件进行了数值分析。
其次,研究了等待空间有限带有止步和中途退出的可修排队系统,其中两个服务员有不同的服务率,故障率和修复率。利用马尔可夫过程的方法,建立稳态概率满足的方程组,求出了系统稳态概率向量的迭代计算公式,并给出了系统的一些性能指标和两个不可靠服务员的可靠性指标。
最后,研究了等待空间有限带有止步和中途退出的休假排队系统,其中两个服务员有不同的服务率,并进行同步多重休假。运用马尔可夫过程的方法,建立了系统的稳态概率方程组,利用矩阵分块的方法,得到了系统稳态概率向量的矩阵解,并且通过计算一些分块矩阵的逆矩阵,给出了系统稳态概率精确的解析表达式,推导出了系统的一些性能指标和两个不同服务员全忙时进入系统并最终接受服务的顾客的条件等待时间分布及条件平均等待时间的精确解析表达式。
The repairable queue and the vacation queue are extension and development of classical queuing theory and have widespread application in communications system ,management system and transportation system. the phenomena of the customers’balking and reneging offen happens in our real life. Thus, the study on the repairable queuing system and the vacation queuing system with balking and reneging has important theoretical significance and actual application value.
In this paper, we consider the queuing models with server breakdowns and the phenomena of balking and reneging, as well as the model with vacation.
Firstly, we investigate a finite waiting room repairable queuing system with balking, reneging and partial server breakdowns, in which there are two types servers ,one is completely reliable and another is unreliable. By Markov process method, we develop the steady-state probability equations, and obtain the simple and obvious iterative formulas of the steady-State probability vectors by using a method of blocking matrix. Then we obtain some performance measures of the system and make numerical analysis by using Matlab software.
Secondly, we study a finite waiting room repairable queuing system with balking and reneging, in which two servers have different service rate, failure rate and repair rate. Using Markov process method, we develop the steady-state probability equations, and derive the iterative formulas of the steady-State probability vectors .we obtain some performance measures of the system and reliability indices of two unreliable servers.
Finally, we study a finite waiting room vacation queuing system with balking and reneging, in which two servers have different service rate and carry on synchronous multiple vacations. By Markov process method, we develop the steady-state probability equations and derive matric solutions of the steady-state probability vectors. Meantime, by computing inverse matrices of some blocking matrices, we obtain the close-form precise expression of steady-state probability and some performance measures of the system. Furthermore, we give the precise close-form expressions of condition waiting time distribution and condition mean time of the arrival customers who finally accept service when two heterogeneous servers are busy.
引文
1唐应辉,唐小我.排队论-基础与分析技术.北京:科学出版社, 2001
2徐光辉.随机服务系统.北京:科学出版社, 1988
3胡运权,郭耀煌.运筹学教程(第二版).北京:清华大学出版社, 2003
4 A. K. Erlang. The Theory of Probabilities and Telephone Conversations. Nyt T idsskrift Matematik, 1909, B20:33-39
5 A. K. Erlang. Solution of some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren, 1917, 13:5-13
6 D. G. Kendall. Some Problems in the Theory of Queues. Journal of Royal Statistic Society, 1951, B13:151-173
7 D. G. Kendall. Stochastic Processes Occurring in the Theory of Queues and their Analysis by the method of imbedded Markov chains. Ann Mathematics Statistic, 1953, 24:338-354
8 Y. Levy, U. Yechiali. Utilization of Idle Time in an M/G/1 Queueing System. Management Science, 1975, 35:708-721
9 B. Doshi. Queueing Systems with Vacation-a Survey. Queueing Systems, 1986, (1): 29-66
10 J. Teghem. Control of the Service process in a Queueing System. European Journal of Operational Research, 1986, 23:141-158
11 Y. Levy, U. Yechiali. An M/M/s Queue with Servers' Vacations. Canadian Journal of Operational Research and Information Processing, 1976, 14(2):153-163
12 B. Vinod. Exponential Queues with Server Vacations. Journal of the Operational Research Society, 1986, 37:1007-1014
13侯玉梅,田乃硕. M/M/C休假排队系统-综述.运筹学学报, 2000, 4(2):88-94
14田乃硕,高作峰,张忠君.异步休假M/M/C排队的稳态理论.应用数学学报, 2001, 4(2):185-194
15田乃硕,张忠君.同步休假GI/M/C稳态理论.运筹学学报, 2001, 1:70-80
16田乃硕.休假随机服务系统.北京:北京大学出版社, 2001:252-320
17 Z. G. Zhang, N. Tian. Analysis on Queuing Systems with Synchronous Vacations of Partial Servers. Performance Evaluation, 2003, 52:269-282
18 Z. G. Zhang, N. Tian. Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers. Queueing Systems, 2003, 45:161-175
19 Z. G. Zhang. On the Three Threshold Policy in the Multi-Server Queuing System with Vacations. Queuing Systems, 2005, 51:173-186
20 N. Tian, X. Xu. The Waiting Time of An M/M/c Queue with Partial Servers Vacations. Operations Research Transactions, 2005, 9(2):1-8
21申利民,金顺福,田乃硕.部分服务台同步策略多重休假的M/M/c排队.工程数学学报, 2004, 21(2):238-244
22刘洺辛,马占友,徐秀丽,田乃硕.部分服务台异步N-策略多重休假M/M/c排队.燕山大学学报(自然科学版), 2006, 30(3):230-234
23 N. Igaki. Exponential Two Server with N-policy and Multiple Vacations. Queuing Systems, 1992, 10:279-294
24 K. C. Mandan, W. Abu-Dayyeh, F. Taiyyan. A Two Server Queue with Bernoulli Schedules and a Single Vacation Policy. Applied Mathematics and Computation, 2003, 145:59-71
25 B. K. Kumar, S. P. Madheswari. An M/M/2 Queueing System with Heterogeneous Servers and Multiple Vacations. Mathematical and Computer Modelling, 2005, 41(13):1415-1429
26 D. Yue, J. Yu, W. Yue. A Markovian Queue with Two Heterogeneous Servers and Multiple Vacations. Journal of Industrial and Management Optimization, 2009, 5(3):453-465
27 H. C. White, L. S. Christie. Queuing with Preemptive Priorities or with Breakdown. Operations Research, 1958, 6:79-95
28 K. Thiruvengauam. Queuing with Breakdowns. Operations Research, 1963, 11(1):62 -71
29 B. Avi-Iizhak, P. Naor. Some Queuing Problems with the Service Station Subject to Breakdown. Operations Research, 1963, 11(3):303-320
30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
31 J. Wang, B. Liu, J. Li. Transient Analysis of an M/G/1 Retrial Queue Subject to Disasters and Server Failures. European Journal of Operational Research, 2008, (189):1118-1132
32 W. Zhou, Y. Deng. On the M/G/1 G-queue with Unreliable Server. Operations Research Transactions, 2006, 10(2):28-36
33朱翼隽,张峰.具有两种不同服务的可修MX/G(M/M)/1排队系统.江苏大学学报(自然科学版), 2005, 26(6A):51-57
34岳德权,吕胜利,李静铂.一个修理工的M/M/N可修排队.燕山大学学报(自然科学版), 2003, 27(3):197-202
35吕胜利,李静铂,岳德权. M/M/N可修排队稳态分布存在条件的一种新形式.系统工程理论与实践, 2005, (8):79-84
36李江华,王金亭.具有不耐烦顾客的M/M/N可修排队系统.北京交通大学学报(自然科学版), 2006, 30(3):84-87
37吕胜利,李静铂,岳德权.两个修理工的M/M/2可修排队系统.数学物理学报, 2008, 28(6):1109-1118
38余君,岳德权,田瑞玲.两个不同服务台的可修排队系统的矩阵几何解.应用数学学报, 2008, 31(4):894-900
39 J-C. Ke, C-H. Lin. Sensitivity Analysis of Machine Repair Problems in Manufacturing Systems with Service Interruptions. Applied Mathematical Modelling, 2008, 32(10):2087-2105
40 P. Singh. Two-server Makovian Queues with Balking, Heterogeneous vs. Homogeneous Servers. Operations Research, 1970, 18(1):145-159
41 A. Montazer-Hzgihighi, J. Medhi, S. G. Mohanty. On a Multiserver Markovian Queueing System with Balking and Reneging. Computers and Operations Research, 1986, 13(4):421-425
42 M. O. Abou-El-Ata, A. M. A. Hariri. The M/M/C/N Queue with Balking and Reneging. Computers and Operations Research, 1992, 19(9):713-716
43 W. Patrick, S. János and M. Hermann. Finite-source M/M/S Retrial Queue with Search for Balking and Impatient Customers from the Orbit. Computer Networks, 2009, 53(8):1264-1273
44 W. Yang and S. Taek. M/M/s Queue with Impatient Customers and Retrials. Applied Mathematical Modelling, 2009, 33(6):2596-2606
45 R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy, S. I. Ammar. Transient Solution of the M/M/c Queue with Balking and Reneging. Computers & Mathematics with Applications, 2009, 57(8):1280-1285
46 K-H. Wang, Y-C. Chang. Cost Analysis of a Finite M/M/R Queuing System with Balking, Reneging and Server Breakdowns. Mathematical Methods of Operations Research, 2002, (56):169-180
47朱翼隽,张继国,王伟.基于可变服务率M/M/S/K+M可修排队的呼叫中心性能分析.江苏大学学报(自然科学版), 2006, 27(4):368-371
48 K-H. Wang, J-B. Kea, J-C. Ke. Profit Analysis of the M/M/R Machine Repair Problem with Balking, Reneging, and Standby Switching Failures. Computers and Operations Research, 2007, 34:835-847
49 K-H. Wang, J-C. Ke. Probabilistic Analysis of a Repairable System with Warm Stanbys Plus Balking and Reneging. Applied Mathematical Modelling, 2003, 27(4):327-336
50 D. Yue, W. Yue. Block-partitioning Matrix Solution of M/M/R/N Queuing System with Balking , Reneging and Server Breakdowns. Journal of Industrial and Management Optimization, 2009, 5(3):417-430
51孙妍平,岳德权.带有止步和中途退出的M/M/S/N同步多重休假排队系统的性能分析.系统工程理论与实践, 2007, 8:152-158
52孙妍平,岳德权.带有止步和中途退出的成批到达的MX/M/R/N同步休假排队系统的性能分析.数学的实践与认识, 2007, 37(19):79-86
53岳德权,孙妍平.带有止步和中途退出的同步N-策略多重休假的M/M/R/K排队系统的性能分析.系统工程理论与实践, 2008, 28(11):94-102
54岳德权,孙妍平.带有止步和中途退出的M/M/C/N部分服务员同步多重休假排队系统的等待时间.系统工程与实践, 2008, (2):89-97
55孙妍平,岳德权.带有止步和中途退出的M/M/C/N部分服务员同步单重休假排队系统的性能分析.海南大学学报(自然科学版), 2008, 26(4):316-322
56 D. Yue, W. Yue, Y. Sun. Performance Analysis of an M/M/c/N Queuing System with Balking, Reneging and Synchronous Vacations of Partial Servers. Asia-Pacific Operations Research Center. The Sixth International Symposium on Operations Research and Its Applications, Xin jiang, China, 2006:128-143
57 D. Yue, W. Yue. Analysis of M/M/c/N Queueing System with Balking, Reneging and Synchronous Vacations. Advanced in Queueing Theory and Network Applications, 2009 :165-180
58马明建,岳德权,余君,马金旺.带有止步和中途退出的修理工有休假的机器维修问题.燕山大学学报, 2009, (2):173-177
59 D. Yue, W. Yue. A Heterogeneous Two-server Queueing System with Balking and Bernoulli Vacation Schedule. Proc. of the 4th International Symposium on Queueing Theory and Network Applications, Fusionololis, Singgapore, 2009:230-244,
60 M. F. Neuts, Y. Takahashi. Asymptotic Behaviour of the Stationary Distribution in the GI/PH/c Queue with Heterogeneous Servers. Probability Theory and Related Fields, 1981, 57(4):441-452
2徐光辉.随机服务系统.北京:科学出版社, 1988
3胡运权,郭耀煌.运筹学教程(第二版).北京:清华大学出版社, 2003
4 A. K. Erlang. The Theory of Probabilities and Telephone Conversations. Nyt T idsskrift Matematik, 1909, B20:33-39
5 A. K. Erlang. Solution of some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren, 1917, 13:5-13
6 D. G. Kendall. Some Problems in the Theory of Queues. Journal of Royal Statistic Society, 1951, B13:151-173
7 D. G. Kendall. Stochastic Processes Occurring in the Theory of Queues and their Analysis by the method of imbedded Markov chains. Ann Mathematics Statistic, 1953, 24:338-354
8 Y. Levy, U. Yechiali. Utilization of Idle Time in an M/G/1 Queueing System. Management Science, 1975, 35:708-721
9 B. Doshi. Queueing Systems with Vacation-a Survey. Queueing Systems, 1986, (1): 29-66
10 J. Teghem. Control of the Service process in a Queueing System. European Journal of Operational Research, 1986, 23:141-158
11 Y. Levy, U. Yechiali. An M/M/s Queue with Servers' Vacations. Canadian Journal of Operational Research and Information Processing, 1976, 14(2):153-163
12 B. Vinod. Exponential Queues with Server Vacations. Journal of the Operational Research Society, 1986, 37:1007-1014
13侯玉梅,田乃硕. M/M/C休假排队系统-综述.运筹学学报, 2000, 4(2):88-94
14田乃硕,高作峰,张忠君.异步休假M/M/C排队的稳态理论.应用数学学报, 2001, 4(2):185-194
15田乃硕,张忠君.同步休假GI/M/C稳态理论.运筹学学报, 2001, 1:70-80
16田乃硕.休假随机服务系统.北京:北京大学出版社, 2001:252-320
17 Z. G. Zhang, N. Tian. Analysis on Queuing Systems with Synchronous Vacations of Partial Servers. Performance Evaluation, 2003, 52:269-282
18 Z. G. Zhang, N. Tian. Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers. Queueing Systems, 2003, 45:161-175
19 Z. G. Zhang. On the Three Threshold Policy in the Multi-Server Queuing System with Vacations. Queuing Systems, 2005, 51:173-186
20 N. Tian, X. Xu. The Waiting Time of An M/M/c Queue with Partial Servers Vacations. Operations Research Transactions, 2005, 9(2):1-8
21申利民,金顺福,田乃硕.部分服务台同步策略多重休假的M/M/c排队.工程数学学报, 2004, 21(2):238-244
22刘洺辛,马占友,徐秀丽,田乃硕.部分服务台异步N-策略多重休假M/M/c排队.燕山大学学报(自然科学版), 2006, 30(3):230-234
23 N. Igaki. Exponential Two Server with N-policy and Multiple Vacations. Queuing Systems, 1992, 10:279-294
24 K. C. Mandan, W. Abu-Dayyeh, F. Taiyyan. A Two Server Queue with Bernoulli Schedules and a Single Vacation Policy. Applied Mathematics and Computation, 2003, 145:59-71
25 B. K. Kumar, S. P. Madheswari. An M/M/2 Queueing System with Heterogeneous Servers and Multiple Vacations. Mathematical and Computer Modelling, 2005, 41(13):1415-1429
26 D. Yue, J. Yu, W. Yue. A Markovian Queue with Two Heterogeneous Servers and Multiple Vacations. Journal of Industrial and Management Optimization, 2009, 5(3):453-465
27 H. C. White, L. S. Christie. Queuing with Preemptive Priorities or with Breakdown. Operations Research, 1958, 6:79-95
28 K. Thiruvengauam. Queuing with Breakdowns. Operations Research, 1963, 11(1):62 -71
29 B. Avi-Iizhak, P. Naor. Some Queuing Problems with the Service Station Subject to Breakdown. Operations Research, 1963, 11(3):303-320
30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
31 J. Wang, B. Liu, J. Li. Transient Analysis of an M/G/1 Retrial Queue Subject to Disasters and Server Failures. European Journal of Operational Research, 2008, (189):1118-1132
32 W. Zhou, Y. Deng. On the M/G/1 G-queue with Unreliable Server. Operations Research Transactions, 2006, 10(2):28-36
33朱翼隽,张峰.具有两种不同服务的可修MX/G(M/M)/1排队系统.江苏大学学报(自然科学版), 2005, 26(6A):51-57
34岳德权,吕胜利,李静铂.一个修理工的M/M/N可修排队.燕山大学学报(自然科学版), 2003, 27(3):197-202
35吕胜利,李静铂,岳德权. M/M/N可修排队稳态分布存在条件的一种新形式.系统工程理论与实践, 2005, (8):79-84
36李江华,王金亭.具有不耐烦顾客的M/M/N可修排队系统.北京交通大学学报(自然科学版), 2006, 30(3):84-87
37吕胜利,李静铂,岳德权.两个修理工的M/M/2可修排队系统.数学物理学报, 2008, 28(6):1109-1118
38余君,岳德权,田瑞玲.两个不同服务台的可修排队系统的矩阵几何解.应用数学学报, 2008, 31(4):894-900
39 J-C. Ke, C-H. Lin. Sensitivity Analysis of Machine Repair Problems in Manufacturing Systems with Service Interruptions. Applied Mathematical Modelling, 2008, 32(10):2087-2105
40 P. Singh. Two-server Makovian Queues with Balking, Heterogeneous vs. Homogeneous Servers. Operations Research, 1970, 18(1):145-159
41 A. Montazer-Hzgihighi, J. Medhi, S. G. Mohanty. On a Multiserver Markovian Queueing System with Balking and Reneging. Computers and Operations Research, 1986, 13(4):421-425
42 M. O. Abou-El-Ata, A. M. A. Hariri. The M/M/C/N Queue with Balking and Reneging. Computers and Operations Research, 1992, 19(9):713-716
43 W. Patrick, S. János and M. Hermann. Finite-source M/M/S Retrial Queue with Search for Balking and Impatient Customers from the Orbit. Computer Networks, 2009, 53(8):1264-1273
44 W. Yang and S. Taek. M/M/s Queue with Impatient Customers and Retrials. Applied Mathematical Modelling, 2009, 33(6):2596-2606
45 R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy, S. I. Ammar. Transient Solution of the M/M/c Queue with Balking and Reneging. Computers & Mathematics with Applications, 2009, 57(8):1280-1285
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