两个不同服务员可能故障或休假的排队系统
摘要
多服务员特别是两个不同服务员的排队系统是排队论中的一类典型模型。与此同时,服务员的故障或休假以及顾客的止步或中途退出等现象对系统的性能指标和经济效益有着重要影响。因此,研究综合上述机制的两个不同服务员的排队系统具有重要的理论意义和应用价值。
论文考虑了两个不同服务员的具有止步或中途退出等现象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且带有止步和中途退出的两个不同服务员的M/M/2/N可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用马尔可夫过程的方法建立了稳态概率满足的方程组,通过分块矩阵的解法求出了稳态概率向量简便而又明显的迭代计算公式,进而得到了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。
其次,研究了等待空间有限且止步依赖于故障过程的两个不同服务员的M/M/2可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用拟生灭过程的方法推导出了系统稳态平衡存在的条件和稳态概率向量的矩阵几何解,进而给出了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。
最后,研究了两个不同服务员的M/M/2异步多重休假的排队系统。利用拟生灭过程的方法,得到了率阵和边界概率向量的明显表达式,还推导出系统队长和服务员状态的联合分布。此外,给出了两个服务员均忙时,系统中等待队长和到达顾客等待时间的解析表达式,并证明了其条件随机分解特性。
The queuing systems of multi-server especially two heterogeneous servers are typical models of the queuing theory. Meantime, the breakdowns or vacations of the servers, and the phenomena of the customers’balking or reneging have a important impact on the performance measures and the economic profit of the system. Thus, the study on the queuing systems of two heterogeneous servers with integrated above mechanism has important theoretical significance and application value.
In this paper, we consider the queuing model of two heterogeneous servers with breakdowns and the phenomena of balking or reneging, as well as the model with vacation.
Firstly, we investigate a finite waiting room M/M/2/N repairable queuing system with balking, reneging and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. 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 reliability indices of the unreliable server. The numerical analysis for the results is also given.
Secondly, we study an infinite waiting room M/M/2 repairable queuing system that balking depends on breakdown process and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. Using the quasi-birth-and-death process method, we derive the existing condition of steady-state equilibrium, and the matrix-geometric solutions of the steady-state probability vectors. We then obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given.
Finally, we study the M/M/2 queuing system with asynchronous multiple vacation and heterogeneous servers. By using the quasi-birth-and-death process method, we obtain the explicit expression of the rate matrix and the boundary probability vectors, and derive the joint distribution of queue length and the state of servers. Furthermore, we give the analytical expression of the waiting queue length and waiting time of the arrival customers when all of the servers are busy, and also prove the conditional stochastic decomposition properties of them.
论文考虑了两个不同服务员的具有止步或中途退出等现象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且带有止步和中途退出的两个不同服务员的M/M/2/N可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用马尔可夫过程的方法建立了稳态概率满足的方程组,通过分块矩阵的解法求出了稳态概率向量简便而又明显的迭代计算公式,进而得到了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。
其次,研究了等待空间有限且止步依赖于故障过程的两个不同服务员的M/M/2可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用拟生灭过程的方法推导出了系统稳态平衡存在的条件和稳态概率向量的矩阵几何解,进而给出了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。
最后,研究了两个不同服务员的M/M/2异步多重休假的排队系统。利用拟生灭过程的方法,得到了率阵和边界概率向量的明显表达式,还推导出系统队长和服务员状态的联合分布。此外,给出了两个服务员均忙时,系统中等待队长和到达顾客等待时间的解析表达式,并证明了其条件随机分解特性。
The queuing systems of multi-server especially two heterogeneous servers are typical models of the queuing theory. Meantime, the breakdowns or vacations of the servers, and the phenomena of the customers’balking or reneging have a important impact on the performance measures and the economic profit of the system. Thus, the study on the queuing systems of two heterogeneous servers with integrated above mechanism has important theoretical significance and application value.
In this paper, we consider the queuing model of two heterogeneous servers with breakdowns and the phenomena of balking or reneging, as well as the model with vacation.
Firstly, we investigate a finite waiting room M/M/2/N repairable queuing system with balking, reneging and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. 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 reliability indices of the unreliable server. The numerical analysis for the results is also given.
Secondly, we study an infinite waiting room M/M/2 repairable queuing system that balking depends on breakdown process and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. Using the quasi-birth-and-death process method, we derive the existing condition of steady-state equilibrium, and the matrix-geometric solutions of the steady-state probability vectors. We then obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given.
Finally, we study the M/M/2 queuing system with asynchronous multiple vacation and heterogeneous servers. By using the quasi-birth-and-death process method, we obtain the explicit expression of the rate matrix and the boundary probability vectors, and derive the joint distribution of queue length and the state of servers. Furthermore, we give the analytical expression of the waiting queue length and waiting time of the arrival customers when all of the servers are busy, and also prove the conditional stochastic decomposition properties of them.
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-基础与应用.成都:电子科技大学出版社, 2000
3 A. K. Erlang. The Theory of Probabilities and Telephone Conversations. Nyt T idsskrift Matematik, 1909, B20:33-39
4 A. K. Erlang. Solution of some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren, 1917, 13:5-13
5 D. G. Kendall. Some Problems in the Theory of Queues. Journal of Royal Statistic Society, 1951, B13:151-173
6 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
7 M. F. Neuts. Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Baltimore, MD: Johns Hopkins University Press, 1981:81-83
8田乃硕.休假随机服务系统.北京:北京大学出版社, 2000
9曹晋华,程侃.可靠性数学引论.北京:科学出版社, 1986:1-226
10 H. C. White, L. S. Christie. Queuing with Preemptive Priorities or with Breakdown. Operations Research, 1958, 6:79-95
11 D. P. Gaver. A Waiting Line with Interrupted Service Including Priorities. Journal of the Royal Statistical Society, 1962, B24:73-90
12 N. K. Jaiswal. Preemptive Resume Priority Queue. Operations Research, 1961, 9(5): 732-742
13 K. Thiruvengauam. Queuing with Breakdowns. Operations Research, 1963, 11(1):62 -71
14 B. Avi-Itzhak, P. Naor. Some Queuing Problems with the Service Station Subject to Breakdown. Operations Research, 1963, 11(3):303-320
15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
16 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
17 W. Zhou, Y. Deng. On the M/G/1 G-queue with Unreliable Server. Operations Research Transactions, 2006, 10(2):28-36
18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
19 I. L. Mitrany, B. Avi-Itzhak. A Many-server Queue with Service Interruptions. Operations Research, 1968, 16(3):628-638
20 M. F. Neuts, D. M. Lucantoni. A Markovian Queue with N Servers Subject to Breakdowns and Repairs. Management Science, 1979, 25(9):849-861
21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
22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
23 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
24 Y. Levy, U. Yechiali. Utilization of Idle Time in an M/G/1 Queueing System. Management Science, 1975, 35:708-721
25 B. Doshi. Queueing Systems with Vacation-a Survey. Queueing Systems, 1986, (1): 29-66
26 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
27 B. Vinod. Exponential Queues with Server Vacations. Journal of the Operational Research Society, 1986, 37:1007-1014
28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
29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
30 Z. G. Zhang, N. Tian. Analysis on Queuing Systems with Synchronous Vacations of Partial Servers. Performance Evaluation, 2003, 52:269-282
31 Z. G. Zhang, N. Tian. Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers. Queueing Systems, 2003, 45:161-175
32 Z. G. Zhang. On the Three Threshold Policy in the Multi-Server Queuing System with Vacations. Queuing Systems, 2005, 51:173-186
33 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
34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
35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
36 N. Igaki. Exponential Two Server with N-policy and Multiple Vacations. Queuing Systems, 1992, 10:279-294
37 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
38 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
39 C. Palm. Etude des delais d’attente. Ericsson technics, 1937, 5:37-56
40 F. A. Haight. Queueing with Balking. Biometrika, 1957, 44:360-369
41 F. A. Haight. Queueing with Reneging. Metrika, 1959, 2:186-197
42 C. J. Ancker, A. V. Gafarian. Some Queuing Problems with Balking and Reneging: I. Operations Research, 1963, 11(1):88-100
43 C. J. Ancker, A. V. Gafarian. Some Queueing Problems with Balking and Reneging: Ⅱ. Operation Research, 1963, 11(6):928-937
44 M. O. Abou-El-Ata. The State-dependent Queue: M/M/1/N with Reneging andGeneral Balk Functions. Microelectronics Reliability, 1991, 31(5):1001-1007
45 D. Yue, Y. Zhang, W. Yue. Optimal Performance Analysis of An M/M/1/N Queue System with Balking, Reneging and Server Vacation. International Journal of Pure and Applied Mathematics, 2006, 28:101-115
46孙妍平,岳德权.带有止步和中途退出的成批到达的Mx/M/1/N多重休假排队系统的性能分析.运筹与管理, 2006, 15(6):60-65
47 S. Drekic, D. G.Woolford. A Preemptive Priority Queue with Balking. European Journal of Operational Research, 2005, 164:387-401
48 S. S. Rao. Queuing Models with Balking, Reneging and Interruptions. Operations Research, 1965, 13:596-608
49 J. C. Ke. Operating Characteristic Analysis on the M[X]/G/1 System with a Variant Vacation Policy and Balking. Applied Mathematical Modelling, 2007, (31):1321- 1337
50程锋,王聚丰.多服务速率多重休假可修排队.工程数学学报, 2005, 22(2):249- 254
51 D. Yue, C. Li. Performance Analysis of the State-Dependent M/Ek/1 Queue with Balking and Server Vacation. Operations Research Society of China. Proceedings of the Eighth National Conference of Operations Research Society of China, Shen zhen, China, 2006:576-583
52田瑞玲,岳德权,胡林敏,赵冰.带有止步和N策略的M/H2/1多重休假排队系统.运筹与管理, 2007, 16(4):56-60
53田乃硕,岳德权.拟生灭过程与矩阵几何解.北京:科学出版社, 2002:16-70
54 C. Latoche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM, Philadelphia, PA:1999
55 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
56 P. Singh. Two-server Makovian Queues with Balking, Heterogeneous vs. Homogeneous Servers. Operations Research, 1970, 18(1):145-159
57 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
58 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
59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
60 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
61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
62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
63 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
64 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唐应辉,唐小我.排队论-基础与应用.成都:电子科技大学出版社, 2000
3 A. K. Erlang. The Theory of Probabilities and Telephone Conversations. Nyt T idsskrift Matematik, 1909, B20:33-39
4 A. K. Erlang. Solution of some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren, 1917, 13:5-13
5 D. G. Kendall. Some Problems in the Theory of Queues. Journal of Royal Statistic Society, 1951, B13:151-173
6 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
7 M. F. Neuts. Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Baltimore, MD: Johns Hopkins University Press, 1981:81-83
8田乃硕.休假随机服务系统.北京:北京大学出版社, 2000
9曹晋华,程侃.可靠性数学引论.北京:科学出版社, 1986:1-226
10 H. C. White, L. S. Christie. Queuing with Preemptive Priorities or with Breakdown. Operations Research, 1958, 6:79-95
11 D. P. Gaver. A Waiting Line with Interrupted Service Including Priorities. Journal of the Royal Statistical Society, 1962, B24:73-90
12 N. K. Jaiswal. Preemptive Resume Priority Queue. Operations Research, 1961, 9(5): 732-742
13 K. Thiruvengauam. Queuing with Breakdowns. Operations Research, 1963, 11(1):62 -71
14 B. Avi-Itzhak, P. Naor. Some Queuing Problems with the Service Station Subject to Breakdown. Operations Research, 1963, 11(3):303-320
15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
16 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
17 W. Zhou, Y. Deng. On the M/G/1 G-queue with Unreliable Server. Operations Research Transactions, 2006, 10(2):28-36
18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
19 I. L. Mitrany, B. Avi-Itzhak. A Many-server Queue with Service Interruptions. Operations Research, 1968, 16(3):628-638
20 M. F. Neuts, D. M. Lucantoni. A Markovian Queue with N Servers Subject to Breakdowns and Repairs. Management Science, 1979, 25(9):849-861
21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
22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
23 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
24 Y. Levy, U. Yechiali. Utilization of Idle Time in an M/G/1 Queueing System. Management Science, 1975, 35:708-721
25 B. Doshi. Queueing Systems with Vacation-a Survey. Queueing Systems, 1986, (1): 29-66
26 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
27 B. Vinod. Exponential Queues with Server Vacations. Journal of the Operational Research Society, 1986, 37:1007-1014
28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
29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
30 Z. G. Zhang, N. Tian. Analysis on Queuing Systems with Synchronous Vacations of Partial Servers. Performance Evaluation, 2003, 52:269-282
31 Z. G. Zhang, N. Tian. Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers. Queueing Systems, 2003, 45:161-175
32 Z. G. Zhang. On the Three Threshold Policy in the Multi-Server Queuing System with Vacations. Queuing Systems, 2005, 51:173-186
33 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
34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
35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
36 N. Igaki. Exponential Two Server with N-policy and Multiple Vacations. Queuing Systems, 1992, 10:279-294
37 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
38 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
39 C. Palm. Etude des delais d’attente. Ericsson technics, 1937, 5:37-56
40 F. A. Haight. Queueing with Balking. Biometrika, 1957, 44:360-369
41 F. A. Haight. Queueing with Reneging. Metrika, 1959, 2:186-197
42 C. J. Ancker, A. V. Gafarian. Some Queuing Problems with Balking and Reneging: I. Operations Research, 1963, 11(1):88-100
43 C. J. Ancker, A. V. Gafarian. Some Queueing Problems with Balking and Reneging: Ⅱ. Operation Research, 1963, 11(6):928-937
44 M. O. Abou-El-Ata. The State-dependent Queue: M/M/1/N with Reneging andGeneral Balk Functions. Microelectronics Reliability, 1991, 31(5):1001-1007
45 D. Yue, Y. Zhang, W. Yue. Optimal Performance Analysis of An M/M/1/N Queue System with Balking, Reneging and Server Vacation. International Journal of Pure and Applied Mathematics, 2006, 28:101-115
46孙妍平,岳德权.带有止步和中途退出的成批到达的Mx/M/1/N多重休假排队系统的性能分析.运筹与管理, 2006, 15(6):60-65
47 S. Drekic, D. G.Woolford. A Preemptive Priority Queue with Balking. European Journal of Operational Research, 2005, 164:387-401
48 S. S. Rao. Queuing Models with Balking, Reneging and Interruptions. Operations Research, 1965, 13:596-608
49 J. C. Ke. Operating Characteristic Analysis on the M[X]/G/1 System with a Variant Vacation Policy and Balking. Applied Mathematical Modelling, 2007, (31):1321- 1337
50程锋,王聚丰.多服务速率多重休假可修排队.工程数学学报, 2005, 22(2):249- 254
51 D. Yue, C. Li. Performance Analysis of the State-Dependent M/Ek/1 Queue with Balking and Server Vacation. Operations Research Society of China. Proceedings of the Eighth National Conference of Operations Research Society of China, Shen zhen, China, 2006:576-583
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