清理策略下有优先权的重试排队系统
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摘要
在有负顾客到达并可清空优先权排队中的全部顾客的假设下,本文研究了清理策略下有优先权的M_2/G_2/1重试排队系统,以及清理策略下有优先权和反馈的M_2/G_2/1重试排队系统两个排队模型。
基于顾客重试的排队理论源自电话话务服务问题的研究。重试排队系统由于其合理的假设,以及在现代通讯网络、计算机网络、电话交换系统及供应链管理等不同领域中广泛的应用背景,近二十年来得到许多专家学者的高度重视。负顾客是一类特殊的顾客,刻画通讯网络中信号干扰、计算机网络中的病毒影响等现象。负顾客和重试排队系统的结合是比较新的研究方向。
此外,带有优先权顾客的重试排队系统近来也受到广泛的关注。由于在通信系统、电子计算机系统中,某些顾客必须获得优先服务,因此有优先权的排队系统的研究有着重要的应用背景。
本文中假设两类顾客的到达分别服从独立的泊松过程,如服务器忙,优先级高的顾客则排队等候服务,而优先级低的顾客只能进入Orbit中进行重试,直到重试成功。此外,假设负顾客的到达服从Poisson过程,当负顾客到达系统时,若发现服务台忙,将带走正在接受服务的顾客及优先权队列中的顾客。若服务台空闲,则负顾客立即消失,对系统没有任何影响。文中应用补充变量及母函数法给出了部分系统排队指标。
In this paper, two models on M_2/G_2/1 retrial queue with negative arrivals andpriority customers are studied under the clearing policy.
Queuing systems based on customers' retrial behavior are stepped from the study of telephone service problems. Due to their reasonable assumptions as well as widely applied background in modern telecommunication networks, computer systems, telephone switching systems and supply chain management, retrial queuing systems have attracted the extensive attention of numerous experts and scholars in the recent twenty years. The negative customer is a kind of special customer, such as signal interference in telecommunication networks and the influence of virus in computer networks. The combination of the negative customer and queuing system is a new research area.
In addition, retrial queuing system with priority subscribers has been attached great attention of many researchers. In the telecommunication system, electrical and computer systems, some customers should be given priority service. Therefore, the study of this kind of queuing system is of great importance.
In the thesis, priority customers and non-priority customers arrive according to two different independent Poisson flows. In the case of blocking the priority customers can be queued whereas the non-priority customers must leave the service area but return after some random period of time to try their luck again. Besides, we also consider the influence of the arrival of negative customers, which delete all priority customers in the system. Using a supplementary variable method, we get some indexes for both queuing and reliability measures of interest.
基于顾客重试的排队理论源自电话话务服务问题的研究。重试排队系统由于其合理的假设,以及在现代通讯网络、计算机网络、电话交换系统及供应链管理等不同领域中广泛的应用背景,近二十年来得到许多专家学者的高度重视。负顾客是一类特殊的顾客,刻画通讯网络中信号干扰、计算机网络中的病毒影响等现象。负顾客和重试排队系统的结合是比较新的研究方向。
此外,带有优先权顾客的重试排队系统近来也受到广泛的关注。由于在通信系统、电子计算机系统中,某些顾客必须获得优先服务,因此有优先权的排队系统的研究有着重要的应用背景。
本文中假设两类顾客的到达分别服从独立的泊松过程,如服务器忙,优先级高的顾客则排队等候服务,而优先级低的顾客只能进入Orbit中进行重试,直到重试成功。此外,假设负顾客的到达服从Poisson过程,当负顾客到达系统时,若发现服务台忙,将带走正在接受服务的顾客及优先权队列中的顾客。若服务台空闲,则负顾客立即消失,对系统没有任何影响。文中应用补充变量及母函数法给出了部分系统排队指标。
In this paper, two models on M_2/G_2/1 retrial queue with negative arrivals andpriority customers are studied under the clearing policy.
Queuing systems based on customers' retrial behavior are stepped from the study of telephone service problems. Due to their reasonable assumptions as well as widely applied background in modern telecommunication networks, computer systems, telephone switching systems and supply chain management, retrial queuing systems have attracted the extensive attention of numerous experts and scholars in the recent twenty years. The negative customer is a kind of special customer, such as signal interference in telecommunication networks and the influence of virus in computer networks. The combination of the negative customer and queuing system is a new research area.
In addition, retrial queuing system with priority subscribers has been attached great attention of many researchers. In the telecommunication system, electrical and computer systems, some customers should be given priority service. Therefore, the study of this kind of queuing system is of great importance.
In the thesis, priority customers and non-priority customers arrive according to two different independent Poisson flows. In the case of blocking the priority customers can be queued whereas the non-priority customers must leave the service area but return after some random period of time to try their luck again. Besides, we also consider the influence of the arrival of negative customers, which delete all priority customers in the system. Using a supplementary variable method, we get some indexes for both queuing and reliability measures of interest.
引文
[1] A.Gomez-Corral. Stochastic analysis of a single server retrial queue with general retrial times ,Naval Research Logistics 46. 1999. 568-581.
[2] Aguir, S. F.Karaesmen. O.Z.Aksin and F.Chanvet. The impact of retrials on call center performance. OR Spectrum. 2004. 26:353-376.
[3] Artalejo, J.R. A classified bibliography of research on retrial queues. Progress in 1990-1999. Top 7. 1999. 187-211.
[4] Artalejo, J. R. and Choudhury. G. Steady state analysis of an M/G/1 queue with repeated attempts and two-phase service. Quality Technology & Quantitative Management. 2004. 1(2). 189-199.
[5] Artalejo, J.R.(Ed.). Retrial Queuing Systems. Mathematical and Computer Modeling 30. 1999. No.3-4. 1-228.
[6] B.D.Choi and K.K.Park. The M/G/1 retrial queue with Bernoulli schedule. Queuing System .1990. (7).219-227.
[7] B.Krishna Kumar, D.Arivudainambi. The M/G/1 Retrial Queue with Bernoulli Schedules and General Retrial times, Computers and Mathematics with Applications[J]. 43. 2002.15-30.
[8] Cao, J. and Cheng, K. Analysis of M/G/1 queuing system with repairable service station. Acta Math.Appl.Sinica.1982. (5)113-127.
[9] Choi, B.D. Chang, Y. Single server retrial queues with priority calls. Math. Comput. Modell. 1999. 30(3-4): 7-32.
[10] Cohen, J.W. Basic problems of telephone traffic theory and the influence of repeated calls.
[11]Cooper R B. Introduction to Queuing Theory. New York. North-Holland. 1981.
[12] Falin G.I. Artalejo, J.R. and Martin, M. On the single server retrial queue with priority customers. Queueing Systems. 1993. (14):439-455.
[13] Falin, G I and J.GC. Templeton. Retrial Queues. Chapman and Hall. London. 1997.
[14]Falin G.I. A survey of retrial queues [J]. Queuing Systems. 1990. (7): 127-167.
[15] Fourneau. J. M, Gelenbe. E. and Souros. R: G-networks with Multiple Classes of Positive and Negative Customers. Theoretical Computer Science. 1996. 15 5:141-15.
[16] Gelenbe. E, Glynn.P. and Sigman. K. Queues with negative arrivals J. Appl. Prob. 1991.28:245-25.
[17] Gelenbe. E. Random Neural Networks with Positive and Negative Signals and Product Form Solution. Neural Computation. 1989. 1 (4):502-510.
[18] Janssens, GK. The quasi-random input queuing system with repeated attempts as a model for a collision-avoidance star local area network. IEEE Transactions on Communications 45.1997. 360-364.
[19] K.C. Madan. An M/G/1 queue with second optional service. Queuing Systems 34. 37-46. (2000).
[20] Krishna Kumar B, Vijayakumar A, Arivudainambi D. An M/G/1 retrial queuing system with two-phase service and preemptive resume. Annals of Operations Research. 2002. 113: 61-79.
[21] Kosten, L. On the influence of repeated calls in the theory of probabilities of blocking. De Ingenieur 59.1947. 1-25.
[22] Kulkarni, V. G. and Choi, B. D. Retrial queues with server subject to breakdowns and repairs. Queueing Systems. 1990. 7. 191-208.
[23] Medhi J. A single server Poisson input queue with a second optional channel. Queuing Systems. 2002. 42: 239-242.
[24] Peter G. Harrison, Edwige Pitel. Sojourn times in Single-Server Queues with Negative Customers, J. Appl. Probab. 1993. 30( no.4): 943-963.
[25] Shi D H. A new method for calculation of the mean failure numbers of a repairable system during (0, t]. Acta Math Appl Sinica. 1985. 8 :101-110.
[26]Wang J. An M/G/1 queue with second optional service and server breakdowns. Computers and Mathematics with Applications. 2004. 47: 1713-1723.
[27] Wang J, Cao J and Li QL. Reliability Analysis of Retrial System with Server Breakdowns and Repairs. Queuing Systems, Vol.38, pp.363-380.2001.
[28] Wang J. On the Single Server Retrial Queues with Priority Subscribers and Server Breakdowns. Jrl Syst Sci & Complexity. 2008. 21:304-315.
[29] Wilkinson, R.I. Theories for toll traffic engineering in the USA. The Bell System Technical Journal 35.1956. 421-514. Philips Telecommunication Review 18. 1957.49-100.
[30] Wu H, Yin X. An M/G/1 Retrial Queue with Individual Removal and Repair. Acta Scientianim Naturalium Universitatis Sunyatseni. 2005. 44. 133-137.
[31] Yang, T. and Li, H. The M/G/1 retrial queue with the server subject to starting failures. Queuing Systems. 1994.(16):83-96.
[32] Yang T. Posner M J M. Templeton J G C. An approximation method for the M/G/1 retrial queue with general retrial times. European Journal of Operational Research. 1994. 76:55 2-562.
[2] Aguir, S. F.Karaesmen. O.Z.Aksin and F.Chanvet. The impact of retrials on call center performance. OR Spectrum. 2004. 26:353-376.
[3] Artalejo, J.R. A classified bibliography of research on retrial queues. Progress in 1990-1999. Top 7. 1999. 187-211.
[4] Artalejo, J. R. and Choudhury. G. Steady state analysis of an M/G/1 queue with repeated attempts and two-phase service. Quality Technology & Quantitative Management. 2004. 1(2). 189-199.
[5] Artalejo, J.R.(Ed.). Retrial Queuing Systems. Mathematical and Computer Modeling 30. 1999. No.3-4. 1-228.
[6] B.D.Choi and K.K.Park. The M/G/1 retrial queue with Bernoulli schedule. Queuing System .1990. (7).219-227.
[7] B.Krishna Kumar, D.Arivudainambi. The M/G/1 Retrial Queue with Bernoulli Schedules and General Retrial times, Computers and Mathematics with Applications[J]. 43. 2002.15-30.
[8] Cao, J. and Cheng, K. Analysis of M/G/1 queuing system with repairable service station. Acta Math.Appl.Sinica.1982. (5)113-127.
[9] Choi, B.D. Chang, Y. Single server retrial queues with priority calls. Math. Comput. Modell. 1999. 30(3-4): 7-32.
[10] Cohen, J.W. Basic problems of telephone traffic theory and the influence of repeated calls.
[11]Cooper R B. Introduction to Queuing Theory. New York. North-Holland. 1981.
[12] Falin G.I. Artalejo, J.R. and Martin, M. On the single server retrial queue with priority customers. Queueing Systems. 1993. (14):439-455.
[13] Falin, G I and J.GC. Templeton. Retrial Queues. Chapman and Hall. London. 1997.
[14]Falin G.I. A survey of retrial queues [J]. Queuing Systems. 1990. (7): 127-167.
[15] Fourneau. J. M, Gelenbe. E. and Souros. R: G-networks with Multiple Classes of Positive and Negative Customers. Theoretical Computer Science. 1996. 15 5:141-15.
[16] Gelenbe. E, Glynn.P. and Sigman. K. Queues with negative arrivals J. Appl. Prob. 1991.28:245-25.
[17] Gelenbe. E. Random Neural Networks with Positive and Negative Signals and Product Form Solution. Neural Computation. 1989. 1 (4):502-510.
[18] Janssens, GK. The quasi-random input queuing system with repeated attempts as a model for a collision-avoidance star local area network. IEEE Transactions on Communications 45.1997. 360-364.
[19] K.C. Madan. An M/G/1 queue with second optional service. Queuing Systems 34. 37-46. (2000).
[20] Krishna Kumar B, Vijayakumar A, Arivudainambi D. An M/G/1 retrial queuing system with two-phase service and preemptive resume. Annals of Operations Research. 2002. 113: 61-79.
[21] Kosten, L. On the influence of repeated calls in the theory of probabilities of blocking. De Ingenieur 59.1947. 1-25.
[22] Kulkarni, V. G. and Choi, B. D. Retrial queues with server subject to breakdowns and repairs. Queueing Systems. 1990. 7. 191-208.
[23] Medhi J. A single server Poisson input queue with a second optional channel. Queuing Systems. 2002. 42: 239-242.
[24] Peter G. Harrison, Edwige Pitel. Sojourn times in Single-Server Queues with Negative Customers, J. Appl. Probab. 1993. 30( no.4): 943-963.
[25] Shi D H. A new method for calculation of the mean failure numbers of a repairable system during (0, t]. Acta Math Appl Sinica. 1985. 8 :101-110.
[26]Wang J. An M/G/1 queue with second optional service and server breakdowns. Computers and Mathematics with Applications. 2004. 47: 1713-1723.
[27] Wang J, Cao J and Li QL. Reliability Analysis of Retrial System with Server Breakdowns and Repairs. Queuing Systems, Vol.38, pp.363-380.2001.
[28] Wang J. On the Single Server Retrial Queues with Priority Subscribers and Server Breakdowns. Jrl Syst Sci & Complexity. 2008. 21:304-315.
[29] Wilkinson, R.I. Theories for toll traffic engineering in the USA. The Bell System Technical Journal 35.1956. 421-514. Philips Telecommunication Review 18. 1957.49-100.
[30] Wu H, Yin X. An M/G/1 Retrial Queue with Individual Removal and Repair. Acta Scientianim Naturalium Universitatis Sunyatseni. 2005. 44. 133-137.
[31] Yang, T. and Li, H. The M/G/1 retrial queue with the server subject to starting failures. Queuing Systems. 1994.(16):83-96.
[32] Yang T. Posner M J M. Templeton J G C. An approximation method for the M/G/1 retrial queue with general retrial times. European Journal of Operational Research. 1994. 76:55 2-562.