有成批到达的离散时间一般重试排队系统
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摘要
离散时间重试排队理论是排队论中的一个重要分支.近年来,由于离散时间排队系统在数字通讯系统和网络等一些相关领域的应用越来越为广泛,更多的学者致力于离散时间排队系统的研究.许多计算机网络和互联网的运作都是以离散时间为基准的,它们内部的所有行为都是发生在一些规则的时间点上.其实研究离散时间排队理论的重要原因之一是在模拟计算机网络和通讯系统时,离散时间排队系统比其对应的连续时间排队系统更为合适,也更贴近于真实的情况.在现实生活中,离散时间排队系统已经广泛应用于模拟计算机和通信网络.
在本文中我们一共分析了两个不同的离散时间重试排队系统模型.第一个模型是有不成功启动、成批到达、可控制进入的离散时间重试排队系统.第二个模型是有不成功启动、成批到达和反馈的离散时间重试排队系统.在每个模型中,我们讨论了在这个离散时间重试排队系统中的马尔可夫链以及它的遍历条件,并计算出了该系统在稳态条件下的一些参数,并说明了一些参数对重试空间平均队长的影响.另外,在一些模型中,本文还给出了两个随机分解法则,并用定理对随机分解法则做了进一步的解释.
Discrete-time queueing systems with repeated customers are important branch of Queueing Theory. Recently, There is a growing interest in the analysis of discrete-time queues due to their applications in communication systems and other related areas. Many computer and communication systems operate on a discrete time basis where events can only happen at regularly spaced epochs. One of the main reasons for analysing discrete-time queues is that these systems are more appropriate than their continuous-time counterparts for modelling computer and telecommunication systems. Now discrete-time queueing systems with repeated customers has been widely used in view of their applicability in the study of many computer and communication systems in which time is slotted.
In this paper, we study two queueing systems, that is, A Discrete-Time Retrial Queue with Starting Failures, Batch Arrive, Admission Control. A Discrete-Time Retrial Queue with Starting Failures, Batch Arrive and Feedback. In each model, we analyse the Markov Chain underlying the regarded queueing system and its ergodicity condition. Then, we present some performance measures of the system in steady-state. Finally, some numerical examples show the influence of the parameters on several performance characteristics. In some models, we give two stochastic decomposition laws and give the two stochastic decomposition for further explaination.
在本文中我们一共分析了两个不同的离散时间重试排队系统模型.第一个模型是有不成功启动、成批到达、可控制进入的离散时间重试排队系统.第二个模型是有不成功启动、成批到达和反馈的离散时间重试排队系统.在每个模型中,我们讨论了在这个离散时间重试排队系统中的马尔可夫链以及它的遍历条件,并计算出了该系统在稳态条件下的一些参数,并说明了一些参数对重试空间平均队长的影响.另外,在一些模型中,本文还给出了两个随机分解法则,并用定理对随机分解法则做了进一步的解释.
Discrete-time queueing systems with repeated customers are important branch of Queueing Theory. Recently, There is a growing interest in the analysis of discrete-time queues due to their applications in communication systems and other related areas. Many computer and communication systems operate on a discrete time basis where events can only happen at regularly spaced epochs. One of the main reasons for analysing discrete-time queues is that these systems are more appropriate than their continuous-time counterparts for modelling computer and telecommunication systems. Now discrete-time queueing systems with repeated customers has been widely used in view of their applicability in the study of many computer and communication systems in which time is slotted.
In this paper, we study two queueing systems, that is, A Discrete-Time Retrial Queue with Starting Failures, Batch Arrive, Admission Control. A Discrete-Time Retrial Queue with Starting Failures, Batch Arrive and Feedback. In each model, we analyse the Markov Chain underlying the regarded queueing system and its ergodicity condition. Then, we present some performance measures of the system in steady-state. Finally, some numerical examples show the influence of the parameters on several performance characteristics. In some models, we give two stochastic decomposition laws and give the two stochastic decomposition for further explaination.
引文
[1]T. Meisling, Discrete time queueing theory, Operations Research 6:96-105 (1958).
[2]J.J. Hunter, Mathematical techniques of applied probability, Vol.2, Discrete-time models: techniques and applications, Academic Press, New York, (1983).
[3]M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, Journal of Applied Probability 21:157-166 (1984).
[4]T. Yang and J.G..C. Templeton, A survey on retrial queues, Queueing Systems 2201-233 (1987).
[5]V.G. Kulkarni, B.D. Choi, Retrial queue with server subject to breakdown and repairs, Queueing Systems 7(2):191-208 (1990).
[6]H. Takagi, Queueing analysis:A foundation of performance evaluation, Discrete-Time Systems, Vol.3,North-Holland, Amsterdam, (1993).
[7]陆传赉,排队论,北京邮电大学出版社,1994.
[8]M.E. Woodward, Communction and computer networks:Modelling with discrete-time queues, IEEE Computer Soc. Press, Los Alamitos, CA,(1994).
[9]J.R. Artalejo and GI. Falin, Stochastic decomposition for retrial queues, Top 2:329-342 (1994).
[10]J.R. Artalejo, New results in retrial queueing systems with breakdown of the servers. Statistica Neerlandica 48:23-26 (1994).
[11]T. Yang and H. Li, The M/G/1 retrial queue with the server subject to starting failures, Queueing Systems 16:83-96 (1994).
[12]T.Yang and H.Li, On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers, Queueing system 21:199-215(1995).
[13]G.I. Falin and J.G.C. Templeton, Retrial Queues, Chapman & Hall, London (1997).
[14]H. Li and T. Yang, Geo/G/1 discrete-time retrial with Bernoulli schedule, European. J. Operate Research 111(3):629-649 (1998).
[15]J.R. Artalejo, A classified bibliography of research on retrial queueus:Progress in 1990-1999, Top 7(2):187-211(1999).
[16]J.R. Artalejo, Accessible bibliography on retrial queueus, Math Computer. Modelling 30:1-6 (1999).
[17]H. li and T. Yang, Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues, Mathematical and Computer Modelling 30:51-63 (1999).
[18]M. Takahashi, H. Osawa and T. Fujisawa, Geo[X]/G/1 retrial queue with non-preemptive priority, Asia-Pacific Journal of Operational Research 16:215-234 (1999).
[19]J. wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems 38:363-380 (2001).
[20]B. Krishna Kumar, S. Pavai Madheswari and A. Vijayakumar, The M/G/1 retrial queue with feedback and starting failures. Applied Mathematical Modelling 26:1057-1075 (2002).
[21]I. Atencia and P. Moreno, Discrete-time Geo[X]/GH/1 retrial queue with Bernoulli feedback, Computers and Mathematics with Applications 47(8-9):1273-1294 (2004).
[22]I. Atencia and P. Moreno, A discrete-time Geo/G/1 retrial queue with general retrial times, Queueing Systems 48:5-21(2004).
[23]J. Wang, An M/G/1 queue with second optional service and server breakdowns. Computers and Mathematics with Applications 47:1713-1723 (2004).
[24]J.R. Artalejo, I. Atencia and P.Moreno, A discrete-time Geo[x]/G/1 retrial queue with control of admission. Applied Mathematical Modelling 29:1100-1120 (2005).
[25]I. Atencia and P. Moreno, A discrete-time Geo/G/1 retrial queue with the server subject to starting failures, Annals of Operations Research 141:85-107 (2006).
[26]P. Moreno, A discrete-time retrial queue with unreliable server and general server lifetime. Journal of Mathematical Science 132(5):643-655 (2006).
[27]Jinting Wang, Qing Zhao, Discrete-time Geo/G/1 retrial queue with general times and starting failures, Mathematical and Computer Modelling 45:853-863 (2007).
[28]A. Aboul, S.I.Rabia, F.A. Taboly, A discrete time Geo/G/1 retrial queue with general times and balking customers, Journal of the Korean Statistical Society 37:335-348 (2008).
[29]田乃硕,徐秀丽,马占友,离散时间排队论,科学出版社,2008.
[30]Atencia, I.Fortes and S.Sanchez, A discrete-time retrial queueing system with starting failure, Bernoulli feedback and general retrial times. Computer & Industrial Engineering 57:1291-1299 (2009).
[2]J.J. Hunter, Mathematical techniques of applied probability, Vol.2, Discrete-time models: techniques and applications, Academic Press, New York, (1983).
[3]M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, Journal of Applied Probability 21:157-166 (1984).
[4]T. Yang and J.G..C. Templeton, A survey on retrial queues, Queueing Systems 2201-233 (1987).
[5]V.G. Kulkarni, B.D. Choi, Retrial queue with server subject to breakdown and repairs, Queueing Systems 7(2):191-208 (1990).
[6]H. Takagi, Queueing analysis:A foundation of performance evaluation, Discrete-Time Systems, Vol.3,North-Holland, Amsterdam, (1993).
[7]陆传赉,排队论,北京邮电大学出版社,1994.
[8]M.E. Woodward, Communction and computer networks:Modelling with discrete-time queues, IEEE Computer Soc. Press, Los Alamitos, CA,(1994).
[9]J.R. Artalejo and GI. Falin, Stochastic decomposition for retrial queues, Top 2:329-342 (1994).
[10]J.R. Artalejo, New results in retrial queueing systems with breakdown of the servers. Statistica Neerlandica 48:23-26 (1994).
[11]T. Yang and H. Li, The M/G/1 retrial queue with the server subject to starting failures, Queueing Systems 16:83-96 (1994).
[12]T.Yang and H.Li, On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers, Queueing system 21:199-215(1995).
[13]G.I. Falin and J.G.C. Templeton, Retrial Queues, Chapman & Hall, London (1997).
[14]H. Li and T. Yang, Geo/G/1 discrete-time retrial with Bernoulli schedule, European. J. Operate Research 111(3):629-649 (1998).
[15]J.R. Artalejo, A classified bibliography of research on retrial queueus:Progress in 1990-1999, Top 7(2):187-211(1999).
[16]J.R. Artalejo, Accessible bibliography on retrial queueus, Math Computer. Modelling 30:1-6 (1999).
[17]H. li and T. Yang, Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues, Mathematical and Computer Modelling 30:51-63 (1999).
[18]M. Takahashi, H. Osawa and T. Fujisawa, Geo[X]/G/1 retrial queue with non-preemptive priority, Asia-Pacific Journal of Operational Research 16:215-234 (1999).
[19]J. wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems 38:363-380 (2001).
[20]B. Krishna Kumar, S. Pavai Madheswari and A. Vijayakumar, The M/G/1 retrial queue with feedback and starting failures. Applied Mathematical Modelling 26:1057-1075 (2002).
[21]I. Atencia and P. Moreno, Discrete-time Geo[X]/GH/1 retrial queue with Bernoulli feedback, Computers and Mathematics with Applications 47(8-9):1273-1294 (2004).
[22]I. Atencia and P. Moreno, A discrete-time Geo/G/1 retrial queue with general retrial times, Queueing Systems 48:5-21(2004).
[23]J. Wang, An M/G/1 queue with second optional service and server breakdowns. Computers and Mathematics with Applications 47:1713-1723 (2004).
[24]J.R. Artalejo, I. Atencia and P.Moreno, A discrete-time Geo[x]/G/1 retrial queue with control of admission. Applied Mathematical Modelling 29:1100-1120 (2005).
[25]I. Atencia and P. Moreno, A discrete-time Geo/G/1 retrial queue with the server subject to starting failures, Annals of Operations Research 141:85-107 (2006).
[26]P. Moreno, A discrete-time retrial queue with unreliable server and general server lifetime. Journal of Mathematical Science 132(5):643-655 (2006).
[27]Jinting Wang, Qing Zhao, Discrete-time Geo/G/1 retrial queue with general times and starting failures, Mathematical and Computer Modelling 45:853-863 (2007).
[28]A. Aboul, S.I.Rabia, F.A. Taboly, A discrete time Geo/G/1 retrial queue with general times and balking customers, Journal of the Korean Statistical Society 37:335-348 (2008).
[29]田乃硕,徐秀丽,马占友,离散时间排队论,科学出版社,2008.
[30]Atencia, I.Fortes and S.Sanchez, A discrete-time retrial queueing system with starting failure, Bernoulli feedback and general retrial times. Computer & Industrial Engineering 57:1291-1299 (2009).
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