Queueing networks
Queueing networks
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DOI：
10.1007/0-387-21748-7_14
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发表时间：
1999
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影响因子：
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通讯作者：
Xiuli Chao
中科院分区：
文献类型：
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作者：
Xiuli Chao
在本章中，我们将继续应用连续映射方法来建立队列的大流量随机过程极限，特别注意极限过程中出现不匹配跳跃的可能性。在第5、8和9章中，我们应用一维反射映射来获得单队列的重流量限制，现在我们应用多维反射映射来获得排队网络的重流量限制。像以前一样，我们省略了一些证明。这些证明和其他支持材料出现在互联网增刊的第8章。关于随机网络的背景，请参见Kelly（1979），Whittle（1986），Walrand（1988）和Serfozo（1999）。关于网络的重流量限制的相关讨论，参见Chen and Mandelbaum（1994 a，B）、Harrison（1988，2000，2001 a，B）、Chen and Yao（2001）和Kushner（2001）。在本章的最后将进一步讨论这些文献。Harrison and Reiman（1981 a，B）和Reiman（1984 a）利用（标准）多维反射映射和连续映射定理，建立了单类开放网络中向量值排队长度、等待时间和工作量随机过程的反射布朗运动极限过程的重载交通极限。由于布朗运动和反射布朗运动具有连续的样本路径，因此有界区间上的一致收敛拓扑可用于这些结果。M1拓扑的变体需要获得具有不连续样本路径的替代随机过程极限，例如反射Lévy过程，当相同样本路径中的不连续性时。
In this chapter we continue applying the continuous-mapping approach to establish heavy-traffic stochastic-process limits for queues, giving special attention to the possibility of having unmatched jumps in the limit process. Paralleling our application of the one-dimensional reflection map to obtain heavy-traffic limits for single queues in Chapters 5, 8 and 9, we now apply the multidimensional reflection map to obtain heavy-traffic limits for queueing networks. As before, we omit some proofs. These proofs plus additional supporting material appear in Chapter 8 of the Internet Supplement.For background on queueing (or stochastic) networks, see Kelly (1979), Whittle (1986), Walrand (1988) and Serfozo (1999). For related discussions of heavytraffic limits for queueing networks, see Chen and Mandelbaum (1994a, b), Harrison (1988, 2000, 2001a, b), Chen and Yao (2001) and Kushner (2001). The literature is discussed further at the end of this chapter. The (standard) multidimensional reflection map was used with the continuous mapping theorem by Harrison and Reiman (1981a, b) and Reiman (1984a) to establish heavy-traffic limits with reflected Brownian motion limit processes for vector-valued queue-length, waiting-time and workload stochastic processes in single-class open queueing networks. Since Brownian motion and reflected Brownian motion have continuous sample paths, the topology of uniform convergence over bounded intervals could be used for those results. Variants of the M1 topologies are needed to obtain alternative stochastic-process limits with discontinuous sample paths, such as reflected Lévy processes, when the discontinuities in the sam-