Block-structured Markov chains and GI/G/s queues and retrial queues

我的目的是研究块结构的马尔可夫链和GI/GI/S队列的各种情况。因为这些对象是相互关联的，相辅相成的。块结构马尔可夫链的特征是将状态空间划分为称为级别的子集，每个级别由多个阶段组成，其转移概率形成一个分块矩阵。我将研究块结构的马尔可夫链，其中块矩阵可以是无限维的，和/或水平相关的，和/或有限维的，但其大小可能随链的水平变化而改变。我打算研究的GI/G/S排队系统是这样的：到达的顾客可能是不同类型的；服务员(S)可能休假；可用服务员的数量可能会改变；阻塞的顾客可能会在随机时间后重试服务，或者可能永远不服务；等候室容量，对于不同类型的顾客，可能是有限的，也可能是无限的。这项研究的预期结果将直接适用于电信网络、无线通信网络、计算机系统和制造系统中出现的各种真实世界的随机模型和排队系统，以及经济和金融等领域。这项研究将开发的技术将提供有关系统性能的信息，这些信息对于系统设计者寻找最优系统配置是必不可少的。从长远来看，这项研究将有助于马尔可夫过程理论和排队和重试排队理论。它还可能导致更具挑战性和实际吸引力的模型，并可能提供在随机建模的其他学科中有用的一般技术。

My objective is to study block-structured Markov chains and various cases of GI/GI/s queues.  These objects are interrelated and complementary to each other. Block-structured Markov chains are characterized by partitioning the state space into subsets called levels, each level consisting of a number of stages, whose transition probabilities form a blocked matrix. I will study block-structured Markov chains where the blocked matrices may be infinite dimensional, and/or level dependent, and/or finite dimensional, but whose sizes may change as the level of the chain changes.    The GI/G/s queueing systems I intend to study are those in which the arriving customers may be of different types; server(s) may take a vacation; the number of available servers may change; blocked customers may retry for service after a random amount of time, or may leave the systems forever without service; and the waiting room capacity may, for different types of customers, be finite or infinite.    The anticipated results of this research will be directly applicable to various real world stochastic models and queueing systems arising in telecommunication networks, wireless communication networks, computer systems and manufacturing systems, and areas such as economics and finance.  The techniques to be developed in this research will provide information on system performance that is essential for system designers seeking out optimal system configurations.  In the longer term, this research will contribute to the theory of Markov processes and the theory of queues and retrial queues. It may also lead to more challenging and practically attractive models, and may provide general techniques useful in other disciplines of stochastic modeling.