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

我的目标是研究块结构的马尔可夫链和GI/GI/s队列的各种情况。这些对象相互联系，互为补充。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，每个级别由许多阶段组成，其转移概率形成块矩阵。我将研究块状结构的马尔可夫链，其中块状矩阵可能是无限维的，和/或水平相关的，和/或有限维的，但其大小可能随着链的水平变化而变化。我要研究的GI/G/s排队系统是指到达的顾客可能是不同类型的；服务人员可以休假；可用服务器的数量可能会改变；被阻塞的客户可能会在一段随机的时间后重试服务，或者可能永远离开没有服务的系统；对于不同类型的顾客，等候室的容量可能是有限的，也可能是无限的。本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以及经济和金融等领域出现的各种现实世界的随机模型和排队系统。在这项研究中开发的技术将提供有关系统性能的信息，这对于系统设计者寻求最佳系统配置至关重要。从长远来看，本研究将有助于马尔可夫过程理论以及排队和重审队列理论的发展。它也可能导致更具挑战性和实际吸引力的模型，并可能提供在其他学科的随机建模有用的一般技术。

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.