We provide contributions to two classical areas of queueing. The first part of this thesis focuses on finding new conditions for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent or geometrically ergodic. Most of our results show that establishing each property listed above is equivalent to finding a good enough feasible solution to a particular optimal stopping problem, and they provide a more complete understanding of the role Foster's criterion plays in the theory of Markov chains. The second and third parts of the thesis involve analyzing queues from a transient, or time-dependent perspective. In part two, we are interested in looking at a queueing system from the perspective of a customer that arrives at a fixed time t. Doing this requires us to use tools from Palm theory. From an intuitive standpoint, Palm probabilities provide us with a way of computing probabilities of events, while conditioning on sets of measure zero. Many studies exist in the literature that deal with Palm probabilities for stationary systems, but very few treat the non-stationary case. As an application of our main results, we show that many classical results from queueing (in particular ASTA and Little's law) can be generalized to a time-dependent setting. In part three, we establish a continuity result for what we refer to as jump processes. From a queueing perspective, we basically show that if the primitives and the initial conditions of a sequence of queueing processes converge weakly, then the corresponding queue-length processes converge weakly as well in some sense. Here the notion of convergence used depends on properties of the limiting process, therefore our results generalize classical continuity results that exist in the literature. The way our results can be used to approximate queueing systems is analogous to the way phase-type random variables can be used to approximate other types of random variables.

近年來興起一個前瞻性無線技術，稱之為無線網狀網路（Wireless Mesh Networks），WMNs 以所費低廉方式提供無線網路最後一哩存取Internet，同時具備ad hoc 網路全部優點，例如自我組織（self-organization）、自我組態（self-configuration）等。儘管WMNs 骨幹部分與平面式ad hoc 網路相似，多網卡（multi-radio）的Mesh Routers 節點卻是固定不動且沒有電力限制。有鑑於此，設計於ad hoc 網路上之路由協定並不適合於WMNs。 WCETT（Weighted Cumulated Expected Transmission Time）metric 考量到intra-flow 干擾，卻未捕捉到inter-flow 干擾；而LBAR（Load-Balanced Ad hoc Routing）協定以計算節點之活躍值（nodal activity）與訊務流量干擾選擇路徑，並無支援多網卡路由，LBAR 可以在單一網卡環境表現良好，在多網卡環境卻不一定。因此，我們提出一個WMNs 上考量到Intra/Inter-flow干擾之多網卡路由協定，稱為MiRii。模擬結果顯示出我們路由協定可以改善網路效能，包含了封包成功傳送率及平均點對點延遲。 / A new promising wireless technology has emerged recently, called wireless mesh networks (WMNs). WMNs are inexpensive way to provide wireless last-mile broadband Internet access and have all the advantages of ad hoc network, such as self-organization, self-configuration. Although WMNs backbone is similar to flat ad hoc network, mesh routers with multi-radio are stationary and have no power constraints. As a result, routing protocols designed for ad hoc network may not be appropriate for WMNs. The WCETT (Weighted Cumulated Expected Transmission Time) metric takes intra-flow interference into consideration, but does not capture inter-flow interference. LBAR (Load-Balanced Ad hoc Routing) calculates nodal activity and traffic interference for choosing a path, but does not supporting multi-radio. LBAR may be perform well in single-radio environment, but do not perform as well in multi-radios. Consequently, we incorporate a new Multi-Interface Routing with Intra/Inter-flow Interference in Wireless Mesh Networks, called MiRii. The simulations show that our routing protocol can improve the network performance including the Packet Delivery Ratio and average end-to-end delay.