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.