In this thesis, we discuss a number of problems in stochastic systems involving queues and Lévy processes. For the more advanced systems in these categories, it is commonly seen that exact analytical results are unachievable and/or that direct numerical computations are too slow. As an alternative, we aim our attention at asymptotic approaches. In some cases, we study the stochastic system as one parameter tends to a threshold value, whereas in other cases it proves useful to asymptotically scale the system as a whole. Three general topics separate the thesis into equally many parts that can be read individually. Part I concerns the effect of the scheduling policy on the asymptotic response time tail in the $M/G/1$ queue. In Part II a scaling approach is adopted to approximate queue-length distributions in a closed product-form network. Lastly, the main content of Part III is the analysis of the (asymptotic) maximum of a Markov additive process.