In this dissertation, we establish large deviations results for partial sum processes on infinite time intervals, and apply them to the characterization of the large deviations behavior of the stationary and transient output processes of a single-server queue with time-varying capacities. We first show that the extension of a partial sum process on the infinite time interval [0, infinity) satisfies the Large Deviations Principle (LDP) in the function space D [0, infinity), provided the partial sum process itself satisfies the LDP in the space D [0,1]. Furthermore, for a stationary random sequence whose associated partial sum process satisfies the LDP in D [0,1], we establish LDP jointly for a partial sum process based on the entire past and future of sequence, a result especially useful in queueing theory. Through a functional approach at the sample path level, the Contraction Principle then enables us to derive the sample path LDP for processes of interest in the study of single-server queues, from that of the inputs. Finally, using our results, we refine the newly introduced notion of effective bandwidths.