| This thesis presents a set of results related to the single-server queue where the distribution of the processing times of jobs is heavy-tailed, in the sense that its moment generating function diverges for any positive parameter. In this setting, asymptotic expressions for relevant measures of performance, such as the probability of buffer overflow, will depend to great extent on the fine structure of the tail distribution of the processing times. Nevertheless, statistically verifying that this structure is present from any given data set is impossible, which raises the question of whether tail asymptotics based on heavy tailed models are useful. The first part of this thesis addresses the question of how robust these asymptotics are when we no longer assume that the heavy-tailed assumptions are satisfied all the way to infinity.; The second part of this thesis is devoted to the analysis of two very popular approximations for the distribution of the waiting time, Winfinity, of the M/G/1 queue: the so-called heavy-traffic approximation and the heavy-tailed asymptotic. If the traffic intensity, rho, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of Winfinity is roughly exponential at scale O((1 - rho)-1). The heavy-tailed asymptotic, on the other hand, describes the distribution of Winfinity for a fixed traffic intensity and states that its tail decays at the same rate as the tail distribution of the processing times, assumed here to be heavy-tailed. The results presented state the exact threshold at which the transition between the two asymptotics occurs for two important classes of heavy-tailed distributions, regularly varying and semiexponential. |
