Transient analysis of Markov-fluid-driven queues

In this paper we study two transient characteristics of a Markov-fluid-driven queue, viz., the busy period and the covariance function of the workload process. Both metrics are captured in terms of their Laplace transforms. Relying on sample-path large deviations we also identify the logarithmic asymptotics of the probability that the busy period lasts longer than t, as t \to\infty. Examples are included that illustrate the theory

On the correlation structure of Gaussian queues

In this paper we study Gaussian queues (that is, queues fed by Gaussian processes, such as fractional Brownian motion (fBm) and the integrated Ornstein-Uhlenbeck (iOU) process), with a focus on the correlation structure of the workload process. The main question is: to what extent does the workload process inherit the correlation properties of the input process? We first present an alternative definition of correlation that allows (in asymptotic regimes) explicit analysis. For the special cases of fBm and iOU we analyze the behavior of this metric under a many-sources scaling. Relying on (the generalized version of) Schilder's theorem, we are able to characterize its decay. We observe that the correlation structure of the input process essentially carries over to the workload process

Transient asymptotics of Lévy-driven queues

With (Qt)t denoting the stationary workload process in a queue fed by a L´evy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB,QTB > qB), for given positive numbers p, q, and a positive determinstic function TB. - We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for B large, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e., TB/B ! 0 as B ! 1) - A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for B large. For various models considered in the literature this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e., TB/B ! 1as B ! 1). Notable exceptions are two ‘heavy-tailed’ cases, viz. the situations in which the L´evy input process corresponds to an -stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB/B2 ! 1. For these input processes we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input we derive intuitively appealing asymptotics, intensively relying on sample-path large deviations results. The regimes obtained have appealing interpretations in terms of most likely paths to overflow