In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q0, Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Lévy inputs.

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

We consider a stochastic fluid network with independent subordinator inputs to the various stations and deterministic internal flow which is of feed-forward type. We show that under suitable conditions the process of fluid contents in the station has a limiting distribution, where the limit holds in total variation and is independent of the initial condition. We also show that this limiting distribution is of product form only for trivial setups.

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.