In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

We study the existence of moments and the tail behavior of the densities of storage processes. We give sufficient conditions for existence and nonexistence of moments using the integrability conditions of submultiplicative functions with respect to Lévy measures. We then study the asymptotical behavior of the tails of these processes using the concave or convex envelope of the release rate function.

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).