In this paper, we aim to investigate the fluid model associated with an open large-scale storage network of non-reliable file servers with finite capacity, where new files can be added, and a file with only one copy can be lost or duplicated. The Skorokhod problem with oblique reflection in a bounded convex domain is used to identify the fluid limits. This analysis involves three regimes: the under-loaded, the critically loaded, and the overloaded regimes. The overloaded regime is of particular importance. To identify the fluid limits, new martingales are derived, and an averaging principle is established. This paper extends the results of El Kharroubi and El Masmari [7].

In this article, we focus on the systemic expected shortfall and marginal expected shortfall in a multivariate continuous-time risk model with a general càdlàg process. Additionally, we conduct our study under a mild moment condition that is easily satisfied when the general càdlàg process is determined by some important investment return processes. In the presence of heavy tails, we derive asymptotic formulas for the systemic expected shortfall and marginal expected shortfall under the framework that includes wide dependence structures among losses, covering pairwise strong quasi-asymptotic independence and multivariate regular variation. Our results quantify how the general càdlàg process, heavy-tailed property of losses, and dependence structures influence the systemic expected shortfall and marginal expected shortfall. To discuss the interplay of dependence structures and heavy-tailedness, we apply an explicit order 3.0 weak scheme to estimate the expectations related to the general càdlàg process. This enables us to validate the moment condition from a numerical perspective and perform numerical studies. Our numerical studies reveal that the asymptotic dependence structure has a significant impact on the systemic expected shortfall and marginal expected shortfall, but heavy-tailedness has a more pronounced effect than the asymptotic dependence structure.

We define the generalized equilibrium distribution, that is the equilibrium distribution of a random variable with support in $\mathbb{R}$. This concept allows us to prove a new probabilistic generalization of Taylor’s theorem. Then, the generalized equilibrium distribution of two ordered random variables is considered and a probabilistic analog of the mean value theorem is proved. Results regarding distortion-based models and mean-median-mode relations are illustrated as well. Conditions for the unimodality of such distributions are obtained. We show that various stochastic orders and aging classes are preserved through the proposed equilibrium transformations. Further applications are provided in actuarial science, aiming to employ the new unimodal equilibrium distributions for some risk measures, such as Value-at-Risk and Conditional Tail Expectation.

We develop anapproximation for the buffer overflow probability of a stable tandem network in dimensions three or more. The overflow event in terms of the constrained random walk representing the network is the following: the sum of the components of the process hits n before hitting 0. This is one of the most commonly studied rare events in the context of queueing systems and the constrained processes representing them. The approximation is valid for almost all initial points of the process and its relative error decays exponentially in n. The analysis is based on an affine transformation of the process and the problem; as $n\rightarrow \infty$ the transformed process converges to an unstable constrained random walk. The approximation formula consists of the probability of the limit unstable process hitting a limit boundary in finite time. We give an explicit formula for this probability in terms of the utilization rates of the nodes of the network.

This paper investigates the time N until a random walk first exceeds some specified barrier. Letting $X_i, i \geq 1,$ be a sequence of independent, identically distributed random variables with a log-concave density or probability mass function, we derive both lower and upper bounds on the probability $P(N \gt n),$ as well as bounds on the expected value $E[N].$ On barriers of the form $a + b \sqrt{k},$ where a is nonnegative, b is positive, and k is the number of steps, we provide additional bounds on $E[N].$