Asymptotic Transient Solutions of Stochastic Fluid Queues Fed By A Single ON-OFF Source

Stochastic Fluid Queues (SFQs) have served many applications in different research fields including understanding the performance measurements of network switches in High-Performance Computing (HPC) environment and understanding the surplus processes in the context of ruin theory. Although many advancements have been made to understand the stationary behavior of SFQs, the transient analysis is an open research area, where Laplace-Stieltjes Transforms (LST) are used to understand time-dependent behavior. However, performing the inversion is impractical in many cases. Chapter 2 and 3 of this work revisits the work of fluid-queues driven by a single "ON-OFF" source in the context of resource allocation in HPC environments. The asymptotic analysis is performed to produce equivalent representations that depict short and long-time behavior. Next, an expansion is proposed that holistically considers these behaviors while also providing a direct correlation between the packets injection rates and state transition rates. The numerical experiments validate our proposed schema, where the results can be adapted to provide better resource estimation of fast-packet switching within HPC environment. Chapter 4 and 5 of this work explores theorizing the collective risk of a single insurance firm during periods of distress, where its current revenues are less than its initial surplus. This behavior can be modeled as a fluid queue fed by a single ON-OFF source, where our contributions are two-fold. First, we found explicit solutions consisting of combinations of modified Bessel functions of the first kind. Next, we obtain asymptotic expansions of our resulting solutions for short and long-time behavior to propose a comprehensive solution that fuses these results. Numerical experiments further demonstrate the viability of our proposed method where only a few terms are needed to effectively approximate the temporal behavior of the surplus probability density function during periods of financial distress. Furthermore, we also show that our asymptotic expansions provide a direct correlation between the firm’s revenue gains, its claim sizes, and the current surplus behavior. The results of this work can be directly applied and extended to audit the solvency of firms operating in adverse financial conditions as well as help effectively identify potential troubled corporations at various stages