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

Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model

We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an alternate representation of the joint steady state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze for large N. We give detailed asymptotic results for the joint distribution, as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These shows how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20

Fluid flow models in performance analysis

We review several developments in fluid flow models: feedback fluid models, linear stochastic fluid networks and bandwidth sharing networks. We also mention some promising new research directions

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Performance analysis of an asynchronous transfer mode multiplexer with Markov modulated inputs

Ankara : Department of Electrical and Electronics Engineering and the Institute of Engineering and Science of Bilkent University, 1993.Thesis (Ph.D.) -- Bilkent Iniversity, 1993.Includes bibliographical references leaves 108-113.Asynchronous Transfer Mode (ATM) networks have inputs which consist of superpositions of correlated cell streams. Markov modulated processes are commonly used to characterize this correlation. The first step through gaining an analytical insight in the performance issues of an ATM network is the analysis of a single channel. One objective of this study is the performance analysis of an ATM multiplexer whose input is a Markov modulated periodic arrival process. Based on the transient behavior of the nD/D/1 queue, we present an approximate method to compute the queue length distribution accurately. The method reduces to the solution of a linear differential equation with variable coefficients. Another general traffic model is the Markov Modulated Poisson Process (MMPP). We employ Pade approximations in transform domain for the deterministic service time distribution in an M MPP/D/1 queue so as to compute the distribution of the buffer occupancy. For both models, we also provide algorithms for analysis in the case of finite queue capacities and for computation of effective bandwidth.Akar, NailPh.D

Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page