Convergence Bounds for Limited Processor Sharing Queue with Impatience for Analyzing Non-Stationary File Transfer in Wireless Network

The data transmission in wireless networks is usually analyzed under the assumption of non-stationary rates. Nevertheless, they strictly depend on the time of day, that is, the intensity of arrival and daily workload profiles confirm this fact. In this article, we consider the process of downloading a file within a single network segment and unsteady speeds—for arrivals, file sizes, and losses due to impatience. To simulate the scenario, a queuing system with elastic traffic with non-stationary intensity is used. Formulas are given for the main characteristics of the model: the probability of blocking a new user, the average number of users in service, and the queue. A method for calculating the boundaries of convergence of the model is proposed, which is based on the logarithmic norm of linear operators. The boundaries of the rate of convergence of the main limiting characteristics of the queue length process were also established. For clarity of the influence of the parameters, a numerical analysis was carried out and presented.

Ergodicity and perturbation bounds for $M_t/M_t/1$ queue with balking, catastrophes, server failures and repairs

In this paper, we display methods for the computation of convergence and perturbation bounds for $M_t/M_t/1$ system with balking, catastrophes, server failures and repairs. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence, perturbation bounds, and the main limiting characteristics of the queue-length process are obtained. Finally, we consider the application of all obtained estimates to a specific model.

Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying M/M/2 queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided.

On the rate of convergence for a class of Markovian queues with group services

There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.

Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method.

Logarithmic norms and regular perturbations of differential equations

In this note we explore the concept of the logarithmic norm of a matrix and illustrate its applicability by using it to find conditions under which the convergence of solutions of regularly perturbed systems of ordinary differential equations is uniform globally in time.