Stationary analysis of a fluid queue driven by an M / M / 1 queue with working vacation

This article examines an system with catastrophic server failures and repairs that feeds a fluid queue. The buffer content's stationary distribution function is expressed explicitly in terms of confluent queueing hyper geometric functions. The expression for the joint probability of the buffer's content and the number of customers in an queueing system has been obtained using continued fraction techniques. The stationary analyses of fluid flow queueing models in a stochastic environment can be effectively scrutinized and applied in the modern telecommunication systems and various advanced computer communication networks. Especially, the fluid queues are used in the performance analyses of data packet transmission networks, traffic flows at high-speed computer networks, and rechargeable batteries in smart mobile phones. A few key performance indicators are acquired. Finally, the numerical work is shown to show how the buffer content distribution behaves and how the performance metrics for the suggested fluid queueing system work.

The geometry of differential difference equations

An analysis of heart rhythm dynamics using a three-coupled oscillator model

Applications of the Lanczos method

On Lie symmetry mechanics for Navier–Stokes equations unified with non-Newtonian fluid model: A classical directory

The purpose of this paper is to derive a means of scaling the dependent variables of a system of linear differential equations so that the normalized system accurately reflects the dependence of the system on initial data. The theory is developed for the special case of a system of first order hyperbolic partial differential equations in two independent variables. Several other systems of differential equations, to which the same normalization could be applied, are also indicated. The scaling is equivalent to row and column scaling commonly used in linear algebra. For the case of positive matrices (that is, for the case where the diagonalizing matrix and its inverse have no zero elements), the theory for the “1” and “$\infty $ ” norms has been thoroughly developed by Bauer [1]. The assumption of positive matrices is too stringent for some applications. Thus, the theory is generalized to show that for irreducible matrices a best scaling exists (for both the “1” and “ $\infty $” norms), while for simply reducible matrices a best scaling exists for at least one of these two norms. Several examples are given.

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained. This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

A system of difference-differential equations for the expected incomes of open Markov queueing networks with different peculiarities is considered. The number of network states and also the number of equations in this system are both infinite. The incoming flows of requests are elementary and independent while their service times have exponential distributions. The incomes from transitions between different states of the network are deterministic functions that depend on its states; the incomes gained by the queuing server systems per unit time under the invariable states also depend on these states only. The system of the difference-differential equations is solved using the modified method of successive approximations combined with the series method. An example of a Markov G-network with signals and the group elimination of positive requests is studied. As demonstrated below, the expected incomes can be increasing and decreasing time-varying functions; can take positive and negative values.

71 - Integrable Combinations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Pade approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.

In the previous chapters we explored varieties of periodic traits of linear difference equations, piece-wise difference equations, rational difference equations, and Max-Type difference equations. First of all, we observed how the periodicity nature varied from the existence of unique periodic cycles to every solution being periodic. Second, we discovered the necessary and sufficient conditions for the existence of periodic cycles. Furthermore, we perceived the disparity between patterns of the even ordered periodic cycles and odd ordered periodic cycles with uniqueness of solutions and alternating patterns. Our next aim is to broaden our developed knowledge to ascertain the periodicity essence of third and higher order difference equations, systems of difference equations, when the periodic sequence \(\{A_{n}\}_{n=0}^{\infty }\) is periodic with period-k, (k ≥ 3) and study of new systems of piecewise difference equations and neural networking models. How will these criteria for the existence of periodic cycles be similar or different compared to our current mastery in previous chapters? What new periodic traits can we expect that will be quite different? How will the delay m ≥ 3 affect the new periodic traits? Will even and odd values of delay m ≥ 3 influence the new periodic traits? We will pose conjectures that will lead to new theorems and results based on our previous studies and will pose new research questions.

The classical Stefan problem proffers a suitable model for determining the temperature regimes as well as conjugate interfacial positions for multiphase problems. Obtaining the solutions to these problems exactly, especially in systems with cylindrical or spherical symmetry, is often an arduous task. This is largely due to inherent nonlinearities in the mathematical statements of Stefan problems. In this paper, a tractable and effective approach is proposed. Subsequent to a recast as a system of differential-difference equations, and a methodical reduction to constant coefficient difference equations, exact similarity solutions are derived for a class of heterogeneous two-phase Stefan problems with cylindrical or spherical symmetry in one spatial dimension, under either Gaussian or hypergeometric perturbations.

We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating of singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without delay. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. In future studies, it is also planned to construct an asymptotic representation of the solution of a control system with delays and degeneracies. Keywords: optimal control system, systems of differential equations with delay, asymptotic solution, turning point.

A variable gravity model of the otolith membrane

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.