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Fluid-flow approximation is an approach to modelling and evaluating the performance of vast computer networks. Due to varying traffic and performance of transmission protocols reacting to traffic overloads, computer networks are in a permanent transient state. The fluid-flow method main advantage is its ability to analyse these transient states. The article reviews and organises several versions of this approach, indicating a few errors. The main reason for these errors is confusion or lack of distinction between the two versions of the Internet Protocol – when the queue of packets at a node is too long, they may be destroyed or only marked as redundant. The paper compares and evaluates these fluid-flow approximation models with mild and aggressive settings of RED parameters. The authors build a software system with hitherto unprecedented capabilities regarding the size of the networks to be analysed and with innovative way of organising the calculations. The paper shows how large differences imprecise assumptions can introduce in quantitative results.

Airport aprons are complex, multi-node operational hubs frequently affected by queue congestion resulting from control handovers, taxi conflicts, and external factors. To enable proactive congestion management, we propose a new and accurate method for apron queue length prediction. The core of our approach is a multi-queue network model in which queues are systematically divided by control position and taxi direction. This framework, which applies the Fluid Flow Approximation and is calibrated with historical data, effectively captures the dynamics of multi-node traffic flow. In a validation case study at Beijing Daxing International Airport (ZBAD), the model achieved high accuracy, with the mean absolute error of queue length prediction averaging 0.5 aircraft. The results demonstrate the model’s ability to characterize queue dynamics on a minute-level scale across a full day.

The article proposes a model in which Diffusion Approximation is used to analyse the TCP/AQM transmission mechanism in a multinode computer network. In order to prevent traffic congestion, routers implement AQM (Active Queue Management) algorithms. We investigate the influence of using RED-based AQM mechanisms and the fractional controller PIγ on the transport layer. Additionally, we examine the cases in which the TCP and the UDP flows occur and analyse their mutual influence. Both transport protocols used are independent and work simultaneously. We compare our solution with the Fluid Flow approximation, demonstrating the advantages of Diffusion Approximation.

The paper addresses two issues: (i) modeling dynamic flows transmitted in vast TCP/IP networks and (ii) modeling the impact of energy-saving algorithms. The approach is based on the fluid-flow approximation, which applies first-order differential equations to analyze the evolution of queues and flows. We demonstrate that the effective implementation of this method overcomes the constraints of storing large data in numerical solutions of transient problems in vast network topologies. The model is implemented and executed directly in a database system. It can analyze transient states in topologies of more than 100,000 nodes, i.e., the size which was not considered until now. We use it to investigate the impact of an energy-saving algorithm on the performance of a vast network. We find that it reduces network congestion and save energy costs but significantly lower network throughput.

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI / G /1 queueing process

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI/G/1 queueing process

Partitioning methods lend themselves very well to implementation on parallel computers. In recent years, branch-and-bound algorithms have been tested on various types of architectures. In this paper, we develop a queueing network model for the analysis of a class of branch-and-bound algorithms on a master–slave architecture. The analysis is based on a fluid flow approximation. Numerical examples illustrate the concepts developed. Finally, related branch-and-bound algorithms are studied using a machine repair queueing model.

The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $\\epsilon \\leq c_0\\mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \\rightarrow \\infty $. For times $t \\gtrsim \\mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of \"2.5 dimensional\" streamwise-independent solutions referred to as streaks.

In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate–pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop (Gervais et al., Lab on a Chip, vol. 6, 2006, pp. 500–507). Gervais et al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate–pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel’s height to its width and of the channel’s height to its length are both assumed small). Our result contains no free parameters and confirms Gervais et al.’s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate–pressure drop relation compares favourably with experimental measurements.