Vol 31, No 4 (2023)

This work is the first part of a large bibliographic review of active queue management algorithms of the Random Early Detection (RED) family, presented in the scientific press from 1993 to 2023. The first part will provide data on algorithms published from 1993 to 2005.

The given article is a continuation of a number of works devoted to the development of models and methods for ranging the filtration rules to prevent a decrease in the firewall performance caused by the use of a sequential scheme for checking packet compliance with the rules, as well as by the heterogeneity and variability of network traffic. The article includes a description of a firewall mathematical model given in the form of a complex system and a queuing system with a phase-type discipline for request servicing, which formalizes the network traffic filtering process with the functionality of ranging the rules. The purpose of modeling is to obtain estimates for major firewall performance metrics for various network traffic behavior scenarios, as well as to evaluate an increase in the firewall performance due to ranging a filtration rule set. Calculation of estimates for the firewall (FW) performance metrics was made using the analytical method for a Poisson request flow. Based on the analysis of the modeling results, conclusions were drawn on the effectiveness of ranging the filtration rules in order to improve the firewall performance for traffic scenarios that are close to real ones.

The problematic of solving stiff boundary value problems permeates numerous scientific and engineering disciplines, demanding novel approaches to surpass the limitations of traditional numerical techniques. This research delves into the implementation of the solution continuation method with respect to the best exponential argument, to address these stiff problems characterized by rapidly evolving integral curves. The investigation was conducted by comparing the efficiency and stability of this novel method against the conventional shooting method, which has been a cornerstone in addressing such problems but struggles with the erratic growth of integral curves. The results indicate a marked elevation in computational efficiency when the problem is transformed using the exponential best argument. This method is particularly pronounced in scenarios where integral curves exhibit exponential growth speed. The main takeaway from this study is the instrumental role of the regularization parameter. Its judicious selection based on the unique attributes of the problem can dictate the efficiency of the solution. In summary, this research not only offers an innovative method to solve stiff boundary value problems but also underscores the nuances involved in method selection, potentially paving the way for further refinements and applications in diverse domains.

Usually, when working with the eikonal equation, reference is made to its derivation in the monograph by Born and Wolf. The derivation of this equation was done rather carelessly. Understanding this derivation requires a certain number of implicit assumptions. For a better understanding of the eikonal approximation and for methodological purposes, the authors decided to repeat the derivation of the eikonal equation, explicating all possible assumptions. Methodically, the following algorithm for deriving the eikonal equation is proposed. The wave equation is derived from Maxwell’s equation. In this case, all conditions are explicitly introduced under which it is possible to do this. Further, from the wave equation, the transition to the Helmholtz equation is carried out. From the Helmholtz equation, with the application of certain assumptions, a transition is made to the eikonal equation. After analyzing all the assumptions and steps, the transition from the Maxwell’s equations to the eikonal equation is actually implemented. When deriving the eikonal equation, several formalisms are used. The standard formalism of vector analysis is used as the first formalism. Maxwell’s equations and the eikonal equation are written as three-dimensional vectors. After that, both the Maxwell’s equations and the eikonal equation use the covariant 4-dimensional formalism. The result of the work is a methodically consistent description of the eikonal equation.