# Modeling a New AQM Model for Internet Chaotic Behavior Using Petri Nets
## 2 Related Work and Motivation  
Chaos means deterministic behavior that is sensitive to its initial condition. The branch of mathematics known as dynamical systems, discrete and continuous, uses bifurcation theory [5] to study chaos in multiple fields, including engineering (see [6; 7; 8; 9; 10]). A bifurcation occurs when a small, smooth change made to the parameter values of a system causes a sudden qualitative or topological change in its general behavior. Generally, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a chaotic system as a function of a bifurcation parameter in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.  
The majority of Internet traffic is generated and controlled by Transmission Control Protocol (TCP), which has chaotic properties: nonlinearity, determinism, order in disorder, sensitivity to initial conditions, and unpredictability. Several decades ago, these major features in Transmission Control Protocol/Internet Protocol (TCP/IP) networks were demonstrated through the phenomenon called the fractal nature of Internet traffic [11]. Thus, it was revealed how extreme sensitivity to initial conditions (also known as the butterfly effect) and odd periodicity are inherent properties in TCP/IP networks that support fundamental applications such as web and electronic mail transmission, respectively, among others. Dynamical systems study internet behavior through bifurcation theory, modeling and reproducing the complexity of chaotic behavior of the TCP-like model for congestion control (see [11; 12; 13; 14; 15; 16; 17]).  
In addition, queue management in routers plays an important role in taking care of congestion from network traffic. Thus, one of the main functions of AQM is to detect emerging congestion early, before the queues in the routers overflow and the packets are dropped [18]. Numerous studies on AQM congestion control schemes have been proposed (see [19; 20; 21; 22]), and among different taxonomies, they can be classified into the queue-based category. These schemes do not directly control the arrival rate at the queue, the congestion is observed by the average or instantaneous queue length, and the aim is to stabilize the queue length. How to stabilize the queue length around an expected target regardless of nonlinear traffic loads requires fine-tuning of when and how to govern buffering and queue management. Moreover, queue-based AQM schemes have been studied from the perspective of chaos theory that also solve congestion control problems by using bifurcation analysis (see [23; 24; 25; 26]).  
Regarding the design of new communication protocols, including AQM schemes, the literature offers various formal description techniques (FDT) [27] that cover all relevant phases from specification to validation. As evidence, the majority of worldwide institutes of standardization in telecommunications, such as the International Organization for Standardization (ISO), the International Telecommunication Union (ITU), the European Telecommunications Standards Institute (ETSI), the Institute of Electrical and Electronics Engineers (IEEE), the Internet Research Task Force (IETF), and others, recognize the wide spectrum of FDT [27]: Specification and Description Language (SDL), Simple ProMeLa Interpreter (Spin), Language Of Temporal Ordering Specifications (LOTOS), Estelle, and Petri nets, among others.  
Many of these FDT offer a complementary approach based on mathematical properties, which can be automatically applied to design and validate the correctness of an algorithm. In this article, we will use Petri nets (PNs) as FDT that are applied to the field of telecommunications engineering (see [28; 29; 30; 31]. Due to space constraints in this article, the reader is directed to [28] for further explanation concerning the mathematical properties of the PNs that are used in Section 4. The next section introduces our dynamical model of queue-based AQM.