Perspective Chapter: Enhancing Network Efficiency Using Ant Colony Optimization and Pareto Optimality for Tuning PI Controller in Congested Routers

Abstract

The objective of this manuscript is to stabilize the queue of the router congestion window by designing an active queue management (AQM). The problem is dealt with under the theory of the command by using the regulator PI for that purpose. The tuning of this controller is based on a new approach of stabilization that relies on an extension of Hermite-Biehler theorem applied to quasi-polynomials. This stabilization method turns out relevant to seek the optimized results achieved within this stability region. For that, the optimization is performed using an improved multi-objective ant colony optimization (ACO) algorithm. The performance of the proposed control scheme is evaluated via a series of numerical simulations in MATLAB and Simulink.

1. Introduction

The exponential increase in size and diversity of communication networks in the current era has become an issue for today’s communities and a fertile field for researchers. In recent decades, researchers have been increasingly motivated to generate new and more reliable computer network structures that can handle the dimensions of current networks.

The TCP/IP protocol, a standard of network languages, is adopted on the Internet to facilitate communication between machines around the world. This universal language ensures that IP packets do not get lost or arrive in duplicate and confirm that the packet has arrived at its destination. However, as traffic increases, the network may become congested, a common phenomenon in routing. Routers may not be able to handle the traffic and lose packets because there is no place to store them. Queue saturation exacerbates congestion, as the time for a packet to reach the head of the queue becomes too long. In addition to queue saturation, these factors contribute to a slowdown of routers [1, 2, 3, 4, 5].

To quantitatively analyze the phenomenon of congestion, the behavior of the TCP protocol was modeled using a system of delayed differential equations based on the analogy of fluid flow. Previous research has shown that this system can be analyzed using the theory of control [6, 7].

Delay times resulting from congestion in Internet management give rise to characteristic functions known as quasi-polynomials. The first research on quasi-polynomials was conducted by Pontryagin [8], who developed a formal and highly relevant mathematical tool for analyzing the stability of time-invariant delayed systems. Pontryagin’s necessary and sufficient condition for the roots of a quasi-polynomial to have a negative real part ensures Hurwitz stability for the quasi-polynomials in terms of a property of roots interlacing.

The issue with stabilizing delayed systems is that the characteristic equations of these models have an infinite number of roots, in contrast to systems without delay. The study of delayed systems presents a significant level of complexity, but a solution to this dilemma has been proposed in the work of [9, 10, 11]. This solution involves adopting an extension of the Hermite–Biehler theorem applied to quasi-polynomials, which is used to stabilize the TCP model using a proportional integrator (PI) controller [12, 13].

According to a survey conducted by the Japan Electric Measuring Instrument Manufacturers Association in 1989, the PI controller is used in more than 90% of industrial processes. This high usage is due to the unique characteristics of the PI controller, which provide stability, regardless of how optimal it may be. The proportional (P) and integral (I) actions of the PI controller complement each other to ensure immediate and rapid correction of any deviation of the quantity being adjusted. The PI controller eliminates large system inertias and residual steady-state error, while also accelerating system response and improving loop stability. Additionally, the PI controller can be combined with transmitters and logic, making it a strong candidate for use in the regulation of information transport, such as in TCP/IP networks.

The manuscript presents a problem that can be considered as a multi-objective optimization or Pareto optimality problem [14, 15]. The aim of the study is to minimize the rise time, settling time, and overshoot rate of the step response of the closed-loop system. However, these performance criteria are often conflicting, making simultaneous optimization challenging. To address this issue, the authors propose a multi-objective ant colony optimization (MOACO) technique based on the concept of Pareto optimality [16]. This technique guarantees reliable data transmission and is commonly used in routing problems. While previous studies have discussed ACO optimization for routing in TCP networks [17, 18, 19], this paper focuses on optimizing the flow of packets in a congested router by tuning the PI controller using the MOACO technique. Finally, the authors present a MATLAB simulation to demonstrate the performance of the proposed method (MOACO) for the PI-AQM control system.

2. Problem statement

2.1 The linearized fluid-flow model of TCP

Transmission control protocol/Internet protocol (TCP/IP) is the most widespread family of protocols that manage routing on the Internet. TCP provides a secure service to deliver packages as IP, it ensures the delivery of these packages. The notoriety of the TCP/IP protocol is due to several factors including its reliability and its ability to allow a significant development of the Internet.

The reliability of transmission in TCP/IP networks is reflected in the use of acknowledgment packets (ACQs) and sequence numbers. When a node receives a packet, it sends an acknowledgment. A second acknowledgment for the same sequence number (duplicate acknowledgment) may be issued, meaning that a previous packet has been destroyed in the network or is delayed. This delay may be the result of increased traffic in the network, which may lead to the rejection of incoming frames in the buffers of the switches; this characterizes the phenomenon of congestion.

To address this issue, several active queue management (AQM) algorithms have been suggested, including random early detection (RED), adaptive RED (ARED) and Blue algorithms [20]. Such mechanisms which are basically heuristic methods tend to bring certain advantages such as drop tail enhancement by focusing on the length of the queue in order to reduce end-to-end delays and losses and, on the other hand, maximize the useful throughput in the network.

In this study, we focus on sharing a communication link between several senders as shown in Figure 1. It is assumed that the IP network has only one bottleneck, and that the link downstream of the router is borrowed by N flows.

Figure 1.

Network topology studied.

The behavior of TCP underwent numerous studies, which have subsequently given rise to a mathematical representation of TCP analogous to fluid flow. This fluid modeling makes it possible to carry out a quantitative analysis of the congestion phenomenon. In this article, we consider the fluid model proposed by [6, 7]. The system is represented by a couple of stochastic differential equations taking into account the evolution of the congestion window but neglecting the mechanisms of slow start and time out. The linearization of the system model around a point of equilibrium [21] gives rise to the following expression:

$$G\left(s\right)=\frac{R^3{C}^3}{2N^2}\frac{e^{-\mathit{Rs}}}{\frac{R^{3}C}{N}{s}^2+\left(R+\frac{R^{2}C}{N}\right)s+2+\mathit{Rs}{e}^{-\mathit{Rs}}}$$E1

2.2 The time-delay system model

Algorithms designed to regulate the length of queues may not always achieve their objectives, resulting in residual gaps. To compensate for these gaps, control theory approaches are employed. The aim of these approaches is to regulate the queue length to a specific threshold.

By reducing the congestion problem to a regulation problem, the system can be approximated as a first-order delay system, which is suitable for quantitative analysis without compromising the relevance of the results. This system model is described by

$$\mathrm{G}\left(s\right)=\frac{K{e}^{-\mathit{Ls}}}{\mathit{Ts}+1}$$E2

Where

$K=\frac{R^3{C}^3}{4N^2}$ is the state gain,

$T=R{\left(\frac{R^4{C}^2}{4N^2}+\frac{2\mathit{RC}}{N}-\frac{2R^{2}C}{N}+2\right)}^{\frac{1}{2}}$ is the constant time, and

$L=\frac{R^{2}C}{2N}+2R-T$ is the time delay of the plant.

3. PI controller description

The focus of this section is on closed-loop stabilization using the PI controller for delayed linear invariant time systems (LTI).

The equation describing the PI regulator is given by

$$C\left(s\right)=K_p+\frac{K_i}{s}$$E3

Where $K_p$represents the proportional gain, and $K_i$ represents the integral gain. The essential point of this study is the computation of the stability region defined by the parameters $K_p$ and $K_i$.

The selection of the PI controller is based on its ability to minimize system error due to its various characteristics. Broadly speaking, the PI regulator achieves regulation by adjusting the system input based on the size of the error (which is the responsibility of the proportional or P action) and the duration for which the error has persisted (which is the responsibility of the integral or I action).

3.1 Hermite-Biehler theorem

This study utilizes an extension of the Hermite-Biehler theorem to quasi-polynomials that are characteristic equations of system models with delay, for stabilizing the system model with the PI regulator after modeling congestion as a first-order delay system. The challenge of studying this class of systems is due to the fact that the number of roots is infinite, unlike polynomials where the number of roots is finite. Early work in this field by Pontryagin [8] formulates a necessary and sufficient condition for the roots of a quasi-polynomial to have a negative real part. This work is later extended and enriched by Batcharaya, Silva, et al. [9, 10, 11, 13] for the study of quasi-polynomials.

This approach has been applied to congestion control and is elaborated upon in [21, 22, 23]. The approach allows for obtaining a complete set of parameters Kp and Ki, which ensures the stability of the closed-loop system, in contrast to conventional methods that rely on a single parameter. This is an essential step for finding optimal stabilizing parameters, which will be discussed in the following section.

3.2 Ant colony optimization

This section focuses on seeking optimal stabilization parameters K_p and K_i from the region of stability computed using the Hermite-Biehler theorem extension approach discussed in the previous section.

When faced with multiple paths between a starting point and an endpoint, the goal is to identify the most efficient route. Ants achieve this by exploring the different paths and leaving a trace (pheromone) along the way. Over time, the preferred paths are marked with more pheromone while less-used paths have their pheromones evaporate (as shown in Figure 2).

Figure 2.

Ants converge to shortest path from their nest to the food.

The ant colony optimization (ACO) approach, introduced by Marco Dorigo et al., is a population-based method inspired by the behavior of real ant colonies in finding food sources. This concept has been applied to complex combinatorial optimization problems that aim to identify optimal solutions within the constraints of the problem, such as congestion control in TCP networks [24, 25].

There are three essential phases in the ant colony algorithm:

1. Initialization: a set of ants is generated at the starting point.

2. Path construction: each ant selects a path to traverse based on the pheromone level and heuristic information.

3. Pheromone update: the amount of pheromone on the path is updated based on the quality of the solution found by the ant.

The ACO algorithm has been used to search for the optimal set of parameters K_p and K_i for the congestion control problem in TCP networks. In this case, the algorithm seeks to identify the best values for K_p and K_i that lead to stable performance within the constraints of the system [24, 25].

In the following, the three steps of ACO algorithm are elaborated.

3.2.1 Initialization

The objective is to identify the shortest Hamiltonian cycle within a graph, wherein each vertex of the graph symbolizes a distinct city. The distance between cities $i$and $j$ is represented by$d_{\mathit{ij}}$, and the pair $\left(i,j\right)$ represents the edge between these two cities. We first initialize the quantity of pheromone on edges with${\tau }_{\mathit{\text{init}}}$, and each ant traverses the graph and constructs a complete path (a solution).

3.2.2 Constructing ant solution

During each phase of the solution-building process, the ant must determine which location to move to, and this determination is made probabilistically based on the levels of pheromones present as well as statistical information. This approach enables the ant to discover a high-quality solution. The probability that an ant $k$ moves from vertex $i$ to vertex$j$, which belongs to a set of vertices that are not yet visited by the ant $k$ denoted by$S_i^k$, is [26]:

$$P_{\mathit{ij}}^k=\frac{{\left({\tau }_{\mathit{ij}}\left(t\right)\right)}^{\alpha }.{\left({\eta }_{\mathit{ij}}\right)}^{\beta }}{{\sum }_{l\in S_i^k}{\left({\tau }_i\left(l\right)\right)}^{\alpha }.{\left({\eta }_{\mathit{il}}\right)}^{\beta }}$$E4

$\alpha$ and $\beta$ are two parameters that influence the importance of the pheromone intensity ${\tau }_{\mathit{ij}}$, and the statistical information called visibility ${\eta }_{\mathit{ij}}$. This value guides the choice of ants to nearby towns and avoids those that are too far away $\left({\eta }_{\mathit{ij}}=\frac{1}{d_{\mathit{ij}}}\right)$. For $\alpha =0$, one takes into account just the visibility, that is to say that the choice will have fallen each time on the nearest city. If$\beta =0$, only the pheromone tracks play on the choice. To avoid too fast selection of a path, a suitable compromise between these two parameters is mandatory.

3.2.3 Updating pheromone

When all the ants have constructed a solution, an amount of pheromones $∆{\tau }_{\mathit{ij}}^k$ is deposited by each ant $k$ on its path.

For any iteration $t$, if the path $\left(i,j\right)$ is in the round of the ant $k$ the quantity of pheromones deposited on this path is [27]:

$$∆{\tau }_{\mathit{ij}}^k\left(t\right)=\frac{Q}{L^k\left(t\right)}$$E5

Where $L^k\left(t\right)$ is the total length of the ant tour $k$, and $Q$ is a constant.

The quantity of pheromones added is contingent upon the quality of the solution achieved, whereby lower pheromone amounts correspond to better solutions. To prevent disregard of inferior solutions and to prevent convergence toward local optima of inferior quality, a parameter is introduced to simulate the evaporation of pheromone trails. This parameter, $\rho ,$called the evaporation rate $0<\rho <1$ is described as follows:

$${\tau }_{\mathit{ij}}\left(t+1\right)\leftarrow \left(1-\rho \right){\tau }_{\mathit{ij}}\left(t\right)+∆{\tau }_{\mathit{ij}}\left(t\right)$$E6

Where $∆{\tau }_{\mathit{ij}}\left(t\right)={\sum }_{k=1}^m∆{\tau }_{\mathit{ij}}^k\left(t\right)$, $t$ represents a given iteration and $m$ the number of ants.

4. Design of PI controller using ACO with multiple objective optimization

In this section, we introduce the PI-MOACO controller, which employs an ACO algorithm to optimize the gains $K_p\text{and}{K}_i$ used in congestion control by AQM, as demonstrated in Figure 3.

Figure 3.

Ant colony optimization flowchart for a PI proportional-integral controller.

The ACO algorithm generates the gains $K_p\text{and}{K}_i$ of the PI controller from the stability region. To leverage the ACO algorithm, it is preferable to represent the optimization problem using a direct path in the form of a construction graph, as illustrated in Figure 4.

Figure 4.

ACO graph.

The population is represented by a matrix (nx2), with the ant selecting the optimal parameters $K_p\text{and}{K}_i$ of the PI controller by minimizing the objective function. Figure 5 depicts the design problem utilizing the ant colony PI algorithm.

Figure 5.

Stabilizing region of (K_p, K_i) for the PI controller in the congestion control.

4.1 Cost functions used for the multi-objective ACO

Each parameter of $K_p\text{and}{K}_i$ is encoded by n nodes in this study. Hence, a solitary node symbolizes the optimal values of $K_p\text{and}{K}_i$. The primary step in utilizing the optimization technique is to select the criteria for optimization that assess fitness. Four performance indices, namely integrated-absolute error (IAE), integrated-squared error (ISE), integrated-time-absolute error (ITAE), and integrated-time-squared error (ITSE), are employed to minimize the error signal e(s). Their expressions are as follows:

$$\left\{\begin{array}{c}\mathit{ISE}=\sum _0^{t_{\mathit{max}}}e{\left(t\right)}^2\\ \mathit{IAE}=\sum _0^{t_{\mathit{max}}}\left|e\left(t\right)\right|\\ \mathit{\text{ITAE}}=\sum _0^{t_{\mathit{max}}}t\left|e\left(t\right)\right|\\ \mathit{\text{ITSE}}=\sum _0^{t_{\mathit{max}}}\mathit{te}{\left(t\right)}^2\end{array}\right.$$E7

where $e\left(t\right)$ is the error signal in time domain.

In the following, the ACO is characterized by a number of ants$m=300$, evaporation rate $\rho =0.7$, and the number of iterations = 50, $\alpha =0.8,$ and$\beta =0.2$ .

5. Simulation results and analysis

In this section, we assess the efficacy of the closed-loop system using the MOACO with PI controller via simulation. The simulation is performed using MATLAB, with an example from the literature model system used to illustrate the study presented throughout the paper, with the aim of optimally stabilizing the system.

$$\widehat{G}\left(s\right)=\frac{2.34.{10}^5{e}^{-0.77s}}{5.03s+1}$$E8

By applying the Hermite–Biehler extension approach, we find that the set of $K_p$stabilizing values is given below:

$$K_p\in \left[-4.27.{10}^{-6};3.89.{10}^{-5}\right].$$E9

Figure 5 provides a two-dimensional representation of the complete set of stabilizing PI controller variables $K_p\text{and}{\mathrm{k}}_i$for the TCP/AQM system. The MOACO method provides the PI controller’s optimum parameters, which are given in Table 1. Figure 6 presents the step response for the different PI controllers optimized by MOACO.

Performances criteria	Kp (10^−5)	Ki (10^−5)
IAE	1.85	1.08
ISE	1.60	1.02
ITAE	2.1	1.11
ITSE	2.35	1.12

Table 1.

Optimum parameters of PI controller.

Figure 6.

Step response by different objectives with MOACO-PI controller.

To analyze the dynamic performance of the system studied, we consider three performance criteria, namely settling time, rise time, and overshoot rate, among others.

Settling time is the time taken to reach the final value of the output to within 5%.

Rise time is the time at which the output signal crosses its asymptote for the first time. The rise time is a relevant parameter for encrypting the speed of a closed-loop system, which is the ultimate objective of our study.

Overshoot rate measures how much the output exceeds the steady-state value before settling.

Table 2 summarizes these standard performance measures.

Performances criteria	Rise time (s)	Settling time (s)	Overshoot (%)
IAE	0.79	9.80	59.15
ISE	0.89	11.14	56.17
ITAE	0.72	10.31	62.15
ITSE	0.66	9.96	65.58

Table 2.

Numerical values of standard performance measures.

The results show a conflict between the performance indices, which is expected since we use an approach based on the Pareto optimality principle. From Table 2, we can see that ITSE provides the best rise time among all the other objectives. The best value of settling time is given by the IAE criterion, and the smallest overshoot rate is provided by the ISE cost function. It has been highlighted that the main purpose of this study is to minimize the time delay occurring through the TCP network, with guaranteed stability. From the table above, we can conclude that these objectives are achieved since we obtain quite small rise and settling times.

6. Conclusion

This paper addresses the problem of congestion, one of the most common challenges encountered in communication networks. The proposed approach involves computing the stability region for a first-order delay system controlled by a PI controller using an extension of the Hermite-Biehler theorem applied to quasi-polynomials. The next step involves using a multi-objective optimization method based on the ant colony optimization (ACO) algorithm to search for optimal PI controller gains K_p and K_i within the stability region. Four performance criteria calculated in closed-loop plant are used as objective functions. Simulation results demonstrate the effectiveness of the proposed method in terms of dynamic performance, including reduction of maximum overshoot, rise time, and settling time. The MOACO algorithm allows for rapid convergence in local research, highlighting the flexibility and efficiency of the proposed approach.

Overall, the simulation results support the effectiveness of the proposed approach in addressing the problem of congestion in communication networks.

Conflict of interest

The author declares no conflict of interest.

Funding

This research received no external funding.