Chebyshev-Gauss Approximation Analysis for Mobile Edge Computing-Aided IoT Networks

Recently, mobile edge computing (MEC) has been widely applied into Internet of Things (IoT) networks, which has attracted a lot of attention from researchers. A critical challenge in the MEC-aided IoT networks is that the performance analysis is often complicated, where it is quite di ﬃ cult for us to obtain some analytical or closed-form solution to the performance analysis, such as outage probability and bit error rate. This has been the bottleneck of the development of MEC-aided IoT networks. To address this challenge, we deeply investigate the Chebyshev-Gauss approximation method and derive the analytical solution to implement this powerful and useful approximation. We then give several examples to show the e ﬀ ectiveness of the Chebyshev-Gauss approximation in the performance analysis for the MEC-aided IoT systems. The results in this work can serve as an important reference and reveal some important inherent mechanisms for the MEC-aided IoT networks.


Introduction
Recently, a lot of wireless nodes cooperate together, to communicate and compute collaboratively, which form the Internet of Things (IoT) networks [1,2]. In such a system, a lot of wireless nodes access the system spectrum, by using orthogonal or nonorthogonal multiple access schemes [3,4]. These nodes can communicate and compute in a collaborative way, when facing some intensive calculating tasks. Besides the communication and calculation, the privacy protection also becomes a key research topic in the study of IoT networks [5,6], where some privacy protection methods from the physical layer to the application layer should be incorporated into the system, in order to enhance the data communication privacy and data calculation privacy, especially for some sensitive data such as medical data and financial data. Some novel techniques have been proposed by researchers to promote the development of IoT networks, among which mobile edge computing (MEC) is a key technology [7,8]. In the MEC-aided IoT networks, some edge nodes have some powerful ability to help calculate the intensive tasks from other nodes, which will be helpful in leading to a smaller delay and power consumption (PoCo). In this area, a lot of studies have been performed to utilize the communication resources as well as calculating resources in the MEC-aided IoT networks, through some conventional optimization methods such as convex optimization or some intelligent algorithms such as deep reinforcement leaning (DRL) algorithms, in order to reduce the system delay and PoCo [9]. This can help make the MEC-aided IoT networks fit the various applications [10][11][12].
Because of restricted regularity sources, cochannel interference has actually been unavoidable in the wireless communication systems. Cochannel interference has restricted the system efficiency seriously, as well as being the traffic jam of the wireless systems. The effect of cochannel interference on the system efficiency of wireless communications was thoroughly examined in the literary works. Some authors examined the dual-hop communicating systems in the existence of cochannel interference, as well as the system data rate and outage possibility. For the secure communicate systems in cochannel interference, the system efficiency might be examined through obtaining the capacity as well as asymptotic privacy outage possibility, whereby the impact of interfering energy on the system efficiency might be exposed.
A critical challenge in the MEC-aided IoT networks is that the performance analysis is often complicated, where it is quite difficult to obtain some analytical or closed-form solution to the performance analysis, such as outage probability and bit error rate. This has been the bottleneck of the development of MEC-aided IoT networks. To address this challenge, we deeply investigate the Chebyshev-Gauss approximation method and derive the analytical solution to implement this powerful and useful approximation. We then give several examples to show the effectiveness of the Chebyshev-Gauss approximation in the performance analysis for the MEC-aided IoT systems. The results in this work can serve as an important reference and reveal some important inherent mechanisms for the MEC-aided IoT networks.

Chebyshev-Gauss Quadrature
In numerical analysis, numerical integration is the method and theory of calculating the value of definite integration. We can use the Leibniz integral rule to calculate the definite integral through the original function. However, it is regularly difficult to calculate the original value of the function. There are few functions that can be expressed by elementary functions, and the integration of most integrable functions cannot be expressed by elementary functions or even analytical expressions. Therefore, in many cases, we can only use numerical integration to calculate the approximate value of the function.
At present, there are many algorithms for calculating definite integral. For instance, these algorithms mainly include the following: Among the above algorithms, the Gaussian quadrature rule with n-point is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes x i and weights w i for i = 1, ⋯, n. Specifically, the Gauss quadrature includes three different forms: (1) Chebyshev-Gauss quadrature (2) Gauss-Hermite quadrature (3) Gauss-Jacobi quadrature Now, we mainly discuss the Chebyshev-Gauss quadrature. Chebyshev-Gauss quadrature is an extension of Gaussian quadrature method, which is used to approximate the following two types of integral value. For the first kind, we can have where x i = cos ðð2i + 1Þπ/2nÞ, the weight w i = π/n, and the approximation error decreases with a larger number of item n.
On the contrary, for the second kind, we have where x i = cos ðiπ/ðn + 1ÞÞ, the weight w i = ðπ/ðn + 1ÞÞ sin 2 ðiπ/ðn + 1ÞÞ, and the approximation error decreases with a larger number of item n.
For a random function f ðxÞ, its integral Ð b a f ðxÞdx can be approximated as where w i is the weight coefficient, and the approximation error decreases with a larger number of item n. Note that the accuracy can be improved by increasing the number of x i or finding the right x i . We can rewrite f ðxÞ as Therefore, (3) can be approximated as where ρðxÞ is the weight function and the approximation error decreases with a larger number of item n. We can use the Chebyshev-Gauss quadrature to approximate (5 Let T n ðxÞ = 0, and the root of the equation can be obtained as where k = 0, 1, ⋯, n − 1, and x k is the Chebyshev node, namely, the root of the Chebyshev polynomials of the first kind. As shown in Figure 1, the Chebyshev node is equivalent to the x-axis coordinates of N equally spaced points on the unit semicircle. The Chebyshev-Gauss quadrature can obtain a relatively approximate solution only when the function f ðxÞ can be approximated by polynomials in the interval ½−1, 1. We use an affine transformation for nodes over an arbitrary interval ½a, b as For (3), let and thus, we have Þy dy: ð10Þ Let y = cos θ, where θ = ðð2k + 1Þπ/2nÞ, k = 0, 1, ⋯, n − 1 , and then, we have We can further write According to the definition of definite integral, we can write By comparing with (5), we can obtain

Numerical and Simulation Results
In this part, we present some numerical and simulation results to verify the convergence effect of the conventional rectangle rule and the Chebyshev-Gauss quadrature. We compare the convergence effect of the following three     log e x − x 2 À Á dx, Figure 2 shows the convergence effect of I 1 . We can find that the Chebyshev-Gauss quadrature and the benchmark method (namely the rectangle rule) both reach the convergent state after 1600 iterations, and the ultimate convergent values are 271.2296 and 272.1258, respectively. Similarly, as shown in Table 1, the Chebyshev-Gauss quadrature method approaches convergence with the convergent value of 271.2469, after about 200 iterations, which is much faster than the benchmark method which approaches the convergent state, after about 1200 iterations.
In Figure 3, the Chebyshev-Gauss quadrature approximation is performed on I 2 to demonstrate the advantage of faster convergence than the benchmark method. From Table 2, we can also find that the Chebyshev-Gauss quadrature method approaches the convergent state with the convergent value of 2.6207, after about 200 iterations, while the benchmark method approaches the convergent state, after about 1200 iterations. The associated ultimate convergent values are 2.6207 and 2.6218 after 1600 iterations, respectively. Figure 4 illustrates the convergence of the function I 3 . It can be seen that the Chebyshev-Gauss quadrature method approaches the convergent state with the convergent value     Table 3. This also shows the advantage that the Chebyshev-Gauss quadrature method converges much faster than the benchmark method. The associated final convergent values are 5.3353 and 2.6218 after 1000 iterations, respectively.

Conclusions
In the MEC-aided IoT networks, a critical challenge was that the performance analysis was often complicated, where it was quite difficult to obtain some analytical or closed-form solution to the performance analysis, such as outage probability and bit error rate. To address this challenge, we deeply investigated the Chebyshev-Gauss approximation method and derived the analytical solution to implement this powerful and useful approximation. We then gave several examples to show the effectiveness of the Chebyshev-Gauss approximation in the performance analysis for the MECaided IoT systems. The results in this work could serve as an important reference and reveal some important inherent mechanisms for the MEC-aided IoT networks.

Data Availability
The data of this paper can be obtained through email to the authors.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this work.