## 1 Introduction  
With continuous progress in the Internet of Things and mobile wireless communication technology, the number of intelligent devices and data they generate is growing rapidly [1]. The rapid iteration and updating of Internet information technology have led to the emergence of new services in smart cities, intelligent manufacturing, intelligent healthcare, and intelligent transportation. However, terminal devices are limited in computing power and cannot support computation-intensive tasks. Many researchers have applied cloud computing (CC) to overcome this challenge. In cloud computing, remote cloud centers provide abundant computing resources to support IoT applications. Users transfer computation-intensive tasks to the cloud for processing and receive the results. However, cloud centers are usually far away from users, and extensive data transmission causes significant latency [2]. For tasks that are sensitive to delay, like real-time monitoring and traffic dispatching, such large task delays are unbearable. Furthermore, massive data transmission can cause network congestion, burdening the core network [3]. In order to solve these problems, edge computing (EC) emerged. By integrating computing, storage, and processing capabilities near users and data sources, edge computing can provide lower network latency, better Quality of Service (QoS), and flexible adaptation to various application scenarios. Task offloading is a core technology of mobile edge computing, which focuses on reducing service delay and device energy consumption by processing computing-intensive tasks on network edge servers. Task offloading strategies play a crucial role in determining which tasks should be offloaded to the edge server for execution, as well as how to select the appropriate edge server. The decision-making process for task offloading can be influenced by various factors, including the type of task, network status, and the load of edge servers. Traditional theoretical performance analysis methods primarily rely on deterministic models to represent the spatial distribution of nodes. The performance analysis process often depends on Monte Carlo simulation or overly idealized assumptions [4, 5]. However, due to the heterogeneity and randomness of node distribution in MEC networks, traditional modeling methods are excessively idealized. Therefore, there is an urgent need for new modeling approaches that can accurately capture the randomness of MEC network topology for precise performance analysis. Stochastic geometric models have been introduced into wireless mobile networks, where researchers model the distribution of base stations and users as a homogeneous Poisson point process (PPP). This approach has been used to validate the accuracy of performance analysis based on stochastic geometric models [6]. To address the issue of heterogeneity, research has further extended these results to multi-layer heterogeneous networks in the literature [7, 8]. In the field of performance analysis, queuing theory plays an important role [9, 10]. It is a mathematical theory and method that studies the phenomenon of the random aggregation and dispersion of systems, as well as the working process of random service systems, also known as the theory of random service systems. Queuing theory provides a theoretical framework for edge computing to analyze and optimize resource allocation, task scheduling, system performance, and other aspects. Through the queuing model, we can gain a better understanding and design the edge computing architecture to meet real-time efficiency, reliability, and other requirements. The main contributions of this paper can be summarized as follows:  
1. We develop a new heterogeneous edge computing network architecture, which is composed of MEC Servers with different transmission power, computing capabilities, number of CPU cores, and waiting queue length. 2.