## 2 Related Work
Supported by software-defined networking and network function virtualization techniques, [PERSON] et al. [51] proposed an MTD route mutation method that dynamically changes the routes for packets. The verification process of the proposed MTD system for these works was achieved using the \"Mininet\" emulator. To sum up, for experiment-based models, most existing literature results and verification were based on a variety of simulation methods that are challenging in terms of the requirements for a long period of time to achieve a steady state. Also, these methods do not provide portability and reuse characteristics. In other words, if an experiment-based assessment is designed for a specific ACD system, it becomes difficult to reuse it or migrate it to another ACD system due to the lack of explicit expression. Another recognized analytical method is assessing the effectiveness of ACD through theoretical models, such as the security models, stochastic processes, and Petri nets. Compared with experiment-based methods limited by the number of samples and complex application scenarios, analytical models explicitly describe the ACDs' effectiveness. They always provide generalized expressions with various metrics such as attack success probability, attack utility, mean time to failure, and defense utility [3]. Such metrics make different ACDs comparable. As a mathematical model describing parallel systems, Petri nets are widely used with many variants [18; 52; 53], such as Colored Petri nets, Generalized stochastic Petri nets (GSPN), and Stochastic Reward Net (SRN). [PERSON] et al. [18] classified system failures into three types, including attrition, pervasion, and exfiltration. The failures and transferring relation were described with SPN to optimize the system configuration, including intrusion detection interval and the redundancy level. However, the costs of defenders were not considered. [PERSON] et al. [53] modeled the MTD using SPN with distributed and parallel applications. They derived a qualitative conclusion that more frequent group transitions lead to a safer system. [PERSON] et al. [24] focused on VM migration scheduling and proposed an SRN for the probability of attack success and availability evaluation of an MTD. The presented results cover VM migration scheduling, VM migration failure probability, and attack success rate. [PERSON] et al. [23] also researched in an environment with a migration-based dynamic platform technique. Concerned with job completion time, they proposed the SRN-based model to describe tradeoffs between performance and security. [PERSON] et al. [54] also presented an SRN for performance evaluation of MTD, but they focused on the dynamic platform technique. They described the performance with job finish time and job fail times. The Markov-based model is one of the most commonly used mathematical models for finding the optimal defending strategy [55; 56; 57] and analyzing security. Related researches also inspire our work. Focusing on the effect of MTD on an enterprise network, [PERSON] et al. [58] built a Markov chain. The probability of a successful intrusion for each node was preliminarily computed in different attack paths. Nevertheless, disturbed by the modeling problems as the increasing network size, they ignored the situations where attackers can be pushed back to any previous node, leading to reduce modeling reliability. [PERSON] et al. [19] introduced a Markov-based method to analyze the relationship between the probability of a successful attack and the cost spent by the adversary in the MTD system. They defined the concept of security capacity as a measurement to analyze the effectiveness of single-target hiding and multiple-target hiding. However, the lack of simulation led to a decrease in modeling reliability. [PERSON] et al. [17] presented a quantitative model for assessing the resource availability and performance of MTDs. They defined a stability metric and devised a method to determine the maximum reconfiguration rate that meets stability constraints.