The overture of constructing the closed-form solution for Petri Nets: begin from the variant k-th order S3PR system

• Main Author: 游宗憲
• Other Authors: 趙玉
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/kpm688

博士 === 國立政治大學 === 資訊管理學系 === 105 === In the light of the rapid innovation of the Internet of Things (IoT), robot systems, and cloud computing systems, we need an efficient methodology to model gradually more and more complicated, real-time resource allocation systems (RAS), constructed using the systems mentioned above, for solving issues such as bottlenecks, deadlocks, and other embedded system-control-related problems. To solve the exponentially increasing complexity in the persistent problem of modeling large systems using Petri nets, which is an NP (nondeterministic polynomial time)-complete problem even when MIP (mixed integer programming) is employed for reachability analysis, Chao broke this barrier by developing the first closed-form solution (CFS) for the number of Control Related States (CRSs) for k-th order and k-net systems. However, the properties of symmetric net structures limit their application range in modeling systems; the inevitable deadlock obstructs the capability of concurrent processing in both systems. To enhance the capability of modeling large dynamic, real-time resource allocation in asymmetric systems, this dissertation extends the research on the CFS of PNs to the so-called Gen-Left k-th order system, the Gen-Left k-net system, and the A-net system, which comprise the three different types of fundamental asymmetric systems. A Gen-Left k-th order (resp. k-net) system is a k-th order (resp. k-net) system containing a non-sharing resource (NSR) at arbitrary locations in the control process, which is the fundamental net structure for modeling contained customized manufacturing processes inside a system. An A-net system is a k-th order system connected to a Top Non-sharing Circle Subnet (TNCS), which is the fundamental net structure to model a shared common manufacturing processing system in real applications. Based upon analyzing the effects of one NSR in the equivalent, the corresponding k-th order (resp. k-net) system, and an equivalent CFS, this dissertation derives the CFS for the Gen-Left k-th order (resp. k-net) system. Due to the independence of the TNCS and the connected Deficient k-th order system, we can first derive the CFS for a Deficient k-th order system just by excluding the number of impossible states from the CFS for its corresponding k-th order system. Then, the CFS of an A-net is constructed by summing the products of the CFS for the two sub-systems in each different case under the condition of the number of tokens inside TNCS. Based on real-time CRS information derived, we can enhance the capability for modeling a large dynamic, real-time resource allocation system in real applications. Employing the proposed deadlock-avoidance algorithm, for instance, we can realize concurrent processing in both k-th order and k-net systems without additional controllers being implemented; and the function of dynamic process allocation in a k-net system without suspending the system’s working flows. In addition to applying siphon computation to construct the fundamental CFS for asymmetric systems, this dissertation pioneers and proposes a new knowledge-based, analysis methodology, called proof by model, to accelerate the construction of the CFS for a PN based upon the validation information from its reverse net. This dissertation opens a new research era for constructing the CFS for PNs beginning from the Variant k-th order system.