# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
## 1 Introduction  
Stability in a traditional continuous dynamic system is a property related to robustness and resilience, and designing a system to have this property is an essential issue of system control and design [1]. A stable system can eventually converge the desired steady state from an arbitrary state. To rephrase, a system can recover its original state even if it has left the steady state because of an exception or abnormality. Although stability has been introduced for continuous systems, discrete event systems, such as manufacturing systems or computer networks, also have similar stability issues such as boundedness, practical stability, and Lagrange stability in the sense of Lyapunov stability or asymptotic stability [2]. A general Petri net, a kind of a discrete event system modeling language, usually analyzes structural boundedness, a special case of Lagrange stability [3]. In that sense, the event graph, also known as a marked graph, is always a bounded Petri net, so it is stable at all times. Therefore, an event graph is rarely necessary to analyze the system's boundedness, but it is useful for modeling a cyclic schedule for mass production, thanks to this stability [3; 4; 5]. However, some cyclic production systems have another type of stability issue even after modeling with event graphs. For example, a cluster tool for chemical vapor deposition (CVD) has a lower time limit, which is the minimum time required to perform a task, as well as an upper limit of wait time until the next task. The constraint forces a cluster tool to regulate fluctuations of delay or wait time between tasks to keep the product quality uniform and predictable. We call this constraint a time window constraint. Many researchers have studied how to model the cyclic schedule of a cluster tool with deterministic time using a timed event graph (TEG) to satisfy the constraint [6; 7]. A TEG is a Petri net that temporizesby introducing the concept of deterministic time to places in an event graph. However, an existing system has persistent time variations or exceptions, so a TEG model analysis result cannot sometimes reflect the real one. Therefore, some studies have considered the predictable time fluctuation range when modeling and analyzing schedulability using the modified TEG [8] or have suggested a method of controlling the system within the time constraint [9]. Nevertheless, if rare exceptions occur such as time fluctuations more significant than expected or abnormality, we must quickly enable the system to converge to its normal state. Analyzing a discrete event system's stability with such time constraints determines whether each task has asymptotically reached desired steadiness that repeats it at fixed intervals rather than analyzing the boundedness. In other words, a system that is not stable does not repeat each task at the same time interval. However, even an unstable cyclic discrete event system repeats a pattern with the time interval. We call the analysis of temporal patterns _cyclicity analysis_[3; 10; 11]. Some researchers have analyzed the cyclicity and stability with regard to the time interval pattern of TEG [3]. Specifically, if each transition repeats firing epochs asymptotically every time a specific integer \(d\)-cycles, the system is considered \(d\)-cyclic, and the one-cyclic system is considered stable. However, it is not intuitive to analyze cyclicity and stability in a TEG. Therefore, these researchers have modeled a system with a TEG and converted it into a matrix form of (max,+) algebra [8; 11; 12; 13]. (max,+) algebra is an idempotent semiring of extended real numbers \((R\cup\{-\infty\},\max,+)\) that can model and analyze the concepts of synchronization and concurrency [11; 14; 15; 16; 17]. It also has a concept of stability like the one we have explained here, so it is easy to analyze the periodicity and stability of the system [10; 18; 19].