Study of Parameterized-Chain networks

📝 Abstract

In areas such as computer software and hardware, manufacturing systems, and transportation, engineers encounter networks with arbitrarily large numbers of isomorphic subprocesses. Parameterized systems provide a framework for modeling such networks. The analysis of parameterized systems is a challenge as some key properties such as nonblocking and deadlock-freedom are undecidable even for the case of a parameterized system with ring topology. In this paper, we introduce \textit{Parameterized-Chain Networks} (PCN) for modeling of networks containing several linear parameterized segments. Since deadlock analysis is undecidable, to achieve a tractable subproblem we limit the behavior of subprocesses of the network using our previously developed mathematical notion `weak invariant simulation.’ We develop a dependency graph for analysis of PCN and show that partial and total deadlocks of the proposed PCN are characterized by full, consistent subgraphs of the dependency graph. We investigate deadlock in a traffic network as an illustrative example. This document contains all the details and proofs of the study.

💡 Analysis

In areas such as computer software and hardware, manufacturing systems, and transportation, engineers encounter networks with arbitrarily large numbers of isomorphic subprocesses. Parameterized systems provide a framework for modeling such networks. The analysis of parameterized systems is a challenge as some key properties such as nonblocking and deadlock-freedom are undecidable even for the case of a parameterized system with ring topology. In this paper, we introduce \textit{Parameterized-Chain Networks} (PCN) for modeling of networks containing several linear parameterized segments. Since deadlock analysis is undecidable, to achieve a tractable subproblem we limit the behavior of subprocesses of the network using our previously developed mathematical notion `weak invariant simulation.’ We develop a dependency graph for analysis of PCN and show that partial and total deadlocks of the proposed PCN are characterized by full, consistent subgraphs of the dependency graph. We investigate deadlock in a traffic network as an illustrative example. This document contains all the details and proofs of the study.

📄 Content

Study of Parameterized-Chain Networks M. H. Zibaeenejad and J. G. Thistle* Abstract In areas such as computer software and hardware, manufacturing systems, and transportation, engineers encounter networks with arbitrarily large numbers of isomorphic subprocesses. Param- eterized systems provide a framework for modeling such networks. The analysis of parameterized systems is a challenge as some key properties such as nonblocking and deadlock-freedom are undecidable even for the case of a parameterized system with ring topology. In this paper, we introduce Parameterized-Chain Networks (PCN) for modeling of networks containing several linear parameterized segments. Since deadlock analysis is undecidable, to achieve a tractable subproblem we limit the behavior of subprocesses of the network using our previously developed mathematical notion ‘weak invariant simulation.’ We develop a dependency graph for analysis of PCN and show that partial and total deadlocks of the proposed PCN are characterized by full, consistent subgraphs of the dependency graph. We investigate deadlock in a traffic network as an illustrative example. I. Introduction A parameterized network is composed of arbitrary finite numbers of isomorphic subpro- cesses. Formally, such systems can be modeled as infinite families of finite-state systems. They are a subclass of the so-called ‘parameterized systems’, whose models incorporate parameters with unspecified values [1]. In the case of parameterized networks, the param- eter is the number of subprocesses in the network. Practical examples of parameterized networks include wireless sensor networks, transportation networks, manufacturing systems and subprocesses in operating systems. Parameterized models are particularly useful when the number of subprocesses is unknown, time-varying, or very large. It is natural to ask how much analysis and control can be done independently of a specific parameter values. Unfortunately, key problems such as checking the nonblocking property for *M. H. Zibaeenejad and J. G. Thistle are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada. mhzibaee,jthistle@uwaterloo.ca parameterized networks are generally undecidable [2]. Parameterized networks have received considerable attention in the model-checking literature [3], [4]. Most recently, the authors of [5] seek to determine whether or not a given safety property holds for all instances of parameterized toroidal mesh networks under process symmetry assumptions. Within control literature, the deadlock analysis of a class of parameterized networks was considered, where subsystems are identical and interact only via events that are shared with all other subsystems [6]. This requires the communication topology of network to be that of a graph-theoretic clique. In previous work [7], the present authors introduced a novel mathematical tool, weak invariant simulation, to support deadlock analysis of parameter- ized networks. Although the deadlock-freedom property is generally undecidable in ring networks, weak invariant simulation relations was used to define a class of these networks in which all the reachable deadlocked states can be calculated [2]. In this paper, we consider Parameterized-Chain Networks (PCN) consisting of multiple linear parameterized segments together with a finite number of finite-state subprocesses having arbitrary structure. In networks consisting of several subprocesses, nontrivial deadlocks often occur in the presence of a circular wait. When a circular wait occurs, the only available action of each subprocess requires a resource that is being held by another subprocess [8], [9]. Graph- theoretic techniques are used to characterize such dependencies in finite-state systems [9], [10]. Unfortunately, these techniques are not directly applicable to the analysis of parame- terized networks. In this paper, we characterize dependencies between subprocesses of any instances of a PCN by means of a single, finite dependency graph. In a preliminary form, the dependency graph was introduced in [11], where it was conjectured it can be used to detect reachable partial deadlocks of a PCN. Here we prove that specific subgraphs of the dependency graph represent reachable generalized circular waits of instances of the PCN. We relate partial and total deadlocks of the PCN to these generalized circular waits. Specifically, we show that the existence of a generalized circular wait is a necessary condition for total deadlock and a sufficient condition for partial deadlock of all but an acyclic subgraph of a PCN. In some applications this yields a necessary and sufficient condition for total deadlock. We illustrate our proposed method by analysis of a traffic network. Section II covers preliminaries. Section III introduces PCN and a running example of a train network. Section IV presents our deadlock analysis method. Section V expresses the main results of the paper: the deadlock analy

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