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Modeling task systems using parameterized partial orders

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Abstract

Inspired by work on model-based design of printers, the notion of a parametrized partial order (PPO) has recently been introduced. PPOs are a simple extension of partial orders, expressive enough to compactly represent large task graphs with finite repetitive behavior. We present a translation of a subclass of PPOs to timed automata and prove that the transition system induced by the Uppaal models is isomorphic to the configuration structure of the original PPO. Moreover, we introduce real-time task systems (RTTSs), a general model for real-time embedded systems that we have used to describe the data paths of realistic printer designs. In an RTTS, tasks are represented as PPOs and the pace of a task instance may vary, depending on the resources that are allocated to it. We describe a translation of a subclass of RTTSs to Uppaal, and establish, for an even smaller subclass, bisimulation equivalence between the timed configuration semantics of an RTTS and the transition system induced by the corresponding Uppaal translation. Lastly, we report on a series of experiments which demonstrates that the resulting Uppaal models are more tractable than handcrafted models of the same systems used in earlier case studies.

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Notes

  1. In this section, we use functions of type \(\mathcal{R}\rightarrow \mathbb{N }\), where \(\mathcal{R}\) is some set of resources. Operations and predicates on \(\mathbb{N }\) are extended to such function by pointwise extension. For instance, for \(f, g : \mathcal{R}\rightarrow \mathbb{N }\), we say that \(f \le g\) iff \(\forall r \in \mathcal{R}: f(r) \le g(r)\), and we define \(f+g : \mathcal{R}\rightarrow \mathbb{N }\) by \((f+g)(r)=f(r)+g(r)\).

  2. Formally, function \(\mathfrak R \) takes as input global states of the Uppaal model described in Sect. 3. We may apply \(\mathfrak R \) to global states of the extended model described in this section by removing all the additional state variables.

  3. The intermediate location \(\mathsf{Update}\) is required since in Uppaal clock guards are not allowed on edges labeled with urgent channels.

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Acknowledgments

An extended abstract of this paper appeared as [1]. This research has been carried out as part of the OCTOPUS project under the responsibility of the Embedded Systems Institute. This project is partially supported by the Netherlands Ministry of Economic Affairs under the Bsik program. This research was also supported by European Community’s FP7 Programme under grant agreement no 214755 (QUASIMODO). We thank Twan Basten, Alexandre David, Martijn Hendriks, Nikola Trčka, Marc Voorhoeve, for inspiring discussions on the topic of this paper. We dedicate this paper to the memory of Marc Voorhoeve, 1950–2011, who devised the notion of a PPO.

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Authors and Affiliations

  1. ASML, Veldhoven, The Netherlands

    Fred Houben

  2. ICIS, MBSD Group, Radboud University Nijmegen, Nijmegen, The Netherlands

    Georgeta Igna & Frits Vaandrager

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  1. Fred Houben
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  2. Georgeta Igna
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Correspondence to Frits Vaandrager.

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An extended abstract of this paper appeared as [1]. The research of Igna and Vaandrager has been carried out as part of the OCTOPUS project under the responsibility of the Embedded Systems Institute. This project is partially supported by the Netherlands Ministry of Economic Affairs under the Bsik program. This research was also supported by European Community’s Seventh Framework Programme under Grant agreement no. 214755 (QUASIMODO)

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Houben, F., Igna, G. & Vaandrager, F. Modeling task systems using parameterized partial orders. Int J Softw Tools Technol Transfer 15, 269–286 (2013). https://doi.org/10.1007/s10009-012-0264-8

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