Abstract
Today’s embedded systems are exposed to variations in load demand due to complex software applications, dynamic hardware platforms, and the impact of the run-time environment. When these variations are large, and efficiency is required, adaptive on-line resource managers may be deployed on the system to control its resource usage. An often neglected problem is whether these resource managers are stable, meaning that the resource usage is controlled under all possible scenarios. In this paper we develop mathematical models for real-time embedded systems and we derive conditions which, if satisfied, lead to stable systems. For the developed system models, we also determine bounds on the worst case response times of tasks. We also give an intuition of what stability means in a real-time context and we show how it can be applied for several resource managers. We also discuss how our results can be extended in various ways.
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Notes
Strictly speaking, V represents a load value. To obtain the accumulation of execution times, one must multiply the function with h.
In addition to this the Karush-Kuhn-Tucker matrix must be non-singular at the starting point, but it can be shown that this condition holds for any \({\overline { \rho }} \in \mathbf {P} \). See Boyd and Vandenberghe (2008) Sect. 10.2 for an in depth treatment of these conditions.
The values \(V ( \overline{x} _{[ k ]} )\) and \(V ( \overline{x} _{[ k +1 ]} )\) depend on the state of the system. If the system did not have its worst-case behavior between t [k] and t [k+1], then the drop in V(.) is higher than β.
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Rafiliu, S., Eles, P. & Peng, Z. Stability of adaptive feedback-based resource managers for systems with execution time variations. Real-Time Syst 49, 367–400 (2013). https://doi.org/10.1007/s11241-013-9176-2
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DOI: https://doi.org/10.1007/s11241-013-9176-2