Abstract
Prior algorithms that have been proposed for the uniprocessor implementation of systems of elastic tasks have computational complexity quadratic (\(O(n^2)\)) in the number of tasks n, for both initialization and for admitting new tasks during run-time. We present a more efficient implementation in which initialization takes quasilinear (\(O(n\log n)\)), and on-line admission control, linear (O(n)), time.
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Notes
For tasks \(\tau _i\) having \(E_i=0\), \(U_i=U_i^{\min }\), and therefore the relationship needs not be satisfied.
All tasks \(\tau _i\) with \(E_i=0\) must have \(U_i \leftarrow U_i^{\max }\) in order to satisfy Expression 1; we assume this is done in a pre-processing step, and the value of \(U_d\) updated to reflect the remaining available utilization.
Observe that \(\varDelta \) equals the amount of utilization that is allocated to the tasks in \(\varGamma _{\textsc {fixed}}\); therefore \((U_d-\varDelta )\) represents the amount available for the tasks in \(\varGamma _{\textsc {variable}}\), and \(\bigl (U_{\textsc {sum}}-(U_d - \varDelta )\bigr )\) the amount by which the cumulative utilizations of these tasks must be reduced from their desired maximums. As shown in the RHS of Expression 2, under elastic scheduling this reduction is shared amongst the tasks in proportion to their elasticity parameters: \(\tau _i\)’s share is \((E_i/E_{\textsc {sum}})\).
References
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Funding
This study was funded by the National Science Foundation.
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Sudvarg, M., Gill, C. & Baruah, S. Linear-time admission control for elastic scheduling. Real-Time Syst 57, 485–490 (2021). https://doi.org/10.1007/s11241-021-09373-4
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DOI: https://doi.org/10.1007/s11241-021-09373-4