Abstract
The generalized multiframe task model (GMF) extends the sporadic task model and multiframe task model. Each frame in the GMF model contains an execution time, a relative deadline, and a minimum inter-arrival time. These parameters are fixed after task specification time in the GMF model. However, multimedia and adaptive control systems may be overloaded and no longer stabilized when the task parameters in such systems are not flexible. In order to address this problem, deadlines and periods of frames may change to alleviate temporal overload, e.g., in the parameter adaptation and elastic scheduling model. In this paper, we propose a new model GMF-PA (the GMF model with parameter adaptation). This model allows task parameters to be flexible in arbitrary-deadline systems. A necessary schedulability test based on mixed-integer linear programming is given to check the schedulability under EDF scheduling and optimally assign frame deadlines and periods at the same time. We also prove that the test is a sufficient and necessary schedulability test when frame deadlines and periods must be integers. An approximation algorithm is also deployed to reduce computational running time and indicates a sufficient schedulability test in general. The speed-up factor of our approximation algorithm is \(1+\epsilon \) where \(\epsilon \) can be arbitrarily small, with respect to the exact schedulability test of GMF-PA tasks under EDF. We also apply the GMF model to self-suspending tasks. By extending recent work on scheduling self-suspending tasks, we remove the assumption that frame deadlines are equally assigned in self-suspending tasks, and the system is extended from constrained-deadline systems to arbitrary-deadline systems. We have done extensive experiments to show that the schedulability ratio is improved using our techniques in our GMF-PA model.
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Notes
Note that in our paper (Peng and Fisher 2016), we define the cycle deadline \(\mathcal {D}_i\) as the upper bound of \(\sum _{j=0}^{N_i-1} D_i^j\). We believe that the new definition is more appropriate for modeling the end-to-end constraint of self-suspending tasks. The change will not affect the evaluation results because the previous work assume \(D_i^j=P_i^j\) (also set for our algorithms in evaluations) which makes \(\sum _{j=0}^{N_i-1} D_i^j\) and \(D_i^{N_i-1}+\sum _{j=0}^{N_i-2}P_i^j\) be equal.
The approximation algorithm is still an MILP (and thus still potentially intractably), but a reduction in constraints leads to a significant improvement in time efficiency as shown in the evaluation section.
We reuse the data generated by Nimmagadda et al. (2010). The systems they used are a robot car and a Linux server. The car is a pioneer 3DX robot with an Intel core2-duo 2 GHz processor and two cameras. The resolution of both cameras is \(640 \times 480\). The server is an Intel Xeon Linux server with eight quad-core 2.33 GHz processors and 8 GB RAM.
In the work (Nimmagadda et al. 2010), the execution of modules are represented by a graph, we choose a reasonable valid sequence from the graph. The aim is to relax the independence of the modules. This sequence is also compatible with the sequences in a more general context (Sivaraman and Trivedi 2013).
In the GMF-PA model, \(E_i^j\) is the j’th frame execution time of task \(\tau _i\). In this section, we omit subscript for simplicity.
Nimmagadda et al. (2010) refer to computation as cycles. For example, \(c_f\) is the total number of cycles needed to extract the features of a picture.
In this case, it is assumed that the three objects have same speed and move parallel, this can be applied in the scenario of tracking cars in different lanes (Sivaraman and Trivedi 2013).
Note that the EDA algorithm in the previous paper (Chen and Liu 2014) only consider one-segment self-suspending tasks, we extend EDA using our MILP. In MILP, we add one more constraint to let the frame deadlines of each task be equal.
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Acknowledgements
This research has been supported in part by the US National Science Foundation (Nos. CNS-0953585, CNS-1618185) and a grant from Wayne State University’s Office of Vice President of Research.
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Peng, B., Fisher, N. Parameter adaptation for generalized multiframe tasks: schedulability analysis, case study, and applications to self-suspending tasks. Real-Time Syst 53, 957–986 (2017). https://doi.org/10.1007/s11241-017-9279-2
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DOI: https://doi.org/10.1007/s11241-017-9279-2