# Speed scaling problems with memory/cache consideration

- 120 Downloads

## Abstract

Speed scaling problems consider energy-efficient job scheduling in processors by adjusting the speed to reduce energy consumption, where power consumption is a convex function of speed (usually, \(P(s) =s^{\alpha }, \alpha =2,3\)). In this work, we study speed scaling problems considering memory/cache. Each job needs some time for memory operation when it is fetched from memory,, and needs less time if fetched from the cache. The objective is to minimize energy consumption while satisfying the time constraints of the jobs. Two models are investigated, the non-cache model and the with-cache model. The non-cache model is a variant of the ideal model, where each job *i* needs a fixed \(c_i\) time for its memory operation; the with-cache model further considers the cache, a memory device with much faster access time but limited space. The uniform with-cache model is a special case of the with-cache model in which all \(c_i\) values are the same. We provide an \(O(n^3)\) time algorithm and an improved \(O(n^2\log n)\) time algorithm to compute the optimal solution in the non-cache model. For the with-cache model, we prove that it is NP-complete to compute the optimal solution. For the uniform with-cache model with agreeable jobs (later-released jobs do not have earlier deadlines), we derive an \(O(n^4)\) time algorithm to compute the optimal schedule, while for the general case we propose a \((2\alpha \frac{g}{g-1})^{\alpha }/2\)-approximation algorithm in a resource augmentation setting in which the memory operation time can accelerate by at most *g* times.

## Keywords

Speed scaling Energy efficiency Scheduling Memory operation time DVS Algorithm design## Notes

### Acknowledgements

This work is supported in part by the National Natural Science Foundation of China under Grants No. 61727809, No. 61572342, No. 61672154, Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU11268616), Natural Science Foundation of Jiangsu Province under Grant No. BK20151240.

## References

- Albers, S. (2010). Energy-efficient algorithms.
*Communications of the ACM*,*53*(5), 86–96.CrossRefGoogle Scholar - Albers, S., & Antoniadis, A. (2014). Race to idle: New algorithms for speed scaling with a sleep state.
*ACM Transactions on Algorithms (TALG)*,*10*(2), 9.Google Scholar - Albers, S., Antoniadis, A., & Greiner, G. (2015). On multi-processor speed scaling with migration.
*Journal of Computer and System Sciences*,*81*(7), 1194–1209.CrossRefGoogle Scholar - Antoniadis, A., Huang, C. C., & Ott, S. (2015). A fully polynomial-time approximation scheme for speed scaling with sleep state. In
*Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms*(pp. 1102–1113).Google Scholar - Aydin, H., Devadas, V., & Zhu, D. (2006). System-level energy management for periodic real-time tasks. In
*Proceedings of the 27th IEEE real-time systems symposium*(pp. 313–322).Google Scholar - Bambagini, M., Marinoni, M., Aydin, H., & Buttazzo, G. (2016). Energy-aware scheduling for real-time systems: A survey.
*ACM Transactions on Embedded Computing Systems (TECS)*,*15*(1), 7.CrossRefGoogle Scholar - Bansal, N., Bunde, D. P., Chan, H. L., & Pruhs, K. (2008). Average rate speed scaling. In
*Proceedings of the 8th Latin American theoretical informatics symposium, volume 4957 of LNCS*(pp. 240–251).Google Scholar - Bansal, N., Kimbrel, T., & Pruhs, K. (2004). Dynamic speed scaling to manage energy and temperature. In
*Proceedings of the 45th annual symposium on foundations of computer science*(pp. 520–529).Google Scholar - Baptiste, P. (1999). An \(O(n^4)\) algorithm for preemptive scheduling of a single machine to minimize the number of late jobs.
*Operations Research Letters*,*24*(4), 175–180.CrossRefGoogle Scholar - Bini, E., Buttazzo, G., & Lipari, G. (2005). Speed modulation in energy-aware real-time systems. In
*IEEE proceedings of the 17th Euromicro conference on real-time systems*(pp. 3–10).Google Scholar - Choi, K., Soma, R., & Pedram, M. (2005). Fine-grained dynamic voltage and frequency scaling for precise energy and performance trade-off based on the ratio of off-chip access to on-chip computation times.
*IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems*,*24*(1), 18–28.CrossRefGoogle Scholar - Hong, I., Qu, G., Potkonjak, M., & Srivastavas, M. B. (1998). Synthesis techniques for low-power hard real-time systems on variable voltage processors. In
*Proceedings of the IEEE real-time systems symposium*(pp. 178–187).Google Scholar - Hsu, C. H., & Feng, W. C. (2004). Effective dynamic voltage scaling through CPU-boundedness detection.
*In the 4th IEEE/ACM workshop on power-aware computing systems*(pp. 135–149).CrossRefGoogle Scholar - Irani, S., Shukla, S., & Gupta, R. K. (2007). Algorithms for power savings.
*Journal ACM Transactions on Algorithms*,*3*(4), 41.CrossRefGoogle Scholar - Ishihara, T., & Yasuura, H. (1998).
*Voltage scheduling problem for dynamically variable voltage processors*. In*Proceedings. 1998 International Symposium on low power electronics and design, 1998*. IEEE.Google Scholar - Li, M., & Yao, F. (2005). An efficient algorithm for computing optimal discrete voltage schedules.
*SIAM Journal on Computing*,*35*(3), 658–671.CrossRefGoogle Scholar - Seth, K., Anantaraman, A., Mueller, F., & Rotenberg, E. (2003). Fast: Frequency-aware static timing analysis. In
*Proceedings of the 24th IEEE real-time system symposium*(pp. 40–51).Google Scholar - Wu, W., Li, M., & Chen, E. (2009). Min-energy scheduling for aligned jobs in accelerate model. In
*Proceedings of 20th international symposium on algorithms and computation (ISAAC 09)*(pp. 462–472).CrossRefGoogle Scholar - Yang, C. Y., Chen, J. J., & Kuo, T. W. (2007). Preemption control for energy-efficient task scheduling in systems with a DVS processor and non-DVS devices. In
*Proceedings of the 13th IEEE international conference on embedded and real-time computing systems and applications*(pp. 293–300).Google Scholar - Yao, F., Demers, A., & Shenker, S. (1995). A scheduling model for reduced CPU energy. In
*Proceedings of IEEE symposium on foundations of computer science (FOCS)*(pp. 374–382).Google Scholar