## Dynamic speed scaling to manage energy and temperature (2004)

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Venue: | In IEEE Syposium on Foundations of Computer Science |

Citations: | 127 - 14 self |

### BibTeX

@INPROCEEDINGS{Bansal04dynamicspeed,

author = {Nikhil Bansal},

title = {Dynamic speed scaling to manage energy and temperature},

booktitle = {In IEEE Syposium on Foundations of Computer Science},

year = {2004},

pages = {520--529}

}

### Years of Citing Articles

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### Abstract

We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed ¡ is ¢¤ £. We provide a tight bound on the competitive ratio of the previously pro-posed Optimal Available algorithm. This improves the best known competitive ratio by a factor � � of. We then introduce a new online algorithm, and show that this algorithm’s competitive ratio is at � £ �� � £ �¨����¥�¥����� � most. This competitive ratio is significantly better and is � ������� approximately for large �. Our result is essentially tight for large �. In particular, as � approaches infinity, we show that any algorithm must have competitive ratio �� � (up to lower order terms). We then turn to the problem of dynamic speed scaling to minimize the maximum temperature that the device ever reaches, again subject to the constraint that all jobs finish by their deadlines. We assume that the device cools according to Fourier’s law. We show how to solve this problem in polynomial time, within any error bound, using the Ellipsoid algorithm. 1.

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Citation Context ...is given by £ £�� ¥���� � ��� £�� ¥���� � ��� ¥�����¥������ . £�� 1.2. Prior Research £�� ¥�§�� ¢�£�� ¥¦��� � £�� ¥ , Theoretical worst-case investigation of speed scaling algorithms was initiated in =-=[17]-=-. The problem is to schedule a given collection of tasks, where each � task has a release ��� time when it arrives into the system, an amount of work � that must be performed to complete the task, and... |

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Citation Context ...5260 kirk@cs.pitt.edu Tracy Kimbrel IBM T. J. Watson Research Center P.O. Box 218 Yorktown Heights, NY, 10598, USA kimbrel@us.ibm.com power densities in microprocessors have doubled every three years =-=[13]-=-. Limiting power requirements is a critical issue for two reasons: Energy: The energy used by a computing device is the � integral over time of power. This is particularly a problem in mobile devices ... |

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Citation Context ...ers [16]. Current estimates are that cooling solutions are rising at $1 to $3 per watt of heat dissipated [13]. Power is now recognized as a first-class design constraint for modern computing devices =-=[9]-=-. There is an extensive literature on power management in computing devices. Overviews can be found in [2, 9, 15]. Both in academic research and practice, dynamic voltage/frequency/speed scaling is th... |

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Citation Context ...[13]. Power is now recognized as a first-class design constraint for modern computing devices [9]. There is an extensive literature on power management in computing devices. Overviews can be found in =-=[2, 9, 15]-=-. Both in academic research and practice, dynamic voltage/frequency/speed scaling is the dominant technique for power management. Speed scaling involves dynamicallyschanging the speed of the processor... |

100 | Algorithms for power savings
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Citation Context ...de any upper bound analysis of OA. [17] state a lower bound of �¦����� on the competitiveness of any online algorithm (presumably for the ��§ case ). This lower bound instance consists of two jobs. � =-=[8]-=- studies online speed scaling algorithms to minimizesenergy usage in a setting where the device also has a sleep state, and gives an offline polynomial-time 2-approximation algorithm. [8] also gives a... |

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Citation Context ...of a video or other multimedia presentation, there may be natural deadlines for the various tasks imposed by the application. In other settings, the system may impose deadlines to better manage tasks =-=[3]-=-. A schedule specifies which task to run at each time, and at what speed that task should be run. The schedule must be feasible with respect to the deadlines; that is, each task must finish by its dea... |

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Citation Context ...[13]. Power is now recognized as a first-class design constraint for modern computing devices [9]. There is an extensive literature on power management in computing devices. Overviews can be found in =-=[2, 9, 15]-=-. Both in academic research and practice, dynamic voltage/frequency/speed scaling is the dominant technique for power management. Speed scaling involves dynamicallyschanging the speed of the processor... |

53 | Getting the best response for your erg
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Citation Context ...root rule holds, using the analysis of AVR from [17], one obtains an algorithm with competitive ratio of at most 653. These results have been extended to the case of multiple slow-down states in [1]. =-=[11]-=- give an efficient algorithm for the problem of minimizing the average flow time of a collection of dynamically released equi-work processes subject to the constraint that a fixed amount of energy is ... |

38 | Optimal power-down strategies
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Citation Context ...cube-root rule holds, using the analysis of AVR from [17], one obtains an algorithm with competitive ratio of at most 653. These results have been extended to the case of multiple slow-down states in =-=[1]-=-. [11] give an efficient algorithm for the problem of minimizing the average flow time of a collection of dynamically released equi-work processes subject to the constraint that a fixed amount of ener... |

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Citation Context ...�� £�� ¥�� and ��� � � � � � Note � £�� ¥ that is non-zero only for � . Thus let us ��� first ��� � ��� � £�� ¥�� consider . We use the following fact that was first proved by Hardy and Littlewood in =-=[6]-=- and later simplified by Gabriel [4]. It can also be found in [7, Theorem 393 and 394]. Fact: � £�� ¥ If are arbitrary non-negative £�� integers, be the maximum over all � , such that � �©��� ¥ � � le... |

15 |
Variational Methods in Optimization
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Citation Context ...ed subject to the constraints that ¢ £�� ��¥�§ ��� � ��� £ � ¥�§ � � � ��� � , and . Here is the derivative of with respect to time � � . This problem falls under the rubric of calculus of variations =-=[14]-=-. Let be the functional £�£ ��� ��� � ¥�����¥������ � � ������ § £ � ��� � ��� � ¥�������� � ����� ��§ � £ � ��� ��� � � ¥�������� � � � � � ��� � ��� � � � ��§�� � � � � . Let be the partial of with ... |

13 |
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Citation Context ...at is maximized if for � all , ����¥ . In this case, � � ¥�� � . Thus, � £ £ � ¥�� � � � ��� � £�� ¥�� ����� � ����� � ��� ��� � � � � ¥�§ � £ � � £ � ¥�� ��� � ��� �� � � � £ � , . Hardy then showed =-=[5]-=- as a special case of Hilbert’s theorem (see [7, Theorem 326]) that ��� � ��� � � Note that in these inequalities the � � � � ��� � ¥�� ��� � £ � £ � £ ����� ¥�� � � � £�� ¥�� . and all that is requir... |

7 |
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Citation Context ...perature is zero. A first order approximation for rate of ��� £�� ¥ change of £�� the over time � is then ��� ¥ temperature � where ¢¤£�� is the supplied power at time ¥ � , and � � and are constants =-=[12]-=-. Equivalently, the power is given ¢¤£�� by £ £�� ¥�¥���� . Thus, if the cube root rule ap¥�§ plies, then the instantaneous speed of the processor is given by £ £�� ¥���� � ��� £�� ¥���� � ��� ¥�����¥... |

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Algorithms for Power Savings
- Gupta, Shukla
- 2003
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Citation Context ...provide any upper bound analysis of OA. [17] state a lower bound of son the competitiveness of any online algorithm (presumably for the case ). This lower bound instance consists of two jobs. =-=[8]-=- studies online speed scaling algorithms to minimize Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04)s0272-5428/04 $20.00 © 2004 IEEEsenergy usage in a settin... |

2 |
An additional proof of a maximal theorem of hardy and littlewood
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Citation Context ...£�� ¥ that is non-zero only for � . Thus let us ��� first ��� � ��� � £�� ¥�� consider . We use the following fact that was first proved by Hardy and Littlewood in [6] and later simplified by Gabriel =-=[4]-=-. It can also be found in [7, Theorem 393 and 394]. Fact: � £�� ¥ If are arbitrary non-negative £�� integers, be the maximum over all � , such that � �©��� ¥ � � let � ��� � £ � of £ ��¥ ���¦����� � ¥... |

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Introductory lectures on convex programming
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Citation Context ...to arbitrary precision in polynomial time (ignoring numerical stability issues) using the Ellipsoid algorithm. For basic information on the use of the Ellipsoid algorithm to solve convex problems see =-=[10]-=-. We assume a constant that is the thermal threshold for the device. The problem is then to determine if there is a schedule that is feasible, and maintains the invariant that the temperature stays be... |