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System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation. Xiliang Zhong and Cheng-Zhong Xu Dept. of Electrical & Computer Engg. Wayne State University Detroit, Michigan http://www.cic.eng.wayne.edu. Outline. Introduction Processor and system energy model

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system wide energy minimization for real time tasks lower bound and approximation

System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation

Xiliang Zhong and Cheng-Zhong Xu

Dept. of Electrical & Computer Engg.

Wayne State University

Detroit, Michigan

http://www.cic.eng.wayne.edu

outline
Outline
  • Introduction
    • Processor and system energy model
  • Related Work
  • System-Wide Energy Optimization for periodic tasks
    • The optimal algorithm
    • A fully polynomial time approximation scheme
    • Performance Evaluation
  • System-Wide Energy Optimization for sporadic Tasks
    • Solution and evaluation
  • Conclusions
introduction
Introduction
  • Mobile/Embedded devices are power critical, with limited battery capacity
  • Software assisted power management
    • Dynamic power management (DPM)
      • Resource shutdown after a timeout
    • Dynamic voltage/frequency scaling (DVS)
      • Processing speed designed for peak performance
      • Slowdown the processor voltage / speed when not fully utilized
dynamic voltage scaling dvs
Dynamic voltage scaling (DVS)
  • The dynamic CPU power is , P ∝ v2f
  • Reducing v also reduce the maximum processors frequency
  • Approximately, energy per cycle∝ f2

Energy per cycle of PXA processor

  • Processor slowdown leads to super-linear energy

savings, while linear execution time increase

system wide energy
System-Wide Energy
  • Processor also has leakage power
  • Applications may use other components such as memory and peripheral devices
    • Can be in active, standby, sleep, and shutdown states
  • System-wide energy consumed in running a task
    • CPU, resource standby and active energy
  • Lowering CPU frequency can increase overall energy expenditure due to prolonged resource standby time of other components
system wide energy cont
System-Wide Energy (cont.)
  • critical speed,thespeed with minimum energy per cycle
    • Not energy efficient using lower speed

Energy per cycle of PXA processor with

different standby power

  • Execute a task at speed no lower than its critical speed, then put the devices into low power state
    • A combined use of slowdown and shutdown
related work
Related Work
  • CPU energy minimization for periodic tasks:
      • Heuristics [Mejia-Alvarez’04], approximations [Chen and Kuo’05]
  • Few studies on system-wide energy minimization
    • Applications w/o deadlines
      • Subject to a performance loss [Choi et al.’04]
    • Real-time periodic tasks on CPU w/ continuous speed levels
      • Heuristics [Zhuo and Chakrabarti’05]
    • Real-time periodic tasks on CPU w/ discrete speed levels
      • Heuristics [Jejurikar and Gupta’04]
  • This work
    • Pseudo-polynomial algorithm for optimal solutions and polynomial approximated schemes
    • Applicable to both offline periodic tasks and online sporadic tasks in processors with practical discrete levels
system wide energy optimization
System-wide energy optimization
  • Periodic Tasks (Offline)
    • : worst case execution time under max speed
    • : task period and deadline
    • : normalized speed of task
  • Sporadic Tasks (Online)
    • Task releases have irregular intervals
    • Online scheduling based on uncompleted tasks, no assumption about future task releases
  • The objective is to minimize
    • overall energy consumption including CPU and all other system components while meeting deadline constraints of all the tasks
energy minimization for periodic tasks
Energy Minimization for PeriodicTasks
  • Minimization of energy consumption for n periodic tasks in a hyper-period,

Feasible constraint under EDF

Boundary constraint

  • Practical processors with discrete speed levels
    • The minimization is an NP-hard Multiple Choice KnapSack (MCKP) problem
    • There exist pseudo-polynomial solutions to MCKP with integer coefficients, not applicable in this problem
an example
An Example
  • Basic idea: first solve subprobs with fewer #tasks
  • A system with an PXA processor with 5 normalized speed [0.15 0.4 0.6 0.8 1]
  • System with memory, flash, and WNIC
  • An example real-time workload w/ 4 periodic tasks
solution to task 1

f: pruned by feasibility condtion

e: pruned by energy condition

(utilization, energy)

Solution to task 1
  • Task 1, execution time 6.4; deadline 16; utilization 0.4
  • Branch on four normalized speeds [0.4 0.6 0.8 1]
  • State pruning
    • Feasibility condition:
      • The 1st node at speed 0.4 removed with utilization already 1
    • Energy condition
      • Task 1 at the smallest speed (2nd , 0.6); tasks 2-4 at the max. Total Energy=7.6 (upper bound)
      • Task 1 at 3rd or 4th speed (0.8 or 1); tasks 2-4 at the min. The required energy exceeds 7.6. The two states can be removed
solution to the first three tasks
Solution to the first three tasks

pairs of (utilization, energy)

f: pruned by feasibility condtion

e: pruned by energy condition

d: pruned by dominance

  • Dominance condition
    • The states (0.867, 9.107) and (0.87, 9.4) of task 3
      • First one leads to smaller utilization
      • Any feasible schedule by the second can also be satisfied by the first
      • First one uses less energy; the second can be removed
slide13

(utilization, energy)

f: pruned by feasibility condtion

e: pruned by energy condition

d: pruned by dominance

optimal state

Maximum state number reduced to 6/4*4*3*3 = 0.4 %

a fully polynomial approximation scheme fptas
A fully polynomial approximation scheme (FPTAS)
  • State # is pseudo-polynomial in task number.
    • can be reduced by providing approximated solutions
  • Approximated with worst case perf. guarantee
    • An algorithm is said to be an approximation scheme if for a given in (0,1), we have
  • A more desirable approximation scheme (FPTAS) has a polynomial running time in both the number of tasks and the performance ratio
a fully polynomial approximation scheme cont
A fully polynomial approximation scheme (cont.)
  • Divide the energy values into a number of groups each of size r,
    • Each value scaled and rounded to
    • Energy values in the same group are treated equally
  • Find the group size r, subject to a given performance bound
    • Energy value of each task introduces an error no larger than group size r
    • Accumulated errors of n tasks no larger than n*r
    • A lower bound of E* is when all tasks run at their critical speeds (Emin), i.e., E*≥ Emin

Solving

derives group size r

performance evaluation
Performance Evaluation
  • Simulation Settings
    • A system with an PXA processor
    • memory: standby power 0.2W, standby time 20%~60% of task execution
    • flash drive: 0.4W and 10%~25%
    • wireless interface: 1W and 5%~20%
  • Periodic Tasks
    • Randomly generated deadlines w/ utilization from 0.1~1
    • Each task randomly chooses a subset of resources
    • Algorithms implemented
      • CPU-DVS, speed control for CPU energy consumption
      • CS-DVS, a heuristic algorithm for system-wide energy savings [Jejurikar and Gupta ISLPED2004],
      • OPT-P, the proposed optimal solution
      • Approximated scheme with perf. bounds 0.01, 0.1, 0.5
performance evaluation periodic tasks
Performance Evaluation (Periodic tasks)

23%

16%

8%

  • Proposed algorithms 23% less energy than CPU-only solutions
  • Energy consumption up to 16% more efficient than CS-DVS
  • Approximation algorithms effectively bound the performance errors
energy minimization for sporadic tasks
Energy Minimization for SporadicTasks
  • Online energy minimization for all uncompleted tasks

n feasible constraints under EDF

boundary constraint

  • On a processor with discrete speed levels
    • Prove the problem is an instance of Multi-dimensional MCKP (NP-hard in the strong sense, any optimal solution has exponential running time)
sporadic tasks cont
Sporadic Tasks (cont.)
  • Consider three tasks released at time 0 with deadlines 3, 5, 7
  • Feasibility of a task (e.g. J2) is not affected by tasks finished later (tasks in a non-decreasing order of deadlines)
  • Satisfy one constraint (e.g. J3) at each iteration
  • Can be solved by a pseudo-polynomial algorithm for the optimal solution and an approximation scheme (FPTAS)
performance evaluation sporadic tasks
Performance Evaluation (Sporadic tasks)
  • Experimental Settings
    • Varied number of tasks
    • Task inter-release times generated by an exponential dist.
    • Algorithms implemented
      • TV-DVS, adaptive speed scaling for CPU energy consumption on processors w/ continuous levels [Zhong and Xu RTSS2005]
      • DVSST, CPU energy consumption with only frequency scaling available (continuous levels) [Qadi et al. RTSS2003]
      • OPT-S, the proposed optimal solution
      • 0.1, 0.5-approximation, approximated solutions with different performance settings
energy consumption sporadic tasks
Energy consumption (Sporadic tasks)

56%

23%

  • Small task number: Energy consumption up to 56% more efficient than TVDVS and DVSST
  • Large task number: 23% more efficient
conclusion
Conclusion
  • System-wide energy minimization for periodictasks
    • pseudo-polynomial algorithm for the optimal solution
    • approximated solution in moderate running time with bounded performance degradation (FPTAS)
  • Minimization for onlinesporadic tasks
    • Pseudo-polynomial algorithm and an FPTAS by exploiting inherent properties of online task scheduling
  • On-going work
    • Implementation of the policies in an embedded system with PXA270 processor
    • Energy/Time overhead voltage and speed switches; overhead in putting a resource into low power state
thank you
Thank you!

System-Wide Energy Minimization for Real-Time Tasks:

Lower Bound and Approximation

algorithm running time
Algorithm running time
  • Running time measured in a Pentium 4 machine with 2 GHz processor
  • OPT-P has a higher complexity than CS-DVS
  • Below 90 ms for systems with up to 50 tasks
  • All approximation algorithms require no more than 0.4 s to finish
  • Algorithm running time for schedules in a 10-minutes run
  • OPT-S has higher running time, but <1% task execution time
  • Comparable time for approximation algorithms with TV-DVS
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