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

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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
- 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

- 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)

- 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

- 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.)

- 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

- 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

- 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 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

- 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

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

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

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)

- 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.)

- 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

- 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)

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 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.)

- 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)

- 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)

56%

23%

- Small task number: Energy consumption up to 56% more efficient than TVDVS and DVSST

- Large task number: 23% more efficient

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

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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