SYSTEM-WIDE ENERGY MANAGEMENT
FOR REAL-TIME TASKS: LOWER BOUND
AND APPROXIMATION
Xiliang Zhong and Cheng-Zhong Xu
ICCAD 2006,
ACM Trans. on Embedded Computing Systems, 2008.
Presenter: Mohammad A Haque
MOTIVATION

Energy cost is becoming one of the greatest
concern for Data Centers and large computer
systems.


Introduction
System Model
Algorithm
Simulation Results
Conclusion
The total energy and cooling cost for servers in the USA are
expected to rise to $50 Billion per year by the end of this
year.
Generated heat is proportional to the energy
consumption. Overheating affects device
performance and lifetime.

CS, GMU cluster HYDRA has more than 50 nodes. Only 18
nodes are kept active at the same time due to overheating
problem.
BACKGROUND



Introduction
System Model
Algorithm
Simulation Results
Conclusion
System-wide energy management means the
power model considers energy consumed by
different system components (memory, wireless
interface, …) along with CPU.
In Real-Time Systems, every task has an
associated deadline. Deadline could be either
Hard or Soft.
This paper deals with system-wide energy
management of Periodic and Sporadic (Aperiodic )
tasks in hard real-time system.
PROBLEM SETTINGS



Introduction
System Model
Algorithm
Simulation Results
Conclusion
We are given a set of periodic/sporadic tasks.
The tasks need to be executed on a single
processor machine with discrete speed levels.
Our goal is to find speed assignments for the
tasks, such that total system-level energy
consumption is minimized while meeting the
deadline constraints.
Introduction
System Model
Algorithm
Simulation Results
Conclusion
TASK MODEL
A set of n tasks are scheduled using Earliest
Deadline First (EDF) policy.
 A task is characterized by a tuple {Ci, Ti, Di},
where Ci is the execution time under maximum
speed. Ti and Di are task period and deadline
respectively. Its common to assume Ti = Di.
 A task utilization is defined by the ratio Ci/Ti.
 A well known result in real-time system is, a task
set can be feasibly scheduled using EDF if and
only if the following condition holds.


n
i 1
(Ci / Ti )  1
ENERGY MODEL
Introduction
System Model
Algorithm
Simulation Results
Conclusion
CPU speed can be adjusted in discrete steps in
range [fmin, fmax]. We define slowdown factor, S =
f/fmax.
 At Si, a task takes Ci/Si time to complete and the
new utilization is Ci/(TiSi).
 The system-level energy consumption for a task i
at slowdown factor Si is,
Si
std S i
Ei ( Si )  Pcpu ( Si )   j Pij
 Eiact
i
Ci
Ci



Pijstd  power consumed by j-th device in standby state.
Eiact  energy consumed by all the devices in active state.
PROBLEM FORMULATION
T
i 1 T Ei (Si )
i
Ci
n
1
subject to i 1
Ti Si
minimize
n
S min_ i  Si  1,1  i  n
Introduction
System Model
Algorithm
Simulation Results
Conclusion
REDUCTION TO 0-1 MCKP
Introduction
System Model
Algorithm
Simulation Results
Conclusion
For each task, i and for each speed level, j we
have a corresponding (utilization, energy) pair
denoted by (uij, eij).
 So the problem can be thought as choosing one
pair from each set,
{(u11, e11), (u12, e12), …, (u1m, e1m)},
{(u21, e21), (u22, e22), …, (u2m, e2m)}, …,
{(un1, en1), (un2, en2), …, (unm, enm)}.
to minimize the sum of eij, while keeping the sum
of uij’s below 1.
 0-1 MCKP is NP hard. Therefore, so is our
problem.

Introduction
System Model
Algorithm
Simulation Results
Conclusion
We know pseudo-polynomial time algorithm for
solving 0-1 MCKP problem. But the traditional
algorithms do not work here, due to a limitation of
the traditional 0-1 Knapsack problem solutions.
What is the biggest limitation with those
solutions??
The traditional solutions for 0-1 Knapsack problem
works only with integer coefficients.
OPTIMAL ALGORITHM
Introduction
System Model
Algorithm
Simulation Results
Conclusion
Start with an empty list, L0 = <(0, 0)>.
 For each task i and for each speed level j, perform
component wise addition of the pair (uij, eij) with the
list Li-1.
 Merge the lists and keep it sorted in non-decreasing
order of energy.
 Delete all infeasible solution from the list in each
step.
 Prune the dominated states from the list. A state is
dominated by another state, if it has a larger
utilization and a larger energy consumption than
the other.

EXAMPLE
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Introduction
System Model
Algorithm
Simulation Results
Conclusion
Consider a task set with 2 tasks and 2 speed levels. The
algorithm works as follows.
Initialization:
L0 = <(0,0)>
1st step:
L1,1 = <(u11, e11)>
L1,2 = <(u12, e12)>
L1 = <(u11, e11), (u12, e12)>
2nd step:
L2,1 = <(u11+u21, e11+e21), (u12,+u21, e12+e21), (u11+u21, e11+e21), (u12+u21, e12+e21)>
L2,2 = <(u11+u22, e11+e22), (u12,+u22, e12+e22), (u11+u22, e11+e22), (u12+u22, e12+e22)>
L2 = <(u11+u21, e11+e21), (u12,+u21, e12+e21), (u11+u21, e11+e21), (u12+u21, e12+e21),
(u11+u22, e11+e22), (u12,+u22, e12+e22), (u11+u22, e11+e22), (u12+u22, e12+e22)>
ANALYSIS
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Introduction
System Model
Algorithm
Simulation Results
Conclusion

The runtime of the algorithm is bounded by O(
where U is the upper bound for the number of
elements in the list and mi is the number speed
levels for task i.
n
i 1
Since the size of the list can grow very large, the
runtime of the algorithm is not polynomially
bounded.
The space requirement for the algorithm is O(U).
miU ),
APPROXIMATION ALGORITHM (FPTAS)
Introduction
System Model
Algorithm
Simulation Results
Conclusion
 eij 
 
 

Choose ε and replace eij with

The maximum error we can have is nε.
rE *
 For a given relative error r, we choose,  
n
where, E* is the energy consumption of the
optimal algorithm.

The number of undominated states in the list
reduces to (Emax/Emin - 1)(n/r).
SPORADIC TASKS
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Introduction
System Model
Algorithm
Simulation Results
Conclusion
Sporadic tasks are released at random time instants and
the task parameters are only known after release.
Whenever a task arrives, we perform an admission control
test to determine whether we can accept the task or not.
The problem reduces to Multi-dimensional MCKP (MMKP),
which is NP-hard in the strong sense.
Similar optimal and approximation algorithm works for the
sporadic task as well.
A key assumption is the feasibility of a task does not
depend on the task that finishes later.
SIMULATION RESULT (PERIODIC)
Introduction
System Model
Algorithm
Simulation Results
Conclusion
SIMULATION RESULT (SPORADIC)
Introduction
System Model
Algorithm
Simulation Results
Conclusion
CONCLUSION


Introduction
System Model
Algorithm
Simulation Results
Conclusion
The paper provides pseudo-polynomial optimal
algorithm and FPTAS for both periodic and
sporadic tasks.
The algorithm is simple yet interesting. However
some more simulation results, specially the effect
of number of speed levels and ratio of Emax/Emin
will be worth experimenting.
THANK YOU