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information: DOI 10.1109/TC.2015.2485229, IEEE Transactions on Computers
IEEE TRANSACTIONS ON COMPUTERS, VOL. X, NO. X, JUNE 2015 3
done on the semi-partitioned scheme which assigns
statically most tasks to one fixed core as in partitioned
scheduling, while a few number of tasks are split
into several subtasks, which are assigned to different
cores. Thus a scheduling algorithm for a multi-core
system in this paper represents a pair of single-core
scheduling algorithm and allocation algorithm. An
example for single-core algorithm is the well-known
RM algorithm or EDF algorithm [19], while FF (First
Fit) or WF (Worst Fit) algorithm can be considered as
examples of allocation algorithm.
Power Consumption Model. Normally the power
consumption of the core is represented through the
speed (frequency) and the voltage of the core. The
core’s power consumption function P consists of 2
parts: a static part existing even when there is no
workload (it’s due to the leakage) and a dynamic part
which is related to the frequency [20] .
P = DV
2
dd
s + P
s
(1)
In the function 1, P
s
is the static part, and s =
k
(V
dd
−V
t
)
2
V
dd
. D, V
t
, V
dd
and k denote respectively the
effective switch capacitance, the threshold voltage,
the supply voltage, and a hardware-design-specific
constant. Note that V
dd
≥ V
t
≥ 0, k > 0, and
D > 0, V
dd
is usually proportional to the speed s. Thus
in the following analysis, we make the simplifying
assumption that the dynamic part scales by a factor of
s
3
, and P
s
is a constant. This simplification is justified
by the close match between the data sheet curves
of real DVS processors and the analytical curves [6].
Actually we can have:
P (s) = Cs
3
+ P
s
(2)
where C and P
s
are constants.
The total power of a multi-core processor is simply
a sum of the power dissipated in each core: P
tot
=
P
P
i
(s). The execution requirement c
i
executed in a
time interval is linear of the core’s speed, and the
energy consumed for a core to execute a task at the
core’s speed s for t time units is t · P (s). Suppose a
task set is assigned to n cores in an identical multi-
core system, and there is not speed change during
time [t
0
, t
1
]. The total energy consumed in this time
interval for the task set is: E
tot
=
P
n
i=1
P
i
(s) ·(t
1
− t
0
).
3 NODE SCALING MODEL STRUCTURE
Today’s multicores are complex systems of cores,
caches, interconnects, memory controllers, multiple-
domain clocking, and other components. The power
consumption of each part in a multi-core processor
must be measured to precisely estimate cores’ power
efficiency. In order to model most of the existing
multi-core processors, Sniper [21] and McPAT [22]
are used to simulate the power consumption of pro-
cessors varying with the processor’s frequency. As a
next generation parallel, high-speed and accurate x86
simulator, Sniper simulator is based on the interval
core model and the Graphite simulation infrastruc-
ture, allowing for fast and accurate simulation when
exploring different homogeneous and heterogeneous
multi-core architectures. Sniper integrates with Mc-
PAT which is a power and area modeling framework
to estimate the program’s and processor’s power
consumption. In this article, the realistic Nehalem
systems are modeled and simulated to act as the target
multi-core system.
For a given real-time task set τ, an existing schedul-
ing algorithm should provide a feasible mapping for
τ on an identical multi-core system π, including the
number of required cores, the core’s speed (normal
equals to s
max
) and the assignment of tasks in each
core. This mapping is defined as the initial state IT
π
=
(m, s), which includes the number of required cores
m and the core’s speed s. Then we extend the number
of cores to m
0
, m
0
≥ m, and reduce the speed of cores
to s
0
in the condition of keeping the schedulability of
the task set τ. The extension results in the extended
state ET
π
= (m
0
, s
0
). Trying to maximize the energy
consumption disparity between the initial state and
the extended state, we can find the suitable m
0
and s
0
which result in the minimal energy consumption for
the given task set τ .
The structure of Node Scaling model is shown in
Fig. 1. We suppose the cores of a multi-core processor
connected by a network with a 2-dimension torus
topology. For a real-time task set τ , the initial state is
represented as the four dark gray cores in the dotted
line rectangle. Then Node Scaling model is used to
compute the extended core number and the reduced
speed. Thus Node Scaling scheduler reserves more
cores (the dark cores in the dashed line rectangle) and
reduces the core’s speed to slow down the system.
Therefore, the conditions to ensure the feasibility of
a task set in the case of scaling the number of cores
and reducing the core’s speed, and the extended state
which results in a maximal system energy saving are
two key problems need to be solved in Node Scaling
model.
4 SCHEDULABILITY TEST
In Node Scaling model, a real-time task set is sup-
posed to execute on the initial state(IT
π
) and the
extended state(ET
π
) in order to maximize the energy
consumption disparity between the two states. We
must find the conditions which ensure all of tasks
can be accomplished before their deadlines without
missing the timing constraints when we extend IT
π
to ET
π
.
Funk et al. [23] studied the problem of exact test
for determining whether a given periodic task set is
feasible in an uniform multiprocessor platform. Since
an identical multi-core system can be considered as
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