(e.g., time 0), and the jitter between sampling time and
release time of a transaction instance is 0. Lastly, the
scheduling algorithms studied in this work are considered
preemptive. Table 1 presents the formal definitions of the
symbols used in this work.
2.3 Problem Statement
Given a set of update transactions, the optimal solution to
the deadline and period assignment problem must mini-
mize the processor workload while maintaining temporal
consistency under EDF. Consequently, the period and
deadline assignment problem for real-time update transac-
tions scheduled under EDF can be described as follows:
Deadline and period assignment problem (DPAP):
Given a set of transactions T¼f
i
g
n
i¼1
with C
i
and V
i
specified
for each
i
, determine T
i
and D
i
for
i
so that the processor
workload UðT Þ is minimized subject to the following constraints:
. Validity constraint: The sum of the period and relative
deadline of transaction
i
is not larger than the validity
interval length V
i
, i.e., T
i
þ D
i
V
i
.
. Schedulability constraint: The transaction set T must
be schedulable under EDF with derived deadlines fD
i
g
n
i¼1
and periods fT
i
g
n
i¼1
.
With EDF scheduling, Baruah et al. proposed an exact,
albeit complex, schedulability test [2], which is later
improved in [4], [32].
Theorem 1 [4], [32]. Given a periodic task set T¼f
i
g
n
i¼1
, T is
schedulable by EDF if and only if U1 and 8t 2 S,
1
hðt; nÞ
1
¼
X
n
j¼1
t D
j
T
j
þ 1
C
j
t; ð1Þ
where S ¼fd
k
jd
k
¼ kT
i
þ D
i
^ d
k
minðL
n
a
;L
n
b
Þ
1
;k2 IN g,
L
n
a
¼ max D
1
; ...;D
n
;
P
n
i¼1
T
i
D
i
ðÞU
i
1 U
ð2Þ
and L
n
b
denotes the synchronous busy period (the length of the
first processor busy period of the synchronous arrival pattern
described in Definition 2 below).
Definition 2 [27]. A synchronous busy period is a processor
busy period in which all tasks are released simultaneously at
the beginning of the processor busy period and ended by the
first processor idle period (the length of such a period can be 0).
By using the notations given in Theorem 1 and the
theorem itself, the Deadline and Period Assignment Problem
can be rephrased as follows:
DPAP. Given a set of transactions T¼f
i
g
n
i¼1
with C
i
and
V
i
specified for each
i
, determine T
i
and D
i
for each
i
in T such
that UðT Þ ¼
P
n
i¼1
C
i
T
i
is minimized subject to
. Validity constraint: T
i
þ D
i
V
i
.
. Schedulability constraint: 8t 2 S; hðt; nÞt.
3GENERAL ALGORITHM FOR THE ASSIGNMENT
PROBLEM
In this section, we first present our general algorithm GE
EDF
by describing Phases 1 (denoted by GE
1
EDF
) and 2 (denoted by
GE
2
EDF
) in Sections 3.1 and 3.2, respectively, and then discuss
the relationship between Phases 1 and 2 in Section 3.3.
3.1 Phase 1: Finding a Solution in Linear Time
One might be tempted to solve DPAP directly. Solving this
constrained optimization problem, however, can be extre-
mely expensive, since verifying the constraint in (1) for all
t values requires high computation complexity. In this
phase, instead of tackling the assignment problem directly,
we approach it circuitously, based on a theorem discussed
shortly. We first present a useful lemma, which states a
necessary condition for any task set to be schedulable under
EDF [5].
Lemma 1. Given a synchronous task set T , let the tasks in T be
sorted in nondecreasing order of deadlines (D
i
) and suppose
that the minimum task deadline, D
min
, is unique. Regardless
of the choices of periods, any task set that is schedulable under
EDF must satisfy the following property:
X
j
i¼1
C
i
D
j
; 8j ¼ 1; ...;n: ð3Þ
To minimize the processor workload, T
i
should be
maximized, which in turn means D
i
should be minimized,
due to the validity constraint (T
i
þ D
i
V
i
). Based on
Lemma 1, it is clear that we should set D
i
¼
P
i
j¼1
C
j
and
T
i
¼V
i
D
i
, which are assumed throughout Section 3.1.
But notice that setting deadlines and periods this way does
not necessarily guarantee schedulability. The following
theorem characterizes a condition under which schedul-
ability is guaranteed, which lays the foundation for GE
1
EDF
’s
assignment of deadlines and periods.
Theorem 2. Given an update transaction set T¼f
i
g
n
i¼1
with
deadlines derived by D
i
¼
P
i
j¼1
C
j
and periods by T
i
¼
V
i
D
i
, if the maximum deadline, i.e., D
max
¼
P
n
j¼1
C
j
,is
not larger than any period in T , then T is guaranteed to be
schedulable under EDF.
Proof. Since D
max
T
k
ð1 k nÞ,weknowforeach
transaction
i
ð1 i nÞ, there is D
i
T
k
ð1 k nÞ;
hence, it is obvious that
LI ET AL.: WORKLOAD-EFFICIENT DEADLINE AND PERIOD ASSIGNMENT FOR MAINTAINING TEMPORAL CONSISTENCY UNDER EDF 1257
TABLE 1
Symbols and Definitions
1. Note that we use hðt; nÞ, rather than h ðtÞ as in [4], for the processor
workload since the notation is used both here and later in Section 3.2.2
where we must consider different subsets of T . For the same reason, we use
L
n
a
and L
n
b
here, as opposed to L
a
and L
b
in [4].