Real-time scheduling of periodic tasks with processing times and deadlines as parametric fuzzy numbers

Abstract

Task scheduling is very important in real-time systems as it accomplishes the crucial goal of devising a feasible schedule of the tasks. However, the uncertainty associated with the timing constrains of the real-time tasks makes the scheduling problem difficult to formulate. This motivates the use of fuzzy numbers to model task deadlines and completion times. In this paper a method for intuitively defining smooth membership functions (MFs) for deadlines and execution times has been proposed using mixed cubic-exponential Hermite interpolation parametric curves. The effect of changes in parameterized MFs on the task schedulability and task priorities are also reported. A new technique is proposed based on the concept of dynamic slack calculation to make the existing model more practical and realistic. Examples are given to demonstrate the more satisfactory performance of the new technique.

Full text

Real-time scheduling of periodic tasks with processing times and deadlines as
parametric fuzzy numbers
Pranab K. Muhuri
a,
*, K.K. Shukla
b
a
Department of Computer Science and Engineering, University of Science and Technology Chittagong (USTC), Foy’s Lake, Chittagong-4202, Bangladesh
b
Department of Computer Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221005, UP, India
1. Introduction
Real-time systems are those computing systems for which a
timely response to the external stimuli within a specified time
frame is a must [1]. Examples of real-time systems occur in
Telecommunication Systems, Defense Systems, Aircraft Flight
Control systems, Air Traffic Control, Space Stations, Nuclear Power
Plants etc. Primary issues in Real-Time Systems are Scheduling,
Resource Allocation and Communication between components
and subsystems etc. Computing applications of real-time systems
i.e. time critical systems, require satisfying explicit timing
constraints of the tasks and hence these timing constraints play
the most important role for the purposeful and safe operation of
real-time systems. These timing constraints may be hard or soft.
Therefore, real-time systems can be classified into hard real-time
systems and soft real-time systems. Computing systems with
hard timing constraints are known as hard real-time systems,
whereas those with soft timing constraints are known as soft real-
time systems. Three main measures that count the merits of real-
time systems are: predictably very rapid response to urgent
events, high degree of schedulability (i.e. surety of feasible
schedule) and stability during transient overload [2].Thegoalof
scheduling real-time tasks in a time critical system is to devise a
feasible schedule, subject to a given set of tasks and task
characteristics, timing constraints, resource constraints, prece-
dence constraints etc. Therefore, task scheduling plays a vital role
in real-time systems. Task scheduling in real-time systems can be
clock driven, weighted round robin and most importantly
priority-driven [4]. In clock driven systems, scheduling decisions
on a particular job are taken only at some specific time. Based on
basic round robin approach, in the weighted round robin
scheduling different weights are given to different tasks instead
of giving all the ready tasks equal weight for processing. In
priority-driven scheduling tasks are prioritized on the basis of
various parameters. Based on these approaches, numerous
scheduling algorithms were proposed, which aim to ensure that
tasks comply with their deadlines [1,3–5]. Few widely studied and
used priority-driven real-time scheduling algorithms are Rate
Monotonic (RM), Earliest Deadline First (EDF), Highest Criticality
First (HCF), Criticality Deadline First (CDF), Least Laxity First (LLF),
First In First Out (FIFO), Last In First Out (LIFO), Shortest Execution
Time First (SETF) etc. [1–6].
The timing constraints of a particular set of tasks comprises of
task deadlines, processing times, task arrival or release times,
intervals between subsequent invocations of tasks i.e. period etc.
The release time of task is the time before which it can not start
execution, whereas the time within which a task must be
completed after it is released is known as the relative deadline of
that particular task. Release time plus the relative deadline gives
Applied Soft Computing 9 (2009) 936–946
ARTICLE INFO
Article history:
Received 14 May 2007
Received in revised form 29 October 2008
Accepted 15 November 2008
Available online 25 November 2008
Keywords:
Fuzzy deadlines
Fuzzy processing times
Parametric representation
Real-time systems
Task scheduling
ABSTRACT
Task scheduling is very important in real-time systems as it accomplishes the crucial goal of devising a
feasible schedule of the tasks. However, the uncertainty associated with the timing constrains of the real-
time tasks makes the scheduling problem difficult to formulate. This motiva tes the use of fuzzy numbers
to model task deadlines and completion times. In this paper a method for intuitively defining smooth
membership functions (MFs) for deadlines and execution times has been proposed using mixed cubic-
exponential Hermite interpolation parametric curves. The effect of changes in parameterized MFs on the
task schedulability and task priorities are also reported. A new technique is proposed based on the
concept of dynamic slack calculation to make the existing model more practical and realistic. Examples
are given to demonstrate the more satisfactory performance of the new technique.
ß2008 Elsevier B.V. All rights reserved.
* Corresponding author.
E-mail addresses: pranabmuhuri@gmail.com (P.K. Muhuri),
kkshukla.cse@itbhu.ac.in (K.K. Shukla).
Contents lists available at ScienceDirect
Applied Soft Computing
journal homepage: www.elsevier.com/locate/asoc
1568-4946/$ – see front matter ß2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.asoc.2008.11.004

the absolute deadline of a task. The time required by the processor
to execute a particular task is known as the processing time or
execution time. Tasks may be periodic, aperiodic or sporadic. When
tasks are released periodically they are periodic tasks. An invocation
of sporadic tasks happens in irregular intervals rather than
periodically whereas aperiodic tasks are not periodic nor carry
any bound on invocation rate [1]. Only an approximate idea of the
real-time tasks and their characteristics are available in the early
phase of real-time system design. As a result, uncertainty or
impreciseness is associated with the timing requirements of real-
time tasks. To address this uncertainty in the timing constraints of
tasks in Real-Time Systems several modeling techniques con-
sidering uncertainty (viz. probability theory, fuzzy set theory etc.)
have been proposed [7]. These techniques provide more realistic
timing analysis in comparison to simple crisp timing constraints.
Among them fuzzy mathematics edges past probabilistic theory in
so many ways as computations are simpler here and expert help
can be taken easily for task modeling. Moreover, it is faster in
computations and provides more flexibility in modeling. Because
without any significant addition in complexity we can choose from
a wide variety of fuzzy membership types for a particular timing
parameter [8–10].
The scheduling of real-time tasks having fuzzy constraints
can be termed as fuzzy real-time scheduling. In [7,11–15] fuzzy
approaches were introduced for various real-time scheduling
algorithms and different models for real-time systems were
proposed. Fuzzy due dates were considered first by Ishii et al.
[15] for a general scheduling problem. An attempt has been
made by Terrier and Chen [16] to apply fuzzy calculus in real-
time task scheduling. They claimed to have considered the task
execution time as fuzzy for the first time. In [17] Lee et al.
proposed a model for fuzzy rule based scheduler for scheduling
real-time tasks. To solve a deterministic type job-shop
scheduling problem, approximate reasoning approach was
introduced by Turksen et al. [18,19].Theyhaveconsidereda
hierarchical scheduling model of five levels using fuzzy rules to
solve the problem. For the formulation of multi-objective fuzzy
scheduling problems, Murata et al. [14] considered the
importance of individual tasks with OWA (ordered weighted
averaging) operator. Most excellent work on fuzzy real-time
scheduling was done by Litoiu et al. [11–13,20] in their proposed
pessimistic model. They have introduced a cost function viz.
satisfaction of schedulability, used before by Ishii et al. [15],and
evaluated the satisfaction of each individual job having fuzzy
deadlines and processing times. Then they have formulated the
problems as the maximization of the minimum satisfaction.
Muhuri and Shukla [24] showed that if the completion time
varies then the optimal tasks schedule changes at different
points for different membership functions of the fuzzy task
deadlines. They also showed that asymmetry in the membership
functions captures the importance of various tasks and the
priorities can be modified naturally based on the importance of
the tasks.
This paper extends the model proposed by Litoiu and Tadei [12]
for general fuzzy numbers in parametric representation. This gives
a wider choice to the designer. Mathematical models for this are
developed from the first principles and used in several test cases
using the Cheddar Real-time Simulator [21]. The effects of the
choice of different types of membership functions and their fuzzy
parameter are studied and simulations results are reported.
Litoiu’s model with triangular membership function appears as
a special case of our parametric model. Again, the technique
followed by Litoiu and Tadei [12] to calculate the minimum
satisfaction is too much pessimistic. To overcome this we propose
to utilize the dynamic slack calculation technique [22] for the first
time in fuzzy real-time scheduling. A real-life example is
considered and discussed in detail to demonstrate the applicability
of the new technique. The rest of the paper is organized as follows.
Section 2introduces the terminology and formulation of the fuzzy
real-time scheduling problem. Section 3gives mathematical
foundations of parametric membership functions including
detailed analysis of deadlines and completion time as parametric
fuzzy numbers. The Fuzzy RTS model with parametric MFs is
reported in Section 4. Finally conclusions are given in Section 5.
2. Fuzzy Real-Time Scheduling
In Real-Time Scheduling (RTS) using Fuzzy Membership
Functions (MFs) to represent deadline and execution times, there
is a need to specify the membership conveniently by the schedule
designer. In the past, simple MFs like Triangular, Trapezoidal have
been used. Although these MFs are simple to understand and
analyze, yet, they contain discontinuities and restrict the designer
in specifying task characteristics. In this paper a method for
intuitively defining smooth MFs for deadlines and execution times
has been proposed using mixed cubic-exponential Hermite
interpolation parametric curves. This method is very versatile
and can model all kind of symmetric and asymmetric MFs, where
Triangular and Trapezoidal MFs appear as special cases. The
usefulness of the method has been demonstrated using simple
numerical examples. The effect of changes in parameterized MFs
on the Satisfaction of Schedulability and task priorities has also
been reported. In the present work, we demonstrate our
techniques of fuzzy scheduling based on Earliest Deadline First
(EDF) policy, where deadlines and execution times are fuzzy
numbers in parametric form.
A real-time system may be considered to have a general model
as per following:
T¼fT
i
ji¼1;2;3;...;ng
be a set of ntasks such that
(1)
Execution time or processing time of task T
i
is e
i
(2)
Period of task T
i
is P
i
(if the tasks under consideration are
periodic)
(3)
Relative deadline of task T
i
is d
i
(4)
Absolute deadline of task T
i
is D
i
(5)
Release time of task T
i
is r
i
(here we have considered set of tasks
with r
i
as zero)
(6)
Phasing of task T
i
is I
i
, i.e., period of the j-th invocation of task T
i
begins at time I
ij
=I
i
+(j1)P
i
and completes by d
ij
=d
i
+I
ij
=-
d
i
+I
i
+(j1)P
i
,j=1,2,...
(7)
At the j-th invocation, the completion time for task T
i
is C
ij
In practical systems, the timing constraints of the real-time
tasks viz. execution times, task deadlines etc. cannot be precisely
defined. This gives scope for the fuzzification of the real-time
scheduling problem. Fuzzification can be done for the real-time
task model using fuzzy constraints, fuzzy processing time, fuzzy
deadlines or a combination of the above three. Fuzzy Set expresses
the degree of compatibility of the same entity with the imprecise
concept represented by that particular Fuzzy Set. Therefore we
consider the deadline d
i
and the completion time C
i
of the task T
i
as
fuzzy number and our interest here is on the parametric form of the
fuzzy numbers.
3. Deadline and completion time as parametric fuzzy numbers
The detailed mathematical foundation of the parametric
representation of the fuzzy number using monotonic interpola-
tions has been given in [23]. Here we are providing the basic
notations for ready reference.
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
937

Continuous fuzzy number or intervals (say, u) can be defined as
any pair (u

,u
+
), where
(1)
(u

,u
+
) are functions given by u

: [0, 1]!R
(2)
u

:a!u

a
2Ris a function which is bounded and mono-
tonically increasing (non-decreasing)
8a
2[0, 1]
(3)
u
þ
:a!u
þ
a
2Ris a function which is bounded and mono-
tonically decreasing (non-increasing)
8a
2[0, 1]
(4)
u

a
u
þ
a
8a2½0;1
In the case of fuzzy intervals, u

1
<u
þ
1
and for fuzzy numbers,
u

1
¼u
þ
1
. The
a
-cut of the fuzzy number ucan be expressed as
u
a
¼½u

a
;u
þ
a
. Conveniently u

and u
+
can be referred as the left and
right branches of the fuzzy number u. The following Fig. 1 shows
the
a
-cut representation of a general fuzzy interval.
If uand vare two fuzzy numbers such that u=(u

,u
+
) and
v¼ðv

;v
þ
Þ, then the addition and subtraction operations among
them in terms of
a
-cuts for
a
2[0, 1] can be given by:
I.
Addition:
ðuþvÞ
a
¼ðu

a
þv

a
;u
þ
a
þv
þ
a
Þ(1)
II.
Subtraction:
ðuvÞ
a
¼ðu

a
v

a
;u
þ
a
v
þ
a
Þ(2)
The use of monotonic splines to approximate fuzzy numbers
was studied and three interpolation techniques, viz. cubic
interpolation, rational cubic interpolation and mixed cubic-
exponential Hermite interpolation were considered. We are giving
here the formulation of the mixed cubic-exponential Hermite
interpolation. The interest here is mainly finding the monotonic
approximations of u

and u
+
with the interpolation of their values
through the decomposition of the interval [0,1] into nsub-intervals
such that 0 =
a
0
<
a
1
<<
a
i1
<
a
i
<<
a
n
= 1. Now if within
a sub-interval [
a
i1
,
a
i
] the values of u

ða
i1
Þ¼u

i1
and u

ða
i
Þ¼
u

i
and their first derivates u
0
a
ða
i1
Þ¼u
0
i1
and u
0
ða
i
Þ¼u
0
i
are
known, then, mixed cubic-exponential Hermite interpolation
technique finds a suitable monotonic function for the interpolation
of the data. The monotonic function has the given derivative at the
respective node of decomposition.
We are particularly interested in the special case when n=1.
That is the case, when the decomposition of the interval [0, 1] has
no internal points. So
a
0
= 0 and
a
1
= 1. This helps to get a
parametric representation of the fuzzy number in terms of
a
-cuts
with 4 pairs of components (comprising all the interpolating points
for the left branch u

and right branch u
+
), as given below:
u¼ðu

0
;u
0
0
;u

1
;u
0
1
;u
þ
0
;u
0
0
þ;u
þ
1
;u
0
1
þÞ (3)
Eq. (3) represents non-linear fuzzy numbers. When u
0
0
¼
u
0
1
¼u

1
u

0
then the linear fuzzy numbers appears as a special
case of the representation (3). When u=(u,0,u,0,u,0,u, 0) then u
is a crisp real number. On the other hand when u=(u
1
,0,u
1
,0,u
2
,0,
u
2
, 0) then it presents a crisp interval [u
1
,u
2
]. Various shapes of
membership functions, which can be used to model task deadlines,
execution times etc., obtained by changing the derivatives at the
points u

0
;u

1
;u
þ
1
;u
þ
0
are shown in Fig. 2. So, the designer has a
wider variety of MFs to choose according to different requirements
and nature of the associated uncertainties in task characteristics.
Using the representation (3) when uand vare two fuzzy
numbers such that u¼ðu

0
;u
0
0
;u

1
;u
0
1
;u
þ
0
;u
0
0
þ;u
þ
1
;u
0
1
þÞ and v¼
ðv

0
;v
0
0
;v

1
;v
0
1
;v
þ
0
;v
0
0
þ;v
þ
1
;v
0
1
þÞ then the addition and subtrac-
tion of uand vcan be given, respectively, by
I.
uþv¼ðu

0
þv

0
;u
0
0
þv
0
0
;u

1
þv

1
;u
0
1
þv
0
1
;u
þ
0
þv
þ
0
;u
0
0
þþ
v
0
0
þ;u
þ
1
þv
þ
1
;u
0
1
þþv
0
1
þÞ
II.
uv¼ðu

0
v
þ
0
;u
0
0
v
0
0
þ;u

1
v
þ
1
;u
0
1
v
0
1
þ;u
þ
0
v

0
;u
0
0
þ
v
0
0
;u
þ
1
v

1
;u
0
1
þv
0
1
Þ
Now if we consider the deadline das the fuzzy number such
that ½u

0
;u
þ
0
is its 0-cut, then, in parametric form we can write,
d¼ðu

0
;u
0
0
;u

1
;u
0
1
;u
þ
0
;u
0
0
þ;u
þ
1
;u
0
1
þÞ. Therefore if d
i
is the dead-
line of the task T
i
, then d
i
¼ðu

0i
;u
0
0i
;u

1i
;u
0
1i
;u
þ
0i
;u
0
0i
þ;u
þ
1i
;u
0
1i
þÞ.
Hence the deadline of the task T
ij
(j-th invocation of the i-th task)
can be expressed as:
d
ij
¼ðu

0i
þI
i
þðj1ÞP
i
;u
0
0i
;u

1i
þI
i
þðj1ÞP
i
;u
0
1i
;u
þ
0i
þI
i
þðj1ÞP
i
;u
0
0i
þ;u
þ
1i
þI
i
þðj1ÞP
i
;u
0
1i
þÞ
¼ðu

0ij
;u
0
0ij
;u

1ij
;u
0
1ij
;u
þ
0ij
;u
0
0ij
þ;u
þ
1ij
;u
0
1ij
þÞ (4)
Similarly considering the processing times as fuzzy numbers in
parametric form we can get the completion time also as fuzzy
numbers in parametric form. This is because, the completion time
as the sum of processing times will be a fuzzy number, governed by
the addition operation mentioned in the previous section, when
processing times are fuzzy. Then the completion time of the task T
i
(considering single invocation) can be written as C
i
¼
ðv

0i
;v
0
0i
;v

1i
;v
0
1i
;v
þ
0i
;v
0
0i
þ;v
þ
1i
;v
0
1i
þÞ with its 0-cut as ½v

0i
;v
þ
0i
.
4. Fuzzy real-time scheduling using parameterized
memberships
Now if
m
(d
ij
) represents the membership function for the fuzzy
deadline d
ij
and C
ij
is the crisp completion time, then satisfaction of
schedulability (S
d
), to see the compliance of the deadlines over all
the periods, can be expressed by
S
d
i
ðC
ij
Þ¼
0ifC
ij
<u

0ij
1R
C
ij
u

0ij
mðd
ij
Þdðd
ij
Þ
R
u
þ
0ij
u

0ij
mðd
ij
Þdðd
ij
Þ
if u

0ij
C
ij
u
þ
0ij
1ifC
ij
>u
þ
0ij
8
>
>
>
>
>
<
>
>
>
>
>
:
(5)
The quantity R
C
ij
u

0ij
mðd
ij
Þdðd
ij
Þ=R
u
þ
0ij
u

0ij
mðd
ij
Þdðd
ij
Þis actually the
measure of dissatisfaction, how far the task completion time is
missing the deadline. If we consider a general bell shaped
membership function with ½u

0ij
;u
þ
0ij
as its 0-cut, then, as shown
in the Fig. 3 below, the denominator is the total area under the
curve, whereas the numerator is the shaded area.
Now by inverting the
a
-cut functions of u

a
and u
þ
a
, as defined in
the previous section, we may get the expression for the
approximated fuzzy membership function,
m
x
,x2R, as given
Fig. 1. General fuzzy interval in its
a
-cut representation (for fuzzy number,
u

1
¼u
þ
1
).
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
938

below:
m
x
¼
0ifxx

m
m

x
if x2½x

m
;x

M

1ifx2½x

M
;x
þ
m

m
þ
x
if x2½x
þ
m
;x
þ
M

0ifxx
þ
M
8
>
>
>
>
<
>
>
>
>
:
(6)
where
(1)
x

m
¼u

0
and x
þ
M
¼u
þ
0
are the corresponding points to the
support of the fuzzy number, whereas the points x

M
¼u

1
and
x
þ
m
¼u
þ
1
gives the interval (number if x

M
¼x
þ
m
) on the x-axis
with membership value 1.
Fig. 2. Various membership functions shapes model using mixed cubic-exponential Hermite interpolation parametric curves. In the respective diagrams different parameters
u¼ðu

0
;u
0
0
;u

1
;u
0
1
;u
þ
0
;u
0
0
þ;u
þ
1
;u
0
1
þÞ are shown.
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
939

(2)
m

x
are increasing (non-decreasing) function such that m

x

m
¼0
and m

x

M
¼1. And m
þ
x
are decreasing (non-increasing) function
such that m
þ
x
þ
m
¼1 and m
þ
x
þ
M
¼0.
Following are the relations that exist between m

x
and m

a
and
between m
þ
x
and m
þ
a
.
(1)
m

x
¼a$u

a
¼x;8a2½0;1;8x2½x

m
;x

M

(2)
m
þ
x
¼a$u
þ
a
¼x;8a2½0;1;8x2½x
þ
m
;x
þ
M

Also, at the end points of the interval the derivatives of m

x
and
m
þ
x
can be given by
(1)
m

x

0
¼
1
u
0
0

at x¼x

m
and m

x

0
¼
1
u
0
1

at x¼x

M
(2)
m
þ
x

0
¼
1
u
0
1
þ
at x¼x
þ
m
and m
þ
x

0
¼
1
u
0
0
þ
at x¼x
þ
M
Now the membership function
m
, can be represented para-
metrically as m¼ðx

m
;m
0
m
;x

M
;m
0
M
;x
þ
m
;m
0þ
m
;x
þ
M
;m
0þ
M
Þ, where the
monotonic components of m

x
and m
þ
x
are approximated, using the
mixed cubic-exponential Hermite interpolation, as given below:
m

x
¼m

m
þm
0
m
1þw

þt
2
ð32tÞm

M
m

m
m
0
m
þm
0
M
1þw


þm
0
M
t
1þw

m
0
m
ð1tÞ
1þw

1þw

(7)
m
þ
x
¼m
þ
m
þm
0þ
m
1þw
þ
þt
2
ð32tÞm
þ
M
m
þ
m
m
0þ
m
þm
0þ
M
1þw
þ

þm
0þ
M
t
1þw
þ
m
0þ
m
ð1tÞ
1þw
þ
1þw
þ
(8)
where m

m
¼0;m

M
¼1;m
þ
m
¼1;m
þ
M
¼0;m
0
m
¼x

M
x

m
=u
0
0
;m
0
M
¼
x

M
x

m
=u
0
1
;m
0þ
m
¼x
þ
M
x
þ
m
=u
0
1
þ;m
0þ
M
¼x
þ
M
x
þ
m
=u
0
0
þ;t

¼
xx

m
=x

M
x

m
and w

¼m
0
m
m
0
M
=m

M
m

m
.
We can re-write Eq. (5), using Fig. 1,as
Now using Eq. (6) for the membership functions and Eqs. (7)
and (8), for the left and right branches of the fuzzy deadlines, we
can express Eq. (9) as
where A
ij
¼Lðu

1ij
ÞLðu

0ij
Þþðu
þ
1ij
u

1ij
ÞþRðu
þ
0ij
ÞRðu
þ
1ij
Þ. And
Lðu

0ij
Þ;Lðu

1ij
Þare respectively the value of L(x) at the points u

0ij
and u
þ
0ij
, while Rðu
þ
0ij
Þ;Rðu
þ
1ij
Þare the same of R(x) at the points u
þ
0ij
and u
þ
1ij
with,
(1)
LðxÞ¼
1
N
m
0
m
þ
u

0ij
D
u

ij

2
3þ
2u

0ij
D
u

ij

!
x
3u

0ij
N
D
u

ij

2
1þ
u

0ij
D
u

ij

x
2
þ
1
N
D
u

ij

2
1þ
2u

0ij
D
u

ij

x
3

x
4
2N
D
u

ij

3
þ
D
u

ij
Nð1þNÞ
m
0
M
xu

0ij
D
u

ij

1þN
þm
0
m
1
xu

0ij
D
u

ij

1þN

(2)
RðxÞ¼ 1þ
1
M
m
0þ
m

u
þ
1ij
D
u
þ
ij

2
3þ
2u
þ
1ij
D
u
þ
ij

! !
xþ
3u
þ
1ij
M
D
u
þ
ij

2
Fig. 3. General fuzzy deadline.
S
d
i
ðC
ij
Þ¼
1ifC
ij
<u

0ij
1R
C
ij
u

0ij
mðd
ij
Þdðd
ij
Þ
R
u
þ
0ij
u

0ij
mðd
ij
Þdðd
ij
Þ
if u

0ij
C
ij
<u

1ij
1R
u

1ij
u

0ij
mðd
ij
Þdðd
ij
ÞþR
C
ij
u

1ij
mðd
ij
Þdðd
ij
Þ
R
u
þ
0ij
u

0ij
mðd
ij
Þdðd
ij
Þ
if u

1ij
C
ij
<u
þ
1ij
1R
u

1ij
u

0ij
mðd
ij
Þdðd
ij
ÞþR
u
þ
1ij
u

1ij
mðd
ij
Þdðd
ij
ÞþR
C
ij
u
þ
1ij
mðd
ij
Þdðd
ij
Þ
R
u
þ
0ij
u

0ij
mðd
ij
Þdðd
ij
Þ
if u
þ
1ij
C
ij
u
þ
0ij
0ifC
ij
>u
þ
0ij
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(9)
S
d
i
ðC
ij
Þ¼
1ifC
ij
<u

0ij
1LðC
ij
ÞLðu

0ij
Þ
A
ij
if u

0ij
C
ij
<u

1ij
1Lðu

1ij
ÞLðu

0ij
ÞþðC
ij
u

1ij
Þ
A
ij
if u

1ij
C
ij
<u
þ
1ij
1Lðu

1ij
ÞLðu

0ij
Þþðu
þ
1ij
u

0ij
ÞþRðC
ij
ÞRðu
þ
1ij
Þ
A
ij
if u
þ
1ij
C
ij
u
þ
0ij
0ifC
ij
>u
þ
0ij
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
(10)
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
940

1þ
u
þ
1ij
D
u
þ
ij

x
2

1
M
D
u
þ
ij

2
1þ
2u
þ
1ij
D
u
þ
ij

x
3
þ
x
4
2M
D
u
þ
ij

3
þ
D
u
þ
ij
Mð1þMÞ
m
0þ
M
xu
þ
1ij
D
u
þ
ij

1þM
þm
0þ
m
1
xu
þ
1ij
D
u
þ
ij

1þM
!
(3)
N¼1þm
0
m
þm
0
M
;Du

ij
¼u

1ij
u

0ij
;M¼1m
0þ
m
m
0þ
M
and
Du
þ
ij
¼u
þ
0ij
u
þ
1ij
.
As a special case, for triangular fuzzy numbers, if we consider
the derivatives in expression (4) as u
0
0ij
¼u

1ij
u

0ij
;u
0
1ij
¼
u

1ij
u

0ij
;u
0
0ij
þ¼u
þ
1ij
u
þ
0ij
and u
0
1ij
þ¼u
þ
1ij
u
þ
0ij
, then we get
respectively, m
0
m
¼1;m
0þ
m
¼1 and m
0þ
m
¼1. This reduces
Eq. (10) to
S
d
i
ðC
ij
Þ¼
1ifC
ij
<u

0ij
1ðC
ij
u

0ij
Þ
2
ðu
þ
0ij
u

0ij
Þðu

1ij
u

0ij
Þif u

0ij
C
ij
u

1ij
ðu
þ
0ij
C
ij
Þ
2
ðu
þ
0ij
u

0ij
Þðu
þ
0ij
u
þ
1ij
Þif u
þ
1ij
C
ij
u
þ
0ij
0ifC
ij
>u
þ
0ij
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
(11)
Fig. 4 shows the variations of the satisfaction function for
various combinations of MFs. It is clear from this figure that
the parameterized method gives considerable flexibility to the
schedule designer for choosing membership function shapes. He
can then model various task deadlines in a realistic manner and
obtain the effect of his choices on satisfaction of schedulability.
This method also gives similar facility for choosing execution time
model.
Now, if the execution times are deterministic then, as their sum,
the completion time C
ij
is also deterministic and algorithms were
proposed for its computation [3]. As fuzzy numbers are continuous
functions, the satisfaction function is certainly continuous and
strictly decreasing. Therefore, denoting Pas the hyperperiod of the
tasks (least common multiple of the task periods) the problem of
scheduling T
i
tasks (with fuzzy deadlines d
ij
and completion time
C
ij
at the j-th invocation) may be summarized as
Maximize S
d
¼min
i;j
S
d
i
ðC
ij
Þ;(12)
finding an optimal assignment of priorities. Where i=1,2,...,n
and j=1,2,...,P/P
i
, as the schedulability should be examined over
P time units.
Similarly considering the processing times as fuzzy numbers in
parametric form, the completion time of the j-th invocation of
the task T
i
can be written, if ½v

0ij
;v
þ
0ij
is the 0-cut, as
C
ij
¼ðv

0ij
;v
0
0ij
;v

1ij
;v
0
1ij
;v
þ
0ij
;v
0þ
0ij
;v
þ
1ij
;v
0þ
1ij
Þ. Then to measure sche-
dulability over the whole task set for all invocations within P, the
satisfaction of schedulability is defined below for the task T
i
for its
j-th invocation.
S
C
i
ðd
ij
Þ¼
0ifd
ij
<v

0ij
R
d
ij
v

0ij
mðC
ij
ÞdðC
ij
Þ
R
v
þ
0ij
v

0ij
mðC
ij
ÞdðC
ij
Þ
if v

0ij
d
ij
v
þ
0ij
1ifd
ij
>v
þ
0ij
8
>
>
>
>
>
<
>
>
>
>
>
:
(13)
Proceeding in the similar way, as for S
d
i
ðC
ij
Þ, if we put, B
ij
¼
Lðv

1ij
ÞLðv

0ij
Þþðv
þ
1ij
v

1ij
ÞþRðv
þ
0ij
ÞRðv
þ
1ij
Þwith Lðv

0ij
Þ;Lðv

1ij
Þ,
as the value of L(x) at the points v

0ij
and v
þ
0ij
, while Rðv
þ
0ij
Þand
Rðv
þ
1ij
Þ, the same of R(x)at the points v
þ
0ij
and v
þ
1ij
, respectively, then
the Eq. (13) for S
C
i
ðd
ij
Þcan be written as
S
C
i
ðd
ij
Þ¼
0ifd
ij
<v

0ij
Lðd
ij
ÞLðv

0ij
Þ
B
ij
if v

0ij
d
ij
<v

1ij
Lðv

1ij
ÞLðv

0ij
Þþðd
ij
v

1ij
Þ
B
ij
if v

1ij
d
ij
<v
þ
1ij
1Rðv
þ
0ij
ÞRðd
ij
Þ
B
ij
if v
þ
1ij
d
ij
v
þ
0ij
1ifd
ij
>v
þ
0ij
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(14)
Fig. 4. Variations of the Satisfaction Function (S
d
) for various combinations of MFs.
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
941

In the interval ½v

0ij
;v
þ
0ij
the satisfaction function S
C
i
ðd
ij
Þis strictly
increasing and continuous. Therefore the problem is summarized
as
Maximize S
C
¼min
i;j
S
C
i
ðd
ij
Þ;(15)
finding an optimal assignment of priorities, where i=1,...,nand
j=1,...,P/P
i
as the schedulability should be examined over Ptime
units.
Now considering both the deadlines and processing times as
fuzzy numbers, problem may be summarized as
Maximize S¼min
i;j
max min
C
ij
;d
ij
ðS
C
i
ðd
ij
Þ;S
d
i
ðC
ij
ÞÞ;(16)
finding an assignment of priorities, where i=1,...,nand
j=1,...,P/P
i
as the schedulability should be examined over P
time units.
Let us now consider, min
i;j
S
d
i
ðC
ij
Þ¼t. Then,
S
d
i
ðC
ij
Þt8i¼1;...;nand j¼1;...;P=P
i
:(17)
From the continuity of the satisfaction function S
d
and its inverse
function, we can therefore write:
C
ij
I
i
þðj1ÞP
i
þd
0
i
ðtÞ;i¼1;...;nand j¼1;...;P=P
i
(18)
where the quantity d
0
i
ðtÞis a crisp quantity that depends on the
minimum satisfaction, t, and is termed as the modified deadline and
the optimal priority assignment of the real-time tasks are
according to the increasing order of this quantity. Now our
interest is to find those values of t(0 t1) for which modified
deadlines of two tasks become equal i.e. those t
ij
that satisfy:
ft
ij
jd
0
i
ðtÞ¼d
0
j
ðtÞ;j¼1;...;n;i¼1;...;n;0t
ij
1g(19)
Now since the modified deadline of different tasks changes at these
t
ij
, so the priorities of the tasks also changes at these points. Hence
we have to check the priorities of the tasks over various intervals
obtained by sequentially placing the quantities t
ij
in the increasing
order, because maximization of min
i;j
S
d
i
ðC
ij
Þis nothing other than
identifying the highest interval on the condition that the resulted
task priorities in that interval gives us a feasible schedule.
Algorithms used by Litoiu et al. [11–13] are as given below:
Algorithm 1.
Algorithm 2.
They used successive search technique to identify the interval
in which satisfaction is maximum. To do that, the fuzzy completion
times of the tasks needs to be calculated. W
i
(C
i
) is the worst fuzzy
completion time, where W
i
ðC
i
Þ¼P
i
j¼1
e
j
½C
i
=P
j
þB
L
i
;B
L
i
being the
time for which a high priority task is blocked by a low priority one.
Considering the worst case situation, when the processor is also
requested by all other tasks, to calculate C
i
/P
j
, the latest extremity
of C
i
i.e. f
i
is taken. In doing this, the fuzzy division is transformed
into a ordinary arithmetic division. Then using the worst fuzzy
completion time the satisfaction is calculated, the minimum of
which gives us the satisfaction of the schedulability.
Example (I).
We took a three task system T = {T
1
,T
2
,T
3
} with the
following characteristics:
T
1
:e
1
= (30 10 40 10 40 10 50 10), P
1
= 170, d
1
= (154 6 160 6
160 6 166 6)
T
2
:e
2
= (65 5 70 5 70 5755), P
2
= 170, d
2
= (155 5 160 5 160
5 165 5)
T
3
:e
3
= (25 5 30 5 30 5355), P
3
= 170, d
3
= (159 2 161 2 161
2 163 2)
The results we got are summarized in the following Table 1 and
Fig. 5 shows the variation of the satisfaction function (S
d
) of the
tasks. The result shows that the task set has three satisfaction
crossover points. Tasks T
1
and T
2
changes their priorities at 0.5, that
between task T
1
and T
3
occurs at 0.2813 while for task T
2
and T
3
the
same happens at 0.2222. The completion times of the tasks observe
the deadlines at 0, 0.2222, 0.2813 and 0.5, but not at 1. Which
shows that the highest satisfaction interval for the tasks set is [0.5,
1] and the satisfaction of schedulability of the task set falls within
this interval.
Therefore, the priorities of the tasks will be [T
1
,T
2
,T
3
]. The
satisfaction of the schedulability of task set is min{1, 1, 0.9987}.
Therefore the feasible and optimal schedule will be [T
1
,T
2
,T
3
] and
the satisfaction of the schedulability of task set is 0.9987.
Example-(II).
Let us now consider the following task set obtained
with variation in the sign of the first derivates u
0
0i
;u
0
1i
;u
0þ
0i
;u
0þ
1i
and
v

0i
;v

1i
;v
þ
0i
;v
þ
1i
T
1
:e
1
= (30 10 40 10 40 10 50 10), P
1
= 170, d
1
= (154 6 160 6
160 6 166 6)
T
2
:e
2
= (65 5 70 5705755), P
2
= 170, d
2
= (155 5 160 5
160 5 165 5)
T
3
:e
3
= (25 5 30 5305355), P
3
= 170, d
3
= (159 2 161 2
161 2 163 2)
The results are summarized in the following Table 2 and Fig. 6.
We can see that the changes in the sign of the derivatives resulted
in the effective changes in the shape of the membership functions,
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
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which in turn changes the respective satisfaction functions. Here
the satisfaction crossover points are t
12
= 0.5, t
13
= 0.2883 and
t
23
= 0.1975. It is seen that changes in the shape of the MFs of the
task deadlines resulted in the changes of the priority crossover
point of two tasks if the peak of the MF for both the tasks are at
different points. On the other hand the priority crossover point for
two tasks remains same if both tasks has peak of the MF at the
same point. If we compare Example-(I) and (II) we see that t
12
are
same but t
13
and t
23
are different. This is because both the task T
1
and T
2
has peak of the MF at 160, while that of the task T
3
is at 161.
Here the satisfaction of schedulability is also different and is found
as 0.9999.
However the above technique is too much pessimistic in the
sense that it considers the worst case execution time (i.e. the right
extremity of the fuzzy processing time) for all the tasks .The
consideration of the worst case execution time (i.e. the right
extremity of the fuzzy processing time) always seems less logical
because there are chances that few task executions may take less
times than their worst case execution times. If a task finishes
execution well before its worst case scenario then some slacks will
result, during this slack time the processor may hardly remain idle
if there are already scheduled tasks in the system. Instead during
this slack the next high priority, yet to be executed, task will get the
processor and begin execution. This will reduce the total execution
time of the tasks and hence will increase the minimum satisfaction
significantly. Therefore incorporation of the dynamic slack
calculation technique, proposed in [22], makes the model more
practical and realistic. To explain, let us consider a system with two
tasks T
a
and T
b
where T
a
is of higher priority than T
b
. We suppose
task T
a
finishes execution x time before than its worst case
completion time. Then task T
b
will start execution x time units
earlier and as a result finish the execution early also by same time
unit. Thus the value of the satisfaction function for both the task
will be effected. To see the possible changes in the schedulability
and satisfaction of the real-time tasks we now consider the
following realistic example:
Example (III).
Industrial Control System
Here we are considering a real life case from [11], where a real-
time system is employed for the control of an industrial plant. As
shown in Fig. 7 above, a Fiber Distributed Data Interface (FDDI).
Network connects five nodes of the system. Tasks at different
nodes are independent and preemptive. For the purpose of
simplification in the analysis we shall consider that there are
negligible amount of communication costs between the tasks.
Here, nodes 1 to 3 are identical and they are dealing with robotic
applications. The nodes 4 and 5 are respectively dedicated to video
monitoring and user console management. We shall investigate
the task schedulability at the each node considering no common
sharable resources. The characteristics and the activities of the
tasks at the nodes 1 to 3 are given in Table 3.
There will be no priority change (no satisfaction crossover
point) among different tasks. Therefore, there is only one
satisfaction interval, which is [0, 1]. The optimal task schedule
is [T
1
,T
3
,T
4
,T
2
]. Within the hyperperiod (360) there are 6
invocations of task T
1
, 2 invocations of task T
2
, 4 invocations of task
T
3
and 3 invocations of task T
4
. The third invocations of task T
4
miss
Table 1
Results for Example (I).
Satisfaction crossover
points (t
ij
)
Satisfaction
intervals
Tasks priority
within the intervals
Interval of highest
satisfaction
Satisfaction of tasks Satisfaction of
schedulability
Optimal
task priority
t
23
= 0.2222 t
13
= 0.2813
t
12
= 0.5000
[0.0000, 0.2222] [T
3
,T
2
,T
1
] [0.5000, 1.0000] T
1
: 1.0000 T
2
: 1.0000 T
3
: 0.9550 0.9987 T
1
:1T
2
:2T
3
:3
[0.2222, 0.2813] [T
2
,T
3
,T
1
]
[0.2813, 0.5000] [T
2
,T
1
,T
3
]
[0.5000, 1.0000] [T
1
,T
2
,T
3
]
Fig. 6. Plot of Satisfaction Function S
d
for the tasks of Example (II).
Table 2
Results for Example (II).
Satisfaction crossover
points (t
ij
)
Satisfaction
intervals
Tasks priority
within the intervals
Interval of highest
satisfaction
Satisfaction of tasks Satisfaction of
schedulability
Optimal
task priority
t
23
= 0.1975 t
13
= 0.2883
t
12
= 0.5000
[0.0000, 0.1975] [T
3
,T
2
,T
1
] [0.5000, 1.0000] T
1
: 1.0000 T
2
: 1.0000 T
3
: 0.9999 0.9999 T
1
:1T
2
:2T
3
:3
[0.1975, 0.2883] [T
2
,T
3
,T
1
]
[0.2883, 0.5000] [T
2
,T
1
,T
3
]
[0.5000, 1.0000] [T
1
,T
2
,T
3
]
Fig. 5. Plot of Satisfaction Function S
d
for the tasks of Example (I).
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
943

the deadlines by 10 units of time. As a result, we get the satisfaction
of schedulability 0.4260. It is shown in Fig. 8(a).
Now let us suppose, different invocations of tasks T
1
,T
2
and T
3
and the first two invocations of task T
4
completes earlier than their
worst case execution time producing significant amount of slack
time in the system. We have studied the scenario with different
amount of probable slack time in the systems as shown in the
following Table 4 given below. Fig. 8 above gives the plot of
satisfaction functions for the third invocation of task T
4
for various
values of slack time. From this figure we see that with the
increasing slack time, the satisfaction of schedulability also
increases because the amount of deadline missing by the third
invocation of task T
4
sharply decreases.
Fig. 9 above shows the plot of slack time versus satisfaction of
schedulability. Therefore we can conclude that the utilization of
the dynamically calculated slack time allows the execution of the
tasks with more satisfaction and hence ensures that the hard task
deadlines are also very nicely met.
As noted earlier, in the system, the node 5 is involved for
console management. There are five tasks those are active at this
node. The tasks are independent and preemptive. Tasks T
1
,T
2
,T
3
,
and T
4
undertake display activities, whereas task T
1
is for control
functions. The details of the task characteristics and their activities
are given in Table 5.
The results we got for the task set active at the node 5 are
summarized in Table 6. There will be no priority change
Table 3
Characteristics and activities of Tasks at nodes 1–3.
Task Processing time Deadline Period Type Activity
T
1
(8, 1, 9, 1, 9, 1, 10, 1) (40, 10, 50, 10, 50, 10, 60, 10) 60 Hard Output control
T
2
(25, 5, 30, 5, 30, 5, 35, 5) (140, 20, 160, 20, 160, 20, 180, 20) 180 Hard Command processing
T
3
(28, 1, 29, 1, 29, 1, 30, 1) (50, 20, 70, 20, 70, 20, 90, 20) 90 Hard Measurement
T
4
(30, 5, 35, 5, 35, 5, 40, 5) (100, 10, 110, 10, 110, 10, 120, 10) 120 Hard Data processing
Fig. 7. Generic high-level overview of a real-time system [11].
Table 4
Variation of the Satisfaction of Schedulability with the Slack Time.
Sl. No. Slack time Satisfaction of schedulability
1 0 0.4240
2 4 0.4847
3 10 0.5740
4 20 0.7941
5 30 0.8107
6 40 0.8935
7 50 0.9527
8 60 0.9882
9 70 1.000
Fig. 9. Slack time versus satisfaction of schedulability for tasks of Node 1.
Fig. 8. Plot of satisfaction functions for the third invocation of task T
4
when, (a) slack = 0, (b) slack = 10, (c)slack = 20, (d) slack = 30, (e) slack = 40, (f)slack = 50, (g) slack = 60,
(h) slack = 70.
P.K. Muhuri, K.K. Shukla / Applied Soft Computing 9 (2009) 936–946
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(satisfaction crossover points) for tasks T
4
and T
5
with others. The
plot of satisfaction function S
d
for the task T
1
,T
2
and T
3
are shown
in the Fig. 10. For tasks T
1
,T
2
and T
3
, two satisfaction crossover
points exists, which are t
12
= 0.98 and t
23
= 0.5. As shown in
Table 6, we therefore get three satisfaction intervals: 0
1
= [0, 0.5],
O
2
= [0.5, 0.98] and O
3
= [0.98, 1]. The satisfaction of schedulability
belongs to the interval O
1
. The priorities of the tasks within this
interval are [T
5
,T
2
,T
3
,T
1
,T
4
].
When the tasks are scheduled according to these priorities, then
it is found that within the first hyperperiod (690) the 2nd and 3rd
invocations of the task T
1
miss the deadlines remarkably, resulting
a very low value of satisfaction. The task T
1
is a soft task and it deals
with the display activities. Therefore missing the deadline, by this
particular task may not be catastrophic in nature. However, these
instances of missing the deadline by the task T
1
causes the hard
periodic task T
5
and other soft tasks T
3
and T
4
to miss their deadline
at their later invocations. For example, 10th and 13th invocations
of task T
5
and 4th invocation of task T
4
will miss their respective
deadlines significantly. Therefore utilization of the dynamically
calculated slack time becomes very much useful.
Let us now consider that the task T
2
,T
3
and T
4
finish execution
respectively 10, 8 and 7 time units earlier than their respective
worst case completion time. This accumulates a slack time of 43
unit before the execution of the 2nd invocation of task T
1
and a
slack time of 68 unit before the execution of the 3rd invocation of
task T
1
. Therefore during these slack times the processor executes
the next high priority tasks and gives us better satisfaction. Then
we get a satisfaction of schedulability 0.7429, which belongs to the
satisfaction interval O
2
. As mentioned in Table 6, the task priorities
within this interval is [T
5
,T
3
,T
2
,T
1
,T
4
]. Therefore the tasks need to
be scheduled according to these priorities.
Table 7 compares the results without and with the considera-
tion of the slack time. Thus we see that utilizing the dynamically
calculated slack time we can get a task schedule, which gives a
better satisfaction of schedulability and also ensures the meeting
of the hard task deadlines.
5. Conclusion
In fuzzy real-time scheduling membership functions (MFs) are
used to represent deadlines and execution times. Although simple
MFs such as Triangular, Trapezoidal etc. have the advantage of
simpler understanding and analysis, they contain discontinuities
and designer gets restricted choice in specifying task characteristics.
It is very likely that different task deadlines (and completion times)
may have different MFs. So finding the task having minimum
satisfactionis not so easy because it is very logical for the designerto
choose different MFs for different tasks based on the nature of the
Table 5
Characteristics and activities of Tasks at node 5.
Task Processing time Deadline Period Type Activity
T
1
(35,10,45,10,45,10,55,10) (170,30,200,30,200,30,230,30) 230 Soft Display
T
2
(15,10,25,10,25,10,35,10) (175,5,180,5,180,5,185,5) 230 Soft Display
T
3
(25,5,30,5,30,5,35,5) (170,10,180,10,180,10,190,10) 230 Soft Display
T
4
(30,5,35,5,35,5,40,5) (300,20,320,20,320,20,340,20) 345 Soft Display
T
5
(35,5,40,5,40,5,45,5) (80,5,85,5,85,5,90,5) 115 Hard Control
Table 6
Results for tasks active at node 5.
Satisfaction crossover
points (t
ij
)
Satisfaction
intervals
Task priority
within the intervals
Interval of highest
satisfaction
Satisfaction of tasks Satisfaction
of schedulability
Task priority
t
23
= 0.50 t
12
= 0.98 [0.00, 0.50] [T
5
,T
2
,T
3
,T
1
,T
4
] [0,0.50] T
1
: 0.4524 T
2
: 1.0000
T
3
: 0.9550 T
4
: 1.0000 T
5
: 1.0000
0.4524 T
1
:4T
2
:2T
3
:3
T
4
:5 T
5
:1[0.50, 0.98] [T
5
,T
3
,T
2
,T
1
,T
4
]
[0.98, 1.00] [T
5
,T
3
,T
1
,T
2
,T
4
]
Fig. 10. Satisfaction plot for task T
1
,T
2
,andT
3
of Node 5.
Table 7
Comparison between results with and without the consideration of slack time.
Dynamically calculated slack time Interval of highest satisfaction Task schedule Satisfaction of tasks Satisfaction of schedulability Task priority
Not considered O
1
: [0.00, 0.50] [T
5
,T
2
,T
3
,T
1
,T
4
]T
1
: 0.4524 0.4524 T
1
:4
T
2
: 1.0000 T
2
:2
T
3
: 1.0000 T
3
:3
T
4
: 1.0000 T
4
:5
T
5
: 1.0000 T
5
:1
Considered O
2
: [0.50, 0.98] [T
5
,T
3
,T
2
,T
1
,T
4
]T
1
: 0.7429 0.7429 T
1
:4
T
2
: 1.0000 T
2
:3
T
3
: 0.9550 T
3
:2
T
4
: 1.0000 T
4
:5
T
5
: 1.0000 T
5
:1
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associated uncertainty of the task deadlines. This makes the
identification process of the task with minimum satisfaction
difficult, as task which has minimum satisfaction with one type
of MF may not be so if it is assigned another MF for its deadline,
which is observed in the plot of the satisfaction function against the
deadlines with different MFs. Therefore the whole optimization
process may become wrong as it maximizes the minimum
satisfaction. Hence choosing the proper MF for a particular task is
very much crucial, so is the task having minimum satisfaction.
The parametric method proposed here is very versatile and has
the ability to model a wide variety of symmetric and asymmetric
MFs as it allows intuitively defining smooth MFs for deadlines and
execution times. Triangular and Trapezoidal MFs appear as special
cases. Simple numerical examples successfully demonstrate the
usefulness of the method and the effect of changes in parameter-
ized MFs on the Satisfaction of Schedulability and task priorities.
The dynamic slack calculation technique helps us to get the actual
task schedule and the value of the satisfaction of schedulability and
removes the pessimistic nature of the Litoiu et al. proposed
technique. A comparison between the Litoiu’s technique and the
proposed approach is given in the Table 8.
Future work will aim towards:(a) Allocation of slack times to the
non-periodic real-time or non real-time tasks. Specifically one may
try to find some relation between the total fuzziness introduced in
the timing constraints of the real-time periodic tasks and the total
slack time of the system. (b) Learning machines like neural networks
(NNs) may be used for proactive scheduling after training them on
several test cases.(c) Performance of the real-time systems with soft
computing techniques mentioned above may be studied under
transient overload conditions, where some aperiodic tasks arrive to
make the problem more challenging for the scheduler. (d)
Theoretical analysis of fuzzy real-time scheduling.
Acknowledgement
First author gratefully acknowledge the moral support of
National Professor of Bangladesh Dr. N. Islam, FVC, USTC and the
financial support of the University Grants Commison, New Delhi,
India.
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Table 8
Comparison between Litoiu’s technique and the proposed approach.
Sr. No. Property Litoiu’s technique Proposed method Remarks
1. Flexibility to the schedule designer. Only trapezoidal and
triangular membership
functions are possible.
Smooth continuos
functions can be chosen;
Litoiu’s functions are
special cases.
Application specific choice of membership
function leads to a more realistic model.
2. Satisfaction of schedulability Medium to high Very high Please see Example-(II). Higher satisfaction
of schedulability means there are more
compliance of the completion time wih the
task deadline.
3. Dynamic slack computation Not considered Taken care of Please see Example-(III).
4. Continuous and differentiable
membership functions
Not considered Includes continuous
and differentiable
membership functions.
In future we intend to carry a theoretical
analysis of Fuzzy real-time scheduling using
analytical optimization techniques. These
will require calculation of 1st and 2nd
differential coefficients of an objective
function which will require continuous and
differentiable membership functions.
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