Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems
Main objective in this paper is to unify results on controllability and observability on time scales and deduce the results of classical theory as a particular case and then resolve the time constraints on concurrency control by incorporating jump operators on time scale dynamical systems.
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
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| Цитувати: | Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems / N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram // Электронное моделирование. — 2010. — Т. 32, № 3. — С. 83-93. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-128302010-10-25T12:02:10Z Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems Ramesh Babu, N. Murty, K.N. Balaram, V.V.S.S.S. Применение методов и средств моделирования Main objective in this paper is to unify results on controllability and observability on time scales and deduce the results of classical theory as a particular case and then resolve the time constraints on concurrency control by incorporating jump operators on time scale dynamical systems. Обобщены результаты исследований управляемости и наблюдаемости при масштабировании во времени. Получены результаты классической теории как частного случая, при этом устранены ограничения по времени при параллельном управлении операторами перехода для динамических масштабируемых систем. Узагальнено результати досліджень керованості та спостережуваності при масштабуванні за часом. Отримано результати класичної теорії як окремого випадку, при цьому усунуто обмеження за часом при паралельному керуванні операторами переходу для динамічних систем, що масштабуються. 2010 Article Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems / N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram // Электронное моделирование. — 2010. — Т. 32, № 3. — С. 83-93. — Бібліогр.: 7 назв. — англ. 0204-3572 http://dspace.nbuv.gov.ua/handle/123456789/12830 en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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Применение методов и средств моделирования Применение методов и средств моделирования |
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Применение методов и средств моделирования Применение методов и средств моделирования Ramesh Babu, N. Murty, K.N. Balaram, V.V.S.S.S. Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
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Main objective in this paper is to unify results on controllability and observability on time scales and deduce the results of classical theory as a particular case and then resolve the time constraints on concurrency control by incorporating jump operators on time scale dynamical systems. |
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Article |
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Ramesh Babu, N. Murty, K.N. Balaram, V.V.S.S.S. |
| author_facet |
Ramesh Babu, N. Murty, K.N. Balaram, V.V.S.S.S. |
| author_sort |
Ramesh Babu, N. |
| title |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
| title_short |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
| title_full |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
| title_fullStr |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
| title_full_unstemmed |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems |
| title_sort |
control of time scale dynamical systems with an application to concurrency control for real-time database systems |
| publisher |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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2010 |
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Применение методов и средств моделирования |
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http://dspace.nbuv.gov.ua/handle/123456789/12830 |
| citation_txt |
Control of Time Scale Dynamical Systems with an Application to Concurrency Control for Real-time Database Systems / N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram // Электронное моделирование. — 2010. — Т. 32, № 3. — С. 83-93. — Бібліогр.: 7 назв. — англ. |
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2025-07-02T14:49:49Z |
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| fulltext |
N. Ramesh Babu, K.N. Murty *, V.V.S.S.S. Balaram
Aurora’s Scientific, Technological and Research Academy
Department of Mathematics and Computer Science
(Bandlaguda, Hyderabad – 500 005, (A.P) India)
Control of Time Scale Dynamical
Systems with an Application to Concurrency
Control for Real-time Database Systems
(Recommended by Prof. E. Dshalalow)
Main objective in this paper is to unify results on controllability and observability on time scales
and deduce the results of classical theory as a particular case and then resolve the time constraints
on concurrency control by incorporating jump operators on time scale dynamical systems.
Îáîáùåíû ðåçóëüòàòû èññëåäîâàíèé óïðàâëÿåìîñòè è íàáëþäàåìîñòè ïðè ìàñøòàáèðî-
âàíèè âî âðåìåíè. Ïîëó÷åíû ðåçóëüòàòû êëàññè÷åñêîé òåîðèè êàê ÷àñòíîãî ñëó÷àÿ, ïðè
ýòîì óñòðàíåíû îãðàíè÷åíèÿ ïî âðåìåíè ïðè ïàðàëëåëüíîì óïðàâëåíèè îïåðàòîðàìè
ïåðåõîäà äëÿ äèíàìè÷åñêèõ ìàñøòàáèðóåìûõ ñèñòåì.
K e y w o r d s: time scale dynamical system, modern control system theory.
1. Introduction. Time scale dynamical system is an interesting area of current
research and a great deal of work has been done by many authors in recent years
[1]. From a modelling point of view, it is perhaps more realistic to model a real
world phenomenon by a time scale dynamical system as it incorporates both
continuous and discrete systems as a particular case [2, 3]. A fascinating fact is
that all the widely different disciplines of application depend on a common core
of Time scale dynamical system of the modern control system theory. These
techniques require real time database systems that run effectively without any
conflicts. In fact the concurrency control method receives certain information
gathered from the transactions made in order to find and resolve conflicts [4].
Further, a real time transaction is a transaction with additional real-time attrac-
tion and importance.
Our main objective in this paper is to unify results on controllability and
observability on time scales and deduce the results of classical theory as a parti-
cular case and then resolve the time constraints on concurrency control by incor-
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 83
* Corresponding author, email : nkanuri@hotmail.com
porating jump operators on time scale dynamical systems. More specifically, the
paper is organized as follows: Section 2 presents some salient features of time
scale dynamical systems that are needed for our later discussion. We introduce
the concepts of controllability and observability for m-input, p-output, n-dimen-
sional linear systems on time scales in section 3. In this section we present a set
of necessary and sufficient conditions for the first order time scale dynamical
system to be completely controllable and observable. For a complete theory on con-
trol on linear system we refer [5]. Section 4 is concerned with Real Time Database
Systems and in fact deals with concurrency control of the database systems.
2. Basic results. In this section, we outline some of the basic notions concern-
ing time scales. A time scale T is a closed subset of R; and examples of time
scales include N, Z, R, cantor’s set fuzzy sets etc. The set Q t R Q t� � � �{ \ , }0 1
are not time scales. Time scales need not necessarily be connected. In order to
overcome this deficiency, we introduce the notion of jump operators. The
mappings �, � : T � T defined by
� ( ) inf { : }t s T s t� � � , � ( ) sup{ : }t s T s t� � ,
are called jump operators.A point t T� is said to be right dense if � ( )t t� , left
dense, if � ( )t t� and left scattered, if � ( )t t . The graininess
: T � [0, �) is
defined by
�( ) ( )t t t� � .
We say that f is rd-continuous if it is continuous in right dense points and if
lim f (s) as s t� exists for all right dense points t T� [1]. A function f : T T� is
said to be differentiable at t T T t T tk� �{ \ ( ( )max ( ), max )}� if
lim
( ( ) ( ))
( )( )�
�
�t s
f t f s
t s�
�
�
,
where s T t� �{ ( )}� exists and is said to be differentiable on T provided it is dif-
ferentiable for each t T k� . A function F : T T� , with F t f t
( ) ( )� for all t T K�
is said to be integrable, if
f F t F s
s
t
( ) ( ) ( )� �
� � � ,
where F is the anti-derivative of f and for all s, t T� . Let f : T T� and if T = R
and a, b T� , then f t f t
( ) ( )� 1 and
f t dt f t t
a
b
a
b
( ) ( )� ��
.
N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram
84 ISSN 0204–3572. Electronic Modeling. 2010. V. 32. ¹ 3
Further, if T = Z (discrete case), then f t f t f t f t
( ) ( ) ( ) ( )� � � �1 and
f t t
f k a b
a b
f k a b
a
b k a
b
( )
( ) ,
,
( ) .
�
�
�
�
�
�
�
if
if
if
1
0
k b
a
�
�
�
�
�
�
��
�
�
�
�
1
If f, g : T � X (X is a Banach space) be differentiable in t T k� . Then for any two
scalars �, �, the mapping� �f g� is differentiable in t and further we have:
1. �� � � �f g t f t g t� � �) ( ) ( ) ( )
;
2. � �f g t f t g t f t g t) ( ) ( ) ( ) ( ( )) ( )
� � ;
3. f t f t t f t( ( )) ( ) ( ) ( )�
� �
;
4. �k f t k f t) ( ) ( )
� , for any scalar k.
Note that if f is
-differentiable, then f is continuous. Further if t is right
scattered and f is continuous at t then
f t
f t f t
t
( )
( ( )) ( )
( )
�
��
.
For a survey on calculus of
-differentiable functions, we refer to Lakshmi-
kantham et. al [5].
3. Controllability and observability criteria for �-differentiable func-
tions. In this section, we shall be concerned with the first order
-differentiable
dynamic system
x t A t x t B t U t
( ) ( ) ( ) ( ) ( )� � , x t x( )0 0� , (1)
y t C t x t( ) ( ) ( )� , (2)
where A t( ) is an ( )n n� square matrix and A : T B Rk n� ( ) is regressive and
rd-continuous. When t = R, (1) is equivalent to
� � �x t A t x t B t U t( ) ( ) ( ) ( ) ( ), x t x( )0 0� , (3)
and when t = z, (1) is equivalent to
x n A n x n B n U n( ) ( ) ( ) ( ) ( )� � �1 , x n x( )0 0� . (4)
The fundamental concepts of controllability and observability for an m-input,
p-output, n-dimentional linear state equation (1) and (2) will be considered in
this section. For a time varying linear state equation (3), the connection of the in-
put signal to the state variables can change with time. Therefore, the concept of
Control of Time Scale Dynamical Systems with an Application
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 85
controllability is tied to a specific finite time interval [t t f0 , ] with, of course
t tf � 0. For a discrete system (4), the connection of the input signal to the next
state variables can change with time.
Definition 1. The
-differentiable dynamic system (1) is said to be control-
lable on [t t f0 , ], if for any given initial state x t x( )0 0� (x n x( )0 0� ), there exists
a continuous (discrete) input signal U (t), such that the corresponding solution of
(1) satisfies x t f( ) �0 (x n f( ) �0).
If time scale dynamical system (1) is controllable for all x0 at t = t0 and for
all xf at t = tf, then the system (1) is said to be completely controllable. We sup-
pose that T a b Tk � �( , ) and the associated homogeneous system is
x t A t x t
( ) ( ) ( )� , x t x( )0 0� .
Let� A t t( , )0 be a fundamental matrix solution of
x t A t x t
( ) ( ) ( )� .
Then any solution x(t) of (1) has the form
x t t t x t s B s U s sA A
t
t
( ) ( , ) ( , ( )) ( ) ( ) ( )� ��� �
0 0
0
� ,
and it is easy to see that
x t t s B s U s sA
t
t
( ) ( , ( )) ( ) ( ) ( )� � �
�
0
is a particular solution of the dynamic system (1) [1].
We are now in a position to develop criteria for the dynamic system (1) to be
completely controllable and observable. We have the following theorem.
Theorem 1. The time scale dynamical system is completely controllable on
the closed interval J t t f� [ , ]0 if and only if the (n n� ) symmetric matrix
W t t t s B s B s t s sf
t
t f
( , ) ( , ( )) ( ) ( ) ( , ( )) ( )* *
0 0
0
� � � �
� �
is non-singular.
P r o o f. We first suppose thatW t t f( , )0 is nonsingular. Then it is claimed
that the dynamic system (1) is completely controllable. For given an (n�1) vec-
tor x0, choose
U t B t t t W t t xf( ) ( ) ( , ( )) ( , )* *� � �� � 1
0 0.
N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram
86 ISSN 0204–3572. Electronic Modeling. 2010. V. 32. ¹ 3
Clearly, the input signal U is continuous on J and the corresponding general
solution of (1) with the initial condition x t x( )0 0� is given by
x t t t x t s B s U s sf f
t
t f
( ) ( , ) ( , ( )) ( ) ( ) ( )� � �� �
0 0 0
0
� .
Substitute for U(t) and using the definition of W t t f( , )0 , we get
x t t t xf f( ) ( , )� �� 0 0
� �� �
( , ( )) ( ) *( ) *( , ( )) ( , ) ( )t s B s B s t s W t t x sf f
t
t
� �0
1
0 0
0
f
t t xf� � �� ( , )0 0
� �� � �( , ) ( , ( )) ( ) *( ) ( , ( )) ( , )t t t s B s B s t s W t t xf f0 0 0
1
0 0� �
( )s
t
t f
0
0� � .
Thus the dynamic system is controllable. This is true for all t t t f0 � � , it follows
that the system (1) is completely controllable.
Next suppose that the dynamic system (1) is completely controllable on J
and suppose that W t t f( , )0 is singular. Then since W t t f( , )0 is non-invertible
there exists a non-zero (n�1) vector y such that
y W t t C y t s B s B s t s y sf* ( , ) * ( , ( )) ( ) *( ) *( , ( )) ( ) (0 0 0� � �
� � s
t
t f
)
0
� .
Because of the fact that the integrand in this expression is non-negative continu-
ous function, we have
y t s B s* ( , ( )) ( )� 0 0� � ,
it follows that
y t s B s* ( , ( )) ( )� 0 0� � , s J� . (5)
Since the state equation is completely controllable on J, choose x0 = y, there
exists a continuous input U (t) such that
0 0
0
� � �� �
( , ) ( , ( )) ( ) ( ) ( )t t y t s B s U s sf f
t
t f
�
or
y t t t s B s U s sf f
t
t f
� � ��
� � �
1
0
0
( , ) ( , ( )) ( ) ( ) ( )�
Control of Time Scale Dynamical Systems with an Application
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 87
� � � �
( , ( )) ( ) ( ) ( )t s B s U s s
t
t f
0
0
� .
Thus,
y y y t s B s U s s
t
t f
* * ( , ( )) ( ) ( ) ( )� � � �
0
0
� ,
and since (5) holds, it follows that y*y = 0, and thus it contradicts the fact that
y � 0. ThusW t t f( , )0 is non-singular and the proof of the theorem is complete.
We now develop an algorithm corresponding to the time scale dynamical
system (1). We define a sequence of (n m� ) matrix functions
K t B t0( ) ( )� , (6)
K t A t K t K tj j j( ) ( ) ( ) ( )� �� �1 1
, j �1 2 3, , , ... . (7)
We observe the following: � � � �( , ( )) ( ) ( ( ))t s t s� �� � , when T = R, � ( )s s�
and � � � �( , ) ( ) ( )t s t s� � . On the other hand when T = Z, � ( )s s� �1 and
� � � �( , ) ( ) ( )t s t s� � ��1 1 . Further
� �
( , ( )) ( ) ( , ( ))t s A t t s� ��
ànd
[ ( , ( )) ] [ ( ) ( ( ))]� � �
t s t s� �� ��1
� � � �� �� � � �( ) [ ( ( )) ( ( ))] ( ) ( ( )) ( ( ))t s A s t s A s1 1� � � � .
Using the iteration idea given in (6), (7), we can easily verify the following:
�� �
( , ( )) ( ( ))] ( , ( )) ( ( ))t s B s t s K s
j
j� � � �� , j �0 1 2, , , ... ,
where
K s A s K s K sj j j( ( )) ( ( )) ( ( )) ( ( ))� � � ��� �� �1 1
, j �0 1 2, , , ... .
When � ( )s s t� � , we have
K t t s B sj s t( ) ( , ) ( )]� ���
, j �0 1 2, , , ... .
Based on the above iterative criteria, we have the following theorem.
Theorem 2. Suppose m is a positive integer such that for all t t t f�[ , ]0 , B is
m times continuously
-differentiable and A is (m – 1) times continuously
-dif-
ferentiable. Then the linear dynamic system (1) is completely controllable on
[ , ]t t f0 if for some t t tc f�[ , ]0 ,
Rank [ ( ), ( ), ..., ( )]K t K t K t nc i c m c0 � . (8)
N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram
88 ISSN 0204–3572. Electronic Modeling. 2010. V. 32. ¹ 3
P r o o f. Suppose rank condition (8) holds for some t t tc f�[ , ]0 . Then it is
claimed that the dynamic system (1) is completely controllable. To the contrary,
suppose the dynamic system (1) is not completely controllable on [t t f0 , ]. Then
by theorem 1, the Grammian matrix W [t t f0 , ] is non-invertible, and hence there
exists an (n � 1) vector y such that
y t t B t* ( , ) ( )� 0 0� , t t t f�( , )0 . (9)
Let y1 be a non-zero vector such that y t t rc1 0�� ( , ) . Then from (9), we have
y t t B tc1 0* ( , ) ( )� � , t t t f�( , )0 . In particular, at t = tc, we have y K tc1 0 0* ( ) � .
Now,
-differentiation with respect to t yields
y t t K tc c1 1 0* ( , ) ( )� � , t t t f�( , )0 .
This implies y K ti c1 0* ( ) � continuing in this way, we get y K tj c1 0* ( ) � for j = 0,
1, 2, …, m. Therefore,
y K t K t K tc i c m c1 0 0* [ ( ), ( ) ... ( )]� � ,
and this contradicts the fact that (8) holds. Thus the proof of the theorem is com-
plete.
We now turn our attention to the concept of observability on a time scale dy-
namical system. It is simpler to consider the case of zero input, and this does not
entail any loss of generability since the concept is not altered in the presence of a
known input signal. Therefore, we consider the unforced dynamical system
x t A t x t
( ) ( ) ( )� , x t x( )0 0� ,
y t C t x t( ) ( ) ( )� .
(10)
Definition 2. The time scale dynamical system (10) is said to be completely
observable on [t t f0 , ], if for any initial state x t x( )0 0� , it is uniquely determined
by the corresponding response y (t) for all t t t f�[ , ]0 .
We now present a necessary and sufficient condition for the system (10) to
be completely observable.
Theorem 3. The time varying time scale dynamical system (10) is com-
pletely observable on [t t f0 , ] if and only if the (n � n) symmetric observability
matrix
M t t s t C s C s s t sf
t
t f
[ , ] ( , ) ( ) ( ) ( , ) ( )* *
0 0 0
0
� � � �
is non-singular.
Control of Time Scale Dynamical Systems with an Application
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 89
P r o o f. Suppose that M t t f[ , ]0 is nonsingular. Then the solution ex-
pression withU t( ) �0 is given by y t C t t t x( ) ( ) ( , )� � 0 0, or
�* *( , ) ( ) ( )t t C t y t0 �� �* *( , ) ( ) ( ) ( , )t t C t C t t t x0 0 0.
Hence
�
* *( , ) ( ) ( ) ( )s t C s y s s
t
t f
0
0
� �
� �� � �
* *( , ) ( ) ( ) ( , ) ( ) ( , )s t C s C s s t x s M t t x
t
t
f
f
0 0 0 0
0
.
Since M is non-singular, x0 can be determined uniquely. Thus the dynamical sys-
tem (7) is completely observable.
Conversely, suppose the dynamic system (10) is completely observable.
Then it can be easily proved as in Theorem 1, that M t t f( , )0 is nonsingular.
4. Real-time database systems. In this section, we shall be concerned with
real time database systems and in fact concurrency control is one of the main is-
sues of real time data base systems. Many real world applications contain time
constraints to data as well as access to data that has temporal validity. Telecom-
munication is an example of an application area, which has database require-
ments that require a real-time database or at least time-cognizant database.
Most database requests are simple reads, with access to few and return to some
value based on the content in the database. Our main concern in this section is,
is there a distributed concurrence control method that is suitable for a real-time
database system.
Traditional databases deal with persistent data. Transactions access this data
while maintaining consistency. The goal of transaction and query processing in
database is to get a good response time. On the other hand, real time systems can
also deal with temporal data; i.e., data that becomes outdated after a certain
amount of time. Due to the temporal character of the data and the response time
requirements forced by the nature, tasks in real time system have time con-
straints. The main purpose of this section is that time goal of real time system is
met with jump operators. These jump operators play a crucial role in updating
data and ignoring the outdated data and in softening the required information and
object information. Efficient concurrency control protocols are required in order
for it to be possible to schedule real-time database transactions. This is achieved
by jump operators to schedule real-time database transactions. We first define
the notion of a «real time». Given t R� , we write � ( )� i t to denote the value of� i
at time t interpreting the clock as a counter. That is � ( ) { / ( ) }� �i it n n t� �Sup .
Here supremum is used in the maximum sense. If the supremum is taken over
N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram
90 ISSN 0204–3572. Electronic Modeling. 2010. V. 32. ¹ 3
an empty set, then it is zero, that is � ( )� i t = 0 if the clock is not ticked at all until
time t. It may be noted that a real-time transaction is a transaction with additional
real-time attributer: deadline, priority and importance. These attributes are used
by scheduling the real time algorithm and concurrency control method [6, 7].
We assume that every site contains a directory containing all objects and
their location. Further, every site contains data structures for keeping transaction
and object information. The transaction data structure contains information of
transaction and object information. The transaction data structure contains infor-
mation of transaction identification, the phase where a transaction is, transac-
tion’s read and write sets, and other information like administration. Before a
transaction can enter the read phase, we must first initialize data by using zero
( )x �0. Now the read phase starts with a begin operation. In the read phase if the
transaction reads an object several checks must be done. We first note that a
transaction requesting the data must be active and not aborted. Secondly, a re-
quested data item must not be marked as an validating object. Finally, if the ob-
ject is not located in the local node, a dead operation must be requested in the ob-
jects local node.
The importance of a real time database is its processing and its approach to
resolve data and resource conflicts. In real-time databases, timely transaction ex-
ecution is more important and both fairness and maximum resource utilization
become secondary goals. Further, the real time databases use the percentage of
transactions that complete within their deadlines. It is usually assumed that a
hard transaction can never come into conflict with any other transaction and hard
transactions cannot be aborted and will always complete successfully. Whereas
soft transactions might be in conflict with other soft transaction, and, if two soft
transactions attempt to obtain a read lock or write lock which violate the lock
compatibility, then the results in late transactions are considered to be in conflict
with each other. We first establish the following theorem which will be used for
further discussion.
Theorem 4. A hard database transaction can never enter a state of deadlock
caused by conflicts with any other database transaction.
P r o o f. Since hard database transaction can never be in conflict with any
other transaction it follows that conflicts can occur only in soft transactions.
These conflicts among soft transactions can be resolved in two ways : (1) If the
conflicting transaction occurs at a lower priority than any other conflicting soft
transaction, and has not entered the committing step, then it is aborted and thus
resolving the conflict. (2) If the conflicting soft transaction is executing at a
lower priority than any other conflicting soft transaction, and has entered the
committing step will be blocked until the transaction is complete and thus releas-
ing all its locks. Since a transaction, which has entered the committing step, can-
Control of Time Scale Dynamical Systems with an Application
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 91
not obtain any further locks, it cannot cause any further conflicts with any other
transaction, the proof is complete.
First, we initialize data and then local validation can be achieved by
SUCCESOR [SUCC (x) = x + 1] function which acts like iterating all objects ac-
cessed by the transaction, finds conflicting operation (if any), and resolves con-
flicts. The adjustment of time stamp intervals iterates through the READ set and
WRITE set of the validating transaction. This is achieved by the objects read and
write time stamp. When access has been made to the same objects both in the
validating transaction and in the active transaction, the temporal interval of the
active transaction is adjusted by the jump operators. Thus we use deferred dy-
namic time adjustment of the serial order.
The following serialization rule applies to each transaction.
Rule 1. A set of executing soft transactions are serialized in the order they
perform the end of transactions. This enables their changes visible for other
transactions.
Rule 2. A hard transaction, reading or writing the value of a data element x,
is serialized before all hard transactions reading or writing the value of x at a later
time. Further, the transaction is serialized before any soft database transaction
obtaining a lock on x at a later time.
N. Ramesh Babu, K.N. Murty, V.V.S.S.S. Balaram
92 ISSN 0204–3572. Electronic Modeling. 2010. V. 32. ¹ 3
T Event Database State Comments
T1 INITIALIZE
BOT
Zero (x)
{x, y}
Zero
T1 starts
T1 W-lock (x) W t
DIFF W t W t
DIFF t t
( )
( ( ), ( ))
( , )
�
�
�
1
Lock is removed
DIFF (x, y) = x – y
SUCC (T1)
W t DIFF t t( ( )) ( ( ), ( ))� � �� � 1
T2
W t DIFF t t( ( )) ( ( ), ( ))� � �� � �1
Goes to the next event T2
T2 Write (x) � x� {x, y} T2 pre-empts T1 and update x.
T2 is serialized after T1.
T3 Write (y)� �y {x�, y�} T3 update y. Since y is not yet
write locked on T1. T3 is serialized
before T1 according to Rule 2.
T1 Upd (y1) �y�� T1 update y however this update is
visible for other transactions.
T1 EOT {x�, y��} T1 ends and releases its lock.
Y11 is now visible.
Rule 3. A hard transaction, updating the value of a data element currently
locked by a soft transaction, is serialized only after that transaction.
We first need to verify that whether or not transactions always read the cor-
rect version of a data element, i.e., the value produced by the last serialized trans-
action updating that particular transaction and no intermediate results produced
by executing transactions are visible to other transactions i.e., we need to verify
consistency of transactions.
Now we consider three transactions T1, T2 and T3 executed in Table.
Note that T1 is a soft transaction T2 and T3 are hard transactions. The algo-
rithm can easily extended to n transactions n 3. In order to avoid monotony we
even omit formulating the algorithm.
From the example we see that the resulting serialization order is T3, T1 and
T2 even thorough the actual order of commit is T2, T3 and T1. This serialization
approach trades a relaxation of serialization for freshness of data.
Óçàãàëüíåíî ðåçóëüòàòè äîñë³äæåíü êåðîâàíîñò³ òà ñïîñòåðåæóâàíîñò³ ïðè ìàñøòàáóâàíí³
çà ÷àñîì. Îòðèìàíî ðåçóëüòàòè êëàñè÷íî¿ òåî𳿠ÿê îêðåìîãî âèïàäêó, ïðè öüîìó óñóíóòî
îáìåæåííÿ çà ÷àñîì ïðè ïàðàëåëüíîìó êåðóâàíí³ îïåðàòîðàìè ïåðåõîäó äëÿ äèíàì³÷íèõ
ñèñòåì, ùî ìàñøòàáóþòüñÿ.
1. Lakshmikantham V., Sivasundaram S., Kaymakelan B. Dynamic Systems on Measure
Chains. — Kluwer Academic Publishers, 1996.
2. Aulbach B., Hilgeu S. A Unified Approach to Continuous and Discrete Dynamics. Qualita-
tive Theory of Differential Equations. — Szeged, Hungary 53. — 1983.
3. Aulbach B., Hilgeu S. Linear Dynamic Process with Inhomogeneous Time Scales. Nonlinear
Dynamic and Dynamical Systems. — Berlin : Academic Verlag, 1980.
4. Amer Abu Ali. An Optimistic Concurrency Control for Real-time Database Systems// Amer.
J. of Applied Sciences. — 2006. — N 3 (2). — P. 1706—1710.
5. Barnett S., Cameron R. G. Introduction to Mathematical Control Theory. — 2nd Edition. —
Oxford University Press, 1985.
6. Ramamritham K. Real-Time Databases.// Int. J. of Distributed and Parallel Databases.—
1993. — 1, N 2. — P. 199—226.
7. Murty K. N., Rao Y. S. Two Point Boundary Value Problems on Inhomogeneous Time Scale
Dynamic Process // J. of Mathematical Analysis and Applications. — 1994. — 184. —
P. 22—34.
Submitted on 30.12.09
Control of Time Scale Dynamical Systems with an Application
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2010. Ò. 32. ¹ 3 93
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/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|