=Paper=
{{Paper
|id=None
|storemode=property
|title=Computer's Analysis Method and Reliability Assessment of
Fault-Tolerance Operation of Information Systems
|pdfUrl=https://ceur-ws.org/Vol-1356/paper_52.pdf
|volume=Vol-1356
|dblpUrl=https://dblp.org/rec/conf/icteri/AtamanyukK15a
}}
==Computer's Analysis Method and Reliability Assessment of
Fault-Tolerance Operation of Information Systems==
https://ceur-ws.org/Vol-1356/paper_52.pdf
Computer’s Analysis Method and Reliability Assessment
of Fault-Tolerance Operation of Information Systems
Igor P. Atamanyuk1, Yuriy P. Kondratenko2
1
Mykolaiv National Agrarian University, Commune of Paris str. 9,
54010 Mykolaiv, Ukraine
atamanyukip@mnau.edu.ua
2
Petro Mohyla Black Sea State University, 68th Desantnykiv Str. 10,
54003 Mykolaiv, Ukraine
yuriy.kondratenko@chdu.edu.ua
Abstract. In this paper there was obtained an calculation method of the assess-
ment of the probability of fail-safe operation of information systems in the fu-
ture instants of time. The method is based on the algorithm for modeling a pos-
teriori nonlinear random sequence of change of values of the controlled parame-
ters which is imposed a limitation of belonging to a certain range of possible
values. The probability of fail-safe operation is defined as the ratio of the num-
ber of realizations that fell in the allowable range to the total number of them,
formed as a result of the numerical experiment. The realization of an a posterio-
ri random sequence is an additive mixture of optimal from the point of view of
mean-square nonlinear estimate of the future value of the parameter analyzed
and of the value of a random variable, which may not be predicted due to the
stochastic nature of the parameters. The model of a posteriori random sequence
is based on the Pugachev's canonical expansion. The calculation method offered
does not impose any significant constraints on the class of random sequences
analyzed (linearity, stationarity, Markov behavior, monotoneness, etc.).
Keywords. calculation method, random sequence, canonical decomposition, es-
timation, probability.
Key Terms. computation, mathematical model.
1 Introduction
One of the most important problems that arises constantly in the process of infor-
mation and communication systems service is the problem of the estimation of system
reliability with the following making decision about the possibility of its further ex-
ploitation on the basis of the information about possible failures in the future [1-5].
The problem becomes especially important in the case when information system is
used for the management of the objects that relate to the class of critical or dangerous
and under the threat of accident objects (aircraft, sea mobile objects, nuclear power
stations, chemical industry plants etc.) [6-8]. The forecasting of failures is the primary
stage of the providing of the dependability of the systems of such class. However,
nowadays the problem of failures forecasting (in overwhelming majority cases) is
solved by means of informal methods and the decision about the possibility of de-
pendable exploitation of the corresponding information system is made on the basis
[9-11]:
─ qualitative and quantitative estimation of its current state;
─ the experience of the exploitation of the given and analogous information systems.
As far as information and communication systems (ICT) become more complicated
and also as far as there is the growth of requirements to the probability of their relia-
ble work, informal methods of making decision are becoming less and less effective.
Hence, the usage of stricter approaches to the estimation of the dependability and
reliability of ICTs functioning based on the quantitative estimations of future state of
experimental information systems is very important.
2 Statement of the problem
Accuracy of the obtained solutions for correspondent class of problems and the
time during which given solutions can be obtained are the most universal parameters
characterizing the quality of information system functioning. Stated parameters (accu-
racy and time) have stochastic character because of the presence of inner defects of
the system and uncontrolled external destabilizing factors. That’s why the problem of
the forecasting control of information system reliability can be formulated in the fol-
lowing way.
Without the restriction of the generality of the next calculations let us assume that
the state of the information system in exhaustive way is determined with scalar pa-
rameter X (accuracy or time of problem solution). The change of the values of the
parameter X in discrete range of points ti , i 1, I is described by random sequence
X X (i), i 1, I . The values of the parameter X must satisfy the condition
a x(i) b, i 1,I (1)
In the case of the crossing by parameter X the limits of acceptable area [a; b] the
failure of information system in the process of its functioning is registered. The state
of the system is controlled periodically in discrete moments of time t , 1, k by
measurement of the values x , 1, k of the parameter X where x , 1, k
is the realization of the random sequence X in the cut set t , 1, k . It is evident
that for the segment of the time [t1; tk ] the inequation a x( ) b, 1, k must be
correct. Otherwise as it follows from (1) under
x i a, i 1,I
or
x i b, i 1,I
on the interval of examination [t1; tk ] the failure would take place that would lead to
the suspension of the process of information system functioning.
The statement of the problem can be formulated in the following way: on the basis
of stated (measured) information about current values of the parameter X on the time
interval [t1; tk ] the conclusion about the usability of the information system to exploi-
tation in future moments of time ti , i k 1, I must be made.
Similarly the problem of providing of dependable functioning and forecasting of
the state of the technical systems or objects to the structure of which information sys-
tems or management-information systems enter in the form of components can be
formulated. Herewith the dependability of functioning and usability of such compli-
cated systems and objects certainly depend on reliability and fail-safety of all their
components..
3 Solution
Given that the value of the controlled parameter X changes randomly within the
forecast region, the probability of fail-safe operation becomes an exhaustive feature
of the safety of functioning of the information system examined.
P(k ) ( I ) P{a X (k ) (i) b, i k 1, I / x( ), 1, k}. (2)
The problem is thus reduced to the determination of the probability of non-falling
of the realization of the a posteriori random sequence
X (k ) (i / x( ), 1, k ),i k 1, I outside the limits of the permissible region [a; b] .
In [12], [13] there was proposed an approach to the estimation of likelihood (2)
through multiple statistical modeling of possible extensions xl (i), i k 1, I , l 1, L
of the random sequence analyzed { X } in the forecast region, verification for each
realization of condition (1) and calculation as a result of the required estimation ex-
periment P*(k ) ( I ) n L ( n is the number of successes). In this method, its canoni-
cal expansion [14] within the range of points analyzed is used as a model of the ran-
dom sequence ti , i 1, I :
i
X (i) M X (i ) V (i ), i 1, I , (3)
1
where V , 1, I - random coefficients: M [V ] 0, M [V V ] 0 for
, M [V ] D ;
2
(i), i, 1, I - nonrandom coordinate function: ( ) 1, (i) 0 under > i .
Elements of the canonical representation (3) are defined by the following recur-
sions:
i 1
V (i) X i M [ X (i)] V (i), i 1, I , (4)
1
i 1
Di M [ X 2 (i)] {M [ X (i)]}2 D 2 (i), i 1, I ; (5)
1
1
(i) {M [ X ( ) X (i)] M [ X ( )]M [ X (i)]
D
(6)
1
D j j ( ) j (i)}, 1, I , i , I .
j 1
Tipping in the expression (3) of known values X ( ) x( ), 1, k converts the a
priori random sequence into the a posteriori one:
i
X ( k ) (i) mx( k ) (i) Vvv (i), i k 1, I , (7)
v k 1
where mx( k ) (i) - linear optimal, by the criterion of mean square minimum of predic-
tion error, estimate of the future value of the random sequence X at the point ti
according to the known initial values of k .
Expressions for finding mx( k ) (i) have two equivalent forms of notation
M [ X (i)], if =0, i =1,I ;
mx( ) (i) ( 1) (8)
mx (i) [ x( ) mx( 1) ( )] (i), =1,k , i = +1,I ;
or
k
mx( k ) (i) M [ X (i)] ( x( ) M [ X ( )]) f ( k ) (i), i = k +1,I ; (9)
j 1
( k 1)
f (i) f (k 1) (k )k (i), k - 1;
f ( k ) (i) (10)
k (i), =k ;
Formation of possible extensions of random sequence { X } by the expression (7) is to
compute estimates mx(k ) (i),i k 1,I , generating one of the known methods of statis-
tical modeling of values of independent random coefficients V , k 1, I with the
required distribution law and transforming of the values obtained by the coordinate
functions (i), i, k 1, I .
The calculation method of forecasting of fail-safe operation of information systems
on the basis of the model (7) covers a fairly wide class of random sequences (non-
markovian, non-stationary, non-monotonic, etc.), but this representation of an a poste-
riori random sequence is optimal only within the framework of linear stochastic prop-
erties, thus reducing significantly the accuracy of prediction of random sequences,
which have non-linear links.
The clearing of this trouble is possible through the use on the basis of estimation
method of the probability of fail-safe operation of an information system of nonlinear
canonical expansion of the random sequence [15], changing values of the parameter
controlled:
i N 1
X (i) M [ X (i)] V( )1( ) (i), i 1, I . (11)
1 1
Elements of the expansion (11) are determined by the following recursions:
1 N 1 1
V( ) X M [ X i ] V( j )
( j)
V( j )
( j)
, 1, I ; (12)
1 j 1 j 1
1 N 1
D M [{ X M [ X ]}2 ] D j ( ){
( j)
}2
1 j 1
(13)
1
D j ( ){ } , 1, I ;
( j) 2
j 1
h( ) (i )
1
{M [ X ( ) X h (i)] M [ X ( )]M [ X h (i)]
D ( )
1 N 1 1
(14)
D j ( )
( j)
( )h( j ) (i ) D j ( )
( j)
( )h(j ) (i )}, 1, h, 1, i.
1 j 1 j 1
In the canonical expansion (11) the random sequence { X } is represented in the range
of points analyzed ti , i 1, I via N 1 the arrays {V ( ) }, 1, N 1 of uncorrelated
centered random coefficients. Vi( ) , 1, N 1, i 1, I . These coefficients contain
information on the values X (i), 1, N 1, i 1, I , and the coordinate functions
h( ) (i), , h 1, N 1, , i 1, I describe probabilistic links of the order h between
the sections t and ti , , i 1, I .
Block-diagram of the procedure for calculating the parameters of the canonical de-
composition is shown in Fig. 1.
The concretization of values X ( ) x ( ), 1, N 1, 1, k allows to move
from the a priori random sequence (11) to the a posteriori one:
i N 1
( ) ( )
X ( k ) (i) mx( k , N 1) (1, i) V 1 (i), i 1, I . (15)
k 1 1
The expression mx(k ,l ) (1, i) M [ X (i) / x ( j ), j 1, k , 1, N 1] is the conditional
expectation of a random sequence providing that values x ( j ), 1, N 1, j 1, k are
known and the process analyzed is fully specified by the discretized moment func-
tions M [ X ( )], M [ X ( ) X h (i)], , h 1, N 1, , i 1, I .
The computing algorithm mx(k ,l ) (1, i) M [ X 1 (i) / x ( j ), j 1, k , 1, N 1] has
two equivalent forms of notation [16]:
M [ X (i )], 0;
h
mx( ,l ) (h, i ) mx( ,l 1) (h, i ) ( xl ( ) mx( ,l 1) (l , ))h(l) (i ), l 1, (16)
mx( 1, N 1) (h, i ) ( x( ) mx( 1, N 1) (1, ))h(1) (i ), l 1
or
k N 1
mx( k , N 1) (1, i) M [ X (i)] x ( j ) F((( kj(N1)(1))
N 1) ) ((i 1)( N 1) 1), (17)
j 1 1
where
( 1)
F ( ) F( 1) ( ) k (i), -1;
F( ) ( ) (18)
( ), = ;
(mod N 1 ( )) ([ / ( N 1)] 1), for k ( N 1);
1,[ /( N 1)]1
( ) (19)
(mod ( ))
1,[ /(NN11)]1 (i), if =(i-1)(N -1) 1.
The simulation procedure of the a posteriori random sequence (15) assumes that
densities of random coefficients Vi( ) , 1, N 1, i 1, I are known. The simplest and
the most effective solution to the problem of determining these one-dimensional den-
sities is to use nonparametric parse type estimates [17]. Together with this the esti-
mate of the required density of distribution f (Vi( ) ) of the random variable Vi( )
according to L of its realization vi(,l ) , l 1, L is represented as
Fig. 1. Block-diagram of the procedure for calculating the parameters of the canonical decom-
position (11).
1 L
f L (Vi( ) ) g (ul ), (20)
dL l 1
where ul d 1 (vi( N ) vi(,N )
l ) , g (ul ) - certain weight function (kernel);
d - constant (blur coefficient).
The estimate (20) at all points of the determination region is obtained unbiased,
consistent and uniformly converges on the desired distribution density f (Vi( ) ) with
probability one, if the weight function fulfils the condition
g (u) 0 ; sup g (u ) ; lim ug (u ) 0 ; g (u )du 1 . (21)
u u
The constant d is selected depending on the number of observations subject to the
conditions
d 0; lim d ( L) 0 ; lim d ( L) L . (22)
L L
When selected as the kernel function g (u ) the uniform density distribution blur
coefficient is determined on the basis of the correlation
d 0,5sup vi(,N ) (N) (N) (N)
l vi,l 1 , vi,l vi ,l 1 , l 2, L. (23)
l
Thus, the offered calculation method of polynomial predictive control of fail-safe
operation of information systems consists of the following stages:
─ construction on the basis of the known a priori information
M [ X ( )], M [ X ( ) X (i)], , h 1, N 1, , i 1, I . of the canonical expan-
h
sion (11) of the random sequence of change of the controlled parameter X ;
─ determination from the formula (16) or (17) the values
mx (1, i) M [ X (i) / x ( j ), j 1, k , 1, N 1]
(k ,l )
of the conditional expectation
[t ...t ]
of the random sequence analyzed within the forecast region k 1 I using the
x ( j ), 1, N 1, j 1, k [t ...t ]
known values on the observation interval 1 k ;
─ multiple simulation of values of random coefficients
Vi( ) , i k 1, I , 1, N 1
under the distribution law (20) and formation using
the expression (15) of the set of possible extensions of the realization of the ran-
[t ...t ]
dom sequence within the forecast range k 1 I ;
─ verification of conditions of non-crossing by the paths obtained of the bounda-
[a; b]
ries of the admissible region of the controlled parameter change X and
determination of the estimate of the probability of fail-safe operation of the in-
formation system as the ratio of the number of successes to the total number of
the experiments conducted.
Increasing the reliability of the estimate of the probability of fail-safe operation on
the basis of the model (15) compared to (7) is achieved by using nonlinear stochastic
properties of the random sequence analyzed: there rises the accuracy of determination
of conditional expectation and reliability of possible paths of a random sequence in
the forecast region through the use in the process of simulation of an additional array
of random coefficients Vi( ) , i k 1, I , 2, N 1 . The gain in accuracy can be
estimated using the expression:
k N 1 k
mx( k , N 1) (1, i ) mx( k ) (i ) { D ( j )(1(j ) (i))2 D j2 (i)}1/2
j 1 1 j 1
e[(ak,)b] . (24)
ba
Let a random sequence X in the discrete row of points ti , i 1, I is set by in-
stant functions: M X l i pl 1 X l 1 i pl 2 ...X 2 i p1 X 1 i ,
l
j N , p j 1, i 1, i 1, I . Decomposition of random sequence X looks like
j 1
[18],[19]:
i 1 N 1
X i M X i V (1) (1)(1) , i V1 i
11(1) 1 1 1
i 1 M ( ) p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... V p(l ) ... p(l ) (l ) ... (l ) (25)
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l 1; 1 l
(1)(l )
p1 ... pl(l )1;1( l ) ...l( l )
, i , i 1, I ,
where
, if N ,
M
N , if N ;
0, if j 1, l 1 or l 1,
p'(j l )
v l j , if j 1, l 1, l 1;
1
'(l ) N l (jl ) , 1, l.
j 1
The casual coefficients of canonical presentation (9) are defined by the expres-
sions:
1 N 1 1
V1 X 1 M X 1 V (1) (1)(1) , V (1) ( )
1 (1) 1
1 (1) 1
1 1 1 1
1 M ( ) p1'( l ) pl'(l1) 1'( l ) l'( l )
( )
(1)1 , ... ... V p(l ) ... p(l ) (l ) ... (l ) (26)
1 2 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l(l ) 1 1 l 1; 1 l
( )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
, , 1, I .
The coefficients V ,...
1 n 1;1 ,... n
which contain information about the values
X n n1 ... X 1 are calculated as
V ,...
1 n 1 ;1 ,... n
X n n1 ... X 1 M X n n1 ... X 1
N 1 ( ,... n 1 ;1 ,... n )
V (1) ( ) (1)1
1
,
1 1(1) 1 1
1 M ( ) p1
'( l )
pl'(l1) 1'( l ) l'( l )
... ... V p (l ) ... p(l ) (l ) ... (l )
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l 1; 1 l
( ,... n 1;1 ,... n )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
,
(27)
n 1 p1'( l ) pl'(l1) 1'( l ) l'(l )
... ... V p(l ) ... p(l ) (l ) ... (l )
l 2 p1( l ) 1 l 1; 1
pl(l )1 pl(l )2 1 1( l ) 1 l( l ) 1 1 l
( ,... n 1 ;1 ,... n )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
,
p1*( n ) pn*(n1) 1*( n ) n*( n )
( ,... ; ,... )
... ... V p ( n ) ... p( n ) ( n ) ... ( n ) p( n1) ... pn(n1) 1( n ) ...n( n ) , .
n 1; 1
pl(n1) pl(n2) 11( n ) 1 n( n ) 1
n n 1; 1
p1( n ) 1 1 1 n
In (27) parameters p1*(n) ,..., pn*(n1) ; 1*(n) ,..., n*(n) are calculated by the following
expressions:
, if 1, p( n)1 , 2, n,
pj ( n) (28)
v l , if p 1 1 , 2, n.
( n)
, if i 1, ( n) , i 2, n;
i 1
i j
j ( n) i 1
(29)
N n i j , if j i 1 , i 2, n.
( n) ( n)
j 1
The values * , 1, n 1; *i , i 1, n, are the indexes of casual coefficient
V * ... *
1 n 1 ; 1... n
* * which proceeds V1...n1;1...n in canonical decomposition
(25) for the moment of time t :
1. * , 1, n 1; *i i , i 1, k 1; *k k 1;
j 1
* j N n j *m , j k 1, n; if k 1, j 1, j k 1, n;
m 1
2. , 1, k 1; *k k 1; * j n j, j k 1, n 1; *i N n i
*
j 1
*m , i 1, n, if i 1, i 1, n; k k 1 1; j j 1 1; j k 1, n 1;
m 1
3. 0; *i 0; V * ... * ; * ... * ( ) 0, if , 1, n 1; i 1, i 1, n.
*
1 n 1 1 n
The expressions for the determination of the dispersion D1 , of casual coeffi-
cients V1 are:
1 N 1 ( )
2
D1 M X 21 M 2 X 1 D (1) (1)1 ,
1 (1)
1
1
1 1
1 1 2
( )
D (1) (1)1 ,
1
1 1
(1) 1
(30)
1 M ( ) p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... D p(l ) ... p(l ) (l ) ... (l )
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l(l ) 1 1 l 1; 1 l
2
( )
(l1) (l ) (l ) ( l ) , , 1, I .
1
p ... p
l 1; 1 ... l
Dispersions D ,...
1 n 1;1 ,... n
of casual coefficients V ,...
1 n 1;1 ,... n
are
defined as
D ,...
1 n 1 ;1 ,... n
M X 2 n n1 ... X 21
1 N 1
M 2 X n n 1 ... X 1 D (1)
1 (1) 1
1 1
2
( ,... ; ,... )
(1)1 n 1 1 n ,
1
1 M ( ) p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... D p(l ) ... p(l ) (l ) ... (l )
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l 1; 1 l
2
( ,... ; ,... )
(l1) n(l)1 (l1) (nl ) ,
1
p ... p
l 1; 1 ... l
(31)
n 1 p '1( l ) p 'l( l)1 '1( l ) 'l( l )
... ... D p(l ) ... p(l ) (l ) ... (l )
l 2 p1( l ) 1 l 1; 1
pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l
2
( ,... ; ,... )
(l1) n(l)1 (l1) (nl ) ,
p1 ... pl 1;1 ...l
p1*( n ) pn*(n1) 1*( n ) n*( n )
... ... D p ( n ) ... p( n ) ( n ) ... ( n )
n 1; 1
pl(n1) pl(n2) 1 1( n ) 1 n( n ) 1
n
p1( n ) 1 1
2
( ,... ; ,... )
( n1 ) n(n1) 1( n ) n( n ) , , 1, I .
1
p ... p
n 1; 1 ... n
The coordinate functions of canonical decomposition (25) are defined by the for-
mulas:
─ to describe the relationship between the value X 1
and X am i bm1 ... X a1 i
D
1 1
(b1...bm 1;a1...am ) , i M X X i b ... X i am
m 1
a1
1
1
M X 1 M X am i bm 1 ... X a1 i
1 N 1 ( )
D (1) (1)1 , (1)1 , i
(b ...bm 1;a1...am )
1 1(1) 1 1 1 1
1 1 (32)
( )
D (1) (1)1 , (1)1 , i
(b ...bm 1 ;a1...am )
1 1
1(1) 1 1
1 M ( ) p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... D p (l ) ... p(l ) (l ) ... (l )
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 1 1( l ) l 1; 1
l( l ) 1 l
( )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
, (pb(1l...) ...bmp(l1);a1(...l )a...m)(l ) , , 1, I .
1 l 1; 1 l
─ to describe the relationship between the value X n n1 ... X 1 and
X am i bm1 ... X a1 i
(b1 ,...bm 1;a1 ,...am ) , i
1 ,... n 1 ;1 ,... n D ,...
1
M X ...X
n
n 1
1
1 n 1 ;1 ,... n
X am i bm 1 ... X a1 i M X n n 1 ... X 1
(33)
M X am i bm 1 ... X a1 i M X n n 1 ... X 1
M X i bm 1 ... X i
am a1
1 N 1 ( ,... n 1 ;1 ,... n )
D ( (1)
j)
(1)1
1 1
, (b(1)1...bm 1;a1...am ) , i
11(1) 1 1
1 M ( ) p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... D p(l ) ... p(l ) (l ) ... (l )
1 l 2 p1( l ) 1 pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l 1; 1 l
( ,... n 1;1 ,... n )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
, (pb(1l ,...
)
bm 1 ;a1 ,...am )
... p ( l ) ( l ) ... ( l )
, i
1 l 1; 1 l
n 1 p '1
(l )
p 'l( l)1 '1( l ) 'l( l )
... ... D p(l ) ... p(l ) (l ) ... (l )
l 2 p1( l ) 1 l 1; 1
pl(l )1 pl(l )2 11( l ) 1 l( l ) 1 1 l
( ,... n 1;1 ,... n )
(l1)
p1 ... pl(l )1;1( l ) ...l( l )
, (pb(1l ,...
)
bm 1; a1 ,...am )
... p ( l ) ( l ) ... ( l )
, i
1 l 1; 1 l
p1*( n ) pn*(n1) 1*( n ) n*( n )
... ... D p( n ) ... p( n ) ( n ) ... ( n )
n 1; 1
pl(n1) pl(n2) 11( n ) 1 n( n ) 1
n
p1( n ) 1 1
( ,... ; ,... ) , 1, I .
( n1) n(n1) 1( n ) n( n ) , (1n ) m(n1) 1( n ) m ( n ) , i
(b ,...b ;a ,...a )
p1 ... pn 1;1 ... n p1 ... pn 1;1 ... n
Tipping in the expression (3) of known values X ( ) x( ), 1, k converts the a
priori random sequence into the a posteriori one:
X i M X i mx( I N 1, I N 2,..., I 1;1,1...1, ) 1, i
i 1 N 1
V (1) (1) , i V1 i
(1)
k 11(1) 1 1 1
(34)
i 1 M ( ) p '1
(l )
p 'l(l)1 '1(l ) 'l(l )
... ... V p(l ) ... p(l ) (l ) ... (l )
k 1 l 2 p(l ) 1 p(l ) p(l ) 1 (l ) 1 (l ) 1 1 l 1; 1 l
1 l 1 l 2 1 l
(1)(l )
p1 ... pl(l )1;1(l ) ...l(l )
, i , i k 1, I .
Values of conditional expectation are defined as
( x i mx( 1...n 1;1...n , ) b1...bm1; a1...am , i
M X am (i b )... X a1 (i ) , 0;
m 1
( *1... *n 1; *1...... *n , )
mx b1...bm1; a1...am , i x n n1 ...x1
m( 1... n 1; 1...... n , ) ... ; ... ,
* * * *
n 1 1
x 1 n
(b ...b ;a ...a )
x i 1... m 1; 1......m , i ,if *1 0,..., *n 0;
1 n 1 1 n (35)
( p '( n 1) ... p'( n 1) ; '( n 1) ... '( n 1) , )
mx 1 n2 1 n 1
b1...bm1; a1...am , i x n n1 ...
1
...x mx
( p1'( n 1) ... pn'( n 21) ;1'( n 1) ... n'(n11) , )
1... n1;1... n ,
1 m 1 1 m , i , if * 0,..., * 0.
( b ...b ; a ...a )
1... n 1;1...... n 1 n
Technology of predictive control of fail-safe operation on the model (35) is the
same as using the expression (15).
4 Conclusion
Thus, there we obtained an calculation method for the estimation of the probability
of fail-safe operation of information systems in the future instants of time. The tech-
nology is based on the model of the canonical expansion of the a posteriori random
sequence of changes of the parameter controlled. Estimation of the probability of fail-
safe operation of a information system based on the results of the numerical experi-
ments is determined as a relative frequency of an event characterized by belonging of
the realization to the allowable region on the forecast interval. The calculation method
offered does not impose any significant constraints on the class of random sequences
analyzed (linearity, stationarity, Markov behaviour, monotoneness, etc.). The only
constraint is the finiteness of variance that is usually performed for real random pro-
cesses. In contrast to the known solutions [12], [13] the suggested estimation proce-
dure for fail-safe operation of information systems allows for nonlinear stochastic
properties of the random sequence analyzed, which improves the accuracy of the pre-
dictive control procedure.
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