=Paper=
{{Paper
|id=Vol-1614/paper_79
|storemode=property
|title=Calculation Methods of the Prognostication of the Computer Systems State under Different Level of Information Uncertainty
|pdfUrl=https://ceur-ws.org/Vol-1614/paper_79.pdf
|volume=Vol-1614
|authors=Igor Atamanyuk,Yuriy Kondratenko,Vyacheslav Shebanin
|dblpUrl=https://dblp.org/rec/conf/icteri/AtamanyukKS16
}}
==Calculation Methods of the Prognostication of the Computer Systems State under Different Level of Information Uncertainty==
https://ceur-ws.org/Vol-1614/paper_79.pdf
Calculation Methods of the Prognostication
of the Computer Systems State under Different Level
of Information Uncertainty
Igor P. Atamanyuk1, Yuriy P. Kondratenko2,3 , Vyacheslav S. Shebanin1
1
Mykolaiv National Agrarian University, Commune of Paris str. 9,
54010 Mykolaiv, Ukraine
atamanyuk_igor@mail.ru, rector@mnau.edu.ua
2
Petro Mohyla Black Sea State University,
68th Desantnykiv Str. 10, 54003 Mykolaiv, Ukraine
3
Cleveland State University,
2121 Euclid Av., 44115, Cleveland, Ohio, USA
y.kondratenko@csuohio.edu, y_kondrat2002@yahoo.com
Abstract. Calculation methods of the prognostication of the computer systems
state under different volume of a priori information and accuracy of the meas-
urement of controlled parameters (under absence and presence of measurement
errors) are obtained in the work. Canonical expansions of random sequences of
the indices characterizing the state of the investigated systems considered as ba-
sic features of the methods. Synthesized methods do not impose any significant
limitations on the qualities of the sequence of the change of the forecast pa-
rameters (linearity, stationarity, Markov behavior, monotoneness, etc.) and al-
low to take into account the stochastic peculiarities of the process of function-
ing of the investigated objects as much as possible. Expressions of the determi-
nation of a mean-square extrapolation error are obtained for solving the prog-
nostication problems specifically concerning the state of computer systems un-
der different level of information uncertainty.
Keywords. calculation method, random sequence, canonical decomposition,
prognostication of the state
Key Terms. computation, mathematical model
ICTERI 2016, Kyiv, Ukraine, June 21-24, 2016
Copyright © 2016 by the paper authors
- 293 -
1 Mathematical statement of the problem of the prognostication of
a technical condition
One of the most important problems that arises constantly in the process of the opera-
tion of computer systems and computerized control systems [1,2,3] is based on quite
evident fact that any decision about the permission of the system operation (of the
realization of a stated problem) is closely connected with the solving of the prognosis
problem. For example, the forecast of the remaining functioning time is a rotating
machinery prognosis, results of which can be used also for forecasting of the reliabil-
ity of machinery components (and additional equipment) as well as for forecasting
future operational conditions. This kind of prognosis is based on the output data of
multi-sensor monitoring system and current results of data processing. The main goal
of such prognosis deals with: (a) reducing downtime of the machinery and corre-
sponding equipment; (b) optimizing spares quantity; (c) decreasing functioning cost;
(d) increasing safety of the machinery maintenance. In [4] authors analysis known
methods of rotating machinery prognosis and classify the approaches to three groups
based on different models: (a) general reliability, (b) environmental conditions, (c)
combining prognostication and reliability.
Special attention should be paid to prognostication of manufacturing and industrial
systems [5]. The results of such prognosis can help to determine the most rational
maintenance modes for long-time functioning of different computer-integrated tech-
nological complexes. Modelling the degradation mechanism and dynamical degrada-
tion monitoring of the most important components of computer-integrated technologi-
cal complexes are the base for prognosis in [5] within an e-maintenance architecture.
Different forecasting methods can be used for short-term electric load prognosis
[6,7] at the enterprises, plants, cities and regions. The surveys of the prognosis meth-
ods based on applying Kalman filter, state space models, linear regression, stochastic
time series, various smoothing algorithms, and artificial intelligence methods as well
as the analysis of their application for solving prognostication problems is presented
in [6,7] with implementation to short-term electric load prognosis.
The uncertainties of functioning conditions, external environment, nonstationary
parameters and working modes and problems with their mathematical formalization
are the main obstacles in using efficient computer models for forecasting future be-
havior of complex technological systems. As example, in [8] authors consider a spe-
cial Bayesian computer model for prognostication of the active hydrocarbon reservoir
future functioning.
Last years, such powerful theoretical-applied tool as the theory of neuro-fuzzy sys-
tems has been introduced successfully for solving different prognostication tasks in
engineering, medicine, investment policy, finance and other fields [9,10,11,12,13].
According to reliability of control systems, it is necessary to note that computers
are the main components of embedded controllers (traditional, fuzzy, neuro, etc.). The
main critical requirement deals with providing efficient functioning such systems and
networks in normal and in failure modes, when any component fails. One of the effi-
cient approach for design process of such control systems is based on the applying
redundant-elements-design-method [14].
- 294 -
Two most important indexes can be taken in to account in solving forecasting tasks
for the e-business systems: (a) insufficient speed of response and (b) preventing fail-
ure of the system. In [15] author consider predictive inputs for the designed prognosis
system as intrinsic (component activity levels, system response time, etc.) and extrin-
sic (time, date, whether, etc.) variables.
The prognostication in computer networks is described in [16] for forecasting the
level of computer virus spread based on two models of viral epidemiology (differen-
tial equation model and the discrete Markov model).
The problem of forecasting control is especially topical for computer systems
which are 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, nu-
clear power stations, chemical industry plants etc.) [17,18,19].
Computer systems are exploited in the conditions of continuous influence on the
great number of external and internal perturbing factors, the influence on the object of
which is random by the moment of origin, duration and intensity. And correspond-
ingly the changes of the system state also turn out to be random and form a random
sequence. Thereupon the extrapolation of the realization of the random sequence de-
scribing the functioning of the investigated system on a certain interval of time is the
mathematical content of the problem of the prognostication of a technical condition.
The most general extrapolation form for the solving of the problem of non-linear
extrapolation is a Kolmogorov-Gabor polynomial [20] but it is very difficult and labo-
rious procedure to find its parameters for the great number of known values and used
order of non-linear relation. Thereupon during the forming of realizable in practice
algorithms of the prognosis different simplifications and restrictions on the qualities
of random sequence are used. For example, a range of suboptimal methods of non-
linear extrapolation with a limited order of stochastic relation on the basis of ap-
proximation of a posteriori density of probabilities of an estimable vector by an
orthogonal expansion by Hermite polynomials or in the form of Edgeworth series
was offered by V.S. Pugachev [21]. Solution of non-stationary equation of A.N. Kol-
mogorov [20] (particular case of differential equation of R. L. Startanovich for the
description of Markovian process) is obtained provided that the drift coefficient is
linear function of state and coefficient of diffusion is equal to constant. Exhaustive
solution of the problem of optimal linear extrapolation for different classes of random
sequences and different level of informational support of the problem of prognosis
(A.N. Kolmogorov equation [22] for stationary random sequences measured without
errors; Kalman method [23] for markovian noisy random sequences; Wiener-Hopf
filter-extrapolator [24] for noisy stationary sequences; algorithms of optimal linear
extrapolation of V. D. Kudritsky [25] on the basis of canonical expansion of V. S.
Pugachev etc.) exists. But maximal accuracy of the prognosis with the help of the
methods of linear extrapolation can be achieved only for Gaussian random sequences.
Thus the development of the new methods of the prognostication of computer sys-
tems state which allow to take into account the information about the investigated
object as much as possible is a topical problem.
Let us assume without restricting the generality that the state of a computer system
is determined in exhaustive way by scalar parameter X the change of the values of
which in discrete range of points ti , i 1, I is described by the discrete sequence
- 295 -
{ X } X (i ), i 1, I . It is necessary to get optimal estimations of future values of a
random sequence under different volume of a priori and a posteriori information.
2 Prognostication under the absence of the errors of measurement
The most universal from the point of view of the limitations that are imposed on the
investigated sequence is the method on the basis of canonical model [26]:
i N
X i M X i W( ) 1( ) i , i 1, I , (1)
1 1
where elements W( ) , h( ) i are determined by recurrent correlations:
1 N 1
W( ) X M X W( j )
( j)
W( j ) ( j ) , 1, I ; (2)
1 j 1 j 1
M W( ) X h i M X h i
1 {M X X h i
( )
i
D
h
M W ( ) 2
1 N
(3)
M X M X h i D j
( j)
h(j ) i
1 j 1
1
D j
( j)
h(j ) i }, 1, h, 1, i, h 1, N , i 1, I .
j 1
D M W( ) M X 2 M 2 X
2
1 N 1 (4)
D j D j ( j ) , =1, N , 1, I ;
( j) 2 2
1 j 1 j 1
The method of extrapolation on the basis of mathematical model (1) has two forms
of notation [27,28,29]
M X h i when 0;
mx( ,l ) h, i mx( ,l 1) h, i xl mx( ,l 1) l , h(l ) i when l 1, (5)
( 1, N )
mx h, i xl mx( 1, N ) l , h(1) i when l 1.
- 296 -
or
k N
mx( k , N ) 1, i M X i x j M X j S((( kNj )1) N ) i 1 N 1, (6)
j 1 1
where
S( 1) S( 1) k i , if -1;
( )
S (7)
, for = ;
/ N ]1 / N 1 , for kN ;
1,[(mod N ( ))
(mod ( )) (8)
1,[ / NN ]1 i , if i -1 N 1.
Mean-square error of extrapolation is determined as
M X i / x j , 1, N 1, j 1, k mx( k , N 1) 1, i M X 2
(9)
N 1
M X M W j( ) 1(j ) i , i k 1, I .
k 2 2
2
j 1 1
Expression mx( ,l ) h, i M X h i / x j , j 1, 1, 1, N ; x , 1, l
for h 1, l N , k is optimal estimation mx( k , N ) 1, i of the future value
x i , i k 1, I provided that for the calculation of the given estimation values
x j , 1, N , j 1, k are used that is the results of the measurements of sequence
X in points t j , j 1, k are known.
3 Prognostication on the basis of a priori information about the
sequence of measurements with errors
Solution of the problem of prognosis (5),(6) presupposes the usage of true values of
random sequence X in the points of discretization t j , j 1, k . But in real situations
the assumption about that that measured values x j , j 1, k are known absolutely
exactly is never carried out. The errors of the determination of the values of the fore-
cast parameter can appear whether as a result of overlay of hindrances in the commu-
nication channel between measuring device and investigated object or as a result of
influence of hindrances on the measuring tools.
Let us assume that as a result of measurements random sequence is observed
- 297 -
Z i X i Y i , i 1, I , (10)
where Y (i ), i 1, I , is a random error of measurement, X (i ), i 1, I , is unobserved
component. It is necessary to obtain optimal (in mean-square sense) estimation of
future values of random sequence X : M X ( ) X h i , , h 1, N , , i 1, I by
the results of measurements z j , j 1, k .
Within the limits of such a statement the simplest nonoptimal solution of the prob-
lem presupposes the usage of algorithms (5),(6) substituting in it the results of meas-
urements
M X h i , 0;
mx( / z,l ) h, i mx( / z,l 1) h, i z l ( ) mx( / z,l 1) l , h(l ) i , l 1; (11)
( 1, N )
mx / z h, i z l ( ) mx( / z1, N ) l , h(1) i , l 1;
k N
mx( k/ z, N ) 1, i M X i z j M Z j S((( kNj 1)) N ) i 1 N 1. (12)
j 1 1
Conditional mathematical expectation remains as before unbiased estimation of fu-
ture values of true extrapolated realization. At the same time the error of a single ex-
trapolation will be written down as:
(x/kz) i mx( k/ z) 1, i x ( k ) i , i k 1, I ,
where x ( k ) (i ), i k 1, I is a true value of extrapolated realization in the area of fore-
cast. These values aren’t known actually and realization x ( k ) i is developing in a
random way in the area of forecast. As a result of this the error of a single extrapola-
tion acquires random character:
i N
x( k/ z) i mx( k/ z) 1, i mx( k ) i W( ) 1( ) i (13)
k 1 1
The application of the operation of mathematical expectation to the last expression
S x( k/ z) i / z j , 1, N , j 1, k mx( k/ z, N ) 1, i mx( k , N ) 1, i
(14)
z j M Z j S((( kNj1)) N ) i 1 N 1
k N
j 1 1
- 298 -
x j M X j S(((kNj1)) N ) i 1 N 1
k N
j 1 1
y j S((( kNj)1) N ) i 1 N 1, i k 1, I .
k N
j 1 1
shows that in the given case (as distinct from an ideal case) a single extrapolation is
accompanied by conditional systematic error.
Correspondingly the dispersion of the error of a single extrapolation from (13),
(14) is determined as
N 1
M x( /kz) i S x( k/ z) i D j 1(j ) i , i k 1, I .
2 i 2
(15)
j k 1 1
With the usage of (13), (14) mean-square error of a single extrapolation will be
written down in the form
Ex/( kz) i / z , 1, k S x( k/ z) i Dx( k ) i , i k 1, I .
2
(16)
As error (16) is conditional averaging (16) by condition that values z , 1, k
are random is necessary for complete characteristic of the accuracy of algorithm (11),
(12). As a result the expression for mean-square error of prognosis is in the form
k N k N
l 1) N ) i 1 N 1
Ex( k/ z) (i ) Dx( k/ z, N ) i M Y l Y j S((( kN )
l 1 1 j 1 1
N 1
(17)
S((( kNj1)) N ) i 1 N 1 D j 1(j ) i , i k 1, I .
i 2
j k 1 1
4 Prognostication with preliminary filtration of the errors of
measurements
Increase of the quality of extrapolation of random sequence X , measured with
noises is possible at the expense of transition from the results of measurement
z , 1, k , k I to estimation.
x* M X 1 F ( ) mx( * 1, N ) 1, F ( ) z , 1, k .
o
(18)
- 299 -
Unbiased estimation of unknown value x being studied as a balanced mean
value of the result of the forecast at -th step mx( *k , N 1) 1, and result - of that
measurement z ( ) .
By means of consecutive substitution with the application of estimation (18) the
algorithm of extrapolation (5) is brought to the form [30]:
M X h (i ) , 0;
mx( * ,l ) h, i mx( * ,l 1) h, i F ( ) z l ( ) mx( * ,l 1) l , h(l ) i , l 1; (19)
( 1, N )
mx*
h, i F ( ) z l ( ) mx( * 1, N ) l , h(1) i , l 1.
Algorithm (19) has equivalent form of notation as following
o
k N
mx( *k , N ) 1, i M X i z j G((( kNj)1) N ) i 1 N 1 , (20)
j 1 1
G( 1) G( 1) k i , -1;
G( ) (21)
, = ;
/ N ]1 / N 1 , for kN ;
F ([ / N ]1) 1,[(mod N 1 ( ))
(22)
/ N ]1 i , if = i -1 N 1.
F ([ / N ]1) 1,[(mod N ( ))
Optimal values of weight coefficients are determined from the condition of mini-
mum of mean-square error of filtration
o
E f k M X * k X k M 1 F ( k ) Z j
k N
2
j 1 1
(23)
2
j 1)( N ) k 1 N 1 F
o o
G(( kN ) (k )
Z k X k .
(k )
After differentiation of this expression on F and solution of the corresponding
equation the expression for calculation of the optimal value of the coefficient is ob-
tained
F1( k ) F2( k ) F3( k )
F (k ) , (24)
F 1
(k )
F2( k ) 2 F3( k ) Dy k
- 300 -
k 1 N o o
F1( k ) Dx k 2 M X j X k G(((( kj 1)1)(NN) ) k 1 N 1
j 1 1
k 1 N k 1 N o o
M X j X l G(((( kj 1)1)NN) ) k 1 N 1
j 1 1 l 1 1
G((((lk1)1)NN) ) k 1 N 1 ;
k 1 N k 1 N
F2( k ) M Y j Y l G(((( kj 1)1)NN) ) k 1 N 1G((((lk1)1)NN) ) k 1 N 1 ;
j 1 1 l 1 1
k 1 N k 1 N o o
M X j X l G(((( kj 1)1)NN) ) k 1 N 1
j 1 1 l 1 1
G((((lk1)1)NN) ) k 1 N 1 ;
k 1 N
F3( k ) M Y j Y k G(((( kj 1)1)NN) ) k 1 N 1 .
j 1 1
Each element of the formula (24) has evident physical sense. Specifically sum-
mand F1( k ) determines contribution to resultant error made by stochastic nature of
random sequence X , summands F2( k ) and F3( k ) are connected with the errors of
past measurements and summand Dy k is dispersion of the last measurement. Algo-
rithm (19),(20) got on the basis of function M X ( ) X h i , , h 1, N , , i 1, I
and results of measurements z ( j ), j 1, k provides minimum of mean-square error of
the prognosis for the given volume of known information about investigated random
sequence as two interconnected consecutive stages (filtration-extrapolation) are ful-
filled in optimal way: weight coefficients of estimation (18) are determined from the
condition of the minimum of mean-square error of approximation to true values and
parameters of extrapolator on the stage of preliminary filtration and further forecast
are optimal which was proved earlier in the theorem.
Mean-square error of extrapolation with the use of the algorithm of polynomial fil-
tration (19),(20) is determined as
o o
k N k N
Ex(*k ) i M X j X l G((( kjN1)) N ) i 1 N 1
j 1 1 l 1 1
S((( kNj 1)) N ) i 1 N 1 G((( kN
l 1) N ) i 1 N 1 S (( l 1) N ) i 1 N 1
) ( kN ) (25)
k N k N
M Y j Y l G((( kNj 1)) N ) i 1 N 1G((( kN
l 1) N ) i 1 N 1
)
j 1 1 l 1 1
- 301 -
Dx( k ) i , i k 1, I ,
5 Prognostication on the basis of complete a priori information
about the sequence measured with errors
Application of the operation of filtration in algorithm (19),(20) allows to decrease
mean-square error of extrapolation compared with (11),(12) as the estimation
x* , =1,k has better accuracy characteristics compared with z , =1, k . But
in algorithm (19),(20) as well as in (11),(12) there is a mismatch between stochastic
qualities of a posteriori information x* , =1, k and parameters of extrapolation
form (5),(6) on the basis of which the method under study is formed.
For the forming of the method of the prognosis by noisy measurements let’s intro-
duce into consideration the mixed random sequence
X Z 1 , Z 2 ,..., Z k , X k 1 ,..., X I combining in itself the results of
measurements till i k , as well as the data about the sequence X for i k 1,I .
The canonical expansion for such a sequence is of the form
i N
X ' i M X ' i U( ) 1( ) i , i 1, I . (26)
1 1
Random coefficients of the canonical decomposition (26) defined by the following
recurrence formulas:
- for observation interval t1 ,..., tk
1 N 1
U( ) Z M Z U ( j )
( j)
U( j ) ( j ) , 1, k; (27)
1 j 1 j 1
- for forecasting interval tk 1 ,..., t I
1 N 1
U( ) X M X U( j )
( j)
U( j ) ( j ) , k 1, I . (28)
1 j 1 j 1
Accordingly, the expression for the dispersion of the random coefficients
U( ) , =1, N , 1, I are of the form:
- for observation interval t1 ,..., tk
D M U( ) M Z 2 M 2 Z
2
(29)
1 N 1
D j D j , 1, k ;
( j) 2 ( j) 2
1 j 1 j 1
- 302 -
- for forecasting interval tk 1 ,..., t I
D M U( ) M X 2 M 2 X
2
1 N 1 (30)
D j D j ( j ) , k 1, I .
( j) 2 2
1 j 1 j 1
The coordinate functions i are calculated using the formulas:
( )
h
- for observation interval t1 ,..., tk (function h( ) i describes the stochastic relation-
ship between the variables Z and Z h i )
h( ) i
1
D
M Z Z h i M Z M Z h i
1 N 1 (31)
D j
( j)
h( j ) i D j ( ) ( j ) h(j ) i , 1, h, 1 i k
1 j 1 j 1
- for description in the canonical decomposition of stochastic correlation between
intervals t1 ,..., tk and tk 1 ,..., t I ( h( ) i describes the relationship between random
variables Z and X h i )
h( ) i
1
D
M Z X h i M Z M X h i
(32)
1 N 1
D j
( j)
h( j ) i D j ( j ) h(j ) i , 1, k , i k 1, I ;
1 j 1 j 1
- for forecasting interval tk 1 ,..., t I (function h( ) i describes the stochastic rela-
tionship between the variables X and X h i )
h( ) i
1
D
M X X i M X M X i
h h
(33)
1 N 1
D j ( j) ( j)
h i D j ( j) ( j)
h i , k i I .
1 j 1 j 1
The necessity of the two expressions (28) and (29) for the determination of the
random coefficients of the canonical expansion (27) is explained with the technology
of the forming of random sequence X : X Z , ti , i=1,k and X X ,
ti , i=k 1,I . The stated peculiarity also results in the increasing compared to the
expansion (1) of the number of formulae for the calculation of the dispersions of the
random coefficients (29), (30) and coordinate functions (31),(32),(33).
- 303 -
In the canonical expansion (26) the random sequence X is presented in the in-
vestigated range of points ti , i 1,I with the help of N arrays U ( ) , 1, N of un-
correlated centered random coefficients U i( ) , i 1, I . The given coefficients contain
information about the values Z i , 1, N , i 1, k and
X i , 1, N , i k 1, I , and coordinate functions i , , h 1, N , , i 1, I ( )
h
describe probabilistic connections of the order h between sections t and
ti , ,i 1,I .
Let us assume that as a result of measurement in the first point of discretization t1
value z 1 becomes known (additive mixture of unobserved true value x 1 and error
y 1 ). Measurement z 1 concretizes random coefficients U1( ) , 1, N for sec-
tion t1 :
1
u1( ) z 1 M Z 1 u1( j ) ( 1j ) 1 , 1, N . (34)
j 1
Substitution of values (34) in canonical expansion (26) and further application of
the operation of mathematical expectation allow to write down the expression for the
estimation of future values x h i with the use of a posteriori information
z l 1 , l 1, N in the following form
mx(1,/ zl ) h, i mx(1,/ zl 1) h, i z l 1 mx(1,/ zl 1) l ,1 h(1l ) i (35)
where mx(1,/ zl ) h, i is optimal (in mean-square sense) estimation of value x h i pro-
vided that for the prognosis values z j 1 , j 1, l are used.
Measurement z 2 leads to the fixation of random coefficients u2( ) , 1, N for
t2 :
N 1
u2( ) z 2 M Z 2 u1( j ) ( 1j ) 2 u2( j ) ( 2j ) 2 , 1, N . (36)
j 1 j 1
Use of the values of random coefficients (36) allows to obtain prognosis algorithm
taking into consideration z l 1 , z l 2 l 1, N :
mx(2,/ zl 1) h, i z l 2 mx(2,/ zl 1) l , 2 h(l2) i , l 1;
mx(2,/ zl ) h, i (37)
mx / z h, i z 2 mx / z 1, 2 h 2 i , l 1.
(1, N ) (1, N ) (1)
- 304 -
For random quantity of measurements z , 1, I the algorithm of optimal ex-
trapolation takes on form:
M X h (i ) , 0;
mx( / z,l ) (h, i ) mx( / z,l 1) (h, i ) z l ( ) mx( / z,l 1) (l , ) h( l) (i ), l 1; (38)
( 1, N )
mx / z (h, i ) z ( ) mx / z (l , ) h (i ), l 1.
l ( 1, N ) (l )
Expression mx( / z,l ) h, i M X h i / z j , j 1, 1, 1, N ; z , for
h 1, l N , k is unbiased optimal estimation mx( k/ z, N 1) 1, i of future value
x i , i k 1, I provided that for the calculation of given estimation values
z j , 1, N , j 1, k are used that is the results of the measurements of sequence
X in points t j , j 1, k are known.
In Fig. 1 the diagram is presented that reflects peculiarities of functioning of the
method of prognosis (38).
Mean-square error of extrapolation with the help of method (39) is determined by the
expression:
M X i / z j , 1, N , j 1, k mx( k/ z, N ) 1, i M X 2 i
2
(39)
M 2 X i D j 1(j ) i , i k 1, I .
k N 2
j 1 1
6 Conclusion
In nowadays, the prognostication of the current state of complex computer systems,
especially, for the class of critical applications, is an important and actual problem for
providing high functioning reliability of the various control objects and decision-
making systems. The proposed approach, based on the mathematical formalization of
the parametrical changes of computer systems using nonlinear canonic models of the
random sequences, allows to take into account the stochastic properties of investi-
gated computer systems. Exhaustive solutions of the prognostication problems are
obtained by authors with the aim of the evaluation of the computer system state and
analysis of its further operational capability in the situations with different volume of
a priori and a posteriori information or with various levels of information uncertainty.
Synthesized prognostication methods as well as assumed, as their basis canonical ex-
pansions do not impose any significant limitations on random sequences
- 305 -
Fig. 1. Diagram of the prognosis of a noise random sequence with the help of extrapolator (38)
- 306 -
of the change of the values of controlled parameters including linearity, stationarity,
Markov behavior, monotoneness, etc. Suggested mathematical expressions for the
determination of mean-square error of extrapolation allow to make a decision about
the choice of the most appropriate method from the totality of the introduced ones for
the solution of the prognostication problem of computer system with prescribed accu-
racy. The specific diagram, presented in the paper, reflects the peculiarities of the
synthesized prognostication methods. Proposed methods are fairly simple in comput-
ing aspects and may be applied for solving computer system prognostication tasks in
real time taking into account that all parameters of the prognostication models can be
defined previously.
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