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Danilenko E.L.
doctor of technical sciences, professor, professor of the department of applied mathematics and information technologies Odessa National Polytechnic University
Даниленко Евгений Леонидович
доктор технически наук, профессор, профессор кафедры прикладной математики и информационных технологий, Одесский национальный политехнический университет
THEORY OF CONTROL OF RANDOM PROCESSES AND APPLICATIONS ТЕОРИЯ КОНТРОЛЯ СЛУЧАЙНЫХ ПРОЦЕССОВ И ПРИЛОЖЕНИЯ
Summary: Mathematical models of control of random processes that are relevant for a wide range of applications are developed, for example, in the management of multi-channel complex and computer networks, the scoring of their reliability efficiency. The bases of construction of control charts and various applications of statistical quality control are described.
Key words: control, random process, controlled set, Markov processes, statistical quality control, control charts.
Аннотация: Разработаны математические модели контроля случайных процессов, которые имеют актуальное значения для широкого круга приложений, например, при управлении многоканальными комплексными и компьютерными сетями, оценке их эффективности надежности. Описаны основы построения контрольных карт и различные приложения статистического контроля качества.
Ключевые слова: контроль, случайный процесс, контролируемое множество, марковские процессы, статистический контроль качества, контрольные карты.
1. Foreword
The creation of a theory of reliability and quality control stimulated the creation scientific schools, the founders of which were outstanding mathematicians and thinkers of the A.N. Kolmogorov, B.V. Gnedenko and their students. For a short period, the theory of statistical quality control was created in the Soviet Union (see the books of Ya.B. Shor, Yu.K. Belyaev, B.V. Gnedenko). The creation of the theory of statistical quality control influenced the following formulation of the problem of developing mathematical models of control of random processes [1-5]. Suppose that a complex system operates in a continuous time t and its state at time t is described by a random process £ (t) with values in the setX. We call a complex system controlled if its state set is divided into a set of controlled states Xo and a set of uncontrolled states Xi, that isX = Xo U Xi, Xo n Xi = 0, Xo + 0, Xi + 0.
A controlled state x0 eXo is a state of the system that corresponds to the pre-established regulations. For example, in control charts [1, 5, 6], such states are those for which the map exponent lies within an admissible region (in control boundaries), and for a computing complex, these can be the states of technical serviceability of its main elements. Note that among uncontrolled states x1e X\ there can be also the rejected states of the system, getting into which means its failure, but it is assumed that such states are recoverable.
We denote the probability of the system staying in the set of controlled states
p (t) = P{%(t) e Xo}, the opposite probability q(t) = 1 - p(t) stay of the system in a set of uncontrolled states Xi . The search problem has a lot of controlled states Xo for a fixed probability pi (t) is called the direct task of controlling a complex system (problem establishment of control boundaries). The task of finding the probability p(t) with a fixed set Xo is called inverse task of control. It is easy to see that calculating probabilities p(t), q(t) depends on the type of random process f (t) and the sets Xo, Xi. The random process is given by its probability measure P(-) (in the particular case, the distribution function F (x, t) or some property of the random process, for example, the property of Markov property or the property of increment independence, the stationarity and ergodicity property). One can discre-tize the time t, that is, represent the time t in the form of a finite or infinite sequence {ti , t2, ... , tn }, {ti , t2 , ... , tn, ... }. There may be a different type of sets of controlled and uncontrolled states. For such different cases (types of random processes f (t), different sets X solve the direct and inverse problems of controlling a complex system, statistical tasks and the theory of control charts and its application in various branches [1,5]. Optimized control models for complex systems and their applications are considered [5].
The basic way of specifying a random process is to construct its canonical probability space (fi , F, P) and it is based on the Kolmogorov theorem, which is the basis of the theory of random processes and whose essence lies in the fact that every family of symmetric compatible distribution functions (a family of finite-dimensional distribution functions) determines a stochas-
tic process uniquely up to equivalence. Finite-dimensional distribution functions play the same role for a random process as the distribution function for a random variable-they contain all the information about the process. Thus, the random process ((t) can be represented by a process X = {X(t), t 6 T}, the given coordinate system is selected by the Gaussian mapping X(t, ш) = X(t, x(-)) = x(t), - set of elementary
events. Then we can talk about the functions of mathematical expectation and covariance of the process, respectively, n(t) = M(x(t)) and C(s,t) = M((X(s) -n(s))(X(t) — n(t)) where M() is the mathematical expectation operator. A real random process X = {X(t),t e Rj is said to be Gaussian if its finite-dimensional distributions are Gaussian. It should be noted that the Gaussian process is given by the mathematical expectation function n(t) and the covariance function C(s,t).
We have a random process £(t) as a family of random variables X = {X (t), t 6 Tj defined on the same probability space (Q, F, P) depending on some parameter t from the set T and taking values in to some other fixed set E. If the set of values of a random process coincides with the real line, E = R, then for a given elementary event ш 6flthe mapping x = x(m) : T ^ R is a function x = {x(t), t 6 7} in the ordinary sense, which is called the trajectory of the random process
The time parameter t 6T can take numeric values from a countable or continuous set, or take values in more complex sets. In this connection, various special concepts are used for random processes. If the set T = N = {1, 2, ...j is a set of positive integers, thenX = {X (n), n 6 Nj is called a random sequence and is denoted by X ={Xn,n 6 N}. If the set T is a real line T = R = (го, or its intervals (a, b), [a, b), (a, b], [a, b], -го < a < b < and is interpreted as time, then we are talking about a random process.
Легко заметить, что при множестве подконтрольных состояний Х0 = [a,b], -<х < а < b < вероятность равна P(^(t) 6 Х0) = p(t) = F(b, t) — F(a, T) и можно просто решать прямую и обратную задачи контроля, пользуясь функцией распределения F (х, t).
2. Control of Markov processes
Let us consider the control of Markov processes [3]. This raises the task of investigating the probabilities of a transition between sets of controlled and uncontrolled states, obtaining ergodic conditions for the controlled system. The ergodicity property of a controlled system is of primary interest for real applications, since it consists in the asymptotic constancy of the probability of stay in sets of controlled and uncontrolled states and the absence of dependence on the initial state. Many studies have been devoted to the substantiation of the use of Markov chains for modeling multi-machine associations (for example, [10]).
For example, in the fairly general case, when £ (t) is a stochastically continuous, regular, time-inhomoge-
neous Markov chain with values in a measurable discrete space (X, B(X)), where B (X) is a Borel algebra of
subsets of X, and a matrix of continuous local probability transitions (infinitesimal matrix)
Q(t) = \\qap(t)\l (a,p) e X2, qap(t) >0, а Ф p, e x Чар (P) = 0, a e X.
We denote by
Q°(t) = \\qap(t)\\, P0(t,s) = \\pap(t,s)\\ (a,p) eXtx Xj (i, j =0, 1);
П(t, s) = J nJap (t, s)||, (а, Ю e X2 ( j =0, 1); Ti
Vap(t.s) = P{m = /3, | S(t) = o},n]ap(t,s) = P[as) = p,ï(u) eXj,t<u <s | S(t) = a}.
We will assume that the behavior of the system, when its functioning starts from a controlled (uncontrolled) state x0 e X0 (x° e X0 ), is known that all states are communicating and with a single probability it is possible to exit from the set Xo (Xi) and return back. A study of the random process control model is proposed to be carried out by constructing two Markov chains (0 (t) u (1(t), t e [0, &>) provided that their state
d
space X remains the same and all states x° £ X0 for (0 (t) , x1 £ X1 for (t) are absorbing. Thus, we establish relations between the probability matrices of the transitions P0 (t, s) h n (t, s),j =0, 1. Note that ^ (t, s) are the matrices of the transition probabilities of the Markov chains %0(t) u (1(t), t £ [0, œ) are solutions of systems from the direct and inverse Kolmogorov equations
-^nj(t,s) = nj(t,s)Q"(s),t< s,
dUAt.s)
dt
U7-(t,s)|t=s
= -Qjj(t)nj(t,s),t < s,
= \fc
ap\\,(a,p)ex2,
where
namely
л -¡!,a=P,
°aP {0, аФр,
ъ
nj(t,s) = exp(J QJJ (u)du =
Zk=o(ftSQJJ(u)du) к!
,j = 0,1.
t
2.1. Let %(t) = a E X0,fi E X0. Then the event existence of such states у E X0,S E X1,e E X1,( E i((s) = P] is representable as an unification of events. X0 and time mates t < и < v < s that there is an event Event {¿(s) = p,3uE (t,s),%(u) E X1] implies the
{%(u -0) = y, E X0 при ш E [t, и)] П [%(u + 0) = S, E X1 при ш E (и, v), - 0) = e] n
$(v + 0) = <;,i(s)=P].
Then, noting that % (t), t E [0, ю) and the above formulas using the for-
n0(t, u)Q00(u), П1(и, v)Q10(v) - probability density mula of total probability, we have of the transition from the set X0 to the set X1 and vice versa, taking into account the markness of the chain
Poo(t,s) = n0(t,s) + U n0(t,u)Q01(u)n1(u,v)Q10 (v)P00(v,s)dudv.
t<u<v<s
This matrix equation is the Volterra integral equation of the second kind with respect to the unknown matrix P00 (t, s). The proof of the existence and uniqueness of its solution can be carried out by standard methods on the basis of the principle of condensed considerations [8, p.88].We solve this equation by means of a resolvent, that is, in the form of a Neumann series of As a result, we have
iterated kernels that obey recurrence relations. We note that the convergence of the solution (the Neumann series) follows from the probabilistic meaning of its terms.
68 Wschodnioeuropejskie Czasopismo Naukowe (East European Scientific Journal) #5(33), 2018 SüSSI P00(t,s) = n0(t,s) + JJ n0(t,u)Q01(u)n1(u,v)Q10 (v)n0(v,s)dudv +
t<U<V<S
JJ JJ n0(t,u)Q01(u)n1(u,v)Q10(v)n0(v,u1)Q01(u1)n1(u1,v1)Q10(v1)
t<u<v<u1<v1<s
x n0(v1,s) dudvdu1dv1 + ■■■
From this expression we see that the matrix P00 (t, s) also satisfies the following integral equation Poo(t,s) = no(t,s) + JJ Poo(t,v)Q01(u)n1(u,v)Q10 (v)Uoo(s,v)dudv,
t<U<V<S
union with the initial integral equation. Assuming in the last equation of
S
L°(u,s) = Q01(u) J n1(u,v)Q10(v)n0(v,s)dv
u
we obtain the integral equation
s
3(t,s) = n0(t,s) + J P00(t,u)L°(u,s)du. t
We note that, in addition to finding exact solutions by means of the resolvent, the approximate solutions of the integral integral equation can also be sought.
2.2. Let Ç(t) = a EX0,p E X1. Similarly to Section 2.1, we obtain
s
P„i(t,s) = no(t,s) + J Poo(t,u) Q01(u)ni(u,s)du.
2.3. Let %(t),t e [0, ro) stochastically continuous regular, time-homogeneous Markov chain with a state space of the form X = Xo U Xi, Xo n Xi = 0, Xo + 0, Xi 0.
Then the matrices of local transition probabilities are independent of time, that is,
Q = \\qap\\,(a,p)eX2;Qii = \\qap\\,(a,p) e Xt x Xj (i,j = 0,1),
and the matrices P0 (t), n, (t), L0 (t) have the form
Po(t) = Po(0,t),n7-(t) = n7-(0,t) = exp (tQi),(i,j) = 0,1,
t
L°(t) = Q01 J n 1(u) Q10(t - u)du.
Jni
0
Then it follows from the equations in Section 2.1 that
t
Poo(t) = no(t) + J Poo(u) L0(t - u)du,
P01 (0 = J Poo(") Q01H1(t - u)du.
o
Let us pass in these integral equations to the Laplace transform
P
t
Poo(z) = rio(z) + Po(z)Z0(z), where Z0(z) = Q01Il1(z)Q10 no(z),
P01 (z) = i501(z)Q01fi1(z). Then from these equations we find the Laplace transform of the unknown probabilities
Poo(z) = flo(z)(£o - Q01fi1(z)Q10 flo(z))-1,
Poi(z) = no(z)(Eo - Ç01Oi(z)Ç10 Oo(z))-1Ç01ni(z).
Similarly we obtain
Pio(z) = Hi(z) (Ei - ÇiorUz)Ço1 Oi(z))-1 ÇioOo(z) ;
In the last four formulas
Pii(z) = (Ei - QioWo(z)Qo1 Hi(z))-1ni(z).
Et = \\SaP\\,(a,p) E Xï,8ap =\Lo aa^pi i = 0,1
We note that the above expression for the inversion of matrices has an obvious meaning in the case of the finiteness of the set of controlled states Xo (it is this set that is of practical interest). If Xo is a countable set, then invertibility must be understood as invertibility in the algebra of operators on the space of bounded sequences. The Laplace transforms uniquely (with an accuracy of the set of measure 0) determine the matrices of the transition probabilities P00(0, P01(t), P10(t), P11(t), which are found by means of Mellin type inversion theorems.
2.4. Suppose that a controlled system is described by a homogeneous locally regular Markov chain £ (t) with continuous time and a finite set of states X = {1 ,...,n} = X0 ®X1,X0 = {1 ,...,m],X1 = {m + 1 ,...,n},m <n and a matrix of local transition probabilities Q = ||qy\\,qy >0,i± j,Y!j=1qij = 0,i = 1 ,...,n.
The partition of the space of states into Xo - controlled and uncontrollable X1 corresponds to the partition of the matrix Q into blocks
Qoo Qio
Qoi
Qii
where Çoo is a square order m matrix.
We call a controlled system regular if sets of controlled and uncontrolled states are communicating, that is,
P[3t > 0:((t) E X1-kK(0) E Xk} > 0,k = 0,1
and from any subset of the )C1 c X1it is possible to go over to its complement X1 \ J(1 c X1 without going into the set Xo, that is
P[3t: Vt: [0,t] ^(t) E X1,^(t) E E Xj > 0.
This condition is interpreted as a condition for a good sub-systemability in uncontrolled states.
Lemma. If the controlled system with a finite set of states is regular, then 1. VA e a(Q11)ReA < 0, where a(Q11) is the spectrum of the matrix Q11; in particular, det Q11 ^ 0. 2. All elements of the matrix Q—1 are negative.
3. The matrix Qo = Qoo - Qo1Q-jLQ1o is of pre-stochastic order m and, as a matrix of local transition probabilities, generates a homogeneous Markov chain ?o(t)eXo.
The proof of the lemma is based on the Perron-Frobenius theory [8, p.339].
Theorem. If a Markov chain (0 (t) of a regular controlled system with a finite set of states is ergodic, then the Markov chain ((t) is also ergodic and the row vector of its final probabilities has the form q = (lkofl|l1)-19ofl, where R = ||£o ! -Qo1Qii1|| is the row vector of the final probabilities of the chain £0(t), Eo is the identity matrix of order m, ||q,0fl||1 is the sum of the elements of the row vector q0R.
Proof of the theorem is available in the author's papers [2-5].
3. Let us consider examples of various random processes described by complex systems.
3.1. Let the random process %(t) e X = {0,1,...} describing the controlled system represent a homogeneous Poisson process [12]. Then the diagonal matrix of local transition probability densities has the form
—X X
0
X X
Q =
where 0 < X < ro is a constant number equal to the intensity of the onset of an event.
Let the sets X0 = {0,1, ...,m- 1} and X1 = {m,m + 1,...}, then the transition probabilities from the set of controlled states Xo to the sets Xo and X and
0
back, obtained from the formulas for the Laplace transforms ¡i00(z),ii01 (z) ,Pioiz),Pii(z) in Section 2.3, using the inverse Laplace transforms, have the form:
P00(r) =
1 A 0 1
||0 0 0 ... 1 II
a*)2 2!
At
(m-1)! (Xr)m-2 (m-2)!
1 At
rn2 2!
P0I(T) = ATe
-Xt
(xt)"
(xt)"
(m-2)! (m-1)!
(xt)"
(m-1)! (m)! (m+1)! (XT)m-2 (XT)m-1 (XT)m (m)!
Ph(t) = e
-Xt
1 At
(xt)2 2!
0 1 AT ..
Pw(T)= 0,
where 0 is the zero matrix.
3.2. The onboard computing system of the spacecraft is considered as a classical mass-service system M/M/1 [11], that is, it consists of one serving computer with a Poisson flow of processing tasks and an exponential distribution law of their execution time. Let us examine the quality of the on-board computing subsystem, which will be characterized by the number of tasks for processing information in the queue.
Let the state {0}, corresponding to the absence of the queue, be controlled, and the remaining states {1, ... , k, ... }- uncontrollable. Then, using the formulas from Section 2.3, we get that the probabilities that in the time t in this airborne subsystem the task queue will increase from zero to k > 0, p0k (t) is uniquely determined by the Laplace transforms
Poo(z) = 7Too o ft00 (z)
i=1 k = 1
Pok(z) = Ho
№YT=1
(z)qk0 T[oo (z))-1 ,k>0.
If the intensity of the assignments is less than the intensity of their performance, the onboard system is an ergodic system, and the stationary state probabilities for all k > 0 are found on the basis of the theorem as pk = Po Yl°=1 R0l ftik (0), where p0 is the stationary probability of the absence of the task queue, which is determined by the intensity of the assignments and their execution.
3.3. The onboard computer system of the spacecraft consists of one working and one backup computing device that are serviced by the repair system. For simplicity, let us assume that the operating time of the main computing device and its recovery time are distributed according to exponential law with parameters
—a
P 0
Q =
a and p, respectively. In a number of cases, this is indeed the case under real conditions [10]. The natural premises allow us to say that it will maintain a uniform Markov chain in time.
Let the set of controlled states of such an onboard systemX0 = {x\, x2}, where x: is one computing device, the second is in reserve, x2 - one computing device is working and the second computing device is serviced by the restoring system, and the set of uncontrolled states X1 consists of one state x3 - two faulty computing device, one computing device recovers (on-board system does not work). Then the corresponding matrices have the form:
0
(a + ¡3) a
P
P
1
''
a
Q01 = 0|,°1o = II0 PII,
n0(t) =
пШ n012(t) n°21(t) nUt)
,^(0 = 11^3(011 .
After simple transformations from the relations transition probabilities py (t) from the state Xi to the for the Laplace transforms T>ij(z), i,j = 1,2 in Section state Xj : 2.3, we obtain Laplace transforms that determine the
Vij(s) = - apftil2(s)Tt^j(s)Tt^}-3(s),i,i = 1,2,
Pl3(s) = aît33 (s)Pl2 (s), P23 (s) = aÏÏ33(s)P22(s).
From the existence of homogeneous Markov chain (0 (t) in the space of controlled states with the matrix of transient local probabilities
м- a a I
Q= \\p -ар(а+р)л1з(0)\
1
it follows that (0) = —. Then under the
33 a(a+P)
hypotheses of the theorem, the stationary probabilities of states x 1, x2 and x3 will be written as
Pi
(a + p)Ti
(a + p)r1 + (a + p + l)r2 2
,i = 1,2;
Pi
(a + P)r1 + (a + p + 1)r2'
where r1 and r2 are stationary probabilities of the renewal chain ^0(t).
4. Mathematico-statistical control
Time series (implementation, trajectory) allows you to visualize the change of any control value in time. It is a graph of the dependence of this quantity on time. Data for its construction can be taken, for example, from control sheets. After its construction, it is possible to identify at what period something happened that affected this value, and determine what it was. For example: wear of equipment, change of subcontractor, use of other material, recruitment of new employees and so on.
When analyzing the graph, it is important to separate insignificant changes that are normal for the process under investigation, from significant ones. It is best to use a time series to detect changes in the mean. When building a graph, it is important not to confuse the sequence.
In Fig. 1 shows an example of a time series of weekly work time losses and shows the situations in which these losses increased
Количество потерянного рабочего времени.час
200
100 -
Поломка компью т ер а
.Л Ср
еднее значение
12345123412345123
Март
Апрель
Июнь
H ед ели месяца
Fig. 1. Time series of weekly working hours losses
In Fig. 2 shows an example of the time series of one of the telemetry parameters of the spacecraft and some control boundaries for it. From the analysis of this figure it can be clearly seen that the telemetry parameter from the 20th observation showed a tendency to punch and rushed to the lower control boundary established by the regulations.
Often the problem of detecting changes in the probabilistic characteristics of the observed processes is solved, or, so-called, the problem of "disruption" [7].
This task covers a wide variety of real situations: disruption is a violation of the uniformity of data, it is a process disturbance or a change in operating modes, impulse noise, a failure in the operation of recording equipment, equipment failure, atmospheric influences in the transmission of radio signals, etc. Let the "disorder" appear at a random time 0 and select an observation method determined by the random time of the stopping time t. For example, the stopping time t = inf {t: 9 (x (t)) > h} is the first time moment when the process 9 (x (t)) exceeds some threshold h. When creating a
monitoring system, that is, giving an alarm signal that the process 9 (x (t)) > h, it is necessary to take into account the natural requirements: 1. The conditional mathematical expectation M (t - 6 / t > 6) —> 0, that is after the appearance of a "breakdown" at the time 0, the alarm must be dropped with as little time as possible. 2. P (t < 0) ^ 0, that is, a "false" alarm (error of the first kind) would be rarely given. It is easy to see that these requirements are contradictory, which leads to a varia-tional problem:
find t * = arg inf M (t - 6 / t > 6), tE {t: P (t < 6)
< a},
where a is a given positive number that limits the probability of a "false" alarm (probability of error of the first kind) P (t < 6). Note that the following mathematical expectations are equal
M (t - 6 /t > 6) P (t > 6) = M (max (0, t - 6)).
This task does not fit into the framework of traditional mathematical statistics, since it requires consecutive observations, which justifies the use of control charts. By analogy with the theory of checking statistical hypotheses, we can introduce the probabilities of errors of the first and second genera, but it is required to divide the time into separate local segments, after each of which one of two decisions is taken: there is "disruption" or there is no "disruption". The efficiency of the algorithm for solving variational problem will depend on the intensity of the "discontinuities" and the a priori value of a, and the original random process £(t) should be stationary broad sense [7].
Control charts are used to track changes in any characteristics of a random process that describes a complex system, for example, statistical estimates such as certain functions of the trajectory of a random process, which in turn is a certain function of time, can be such. Practically control charts are used for statistical control and regulation of technological processes. The control card is given the values of some statistical estimate (characteristics) in the form of a point at a fixed time, calculated from the data of the trajectories in the order of their receipt, the upper and lower control boundaries, and the upper and lower limits of technical tolerances (if any) that are taken from technical regulations. Sometimes warnings are also used. An example of a control chart of the arithmetic mean as an unbiased estimate of the mathematical expectation is shown in Fig. 3, 4. The upper and lower control boundaries, as well as warning boundaries, are calculated for a stationary Gaussian random process (stationarity in the broad sense and narrow sense for Gaussian processes coincides) according to the standard formula: Ke,n = ^ ± upo, where the mathematical expectation of the random process M£(t) = ^, the variance D£(t) = o2, Up - quantile of the Gaussian distribution, which depends on the confidence probability p = i - a, a is the level of significance (note that the values of a in variational problem are fundamentally different here too.) Quantiles for warning boundaries uo.95 ~ 2 are often taken, and for control boundaries uo,99 ~ 3, which is known as a rule of "two sigma" and the rule of "three sigma".
Время
Fig. 3. Control chart of average values of accuracy of definitions
Fig. 4. Control card of average values of signal amplitude
According to the position of the points with respect to the boundaries, it is judged that the technological process has been adjusted or broken down. Usually the process is considered to be diluted in the following cases: 1. Some points go beyond the control limits. 2. A series of seven points is on one side of the middle line. In addition, if on one side of the middle line there are: a) ten from a series of eleven points, b) twelve of fourteen points, c) sixteen of twenty points. 3. There is a trend (drift), that is, the points form a continuously rising or continuously falling curve. 4. Two or three points turn out to be precautionary two-sigma boundaries. 5. Approaching the center line. If most of the points are inside the half-sarsigram lines, this means that the data from different distributions are mixed in the subgroups. 6. There is a periodicity, i.e. then rise,
then decline with approximately the same time intervals. 7. The control limits are wider than the tolerance. Ideally, it is sufficient that the control limits are 3/4 of the tolerance value.
If the control card shows that the technological process is dilapidated, find the causes of the breakdown and make adjustments.
As a measure of the control card, any statistical estimates can be taken, but the most common mean arithmetic mean and standard deviation, as the characteristics of the most probable value and variation of the random process (Fig. 5). The simultaneous representation of the estimates calculated from the realizations (trajectories), the mean value of x, and the standard deviation s gives an almost complete picture for the decision.
Fig. 5. A control chart of estimates of mean values and rms values of the amplitude of a radio signal
Both parametric and nonparametric statistics are used. The best in terms of their relevance are the control charts of histograms [5], built on the basis of trajectories. Histogram - a method of graphical representation of tabular data, which is a graphical representation of the dependence of the frequency of trajectory elements (sampling) from the corresponding grouping interval.
Control charts of individual values, median maps, span, tolerant intervals (an example of a tolerant control chart is shown in Fig. 6), asymmetry and kurtosis coefficients, covariance estimates, regression model estimates, nonparametric coupling parameters, sign criteria, series criterion, Wilcoxon test and others (for example, Student, Pearson, Kolmogorov- Smirnov, Fisher) and others [1, 5].
Fig. 6. Tolerant control chart of the relative dosing error (1 - tolerant interval, 2 - regulatory control
boundaries, 3 - graph of mean values)
Список литературы:
1. Даниленко, Е.Л. Математико-статистиче-ские методы оперативного контроля случайных процессов / Е.Л. Даниленко // Исследование операций и АСУ. — Киев: Вища школа, 1982. — Вып. 19.
— С. 31-39.
2. Даниленко Е.Л. Вероятностные модели оперативного контроля дискретной системы / Е.Л. Да-ниленко // Автоматика и вычислительная техника -1983. - N01. - С. 66-71.
3. Даниленко Е.Л. Марковская модель оперативного контроля сложной системы / Е.Л. Даниленко // Известия АН СССР. Техническая кибернетика. - 1983. N0 6. - С.176-182.
4. Даниленко Е.Л. Моделирование контроля сложной системы / Е.Л. Даниленко // Информатика и математические методы в моделировании. / Одесский национальный политехнический университет.
- 2012. - Том 2. - N0 4. - С. 363-373.
5. Даниленко Е.Л. Теория и практика контроля сложных систем/ Е.Л. Даниленко. - Одесса: Освита Украины, 2014. - 44с.
6. Даниленко Е.Л. Эффективное применение математико-статистических методов / Е.Л. Дани-ленко // Информатика и математические методы в моделировании. / Одесский национальный политехнический университет. - 2013. - Том 3. - N0 2. -С. 132-145.
7. Ширяев, А.Н. Статистический последовательный анализ / А.Н. Ширяев. - М.: Наука, 1976. -232 с.
8. Колмогоров А.Н. Элементы теории функций и функционального анализа/ А.Н. Колмогоров, С.В. Фомин. -М: Наука, 1976. 544 с.
9. Гантмахер Ф.Р. Теория матриц/ Ф.Р. Гантма-хер. - М.: Наука, 1967. - 576 с.
10. Авен О.И. Управление вычислительным процесом в ЭВМ/ О.И. Авен, Я.А. Коган. - М.: Энергия, 1978. - 240 с.
11. Клейнрок Л. Теория массового обслуживания/ Л. Клейнрок. - М.: Машиностроение, 1979. -432 с.
12. Кингман Дж. Пуассоновские процессы/ Дж. Кингман. М.: МцНМО, 2007. - 136 с.
