=Paper=
{{Paper
|id=Vol-3179/Paper_17.pdf
|storemode=property
|title=Taking into Account a Priori Uncertainty in the Model of Maintenance of Objects with Time Redundanc
|pdfUrl=https://ceur-ws.org/Vol-3179/Paper_17.pdf
|volume=Vol-3179
|authors=Borys Kredentser,Dmytro Mogylevych,Ihor Subach,Iryna Kononova
|dblpUrl=https://dblp.org/rec/conf/iti2/KredentserMSK21
}}
==Taking into Account a Priori Uncertainty in the Model of Maintenance of Objects with Time Redundanc==
https://ceur-ws.org/Vol-3179/Paper_17.pdf
Taking into Account a Priori Uncertainty in the Model of
Maintenance of Objects with Time Redundanc
Borys Kredentsera, Dmytro Mogylevychb, Ihor Subachc and Iryna Kononovab
a
Military Institute of Telecommunications and Informatization named after Heroes Krut, Moskovskaya str. 45/1,
Kyiv, 01011, Ukraine
b
The Institute of Telecommunication Systems of the National Technical University of Ukraine "Kyiv Polytechnic
Institute", Industrialnyi alley 2, Kyiv, 03056, Ukraine
c
National Technical University of Ukraine "Kyiv Polytechnic Institute named after Igor Sikorsky", st.
Verkhnoklyuchova, 4, Kyiv, 03056, Ukraine
Abstract
The article presents the results of a theoretical study of maintenance of restored objects of
continuous use, which provides for temporary redundancy − the permissible time for
performing current repairs and maintenance. The formulation of the problem is formulated
and a formalized description of the process of servicing these objects is given, a distinctive
feature of which is the presence of incomplete information about the distribution functions
of the recovery time of an object after a failure and the time for performing maintenance
(only two initial moments of these random variables are known).
A general approach to solving the formulated problem is proposed, based on the sequential
solution of a number of particular interrelated problems: obtaining formulas for indicators
of the quality of maintenance with complete initial information, which contain functionals
depending on the type of distribution functions, information about which is limited;
obtaining two-sided estimates of these functionals; synthesis of particular results using the
proposed method for calculating the boundary values of the optimal frequency and
bilateral estimates of service quality indicators − the coefficient of technical utilization and
average unit costs. A numerical example is considered to illustrate the practical application
of the results obtained.
Keywords 1
Maintenance, prior uncertainty, temporary redundancy.
1. Introduction
Ensuring high reliability of the functioning of modern complex technical systems includes a wide
range of problems, which are addressed by the efforts of many specialists of various profiles in the
design, manufacture, testing and operation of such systems. Among these problems, an important place
is occupied by the organization of effective maintenance, the main task of which is to maintain the
required level of reliability of complex systems during operation.
However, the analysis of a number of publications of domestic and foreign specialists indicates a
decrease in attention to the issues of researching the reliability and maintenance of complex technical
systems, in particular telecommunications equipment of communication networks [1]. The results
known in this subject area are devoted to the study of mainly traditional methods of maintenance and
are obtained, as a rule, without a complex consideration of a set of factors that significantly affect the
efficiency of maintenance. These factors include the usage of various types of redundancy, which is a
necessary condition for ensuring the reliability of the functioning of almost any technical system. In
particular, the usage of temporary redundancy imposes certain conditions on the organization and
Information Technology and Implementation (IT&I-2021), December 01–03, 2021, Kyiv, Ukraine
EMAIL: kredenzerbp@gmail.com (A.1); mogilev11@ukr.net (А.2); igor_subach@ukr.net (А.3); viti21@ukr.net (А.4)
ORCID: 0000-0003-1920-2107 (A.1); 0000-0002-4323-0709 (А.2); 0000-0002-9344-713X (А.3); 0000-0001-6945-0323 (А.4);
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
180
conduct of such operational activities as maintenance, routine repairs and others, therefore, this factor
must be taken into account when organizing the operation of technical facilities, since a time reserve
objectively exists in many real systems or can be provided by carrying out organizational and technical
measures. In well-known publications, such factors are not always taken into account that characterize
the modes of usage of objects for their intended purpose, as well as the parameters of the flows of
failures and equipment recovery. All this leads to results that are difficult to use in practice.
The subject of the study are technical facilities with time redundancy (systems "object time"), which
provide for periodic maintenance [2, 3]. In contrast to the work [4 − 7], the article considers continuous
use systems, examples of which are telecommunication systems of communication networks,
automated control systems (ACS) for various purposes and other systems. In addition, the study of these
systems is carried out in conditions of a priori uncertainty, which means incomplete source information
about the laws of distribution of determinants of random variables, when the exact type of these
distribution functions is not specified, and only some numerical characteristics of random variables are
known. The purpose of the article is to substantiate a general approach to the study of maintenance of
the mentioned above class of technical objects under conditions of limited initial information about the
distribution function of the object recovery time and the distribution function of the maintenance
execution time (only two initial moments of these random variables are known), which allows one to
obtain design relations for two-way estimates (lower and upper bounds) of the optimal frequency of
maintenance and the corresponding extreme values of quality indicators, values of service quality
indicators: the maximum value of the coefficient of technical utilization of the facility and the minimum
value of the average unit costs.
2. Statement of the problem
We will assume that the object is represented by one structural element, the time to failure of which
t 0 is distributed according to an arbitrary law F t Pt0 t with the ending mathematical expectation
t0 , and failures are manifested at the time of their occurrence. The system provides for the
implementation of two types of restoration work: periodic maintenance, which is based on the
implementation of planned preventive maintenance over time T const (frequency of maintenance)
and emergency (current) repairs after the failure of the object. Maintenance duration is a random
variable t m with an arbitrary distribution function Fm t and the ending mathematical expectation. If
the maintenance is performed within the allowable time ta1 const , which determines the time reserve
used in the system, it refers to the useful time of operation of the system, otherwise (at tm ta1 ) – to
downtime. hen a failure occurs, the object begins to recover, the duration of which is a random variable
t R with an arbitrary distribution function FR t with the ending mathematical expectation t R . If the
object is restored for a time not exceeding the allowable value ta const ( ta t R ), then it refers to
useful time, otherwise (at t R ta ) – to system downtime. Note that in the particular case of magnitude
t a and ta1 can have the same values [8]. Let the exact type of the function of distribution of recovery
time FR t and maintenance time Fm t of the object is unknown, and only two of their initial moments
are known:
1 t R tdFR t ,
0
(1)
2 t dFR t ,
2
0
12 2 ,
s1 tm tdFm t ,
0
s2 t 2 dFm t , (2)
0
s12 s2 .
181
Let's also assume that when you restore an object, the average cost per unit time is cR , and when
performing maintenance – cm . After completing any kind of restoration work, the system (object +
time reserve) is completely restored, the moment of the next maintenance is rescheduled, and the whole
process is repeated. In this case, we will assume that during the period of operation of the object under
consideration, the planned types of repairs are not carried out.
To assess the quality of maintenance of the object, we use two indicators: a comprehensive indicator
of the reliability of the system "object time", which is considered – the coefficient of technical use
K tu T and cost indicator – average costs per unit time of the system in working order – average unit
costs C T [1, 10]. For the above conditions of the system, it is necessary to obtain analytical ratios to
determine bilateral estimates (limit values) of the optimal frequency of maintenance and the
corresponding extreme values of service quality indicators: the maximum value for the coefficient of
technical use of the object and the minimum for average unit costs.
2.1. A general approach to solving a problem
The analysis showed that to obtain in the general case the exact values of maintenance quality
indicators in terms of the above problem is not possible, and the task is to find two-way estimates (exact
upper and lower limits) of these indicators, when the unknown distribution functions FR t and Fm t
belong to some fixed class of distributions K [11].
To solve this problem, it is advisable to use a method that takes into account the distinctive feature
of the calculated ratios for quality indicators of maintenance of objects with time redundancy, obtained
with complete source information. This feature is that these formulas include functionals of a special
type, the values of which depend on the type of distribution functions of the original random variables,
a priori information about which is limited.
In accordance with the formulation of our problem in the formulas for quality indicators of
maintenance (coefficient of technical usage K tu T and average unit costs C T ) should include
functionals that characterize the average time of restoration work (repair and maintenance), which are
completed before the backup (allowable time) t a and ta1 respectively. This time refers to the useful
time of using the system with time redundancy and helps to increase the efficiency of its operation. The
above functionals for known (or given) distribution functions FR t and Fm t look like:
ta
I FR 1 FR t td g t dFR t , (3)
0 0
t at 0 t ta ,
where g t
ta at t ta ;
t a1
I Fm 1 Fm t td ht dFm t , (4)
0 0
t at 0 t ta1 ,
where ht
ta1 at t ta1;
If we obtain analytical relations for the exact lower and upper limits (for bilateral estimates) of these
functionals and present these values in the formulas for maintenance quality indicators obtained with
complete initial information, it is possible to construct bilateral estimates of quality indicators in a priori
uncertainty.
This approach is the basis for solving the formulated problem. It includes the sequential solution of
the following partial problems:
– building a model of maintenance of objects with complete source information;
– obtaining bilateral assessments of functionals I FR and I Fm that characterize the quality of
maintenance;
182
– obtaining the limit values of the optimal frequency of maintenance and the corresponding extreme
values of quality indicators.
Synthesis of the results of solving these problems allows to obtain general solutions to the problem.
2.2. Construction of a service model with complete source information
We will keep the initial conditions of our problem, but we will assume that the functions of
distribution of recovery time FR t and maintenance duration Fm t are known or given, and we will
build a model of maintenance of objects of continuous use.
Under the maintenance model we will understand the mathematical (in our case analytical) model
that establishes the relationship between the quality of maintenance of the object and the characteristics
of its reliability, maintenance parameters and the process of operation of the object. Construction of
such model will allow to receive settlement ratios for indicators of quality of maintenance at the full
initial information [12]. It is easy to see that the process of functioning of the object of continuous use
with time redundancy can be described using the model of the regenerating random process xt , the
graph of states and transitions of which is shown in Fig. 1.
E
E
Figure 1: Graph of states and transitions of a random process x(t ) , which describes the functioning of
the system of continuous use
According to fig. 1, the process xt at any time may be in one of the following states:
e0 – the condition in which the facility is operational;
e1 – the condition in which the object is inoperable, but its restoration is carried out over time
t R ta ;
e2 – the condition in which the facility is maintained for time tm ta1 ;
e3 – the condition in which the facility is serviced for a time exceeding the allowable value ta1 ;
e4 – the condition in which the object restores its operability for a time exceeding the allowable
value t a .
In fig. 1 through E and E marked areas (subsets) of working and inoperable states of the system,
respectively. Note that in accordance with the accepted initial conditions of the states e1 and e2 , in
which restoration works (repair and service) for time less than admissible are carried out on object, we
carry to area E .
It is important to note that for this service strategy at the time of completion of restoration work
(moments of time t k k 0,1,2... transition to the state e0 ) further course of the process xt does not
depend on the past, as in these moments the system (object + time reserve) is completely updated, and
the next service is planned. These moments t k are moments of regeneration of the process xt , and a
183
sequence of time intervals X k tk tk 1 k 1,2... between the regeneration points forms a recurrent
~
~
recovery process. The values X k is a sequence of positive, equally distributed, mutually independent
random variables. Whereas the process xt has regeneration points, then random variables X ki ,
showing how much time the system has spent in the state ei by k period between the moments of
regeneration, are mutually independent and equally distributed random variables.
With prolonged usage, the proportion of time K i , which the system conducts in the state ei , equal
to the ratio of the average regeneration time МX i , conducted in the state ei for the period between the
~
moments of regeneration, to the average duration МX of this period.
Therefore, for the system under consideration, the operation of which is described by a random
process xt (Fig. 1), the coefficient of technical usage can be represented as follows:
i
МX
iE
K tu ~ , (5)
МX
where E e0 , e1,e2 is a subset of the operating states of the system.
Average time МX 0 of the system is in a state e0 is defined as the mathematical expectation of the
minimum of two quantities: operating time before the failure of the object t 0 and time before scheduled
maintenance (periodicity τ ), namely
МX 0 Мmin t0 , T 1 F t dt .
T
(6)
0
Average time MX 1 of the system is in a state e1 is equal to the product of the probability of
fulfillment of the condition t0 t (i.e. the probability that the object will fail before the start of
scheduled maintenance) on the mathematical expectation of the minimum of two values: the duration
of recovery t R and allowable time t a , namely
ta
МX 1 Pt0 T Мmin t R ,t a F t 1 FR t d t . (7)
0
Average time МX 2 of the system is in a state e2 is equal to the product of the probability that the
object will not refuse until the start of scheduled maintenance on the mathematical expectation of the
minimum of two values: the duration of maintenance t m and allowable time τ a1 , namely
ta
МX 2 Pt0 T Мmin tm ,t a1 1 F t 1 Fm t d t . (8)
0
Time interval X between adjacent system upgrade points (between process xt regeneration
~
points) xt ) consists of two intervals: the interval from the end of the previous update to the beginning
of the restoration work, which is equal to min t0 , τ , and a recovery interval representing some random
variable . According to the formula of complete mathematical expectation we obtain:
МX Мmin t0 , T М Мmin t0 , T t R Pt0 T tm Pt0 T
~
(9)
1 F t dt t R F T tm 1 F T .
T
0
Substituting formulas (6) – (9) in (5), can be obtained:
t a1
1 F t dt F T 1 FR t dt 1 F T 1 Fm t dt
T T
Ktu 0 0 0
. (10)
1 FR t dt t R F T tm 1 F t
T
0
Consider further the cost indicator – the average cost of restoration work per unit time of the system
184
in working order C T .
The average residence time of the system in the field of working conditions for the period between
the points of regeneration of the process xt (Fig. 1) is determined by the formula:
ta t a1
МX 0 МX 1 МX 2 1 F t dt F T 1 FR t dt 1 F T 1 Fm t dt.
T
0 0 0
Average costs for the same period are determined by the formula of full mathematical expectation:
cRt R Pt0 T cmtm Pt0 T cRt R F T cmtm 1 F T .
Therefore, the expression for average unit costs C T will look like:
cRt R F T cmtm 1 F T
C T T ta t a1
. (11)
1 F t dt F T 1 F t dt 1 F T 1 Fm t dt
0 0 0
Thus, the calculated relations (formulas (10) and (11)) for indicators of quality of maintenance
(objective functions) at the complete initial information which include linear functionals I FR and
I Fm (formulas (3) and (4)) are received.
We obtain formulas for determining the optimal values of the frequency of maintenance T and
T1 , that provide extreme values of quality indicators Ktu T and C T1 .
Differentiating expression (10) by T and equating the derivative to zero, we obtain the following
equation to find the optimal value of the frequency of maintenance T :
tm I Fm T t R I Fm tm I FR
F t λT 1 F t dt
tR I FR tm I Fm
, (12)
tR I FR tm I Fm 0
F T
where λ T – the failure rate of the object.
1 F T
Equation (12) is a necessary condition for the existence of the optimal value of the frequency of
maintenance, at which the coefficient of technical use of the system takes the maximum value. We
obtain formulas for checking the fulfillment of a sufficient extremum condition of the objective function
(10). Assume that tm t R and λt 0 and denote by V T the right part of the equation (12). It is
easy to see that when T 0 we receive V 0 0 and fair inequality:
tm I Fm
V 0 .
tR I FR tm I Fm
Therefore, for T 0 the derivative of function (10) is positive.
Let now T . If at the same time
tm I Fm
lim V T , (13)
tR I FR tm I Fm T
then equation (12) will have at least one root, and the objective function (10) is the absolute maximum
at T 0, .
Considering that lim λT λ exists, from expression (12) we obtain:
T
t R I Fm tm I FR
lim V T 1 λ t0
tR I FR tm I Fm
, (14)
T
where t0 – the average operating time of the object to failure.
Then inequality (13) takes the form:
185
tm I Fm t R I Fm tm I FR
1 λt0
tR I FR tm I Fm
. (15)
tR I FR tm I Fm
Inequality (15) is a sufficient condition for the existence of the absolute maximum of function (10)
for T 0, . From this formula it is seen that when λ inequality (15) holds for any values of
other parameters. If λ , it is necessary to perform a calculation according to formula (15) for a
given initial data and when the condition under test, to draw conclusions about the feasibility of
maintenance after a finite time. Note that the calculation of values T using formula (12) it is advisable
to perform the graphical method. Let T – the point at which the absolute maximum of the objective
function is reached. Then to determine this maximum, you can use the formulas that are obtained
provided that the value of T satisfies equation (12):
tm I FR
t t , T ,
m R
max Ktu T Ktu T
(16)
1 λ T I FR I Fm , T .
T
1 λ T t R tm
From formula (16) it is seen that if T , then it is impractical to perform maintenance, and we
get a formula for the system readiness factor.
Next, we investigate the objective function (11) (cost quality of maintenance) at a minimum of T .
Differentiating expression (11) by T and equating the derivative to zero, can be obtained the equation
to determine the optimal values of the frequency of maintenance T1 :
T c t I F cmtm I FR
F T T 1 F T dt R R m
cmtm
. (17)
cRt R cmtm 0 cRt R cmtm
In its structure, equation (17) coincides with equation (12), so, repeating the previous reasoning, we
arrive at the following inequality:
c t I F cm t m I FR
1 t 0 R R m
cm t m
, (18)
c R t R cm t m c R t R cm t m
which is a sufficient condition for the existence of the optimal frequency of maintenance T1 in terms
of average unit costs (formula (11)).
If T1 – the point at which the absolute minimum of function (11) is reached, then given that when
T1 equation (17) holds, we obtain
cR t R
t I F , T1 ,
0
min C T C T1
R
(19)
T1 cRt R cmtm , T .
T
1 T1 I FR I Fm 1
Next, to solve this problem it is necessary to find bilateral evaluations of the functionals I FR and
I Fm , characterizing the quality of service.
2.3. Bilateral evaluations of functionals I FR and I Fm
At the preliminary stage of the solution for the case of complete initial information the calculated
relations for determination of optimum values of periodicity of maintenance T and T1 are received
(formulas (12) and (17)), to verify the fulfillment of sufficient conditions for the existence of the
extremum of maintenance quality indicators (formulas (15) and (18)) and to calculate the extreme
values of maintenance quality indicators (formulas (16) and (19)).
186
These formulas include linear functionals I FR and I Fm (expressions (3) and (4)), the value of
which depends on the type of function of the recovery time of the object FR t and duration of
maintenance Fm t , the exact form of which is unknown, and only two initial points are known 1 , 2
and s1 , s2 (formulas (1) and (2)). Let the distribution function FR t belongs to the set of distribution
functions K 2 , satisfying the restriction (1), and the distribution function Fm t – set of distribution
functions L2 , satisfying constraint (2). For the above conditions, it is necessary to obtain accurate lower
and upper estimates of the functional I FR :
I FR inf I FR , I FR sup I FR ;
FR K 2 FR K 2
provided that
dFR t 1 , 1 I FR dt 1 , t dFR t 2 , 0 1 2 ;
2 2
0 0 0
and functional I Fm :
I Fm inf I Fm , I Fm sup I Fm ;
Fm L2 Fm L2
provided that
dFm t 1 , 1 I Fm dt s1 , t dFm t s2 , 0 s1 s2 .
2 2
0 0 0
Record FR K 2 (or Fm L2 ) means that the distribution function is unknown FR t (or Fm t )
belongs to the set of distribution functions K 2 1 , 2 (or L2 s1 , s2 ) positive random variables with
fixed initial moments 1 , 2 (or s1 , s2 ). A general approach to the analytical solution of such problems
was developed by L.S. Stoykova [13]. The main theorems underlying this approach, which determine
the necessary and sufficient conditions for the existence of an extremum (exact upper and lower bounds)
of some functionals, are given in the paper [14]. Here we obtain two-sided estimates of the linear
functional, which for our case have the form:
t a 12 2
at t a 2 21 ,
I * FR inf I FR (20)
FR K 2
0,5 1 t a t a2 2t a 1 2 at t a 2 21 ,
t at t a 1 ,
I * FR sup I FR a (21)
FRK 2 1 at t a 1 ,
s12 s
t a1 at t a1 2 ,
s2
I * Fm inf I Fm
2s1
(22)
FmL2
0,5 s t t 2 2t s s at t s 2 ,
1 a1 a1 a1 1 2
a1
2s1
t at ta1 s1,
I * Fm sup I Fm a1 (23)
Fm L2 s1 at ta1 s1.
3. Method of calculating the limit values of the optimal periodicity and
bilateral assessments of service quality indicators
We now turn to obtain the calculated relations that determine the overall solution of the problem. To
do this, we use the results of solving two parts of the problems obtained in the previous stages of the
study. If substitute the limit values I* FR , I * FR and I* Fm , I * Fm functionals I FR and I Fm
(expressions (20) – (23)) in formulas (12), (15 – 19) it is easy to obtain expressions to verify the
conditions of existence of the extremum of the lower and upper estimates of quality indicators (Table 1),
187
the equation for determining bilateral estimates of optimal periodicity maintenance (Table 2) and the
calculated ratios for two-way assessments of extreme values of quality indicators (Table 3) [15, 16, 17].
Table 1
Formulas for checking the fulfillment of sufficient conditions for the existence of an extremum of
quality indicators
Time Quality Formulas for checking the Formulas for checking the sufficient
reserve indicator sufficient condition for the existence condition for the existence of the
of the extremum extremum of
of the lower estimate of the the upper estimate of the indicator
indicator
tm I* Fm tm I Fm
*
1
tR I* FR tm I* Fm
t R I FR tm I Fm
* *
1
K tu T t R I* Fm tm I* FR t R I Fm tm I FR
t0
* *
t0
ta 0 t R I* FR tm I* Fm
t R I FR tm I Fm
* *
ta1 0
сmtm
1 сmtm
сRt R сmtm 1
C T сR t R сR t R
c t I Fm сmtm I FR
* *
t0 R R c t I F сmtm I* FR
сRt R сmtm
t0 R R * m
сRt R сmtm
Note: tR 1 xdFR x ; tm s1 xdFm x ; t0 xdFx ; I* FR – formula (20); I * FR –
0 0 0
F t
formula (21); I* Fm – formula (22); I * Fm – formula (23); t .
1 F t
Note: tR 1 xdFR x ; tm s1 xdFm x ; t0 xdFx ; I* FR – formula (20); I * FR –
0 0 0
F t
formula (21); I* Fm – formula (22); I * Fm – formula (23); t .
1 F t
Note: I* FR – formula (20); I * FR – formula (21); I* Fm – formula (22); I * Fm – formula (23);
F t
t R 1 xdFR x ; tm s1 xdFm x ; t ; t0 xdFx
0 0 F t 0
Thus, the following method can be used to calculate the limit values of the optimal periodicity and
two-way evaluations of service quality indicators.
1. Initial data for calculation:
– distribution function F t (or F t 1 F t ) operating time t 0 object to failure with a
mathematical expectation t 0 ;
– distribution density f t F t of random variable t 0 ;
F t f t
– the failure rate of the object t = ;
F t F t
– the first 1 t R and the second 2 initial moments of recovery time FR t ;
– the first s1 tm and the second s 2 initial moments of maintenance duration;
– the amount of time reserve used in recovery t a and when performing maintenance ta1 ;
– average costs per unit of time for recovery сR and when performing maintenance сm .
188
Table 2
Equation for determining bilateral estimates (lower and upper limits) of the optimal frequency of
maintenance
Time Quality Equation to determine the lower limits of Equation to determine the upper
reserve indicator the optimal frequency of maintenance limits of the optimal frequency of
*
T and T 1
* maintenance
T * and T1*
Ktu T tm I* Fm tm I * Fm
t R I* FR tm I* Fm
t R I FR tm I * Fm
*
ta 0 F T T F T T
t a1 0 T t R I* Fm tm I* FR T t R I * Fm tm I * FR
F t dt F t dt
0 t R I* FR tm I* Fm 0
t R I * FR tm I * Fm
C T сmtm
F T T сmtm
сRt R сmtm F T T
сRt R сmtm
T c t I * Fm сmtm I * FR
F t dt R R T c t I F сmtm I* FR
0 сRt R сmtm
F t dt R R * m
0 сRt R сmtm
Table 3
Bilateral assessments of extreme values of maintenance quality indicators
Time Quality The lower limit of the extreme The upper limit of the extreme value
reserve indicator value of quality indicators of quality indicators
*
min K tu T , min C T 1
*
max K tu T * , max C T1*
K tu T * t0 I* ( FR ) t0 I * FR
T* T*
t t
0 R t0 t R
ta 0
*
1 T I* ( FR ) I* ( Fm ) T *
1 T * I * ( FR ) I * Fm
t a1 0
1 T tR tm
*
*
T
1 T t R tm
*
C T1 cRtR
t I * F
cR t R
T1* T1*
0 t
0 * R
I F
R
T 1 сRtR cmtm
*
T 1 с R t R cm t m
*
*
T1
T1
*
1 T 1 I FR I Fm
* * *
1 T 1 I * FR I * Fm
*
2. Restrictions and assumptions:
– after the completion of any of the restoration work, the system (object + time reserve) is completely
updated;
– it is necessary to comply with the following conditions:
tR I* FR tm I* Fm ; tR I * FR tm I * Fm ; cRtR cmtm ; tа 2 ; ta 1 ; ta1 s2 ; ta1 s1
21 2s1
3. Calculation of limit values of quality indicators.
3.1. Calculation of limit values I* FR , I * FR and I* Fm , I * Fm functionals I FR and I Fm
(formulas (20) – (23)).
3.2. Checking the fulfillment of sufficient conditions for the existence of the extremum of quality
indicators using the inequalities listed in table. 1.
Execution of inequalities indicates the existence of finite values of the optimal periodicity of
189
maintenance, providing extremes of quality indicators. Otherwise it is necessary to draw a conclusion
that at the set values of initial data carrying out service is inexpedient T * .
* * * *
3.3. Definition of bilateral assessments (lower T , T 1 and upper T , T 1 boundaries) optimal
frequency of maintenance using the equations given in table. 2.
*
*
3.4. Calculation of bilateral estimates (lower min Ktu T * , min C T 1 and upper max Ktu T ,
max C T1* boundaries) of extreme values of quality indicators using the formulas of table. 3.
Numerical example. Consider a system of continuous use with time redundancy, which has the
following characteristics:
– the operating time of the object before failure is distributed according to Erlang's law of the and
order
F t 1 e t 1 t
with the parameter 0,02 1 h and mathematical expectation t0 2 100 h;
– the failure rate of the object
2t 4 10 4 t
t 1h;
1 t 1 0,02 t
– the first and second initial moments of the recovery time 1 t R 2,0 h , 2 8 h 2 and
maintenance time s1 tm 0,5 h , s1 0,5 h 2 ;
– allowable recovery time t a 1,0 h and maintenance t a1 0,2 h ;
– average costs per unit of time for recovery cR 60 c.y./h and when performing maintenance
cm 40 c.y./h .
For these initial data we will define limit values of optimum periodicity of carrying out maintenance
and corresponding bilateral estimations of indicators of quality – a factor of technical use and average
specific expenses. Solution. Consider first the solution for the coefficient of technical use of the system.
1. We calculate bilateral estimates I* FR , I * FR and I* Fm , I * Fm functionals I FR and I Fm
(formulas (20) – (23)).
8
Whereas tа 1,0 h less than value 2 2 h, so
21 4
12 1 4
I* FR tа 0,5 h; (24)
2 8
value I * FR can be found at tа 1 :
I * FR tа 1,0 h . (25)
It can be defined similarly
s2
I* Fm tа1 1 0,2
0,25
0,1 h, (26)
s2 0,5
I * Fm tа1 0,2 h. (27)
Using formulas (24) – (27), we make sure that the conditions specified in the restrictions and
assumptions are met.
2. Whereas lim t 0,02 – the final value, then check the fulfillment of a sufficient condition for
t
the existence of the optimal frequency of maintenance, using the formulas of Table 1 for the indicator
K tu T . Calculate the left and right parts of the formula, the lower score: the left part, which is equal to
0.364, less than the right part (0.999). Similar calculations are performed for the upper assessment of
the indicator K tu T (Table 1): the left part is 0.428, and the right – 0.997. Therefore, sufficient
conditions for the existence of the optimal frequency of maintenance are met for the lower and upper
estimates of the indicator K tu T .
3. Using the formulas of Table. 2 for the indicator, write the equation to determine the limit values
( T and T ) of optimal frequency of maintenance. Denote by V T f1 T and V T f 2 T right
* *
190
parts of these equations.
Then
T t R I * Fm t m I * FR
V T F T T F t dt 1 e
0, 02T
1 0,02T
0 t R I * FR t m I * F
m
(28)
4 10 4 T
1 0,02T
100 1 e 0,02T Te 0,02T
2 0,1 0,5 0,5
2 0,5 0,5 0,1
(recall that the left side of the equation is 0.364).
T t R I * Fm tm I * FR 4 10 4 T
V T F T T F t dt 1 e
0, 02T
1 0,02T
0
t R I * FR tm I * Fm 1 0,02T
(29)
100 1 e 0,02T Te 0,02T
2 0,2 0,5 1
2 1 0,5 0,2
(the left part of the equation is 0.428).
*
4. Solving these equations graphically (Fig. 2), we obtain T 122 h and T 96 h .
*
Substitute the found limit values of the optimal frequency of maintenance in the formulas of table. 3
and get the lower and upper estimates of the coefficient of technical use min Ktu T * 0,9858 ;
max Ktu T 0,991 . When the maintenance is not performed T and the time reserve in the
*
system is not provided ta ta1 0 , the coefficient of technical usage is converted into a coefficient
of readiness K h :
t 100
Ktu K h 0 0,98 .
t0 tR 100 2
We move on to the definition of bilateral estimates of the cost indicator of С T . Using the formulas
of table. 1, calculate the value of the right-hand sides of the inequalities of this indicator and obtain a
value close to unity. The left part of these inequalities is equal to 0.2. As we can see, the sufficient
condition to be checked is fulfilled, so there are finite limits of the optimal periodicity of service, which
provide extreme values of bilateral estimates of average unit costs [15, 16].
Using the formulas of table. 2 for the indicator С T , calculate the right-hand sides of the equations
to determine the limit values ( T 1 і T1 ) of optimal frequency of service:
V 1 T 1 e 0,02T 1 0,02T
4 10 4 T
1 0,02T
100 1 e 0,02T Te 0,02T
60 2 0,2 40 0,5 1
60 2 40 0,5
;
V1 T 1 e 0,02T 1 0,02T
4 10 4 T
1 0,02T
100 1 e 0,02T Te 0,02T
60 2 0,2 40 0,5 0,5
60 2 40 0,5 ;
while the left parts of these equations are 0.2.
It is easy to make sure that for the received source data V 1T V1T . Therefore, we construct one
dependence curve V1 T f T (Fig. 3) and determine the value T1* 50 ч. Substituting this value
of the optimal frequency of service in the formulas of table. 3 for the indicator C T1 , we get the bottom
( min C T1* )estimates of average unit costs:
) and the top ( max C T1*
min C T 0,992 c.y./h ; max C T 0,996 c.y./h.
1
*
1
*
For comparison, we determine the average unit costs for the case when maintenance is not performed
T and there is no time reserve ta ta1 0 . In this case
C T1* ct t 60100 2 1,2 c.y./h .
R R
0
191
4. Summary
The given theoretical research of process of maintenance of objects with time reservation allows to
draw the following conclusions:
1. The proposed approach to take into account the a priori uncertainty (incompleteness of the source
information) when building a model of maintenance of objects of continuous use allowed to obtain
relatively easy analytical formulas for quality indicators of maintenance, convenient for practical use.
These relationships establish a relationship between the quality indicators and reliability characteristics
of the object, the values of periodicity and duration and the parameters of time redundancy.
2. Taking into account in the developed model the parameters of time redundancy (allowable time
of recovery and maintenance) the ability of objects to function normally under the influence of various
destabilizing factors (failures, equipment failures, etc.). It is the presence of time reserve, which is a
system parameter, that can in many cases explain why objects perform their functions more successfully
than is the result of equipment failure
,
Figure 2: Graph of functions V T and V T to determine the limit values of the optimal frequency
of maintenance
Figure 3: Graph of the function V T to determine the value of the optimal frequency of
maintenance T1*
192
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