=Paper=
{{Paper
|id=Vol-2845/Paper_28.pdf
|storemode=property
|title=Analytical Model with Interruption of Service of Short-term Objects with Temporary Reservation
|pdfUrl=https://ceur-ws.org/Vol-2845/Paper_28.pdf
|volume=Vol-2845
|authors=Borys Kredentser,Dmytro Mogylevych,Iryna Kononova,Vadym Mohylevych
|dblpUrl=https://dblp.org/rec/conf/iti2/KredentserMKM20
}}
==Analytical Model with Interruption of Service of Short-term Objects with Temporary Reservation==
https://ceur-ws.org/Vol-2845/Paper_28.pdf
Analytical Model with Interruption of Service of Short-term
Objects with Temporary Reservation
Borys Kredentsera, Dmytro Mogylevychb, Iryna Kononovab, Vadym Mohylevychс
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
Taras Shevchenko National University of Kyiv, Bohdan Hawrylyshyn str. 24, Kyiv, 04116, Ukraine
Abstract
The article presents an analytical model for the maintenance of short-term facilities with
temporary redundancy, which are used not continuously, but occasionally. These systems
perform tasks that arrive at random times and take some time to complete. Examples of such
systems are communication systems as well as various automated control systems. In the
developed model, various factors of the real functioning of this class of systems are taken
into account: the possibility of interrupting the maintenance of an object upon receipt of a
task and two components of the replenished reserve of time, one of which is provided in the
system itself, and the second is due to the random nature of the receipt of tasks.
A method is proposed for determining the optimal maintenance frequency and extreme
values of the indicators of the systems under consideration: a complex reliability indicator −
the coefficient of technical utilization and a cost indicator − the average costs per unit time of
the object's stay in a working condition (average unit costs).
Keywords 1
Information and communication networks, refusal, failure, readiness index, reliability index.
1. Introduction
The need for maintenance arises at the stage of operation of any technical system. The main
purpose of maintenance is to ensure the maximum efficiency from the usage of the system during
operation. This goal can be achieved by purposeful intervention in the functioning of the system
through a rational choice of the type of maintenance, justification of the optimal timing and content of
maintenance. Therefore, the solution to the problem of optimal maintenance, in which the extreme
values of the selected indicators are provided, is relevant.
2. Formulation of the problem
Consider a system with a replenishment reserve of time (system “object-time”), in which an object
is represented by one generalized structural element [1, 2]. Let the operating time of an object to
failure t0 be distributed according to an arbitrary law F t Pt0 t with a finite mathematical
expectation t0 , and a failure manifests itself at the moment of its occurrence (a system with instant
indication of failures). The system performs tasks arriving at random moment of time, and the time
intervals between the times of arrival are distributed exponentially with a parameter f . We will
IT&I-2020 Information Technology and Interactions, December 02–03, 2020, KNU Taras Shevchenko, Kyiv, Ukraine
EMAIL: kredenzerbp@gmail.com (B. Kredentser); mogilev11@ukr.net (D. Mogylevych); viti21@ukr.net (I. Kononova);
vadym.vadym63@gmail.com (V. Mohylevych)
ORCID: 0000-0003-1920-2107 (B. Kredentser); 0000-0003-2937-5327 (D. Mogylevych); 0000-0001-6945-0323 (I. Kononova); 0000-
0003-2937-5327 (V. Mohylevych)
©️ 2020 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)
295
assume that the duration of the task is so short in comparison with the average operating time of the
object to failure that it can be practically neglected (short-term system) [3]. This means that if an
incoming task finds the system in a working state, then it is executed with a probability of one.
Suppose that at the initial moment of time t 0 the object is operational and a scheduled
preventive maintenance is assigned at a deterministic time T , which determines the frequency of
maintenance. Before the start of service execution, one of two independent events can occur: object
failure or task arrival. Let the object fail is first. In this case, the restoration of its performance
immediately begins, the duration of which is a random variable t R , distributed according to an
exponential law with a parameter and a finite mathematical expectation t R 1 . If a task is
received during the recovery process, then a delay (delay) in its execution for a time a is allowed,
which can be a random variable, exponentially distributed with a parameter , or a deterministic
value ta const . If the restoration of the object's operability is completed before the use of the time
reserve a (or t a ) that determines the permissible lag time, then the task is executed with probability
one, otherwise, at the moment the condition t R a (or t R ta ) is fulfilled, the system fails (task
execution failure) [4]. Let the object fail until the moment T (an event t0 T has occurred). Then, at
the appointed time T , maintenance starts, the duration of which is exponentially distributed with a
parameter . If a task arrives in the process of servicing, then servicing is terminated and the object is
transferred to the main (operating) mode in a random time t z , exponentially distributed with a
parameter z . At the same time, the system provides for a permissible time for bringing an object to
readiness to perform a task a1 , which can be a random variable with an exponential distribution law
with a parameter 1 or a deterministic value ta1 const . If t z a1 (or t z ta1 ), then the received task
is executed with probability one, after which the object is transferred to the maintenance completion
mode. Otherwise (at t z a1 or t z t a1 ) at the moment when the time reserve is used up, the system
malfunctioning (refusal to complete the task) occurs. Thus, the system uses two components of the
replenished reserve of time: one component is provided in the system itself ( a and a1 or t a and ta1
), and the other is due to the random nature of the arrival of tasks (the time from the moment of object
failure or the start of maintenance to the moment the task is received) [5,6]. It is necessary to obtain
analytical expressions for the service quality indicators [7]: technical usage coefficient K tu Т and
average unit costs С Т (objective functions) and determine the optimal values of the service
frequency Т
*
and Т 1* , at which the quality indicators take extreme values: the maximum value
Ktu Т * for the technical utilization factor and the minimum value С Т1* for the average unit costs.
2.1. Obtaining target functions
Let us introduce into consideration a random process xt describing the functioning of the system,
the graph of states and transitions of which is shown in Fig. 1, where:
е0 – the state in which the object is operational;
е1 – the state in which the object is undergoing maintenance and there are no tasks;
е2 – the state in which the object is being restored and there are no tasks;
е3 – the state in which the object is inoperable and the received task is waiting for the end of the
restoration of operability within the admissible time a (or t a );
е4 – the state in which the received task is waiting for the end of the transfer of the object from the
service mode to the operating mode within a permissible time a1 (or ta1 );
е5 , е6 – the states in which the system is, respectively, under recovery or under maintenance after
spending the replenished reserve of time a or a1 ( t a or ta1 );
296
Е , Е – subsets of healthy and faulty states of the system, respectively.
Е+
е1 е0
е2
е4
е3
Е−
е6 е5
Figure 1: The graph of states and transitions of a random process xt describing the operation of a
system with service interruption
The technical usage coefficient K tu Т is defined as the stationary probability of the xt process
staying in the subset of operable states Е
1
4
6
K tu Т і aі і aі , (1)
і 0 і 0
where π і the stationary probabilities of the embedded Markov chain π і і 0,6 ; aі − average
residence time of process xt in state eі , i 0,6 [8].
The probability і і 0,6 can be found using the well-known system of equations [9]:
π і pij π j ,
jE
where pij stationary transition probabilities of the embedded Markov chain.
This system of equations taking into account the graph of states and transitions of process xt
(Fig. 1) takes the form:
π 0 1 p10 π 2 p20 π 3 p30 π5 p50 ,
π1 π 0 p01 π 4 p41 π 6 p61,
π 2 π 0 p02 ,
π3 π 2 p23 , (2)
π 4 π1 p14 ,
π5 π 3 p35 ,
π 6 π 4 p46
6
taking into account the normalization condition і 1 .
і 0
Solving this system, we get
p
0 В ; 1 01 В; 2 p02 В ; 3 p02 p23В;
p10
(3)
p p
4 p14 01 В ; 5 p02 p23 p35 В ; 6 p14 p46 01 В,
p10 p10
297
1
p
где
where В 1 01 p14 1 p46 p02 1 p23 p02 p23 p35 .
p10
Let us now substitute the found values of і і 0,6 into the general formula (1) and obtain an
expression for the coefficient of technical use in general form:
p
a0 p02 a2 p23a3 01 a1 p14a4
K tu Т p10 . (4)
p01
a0 a1 p14 a4 p46a6 p02 a2 p23 a3 p35a 5
p10
Using formulas aі xdFi x and pij lim Pi t [10] and taking into account the initial
t
0
conditions of the problem and the accepted assumptions and constraints, we obtain the following
expression for the stationary transition probabilities ij and average sojourn times aі in states:
еі , і 0,6; і j
f
p01 F T ; p02 F T ; p10
; p14 ; p20 ;
f f f
f z 1
p23 ; p30 ; p35 ; p41 ; p46 ;
f z 1 z 1
Т
(5)
p50 p61 1; a0 F t dt ; a1
1 1 1
; a2 ; a3 ;
f f
0
1 1 1
a4 ; a5 ; a6 .
z 1 z
After substituting the formulas for pij and aі , і 0,6; і j in (4), obtain the final calculated ratio
for the coefficient of technical utilization of the system under consideration:
Т
F T f F T f
F t dt 1 1
f 1
K tu Т
0 z
, (6)
Т
F T F T f
F t dt 1
0 z
where F t 1 F T .
In formula (6), F t denotes in general form the distribution function of the operating time of the
object to failure. For concrete forms of this function, various particular solutions of practical interest
can be obtained from formula (6). So, for the case when F t has an exponential distribution with a
1
parameter , we have
t0
T
F t 1 et , F t dt
1
1 e T , (7)
0
and for the case when F t has a second-order Erlang distribution with a parameter
2
, can be
tv
obtained
T
F t 1 e t 1 t , F t dt
2
1 e T Te T . (8)
0
Substituting expressions (7) or (8) into formula (6), we obtain the calculated ratios for the
coefficient of technical use:
298
a) for the case of exponential distribution of mean time to failure F t :
1 e 1 1 1 1 1 e
T f f T
f f 1
K tu Т
z
; (9)
1 1 f
1 e
T 1
1 e T
z
b) for the case when F t has a 2nd order Erlang distribution:
x
f
y
1 Т z f 1
Т e T
Ktu Т
f z 1
1
x y
1 Т f z
T
Т e
, (10)
z
where x
2
1 e T ; y 1 eT 1 Т .
The obtained design ratios (6), (9) and (10) are target functions for optimizing the frequency of
maintenance in terms of the complex reliability indicator − the coefficient of system technical use.
We now obtain an objective function for optimizing the frequency of servicing according to the
criterion of the minimum average unit costs С Т [11].
In the problem under consideration, the average unit costs С Т can be defined as the ratio of the
sum of the average costs of restoring the facility's operability, performing maintenance, and making
the facility ready from the maintenance mode to the operating mode to the average residence time of
the system in a subset of operable states [12, 13].
Proceeding from this and using the graph of states and transitions of the system (Fig.1), you can
write the following formula for С Т :
с a 3a3 5a5 сm 1a1 сz 4a4 6a6
С T R 2 2
0a0 1a1 2a2 3a3 4a4
or, taking into account expressions (3) and (5), after simple transformations, we obtain:
F T F T
сR cm cz f
z
С Т Т , (11)
f z 1 f
F t dt F T F T
0 f z 1
where сR , сm , сz are the average costs per unit of time, respectively, when restoring operability,
when performing maintenance and when transferring an object from maintenance mode to operating
mode.
The resulting relationship (11) is an objective function for optimizing the frequency of
maintenance for the selected cost indicator.
2.2. Optimization problem solution
Let us bring formula (6) to the following form:
Т
F t dt F T A B B
K tu Т 0 T , (12)
F t dt F T D E
0
where
A
f
; B
z 1 f
; D
z z f ; E . z f
f
(13)
z 1 z z
299
Formulas (13) for A and B are written for the case of exponential distributions of quantities a
and a1 with parameters and 1 , respectively. In the case of degenerate distribution functions (the
quantities t a and ta1 are deterministic), we obtain
f 1 q z f 1 q1
A
f ; B
z
, (14)
where
q PtR ta еt a ; q1 Pt z ta1 е z t a1 . (15)
Differentiating expressions (12) with respect to Т and equating the derivative to zero, we obtain the
following equation for finding the optimal value of the service frequency:
ЕВ T DB A B E
F T T F t dt , (16)
D A B 0 D A B
where the coefficients A , B , D , E are expressed by formulas (13) − (15), and is the object failure
rate, i.e.
F ' T f T
Т , F T 1 F T .
F T F T
Equation (16) is a necessary condition for the existence of an optimal value of the service
periodicity.
Let us investigate the question of the existence of roots of equation (16). Therefore, it can be
argued that if for T 0
z f z 1 f
EB z z 1
0
D A B z z f f
(17)
z 1 f
z f z 1
and if lim (t ) , then equation (16) has at least one root. This can be verified by comparing the
T
left and right sides of the equation when T 0 and T .
If, in addition, the monotonicity condition T 0 is added, then the absolute maximum of the
function K tu Т is attained at the smallest root of equation (16).
For the case when lim (t ) it is necessary to check the fulfillment of the inequality, which is
T
a sufficient condition for the existence of an extremum of the objective function (12) [14, 15]:
ЕВ
lim V T , (18)
D A B T
where V T is the right side of equation (16).
If condition (18) is satisfied, then it can be concluded that it is expedient to carry out maintenance
after a finite time. Let Т * be the point at which the absolute maximum of the objective function (12)
is achieved. Then, to determine this maximum, following formulas can be used:
t0 A
, T * ;
1
t0
max K tu Т K tu Т * (19)
T
1 T * A B
, T * ,
1 T * D
where the coefficients A , B , D are determined using formulas (13) − (15).
Let us now turn to the study of the cost indicator C Т − the average costs per unit time of the
system's stay in the subset of operable states (formula (11)).
Transform the formula (11) to the following form:
300
F T K L
C Т T , (20)
F t dt F T A B B
0
where
cR cm cz f
K , (21)
z
cто cпг з
L , (22)
пг
and A , B are determined using formulas (13) − (15).
Differentiating expression (20) and equating the derivative to zero, we obtain an equation for
determining the optimal value of the maintenance frequency:
L T
F T T F t dt B
A B L .
(23)
K 0 K
By its structure, equation (23) coincides with equation (16), therefore, repeating the previous
reasoning, we arrive at the following inequality:
L
1 tv B
A BL , (24)
K K
which is a sufficient condition for the existence of a finite value of the optimal frequency of servicing
in terms of average unit costs.
If T1* is the point at which the absolute minimum of the function (20) is reached, then, taking into
account that for T1* , the quantity T1* satisfies equation (23), we obtain:
cR
, T * ;
*
min С T С T1 0t A (25)
T
T* K
, T * ;
1 T * A B
where the coefficients A, B, K are determined by formulas (13) − (15) and (21).
3. Methodology for determining the optimal frequency of maintenance and
extreme values of quality indicators
Required initial data for the calculation:
distribution function of the operating time t 0 of the object to failure F t (or F t 1 F t )
with mathematical expectation t0 ;
density of distribution f t F ' t of a random variable t0 ;
F ' t f t
the rate of object failures t ;
F t F t
intensity of maintenance 1 tm ;
the intensity of transferring the object to the main (working) mode z 1 t z ;
intensity of receipt of the task f ;
parameter of the exponential distribution law of the admissible recovery time of the object's
operability 1 a or the value of the reserve time ta const , if the reserve time is a deterministic
value;
parameter of the exponential law of distribution of the allowable time for transferring the
301
object to the main (operating) mode 1 1 a1 or the value of the reserve time ta1 const , if the
reserve time is a deterministic value;
average costs per unit of time, respectively, when restoring the facility's operability с R , when
performing maintenance с m and when transferring the facility to operating mode с z .
1. Limitations and assumptions:
after the end of maintenance and restoration of operability, the system (object + standby time)
is completely updated;
the average recovery time is greater than the average service duration ( t R tm , where t R 1 ,
tm 1 ).
the time to complete the task is so short compared to the average operating time of the object
to failure that it can be practically neglected;
the following conditions must be met:
cz f
D A B ; R m
c c
.
z
2. Formulas for calculating the optimal values of the frequency of maintenance and extreme values
of the coefficient of technical use. Verification of the fulfillment of the accepted restrictions and
assumptions. Checking the fulfillment of a sufficient condition for the existence of an optimal
periodicity of servicing in a finite time (inequality (18)). Determination of the optimal value of the
service frequency T * as a root of the equation:
EB T DB A B E
F T T F t dt , (26)
D A B 0 D A B
where the coefficients A , B , D , E are determined by formulas (13) − (15).
Note that if the facility failure rate t is an unboundedly increasing monotonic function (
т.е. lim t ), then equation (26) has a single root. Calculation of the maximum value of the
t
coefficient of technical use. If equation (26) has one root T * , then the maximum value of the
coefficient of technical use is determined by the formula:
max K tu Т K tu Т *
1 Т * А В
T
1 Т* D
. (27)
If the equation has no roots, then T * and:
t A
max Ktu Т Ktu K h 0 . (28)
T 1
t0
This means that the absolute maximum of the comprehensive reliability index is achieved when
the facility is not undergoing maintenance. This indicator is the system availability factor K h .
If equation (26) has several roots T1* , T2* ,...,Т k* , then the value of the optimal periodicity will be
numerically equal to one of these roots or T * depending on the point at which the absolute
maximum of the function Ktu Т * is reached. It is easy to find out by comparing the values
.
Ktu Т і* , і 1,2,...,k , and K tu
3. Formulas for calculating the optimal values of the frequency of maintenance T * and the
minimum values of the average unit costs. Verification of the fulfillment of the accepted restrictions
and assumptions. Inequality check:
сm сz f
z L A B
1 t0 B , (29)
сR сm сz з K
z
302
which is a sufficient condition for the existence of an extremum (minimum) of the objective function
(20). The fulfillment of this inequality testifies to the fact that the optimal frequency of servicing T1*
has a finite value.
Determination of the optimal value of the periodicity T1* as a root of the equation:
сm с z з
z T
F T T F t dt B
A B L ,
(30)
с R сm с z з 0 K
z
where the coefficients A , B , K , L are determined by formulas (13) − (15), (21), (22).
If the equation has one root T1* , then the minimum value of the average unit cost is determined
using the expression:
c c c
T1* R m z f
z
T
min C T C T1*
1 Т1* А В
, (31)
if equation (30) has no roots, then T1* and
min С T С
сR . (32)
Т t0 A
This corresponds to the case where maintenance of the facility is impractical. If equation (30) has
several roots, then the value of the optimal servicing frequency will be numerically equal to one of
them, depending on the point at which the absolute minimum of the function С Т * is achieved.
Note that formulas (27) and (31) for calculating the extreme values of the quality indices Ktu Т *
and С Т1* are obtained under the assumption that the optimal values of the servicing frequency T *
and T1* satisfy equations (26) and (30), respectively.
4. Numerical example.
Consider a system characterized by the following data:
the operating time of the object to failure is distributed according to the second order Erlang
law F T 1 exp t 1 t with a parameter 0,01 h 1 ;
density of distribution of operating time to failure f t F ' t 2t exp t ;
f t 2t
the rate of object failures t ;
1 F t 1 t
the intensity of restoration of the facility's operability 1 tR 1 6 h 1 and performance of
maintenance 1 tm 1 4 h 1 ;
the intensity of the transfer of the object to the main (working) mode z 1 t z 1,0 h 1 ;
the intensity of the receipt of tasks f 0,5 h 1 ;
values 1 a 0,5 h 1 and 1 1 д1 0,5 h 1 ;
average costs per unit of time when restoring operability сR 50 у.е. / ч , when performing
maintenance сm 20 c.u./ h and when transferring an object to an operating mode сz 10 c.u./ h .
For these initial data, we will determine the optimal values of the frequency of maintenance and
the corresponding extreme values of reliability indicators Ktu Т * and average unit costs С Т1* .
Solution. We substitute the initial data into the equation (26) and get:
1,25 V T , (33)
where V T denotes the function
303
10 4 Т
V T е 0.01Т 1 0,01Т
1 0,01Т
200 1 е 0,01Т Те 0,01Т 6 . (34)
Since for the accepted initial data lim T , we check the sufficient condition for the
T
existence of the extremum of the function (12):
1,25 lim V T 2,06.
T
Consequently, equation (33) has a single finite root. The root of the equation (33) is determined
from the graph of the function V T on the right side of the equation, which is shown in Figure 4.4.
For the accepted initial data, the optimal value of the maintenance frequency T * 100 h .
Next, we find the maximum value of the coefficient of technical utilization by the formula (27):
f 1 f
1 Т*
z
f z 1
K tu Т
*
* z z f
1 Т z
10 4 100 0,17 0,5 0,5 1,0 0,5 0,5
1
1 10 100 0,17 0,50,17 0,5 0,251,0 0,5
2
0,992 .
10 4 100 1,0 0,25 0,17 1,0 0,5
1
1 10 2 100 0,17 1,0 0,25
For comparison, we present the value of the coefficient of technical use for the case when
maintenance is not carried out T * :
f 0,17 0,5 0,5
t0
f 200 0,17 0,50,17 0,5
K tu K h 0,988 .
t0
1 200 6
minimum value of the average unit costs. Substituting the initial data into equation (30), we get:
1,5 V1 T , (35)
4
where V1 T e 0,01T 1 0,01T
10 T
200 1 e 0,01T Te 0,01T 6,69 .
1 0,01T
Further check that the sufficient condition of an extremum of the function is executed (20)
1,5 lim V1 T 2,0669 . (36)
T
Let us now determine the optimal value of the frequency of maintenance, which ensures the
Figure 2: Graphical determination of optimal values of maintenance intervals
From inequality (36) it follows that equation (35) has one root, which can be determined from the
304
graph of the function V1 T (Fig. 2). For the accepted initial data, the optimal value of the frequency
of maintenance Т1* 240 h . To calculate the minimum value of the average unit costs С T1* , we use
the formula (31), into which we substitute the value
Т1*
2Т1*
1 Т1*
.
As a result,
2Т1* сR сm сz f
*
С T1*
1
2 *
Т 1
z
Т1 f z 1 f
1
1 Т1* f z 1
10 4 240 20 10 0,5
2 50 6
1 10 240 0,25 0,25 10
1,46 c.u./h
10 4 240 0,17 0,5 0,5 1,0 0,5 0,5
1
1 10 2 240 0,17 0,50,17 0,5 0,251,0 0,5
For comparison, we present the value of the average unit costs for the case when the system is not
serviced Т1* :
сR
50 6
С 1,481 c.u./h.
f 0,17 0,5 0,5
t0 200
f 0,17 0,50,17 0,5
Let us analyze some of the results of a theoretical study of the technical utilization factor K tu Т
of the considered short-term system with service interruption, which provides for temporary
redundancy and periodic maintenance. In fig. 3 and fig.4 show graphs of the dependence of the
maintenance factor on the frequency of maintenance Т at different values z t a1 (Fig. 3) and at
different values of the failure rate (Fig. 4). The solid curves correspond to the case when the
operating time of the object to failure t0 is distributed according to the second order Erlang law, and
the dotted curves correspond to the case of the exponential distribution of a random variable t0 (the
calculations were carried out on the same mathematical expectations of the operating time to failure).
4. Summary
The studies carried out and the results obtained allow us to draw the following conclusions:
For the same values of the time for performing maintenance and restoring the object's operability
1 , the considered maintenance strategy provides a sufficiently high efficiency of the types of
restoration work used, and the value of the technical utilization factor depends significantly on the
f
ratio of parameters z ta1 , f , , that determine the amount of time reserve and the efficiency of its
z
use. In particular, from Fig. 3 it can be seen that an increase in the intensity z of the transfer of the
object from the maintenance mode to the main mode (with an increase in z ta1 and ta1 const ) by
only two times leads to a significant increase in the efficiency of maintenance. In this case, the
optimal value of the service frequency is shifted to the left along the abscissa axis. This is due to the
fact that with an increase in z ta1 system downtime due to maintenance, they are more and more
compensated for by the time reserve existing in the system, while the amount of downtime
compensation when restoring the object's operability does not change.
In the case when the value Т (at const ), in the system, only the restoration of
305
operability after failures is carried out and all six curves (Fig. 3) asymptotically tend to the value of
the stationary availability factor of the system K h in which maintenance is not carried out.
Fig. 4. it can be seen that the value of the technical utilization factor is also significantly influenced
by the failure rate of the object under consideration. With a decrease , the optimal service
frequency shifts to the right along the abscissa, while the curves in the region of maximum values
K tu Т become flatter. Of considerable interest is the study of the influence of the type of the law of
distribution of operating time to failure F t on the value of the coefficient of technical use. The
graphs shown in Fig. 3 and Fig. 4 show that in the case of the distribution of a random variable t0
according to the exponential law (dotted curves) and according to the second-order Erlang law (solid
curves) at the same values of the parameters, the shape of the curves K tu Т is different. At the same
time, for the Erlang law of the second order, at certain values of the parameters, there is an extreme
value of the coefficient of technical utilization when changing the values of the service frequency
within the interval 0, , while the extreme value in the case of an exponential distribution is reached
at the edges of this interval. The exponential distribution of a random variable t0 gives the lower
bound for the considered complex reliability indicator.
Ktu(λT)
0,9
0,8
0,7
0,6
0 1,25 2,5 3,75 λT
Figure 3: Graphs of the dependence of the coefficient of technical use on Т at various values z t a1
at 1 , f 1
Ktu(λT)
λ=0,01
0,9
λ=0,1
0,8
λ=0,2
0,7
0,6
0 10 20 30 T, h
Figure 4: Graphs of the dependence of the coefficient of technical use on the frequency of maintenance
at various values at 1 , z t a1 1,6 , f 1 (where 1 − 0,01 ; 2 − 0,1 ; 3 − 0,2 )
306
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