Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution
Design procedures of sufficiency of the ZIP, quantities of nonrestorable spare elements on the basis of use as theoretical model of refusals of elements and a product of two-parametrical DN -distribution are presented.
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Інститут проблем математичних машин і систем НАН України
2009
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| Назва видання: | Математичні машини і системи |
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| Цитувати: | Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution / V.P. Strelnikov, Shou Weichao // Мат. машини і системи. — 2009. — № 2. — С. 139–144. — англ. |
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irk-123456789-470432013-07-09T03:04:01Z Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution Strelnikov, V.P. Shou Weichao Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення Design procedures of sufficiency of the ZIP, quantities of nonrestorable spare elements on the basis of use as theoretical model of refusals of elements and a product of two-parametrical DN -distribution are presented. Представлено методики розрахунку достатності ЗІП, кількості невідновлюваних запасних елементів на основі використання як теоретичної моделі відмов елементів і виробу двопараметричного DN -розподілу. Представлены методики расчета достаточности ЗИП, количества невосстанавливаемых запасных элементов на основе использования в качестве теоретической модели отказов элементов и изделия двухпараметрического DN -распределения. 2009 Article Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution / V.P. Strelnikov, Shou Weichao // Мат. машини і системи. — 2009. — № 2. — С. 139–144. — англ. 1028-9763 http://dspace.nbuv.gov.ua/handle/123456789/47043 62.019 en Математичні машини і системи Інститут проблем математичних машин і систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
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Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення |
| spellingShingle |
Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення Strelnikov, V.P. Shou Weichao Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution Математичні машини і системи |
| description |
Design procedures of sufficiency of the ZIP, quantities of nonrestorable spare elements on the basis of use as theoretical model of refusals of elements and a product of two-parametrical DN -distribution are presented. |
| format |
Article |
| author |
Strelnikov, V.P. Shou Weichao |
| author_facet |
Strelnikov, V.P. Shou Weichao |
| author_sort |
Strelnikov, V.P. |
| title |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution |
| title_short |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution |
| title_full |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution |
| title_fullStr |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution |
| title_full_unstemmed |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution |
| title_sort |
calculation of spare elements in complete sets of the zip on the basis of dn-distribution |
| publisher |
Інститут проблем математичних машин і систем НАН України |
| publishDate |
2009 |
| topic_facet |
Якість, надійність і сертифікація обчислювальної техніки і програмного забезпечення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/47043 |
| citation_txt |
Calculation of spare elements in complete sets of the ZIP on the basis of DN-distribution / V.P. Strelnikov, Shou Weichao // Мат. машини і системи. — 2009. — № 2. — С. 139–144. — англ. |
| series |
Математичні машини і системи |
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AT strelnikovvp calculationofspareelementsincompletesetsoftheziponthebasisofdndistribution AT shouweichao calculationofspareelementsincompletesetsoftheziponthebasisofdndistribution |
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2025-07-04T06:40:26Z |
| last_indexed |
2025-07-04T06:40:26Z |
| _version_ |
1836697493609381888 |
| fulltext |
© Strelnikov V.P., Weichao Shov, 2009 139
ISSN 1028-9763. Математичні машини і системи, 2009, № 2
ЯКІСТЬ, НАДІЙНІСТЬ І СЕРТИФІКАЦІЯ ОБЧИСЛЮВАЛЬНОЇ
ТЕХНІКИ І ПРОГРАМНОГО ЗАБЕЗПЕЧЕННЯ
UDC 62.019
V.P. STRELNIKOV, SHOU WEICHAO
CALCULATION OF SPARE ELEMENTS IN COMPLETE SETS OF THE ZIP ON THE
BASIS OF DN-DISTRIBUTION
Abstract: Design procedures of sufficiency of the ZIP, quantities of nonrestorable spare elements on the basis of use
as theoretical model of refusals of elements and a product of two-parametrical DN-distribution are presented.
Key words: probability of non-failure operation, spare elements, the period of updating the ZIP, sufficiency of the ZIP.
Анотація: Представлено методики розрахунку достатності ЗІП, кількості невідновлюваних запасних
елементів на основі використання як теоретичної моделі відмов елементів і виробу двопараметричного
DN-розподілу.
Ключові слова: імовірність безвідмовної роботи, запасні елементи, період поповнення ЗІП, достатність
ЗІП.
Аннотация: Представлены методики расчета достаточности ЗИП, количества невосстанавливаемых
запасных элементов на основе использования в качестве теоретической модели отказов элементов и
изделия двухпараметрического DN-распределения.
Ключевые слова: вероятность безотказной работы, запасные элементы, период пополнения ЗИП,
достаточность ЗИП.
1. Introduction
The complete set of spare parts, tools and accessories (ZIP) should include the spare parts necessary for
repair and maintenance of an efficient condition of products during certain time (the period of updating the
ZIP) and maintenance of a demanded level of reliability of the last. One of the major aprioristic information
defining finally volume of spare parts, the theoretical model of refusals accepted at calculations of number
of refusals is represented. In the present work it is accepted, that distribution of an operating time to
refusal (on refusal) elements is described by DN-distribution which is represented the most universal and
adequate model of refusals of electro technical products.
By development of a design procedure of spare elements in single complete sets the it ZIP
following assumptions are accepted: the equipment consists of nonrestorable elements consistently
connected reliability; reliability of working and spare elements of each standard rating is identical; at
storage spare elements do not refuse; all working (reserve, spare) elements refuse independently.
Depending on purpose of the equipment, system of its maintenance service and repair,
requirements to the equipment on reliability in coordination with the customer the period of updating ( )ПЗT
is accepted to equal one, two or three years. The present technique allows to count volume the ZIP and
on more long periods (10–20 and more years).
2. Estimation of reliability of a product and parameters of sufficiency the ZIP
Define a total operating time of products ∑t at the moment of the beginning of the period of updatings in
view of factor of intensity of operation during this period. Establish a demanded level of a parameter of
reliability of a product at the moment of the ending of the period of updating: probability of non-failure
operation тр
издR .
ISSN 1028-9763. Математичні машини і системи, 2009, № 2 140
Parameters of sufficiency define the ZIP дπ on the basis of the analysis of expected and
demanded parameters of reliability ( )(/ ПЗизд
тр
издд ТRR≥π , approximating up to the nearest values of
some: .9999,0;9995,0;999,0;995,0;99,0;95,0;9,0
The note. The parameter of sufficiency of the ZIP дπ can be set by the customer irrespective of
expected parameters of reliability of a product.
Nitial data for calculation of the nomenclature of the complete set the ZIP, quantities of spare parts
are: structure and composition of a system; the nomenclature of making elements m (standard nominals )
and number of elements of each type in ),1( mi = ; parameters of reliability of each standard nominal
oioi VT ,
.
The task of requirements to parameters of sufficiency of the ZIP for products with nonrestorable spare
elements
The ZIP of products with nonrestorable spare elements applies дπ to an estimation of sufficiency of
complete sets probability of that during ПЗT work of a product there it will be no refusal the ZIP. The
probability дπ is applied to an estimation of sufficiency of complete sets the ZIP provided that all stocks in
this complete set replenish periodically with the identical periods and a parameter of reliability of a product
the probability of non-failure operation serves. Initial data for calculation of parameters of sufficiency дπ of
the ZIP are expected probability of non-failure operation of a product )(0 ПЗТR at the moment of the
ending of the period of updating the ZIP and requirements to parameters of reliability of a product .тр
издR
If it is not known )(0 ПЗТR , calculate
.
11
)(
0
2
0
0
2
0
+−Φ−
−Φ=
−
xV
x
e
xV
x
ТR V
ПЗ
Where 0ТTx ПЗ= ; 0T – an average operating time to refusal of a product; ( )⋅Φ – function of normal
distribution.
The note. In last formula value of an operating time ПЗTt = that corresponds to the beginning of
operation of a product is accepted. Generally, when count )(0 ПЗТR , consider an operating time Σt prior
to the beginning of the next period of updating the ZIP, ПЗTtt += Σ .
Estimation of demanded probability of non-failure operation of elements
Accept the value тр
издR , satisfying to a parity тр
издR ≤ )(0 ПЗТR and according to values of the aforesaid of
some. Define a parameter of sufficiency of the complete set the ZIP from a ratio )(/ 0 ПЗ
тр
издд ТRR≥π . If
)(0 ПЗТR less than the smallest value of number from recommended of some probabilities accept value
ISSN 1028-9763. Математичні машини і системи, 2009, № 2 141
тр
издд R=π from these of some from reasons of importance of carried out functions and economic
feasibility.
Calculate a demanded level of probability of non-failure operation тр
iR for set of elements of i -th
standard rating
= mтр
изд
тр
i RR
1
)( , approximating up to the nearest value from recommended.
Estimation of probability of non-failure operation of elements
Settle an invoice iR (probabilities of non-failure operation of set in of elements of i -th type for an interval
( ПЗТt +Σ,0 )). On values
5,0
1
+
=
in
F and oiV=ν from corresponding tables of DN -distribution
define or deciding the equation:
+−Φ
+
−Φ=
x
x
x
x
F
ννν
12
exp
1
2
.
Further determine an operating time to first failure elements of i -th type under the formula
iii Txt 01 = . Calculate ( ) iПЗi tTtx 1/+= Σ
∗ . On values ∗
ix again from the same tables already a return
entrance define (probability of occurrence of refusal for an interval ( ПЗTt +Σ ) i -th type of elements) or
deciding the above-stated equation for ∗= ixx rather F . Calculate the required value ii FR −= 1 .
On values iR also тр
iR define a parameter of sufficiency тр
iдπ of maintenance of nonrestorable
spare making elements of i -th type (approximating up to the nearest greater value from some
recommended) under the formula (under condition of тр
ii RR < ):
( )
( ) ( )
≥
<
−
−−
=
.,
;,
)1(
)1(
1
11
1
m
дi
m
д
m
дi
i
тр
i
тр
дi
Rпри
Rпри
R
R
ππ
π
π
The note. If iR ≥ тр
iR , elements of the given type are not included into the nomenclature of the
ZIP (accept 0=iz for i =1, 2, …, q ).
Calculation of factor of recalculation
Values of factor of the recalculation дiθ providing the set level of sufficiency тр
дiπ of nonrestorable
elements of i -th type, is calculated under the formula:
+−Φ
+
−Φ=
∗∗∗
дii
дi
iдli
дiтр
дi
VVV θ
θ
θ
θπ
0
2
00
1
)(
2
exp
1
.
ISSN 1028-9763. Математичні машини і системи, 2009, № 2 142
Where
i
i
i
V
V
α
0
0 =∗ ; [ ]1+= ii aINTα ; ia – the average expected value of number of refusals for
considered{examined} time.
At use of corresponding tables value дiθ is defined on values =F тр
дiπ and =ν ∗
iV0 , i.e.
tabulared value x is equal to factor of recalculation ( )∗= i
тр
дiдi Vx 0;πθ . For example, for 95,0=тр
дiπ and
∗
iV0 =0,7 is defined from tables of DN -distribution: ( ) ( ) .3634,27,0;95,0; 0 === ∗ xVx i
тр
дiдi πθ
3. Calculation of quantity of spare elements
Calculation of quantity of spare nonrestorable elements iz , making out in the form of tabl. 1, make in the
following order:
1. Define average operating time to refusal of making elements iT0 :
– if average operating time to refusal of elements accept ii TT 00 = is presented;
– if i0λ given, calculate iT0 as follows:
– if i0λ > 510− , iT0 =
i0
1
λ
;
– if 510− ≥ i0λ > 910− , iT0 = [ ]21
0
1
0 )(ln02443,0)(ln0971,120568,2exp −− −+ ii λλ ;
– if 910− ≥ i0λ , iT0 =
i0
002,0
λ
.
2. Using the known information, establish values of factors of a variation of an operating time of elements
oiV . For products of electronic techniques (semi-conductor devices, integrated microcircuits, resistors,
condensers, etc.) accept oiV =1 if there is no the information specifying this value.
Initial data (the type of an element i , number of elements of i -th type in , an average operating
time to refusal of i -th type of elements oiT , factor of a variation of an operating time of elements oiV )
writes down in first four columns. Calculate values iR and write down in 5 column of tabl. 1. Calculate
values тр
iR and write down in 6 column of тabl.1. On values iR also тр
iR define a parameter of
sufficiency for corresponding elements тр
iдπ and write down results in 7 column, approximating (if
necessary) result up to tabulated values according to some accepted.
Calculation of the expected number of refusals of nonrestorable elements
If the expected number of refusals is not enough, at 5,0
0
2 ≤+= Σ
i
ПЗ
i T
Tt
x , calculate the average
expected number of refusals ia as follows. Calculate probability of refusal of elements of i -th type at the
moment of the beginning of the period of updating 1iF :
ISSN 1028-9763. Математичні машини і системи, 2009, № 2 143
+−Φ
+
−Φ=
10
1
2
010
1
1
12
exp
1
ii
i
iii
i
i
xV
x
VxV
x
F .
If Σt =0, 1iF =0.
Calculate probability of refusal of elements of i -th type at the moment of the ending of the period
of updating 2iF :
.
12
exp
1
20
2
2
020
2
2
+−Φ
+
−Φ=
ii
i
iii
i
i
xV
x
VxV
x
F
Calculate the average expected value of refusals under the formula:
)( 12 iiii FFna −=
.
The calculated values of the expected average of refusals ia bring in 8 column of tabl. 1.
If a relative operating time 5,02 >ix , repeated refusals a population mean of number of refusals
elements of i -th type of the beginning for a moment of the period of updating )( 11 ii xΩ calculate are
probable, using corresponding{meeting} tables, or under the formula:
.
2
exp)(
5
1 10
1
2
010
1
11 ∑
=
+−Φ
+
−Φ=Ω
k ii
i
iii
i
ii
xV
kx
V
k
xV
kx
x
If Σt =0, )( 11 ii xΩ =0.
Calculate a population mean of number of refusals of elements of i -th type at the moment of the
ending of the period of updating )( 22 ii xΩ :
.
2
exp)(
5
1 20
2
2
020
2
22 ∑
=
+−Φ
+
−Φ=Ω
k ii
i
iii
i
ii
xV
kx
V
k
xV
kx
x
Calculate average expected value of refusals under the formula:
[ ])()( 1122 iiiiii xxna Ω−Ω= .
The calculated values of an expected average of refusals ia , write down in a column 8 tabl. 1. On
the basis of the received values тр
iдπ , ia and iα define factor iдθ and result write down in 9 column of
tabl. 1.
Define number of spare parts under the formula iдii az θ= (in case of presence of not loaded
number reserve ( ir – elements of i -th type) and ii ra > calculate number of spare parts under the
ISSN 1028-9763. Математичні машини і системи, 2009, № 2 144
formula )( iiдii raz −= θ ). A rounding of settlement value iz make up to an integer in greater side, for
05,0<iz accept 0=iz .
The received values iz bring in 10 column of tabl. 1.
Table 1
Type of an
element, i
in ,
pieces
oiT ,
h.
iV 0
iR тр
iR тр
iдπ ia
iдθ iz
1 2 3 4 5 6 7 8 9 10
4. The conclusion
Design procedures of parameters of sufficiency, calculation of number of nonrestorable elements are
presented to the ZIP on the basis of the most adequate two-parametrical function of distribution of an
operating time to refusal (on refusal) of Diffusion nonmonotonic distribution ( DN -distribution). More
precise definition of volume of the ZIP has the important practical value for maintenance of reliability at
operation of technical systems.
Стаття надійшла до редакції 20.03.2009
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