Mathematical models for evaluation of operational readiness of periodically inspected electronic systems
The mathematical models and theorems that identify stationary and non-stationary coefficients of operational readiness are developed for an arbitrary and exponential distribution of time to failure of avionics systems. Proven expressions, in contrast to all other known, take into account the trustwo...
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| Cite this: | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems / V.V. Ulansky, I.A. Machalin // Мат. машини і системи. — 2012. — № 1. — С. 119-128. — Бібліогр.: 13 назв. — англ. |
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| citation_txt | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems / V.V. Ulansky, I.A. Machalin // Мат. машини і системи. — 2012. — № 1. — С. 119-128. — Бібліогр.: 13 назв. — англ. |
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| description | The mathematical models and theorems that identify stationary and non-stationary coefficients of operational readiness are developed for an arbitrary and exponential distribution of time to failure of avionics systems. Proven expressions, in contrast to all other known, take into account the trustworthiness characteristics of multiple inspections. It is shown that the mean time between unscheduled repairs of the system is largely determined by the probability of a false failure.
Розроблено математичні моделі та доведено теореми, що дозволяють визначити стаціонарний і нестаціонарний коефіцієнти оперативної готовності при довільному і експоненційному законах розподілу напрацювання системи до відмови. Доведені вирази, які, на відміну від відомих, враховують характеристики достовірності багаторазового контролю працездатності. Показано, що середній час напрацювання на незапланований ремонт системи багато в чому визначається ймовірністю помилкової відмови.
Разработаны математические модели и доказаны теоремы, позволяющие определить стационарный и нестационарный коэффициенты оперативной готовности при произвольном и экспоненциальном законах распределения наработки системы до отказа. Доказанные выражения, в отличие от известных, учитывают характеристики достоверности многократного контроля работоспособности. Показано, что среднее время наработки на незапланированный ремонт системы во многом определяется вероятностью ложного отказа.
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© Ulansky V.V., Machalin I.А., 2012 119
ISSN 1028-9763. Математичні машини і системи, 2012, № 1
UDC 629.735.05:621.3(045)
V.V. ULANSKY, I.A. MACHALIN
MATHEMATICAL MODELS FOR EVALUATION OF OPERATIONAL READINESS
OF PERIODICALLY INSPECTED ELECTRONIC SYSTEMS
Анотація. Розроблено математичні моделі та доведено теореми, що дозволяють визначити
стаціонарний і нестаціонарний коефіцієнти оперативної готовності при довільному і експонен-
ційному законах розподілу напрацювання системи до відмови. Доведені вирази, які, на відміну від
відомих, враховують характеристики достовірності багаторазового контролю працездатності.
Показано, що середній час напрацювання на незапланований ремонт системи багато в чому визна-
чається ймовірністю помилкової відмови.
Ключові слова: системи авіоніки, технічне обслуговування, надійність, періодичний контроль пра-
цездатності, функція оперативної готовності, коефіцієнт оперативної готовності, помилки ко-
нтролю.
Аннотация. Разработаны математические модели и доказаны теоремы, позволяющие опреде-
лить стационарный и нестационарный коэффициенты оперативной готовности при произволь-
ном и экспоненциальном законах распределения наработки системы до отказа. Доказанные вы-
ражения, в отличие от известных, учитывают характеристики достоверности многократного
контроля работоспособности. Показано, что среднее время наработки на незапланированный
ремонт системы во многом определяется вероятностью ложного отказа.
Ключевые слова: системы авионики, техническое обслуживание, надежность, периодический ко-
нтроль работоспособности, функция оперативной готовности, коэффициент оперативной гото-
вности, ошибки контроля.
Abstract. The mathematical models and theorems that identify stationary and non-stationary coefficients
of operational readiness are developed for an arbitrary and exponential distribution of time to failure of
avionics systems. Proven expressions, in contrast to all other known, take into account the trustworthiness
characteristics of multiple inspections. It is shown that the mean time between unscheduled repairs of the
system is largely determined by the probability of a false failure.
Keywords: avionics systems, maintenance operations, reliability, periodic control of working ability,
function of operational readiness, coefficient of operational readiness, inspection errors.
1. Introduction
Statement of the problem. The effectiveness of some aircraft electronic systems is largely
determined by the operational readiness of the systems to perform their functions within a
specified time at a certain range of flight operations. In this paper, the operational readiness is
defined as the property of a given system to be available when it demanded and to operate within
the tolerances for a specified period of time. The measures of the operational readiness of aircraft
electronic systems determine the level of safety and regularity of the aircraft operations. One of
the major challenges currently facing the airlines is the development of optimal maintenance
programs of operated aircraft fleet. To optimize the maintenance program it is necessary to
develop mathematical models for evaluating the measures of the operational readiness of
electronic systems. The operational readiness can be characterized by different measures. The
most widely used measures are the non-stationary and stationary operational readiness
coefficients. The operational readiness coefficient is defined as the probability that the system
will be in the operable state at any time, except for the scheduled periods during which the use of
the system on the appointment is not required, and from that moment will work reliably within a
specified interval of time. The non-stationary and stationary coefficients of operational readiness
are considered, respectively, on a finite and an infinite interval of service planning.
Analysis of the recent research and publications. A number of papers, for example
120 ISSN 1028-9763. Математичні машини і системи, 2012, № 1
[1−4], have been devoted to performance evaluation of operational readiness of technical
systems. In these studies as an integral index of reliability and availability of systems the coeffi-
cient of operational readiness is proposed to use. For avionics systems, the values of this
coefficient depend not only on the faultlessness and maintainability characteristics but also on the
trustworthiness characteristics of the onboard built-in test equipment (BITE), since due to an
error checking the adoption of erroneous decisions is possible. However, the well-known
expressions do not include the trustworthiness indexes of multiple inspections, so do not provide
an adequate assessment of the operational readiness of onboard electronic systems.
Thus, the aim of the article is to develop the mathematical models for determining the
non-stationary and stationary coefficients of operational readiness taking into account the
characteristics of trustworthiness of multiple inspections, faultlessness, maintainability, and
spare part system sufficiency.
Problem statement and description of maintenance strategy. Currently, aircraft
avionics meets the requirements of ARINC 700 [5]. Each avionics system is represented by a set
of redundant and easily replaced blocks called the line replaceable units (LRUs). Each LRU is a
single-block system consisting of several modules and having the BITE. The modular design of
LRUs provides easy access to circuits and components for testing and replacement in case of
failures. Each LRU operates till safety failure, which is recorded during the flight or at the base
airport after landing the aircraft. Rejected LRUs are replaced in the base airport by the spare
LRUs from the warehouse. Since all avionic systems are redundant, the failure of any LRU does
not lead to the failure of the corresponding system. Therefore, this strategy is called the strategy
till safety failure or breakdown maintenance strategy. It is necessary to construct a mathematical
model that takes into account the main parameters of the exploitation process to evaluate the
operational readiness measures.
For determining the operational readiness measures we will use the well-known property
of the regenerative processes [6, 7], which consists in the fact that the fraction of time during
which the system was in the state ),1( rE =µµ equals the ratio of average time spent in the state
µE per regeneration cycle to the average duration of the cycle.
Let us consider the maintenance model of an avionics system consisting of a single LRU,
i.e., the system structure in terms of reliability is not considered. This is due to the fact that the
LRU is the smallest removable onboard part of any system of avionics. The LRU operation
process can be considered on a finite or an infinite time interval as a sequence of changing
various LRU states. For definiteness, we first consider a finite interval of the planning service.
Therefore, the LRU behavior in the range of service planning ),0( T can be described by a
stochastic process )(tL with a finite space of states U
i
iEE = . The process )(tL changes only
stepwise, with each jump due to the transition of the LRU in one of possible states. It is assumed
that a regenerative stochastic process, having the property will always return to the point of
regeneration, from which further development of the process does not depend on its past behavior
and is a copy of the probabilistic process that began at the moment t .
Suppose that at time 0=t the operation of a new LRU begins and the periodic LRU
inspections are planned in the moments ττττ N,...3,2, , where 1−=
τ
T
N is the number of the
LRU inspections on the ),0( T range. Assume that the checking of the LRU is run before each
aircraft departure. By the LRU checking results at the moment ),1( Nkk =τ the following
decisions are possible:
− to use the LRU until the next inspection, if it is recognized as operable;
− to repair the LRU, if it is recognized inoperable, and then allow its further use;
ISSN 1028-9763. Математичні машини і системи, 2012, № 1 121
− to nominate next inspection moment of the LRU in the moment τ+ )1(k .
Let us define the random process )(tL . At any given time t the LRU can be in one of the
following states [8−12]: ,1E if at the moment t the LRU is used as intended and is in the operable
state; ,2E if at the moment t the LRU is used as intended and is in an inoperable state (latent
failure); 3E , if at the moment t the LRU is not used for its intended purpose because of
operability checking; 4E , if at the moment t the LRU is not used as intended and its removal is
held from the board of an aircraft; 5E , if at the moment t defective LRU is in the state of
unscheduled idle on the board of an aircraft at the base airport because of dissatisfaction with the
application to a spare LRU from the warehouse; 6E , if at the moment t the LRU is not used for
its intended purpose because the "false recovery" is performed; 7E , if at the moment t the LRU
is not used for its intended purpose because of the "proper recovery"; 8E , if at the moment t the
LRU is not used as intended and performed its installation on the board of an aircraft.
Let iS be the time in the state )8,1( =iEi . Obviously, iS is a random variable with ex-
pected mean time [ ] .ii MSEM = Define Ξ as the time to failure. The uncertainty in the values
that Ξ can take is described through a failure distribution function )ξ(F which characterizes the
probability of { }ξ≤ΞP . The equations for 81, MSMS were published in [8, 12]. The average
regeneration cycle of the LRU is determined by the formula:
∑
=
=
8
1
0
i
iMSMS .
2. Non-stationary operational readiness coefficient
Let ( )θτ,kP be the probability that the LRU will be operable at time ( )Nkk ,0=χ+τ and it will
work faultlessly within a specified time θ starting from the moment χτ +k , where θτχ −≤<0 ,
and θ is the time of the task. Suppose χ is a random variable with a uniform distribution on the
interval from τk to ( ) θ−τ+1k , and its distribution is described by the probability density function
( )
θ−τ
= 1
xf . (1)
Theorem 1. The following formula holds for the non-stationary function of operational
readiness:
( ) ( ) ( )
( ) ( ) ( )
( )
∫ ∫∑
∫ ∫
− ∞
++−=
− ∞
++
ϑϑϑ−−−
−
+
+ϑϑϑ−
−
==
θτ
xjk
PO
k
j
R
xkτ
PO
,dxdkjkPjP
dxdk;)(k,P,kP
0 θτ1
θτ
0 θ
ωτj)(;τ1)(τ,τ
θτ
1
ωττ1τ
θτ
1
θτ
(2)
where ( )ξττ1τ k;)(k,PPO − is the conditional probability of the event “properly operable”, defined
as the probability of co-occurrence of the following events: in the operability checking at the in-
stants ττ k, the LRU was recognized operable, on condition that ξΞ = and ( )τ1ξτ +≤< kk ; Ξ is
the LRU random operation time to failure; ( )τjPR is the probability of the LRU recovery at in-
stant τj ; ( )ξω is the probability density function of the random variable Ξ .
Proof. The non-stationary function of operational readiness can be defined as the
122 ISSN 1028-9763. Математичні машини і системи, 2012, № 1
probability that the interval of a trouble-free LRU operation ]θ,[ +χχ entirely falls within one of
the intervals between inspections ( ) Nkkk ,0,τ)1(,τ =+ . It should be taken into account that at
any of the moments τk the LRU can be recovered (properly or falsely).
The LRU will work faultlessly in the interval [ ]θ+χ+τχ+τ kk , , if one of the following
events occurs:
– the time to failure of the system is more than θ+χ+τk and by inspection results at
instants ττ k, the LRU has been acknowledged as operational, i.e.,
( )I I
>+χ+>=∆
=
∗
k
i
i ik
1
1 τΞθτΞ , (3)
where ∗
iΞ is a random assessment of the operating time to failure Ξ by the checking results of the
LRU at the time τi [13];
– the last LRU recovery occurred at the moment ( )kjj ,1=τ , and then the LRU
no longer failed. When checking the operability at the moments ( ) ττ+ kj ,1 the LRU was
recognized operable, i.e.
( ) ( )( ) ( ) ,τΞθτΞτ
1 1
2 U I II
k
j
k
ji
i jijkjB
= +=
∗
−>+χ+−>=∆ (4)
where ( )τjB is the event consisting in the LRU recovery at the instant τj .
The probability of the LRU recovery is calculated as
( ){ } )τ()τ(τ)( jPjPjBPjP PRFRR +==τ , (5)
where ( )τjPFR and ( )τjPPR are, respectively, the probability of a false and proper LRU recovery
[9, 10].
The probability of the event (3) we find by integrating the probability density
( )** ,; k10 ξξξω
of scalar random variables Ξ,Ξ,Ξ **
1 k [13] within appropriate limits:
( ) ( ) ( ) ( ) dxdkkPPkP
xk
PO ϑϑϑ−
−
==+ ∫ ∫
− ∞
++
θτ
0 θτ
1 ωτ;1)τ(τ,
θτ
1
∆θτ . (6)
Using the addition theorem of probability and the probability density functions
( )**
10 ,; kξξξω and (1), we find the probability of the event (4):
( ) ( ) ( ),θτττ
1
2 +−=∆ ∑
=
jkPjPP
k
j
R (7)
where
( ) ( ) ( )
( )
.ωτ)(;τ)1(,τ
θτ
1
θττ
θτ
0 θτ
dxdjkjkPjkP
xjk
PO ϑϑϑ−−−
−
=+− ∫ ∫
− ∞
++−
(8)
On the basis of the addition theorem probability for mutually exclusive events we can
write:
( ) ( ) ( )21, ∆+∆=θτ PPkP . (9)
Substituting expressions (6) and (7) into (9) gives (2). This proves the theorem.
ISSN 1028-9763. Математичні машини і системи, 2012, № 1 123
Equation (2) can be simplified by setting ( ) 10 =RP . Then
( ) ( ) ( ) ( )
( )
∫ ∫∑
− ∞
++−=
ϑϑ−−−
−
=
θτ
0 θτ0
ωτ)(;τ)1(,ττ
θτ
1
θ,τ
xjk
PO
k
j
R jkjkPjPkP . (10)
Corollary 1. If the LRU has an exponential distribution of time to failure, then the
following relations hold:
( )( )
( ) ( ) ( )∑
=
λτ−−−
θ−τλ−λθ−
α−τ
θ−τλ
−=θτ
k
j
jkjk
R ejP
ee
kP
0
1)(
1
),( , (11)
( )∑
−
=ν
−ν−τν−λ− α−ντα=τ
1
0
1)( 1)()(
j
jj
RFR ePjP , (12)
( ) ( ) ( ) ( ) ( )( )( )
βα−−ντβ−=τ ∑∑
−
ν=µ
µ−−ν−µλτν−+µ−λτν−µ−
−
=ν
1
11
1
0
11
j
j
j
RPR eePjP , (13)
where α and β are, respectively, the conditional probabilities of false rejection and undetected
failure for the LRU operability checking. The proof of corollary is omitted because of its bulki-
ness.
Example 1. Calculate the non-stationary function of operational readiness, if the LRU has
[ ] h10000/1Ξ =λ== MMTBF , warranty service life h5000=T , average duration between the
LRU operability checking h10=τ , time of the task h1θ = , conditional probability of “false re-
jection” and “undetected failure” during operability checking by the built-in test equipment
005,0=β=α .
Using (5) and (10)-(12), we calculate the values of the probabilities RP , FRP , PRP and
( )θτ,kP . They are listed in Table 1.
As can be seen from Table 1, starting from the sixth checkout all probabilities reached sta-
tionary values. It should be noted that the probability of false recovery in five times as much the
probability of proper recovery.
Table 1. Values of the calculated probabilities, depending on the number of checking operability
j
RP FRP PRP ( )θτ ,kP
0 1,0 0 1,0 0,999400221605680
1 0,005989505164959 0,004995002499167 0,000994502665792 0,999395227102240
2 0,005994447745731 0,004994977536638 0,000999470209093 0,999395202154683
3 0,005994472433934 0,004994977411327 0,000999495021984 0,999395202030070
4 0,005994472557252 0,004994977411327 0,000999495145925 0,999395202029444
5 0,005994472557868 0,004994977411324 0,000999495146544 0,999395202029444
6 0,005994472557871 0,004994977411324 0,000999495146547 0,999395202029444
7 0,005994472557871 0,004994977411324 0,000999495146547 0,999395202029444
500 0,005994472557871 0,004994977411324 0,000999495146547 0,999395202029444
In general, the non-stationary coefficient of operational readiness is determined by
averaging expression (10) on the interval ( )T,0
( ) ( )( ) ( ) ( ) ( )
( )
∑ ∫ ∫∑
=
− ∞
++−=
ϑϑϑ−−−
−+
=
N
k xjk
PO
k
j
ROR dxdjkjkPjP
N
TK
0
θτ
0 θτ0
ωτ)(;τ)1(,ττ
θτ1
1
θ, . (14)
124 ISSN 1028-9763. Математичні машини і системи, 2012, № 1
With an exponential distribution of time to failure we get from formula (14)
( )
( )( )
( ) ( ) ( ) ( )∑∑
= =
−−−
−−−
−
−+
−=
N
k
k
j
jkjk
ROR ejP
N
ee
TK
0 0
λτ
θτλλθ
α1)τ(
θτλ1
1
θ, . (15)
Example 2. Calculate the non-stationary coefficient of operational readiness for the initial
data of Example 1.
By substituting the initial data in (15), we obtain ( ) 0,999395, =θTKOR .
3. Stationary operational readiness coefficient
The stationary coefficient of operational readiness is used in the case of infinite time of service
planning ( )∞,0 . The following theorem determines the stationary coefficient of operational rea-
diness.
Theorem 2. For the stationary coefficient of operational readiness the following formula
holds:
( ) ( ) ( ) ( ) dxdkkP
MSMS
K
k xk
POOR ϑϑϑ−
−
−= ∑ ∫ ∫
∞
=
− ∞
++
ωτ;τ)1(,τ
θττ
θ
0
θτ
0 θτ30
. (16)
Proof. Suppose as before that χ is a random variable with the uniform distribution in the
interval between τk and ( )τ+1k and the probability density function determined by (1). To prove
(16) we express the probability ( )θτ,kP through the renewal density function, and then proceed to
the limit
( ) ( )θτθ ,lim kPK
k
OR
∞→
= .
The probability ( )θ+τkP is given by (6).
Since the LRU recovery is only possible at discrete moments of time ...,, ττ 2 , the renewal
density function can be expressed through the δ -function:
( ) ( ) ( )τηδτη
1
jjPh
k
j
R −=∑
=
. (17)
Using (17), expression (7) can be represented in the integral form:
( ) ( ) ( )∫
τ
ηηθ+η−τ=∆
k
dhkPP
0
2 .
On the basis of the addition theorem of probability for the mutually exclusive events
we can write:
( ) ( ) ( ) ( ) ( ) ( )∫
τ
ηηθ+η−τ+θ+τ=∆+∆=θτ
k
dhkPkPPPkP
0
21 .,
Since ( ) 1ξτ;τ)1(,τ ≤− kkPPO , then ( ) ( )θ+τ−≤θ+τ kFkP 1 , so ( ) 0lim =θ+τ
∞→
kP
k
.
Further, since the function )( θ+τkP is not negative, it is of limited variation on the semi-
axis ( )∞,0 and it satisfies the inequality
∫ ∫
∞ ∞
∞<−≤
0 0
)](1[)( dyyFdyyP ,
ISSN 1028-9763. Математичні машини і системи, 2012, № 1 125
then according to Smith's theorem in the case of lattice random variable [7] we have:
∑∫
∞
=∞→
+=+−
0
τ
0
θ)τ(
µ
τ
η)η()θητ(lim
k
k
k
kPdhkP (18)
where µ is the average time between the LRU recovery.
Since for the calculation of the stationary coefficient of operational readiness the scheduled
maintenance is not taken into account, then
.30 MSMS −=µ (19)
Substituting expressions (6) and (19) into (18) gives (16). The theorem is proved.
Corollary 2. If there are no errors at the operability checking, then
( )
( ) ( )
( ) ( )[ ] ( ){ } МFRPRD
k
k xk
OR
tttMStkFkFk
dxd
K
+++++−++
ϑϑ−
=
∑
∑ ∫ ∫
∞
=
∞
=
− ∞
++
5
0
0
θτ
0 θτ
ττ11τ
ωθττ
θ , (20)
where 5MS is given by
)( CDMSP5 ttttMS −++∆Ψ= ,
( )
( )
++∆<
++∆≥
=Ψ
;,
;,
MDSPС
MDSPС
ttttif1
ttttif0
SPt∆ is the average delay time of an application for a spare LRU;
M
t is the average installation
time of the LRU on the board of an aircraft; Dt is average time of dismantling the LRU from the
board of an aircraft; PRt and FRt is the average time of the LRU proper and false recovery respec-
tively; Ψ – indicator function; Ct is the average scheduled time of technical parking of an air-
craft.
Proof. In the absence of errors in the checks of operability the probabili-
ty ( ) 1ξτ;τ)1(,τ =− kkPPO , therefore
( ) ( ) ( )∑ ∫ ∫∑ ∫ ∫
∞
=
θ−τ ∞
θ++τ
∞
=
θ−τ ∞
θ++τ
ϑϑω=ϑϑωϑττ−τ
0 00 0
;)1(,
k xkk xk
PO dxddxdkkP .
Furthermore, from [8, 12] follows that in the case of ideal operability checking
=+ 21 MSMS ( ) ( )[ ] ( ){ }∑
∞
=
τ−τ++τ
0
11
k
kFkFk ; (21)
( ) ( )[ ] ( ){ } .11 5
0
876542130
MFRPRD
k
tttMStkFkFk
MSMSMSMSMSMSMSMSMS
+++++τ−τ++τ=
=++++++=−
∑
∞
=
(22)
Substitution of (21) and (22) into (16) gives (20). The corollary is proved.
Corollary 3. If the LRU has an exponential distribution of time to failure, then the
following formula holds:
( )
( )( )
( ) ( )[ ]( )30
λτ
θτλλθ
α11θτλ
1τ
θ
MSMSe
ee
KOR −−−−
−= −
−−−
. (23)
126 ISSN 1028-9763. Математичні машини і системи, 2012, № 1
Proof. With the exponential distribution law of time to failure we can get
( ) ( ) ( ) =ϑλα−=ϑϑωϑττ−τ ∑ ∫ ∫∑ ∫ ∫
∞
=
θ−τ ∞
θ++τ
λϑ−
∞
=
θ−τ ∞
θ++τ
dxdedxdkkP
k xk
k
k xk
PO
0 00 0
1;)1(,
( )( ) ( )[ ] ( )( )
( )[ ]λτ
θτλλθ
0
kλτ
θτλλθ
α11λ
1
α1
λ
e1
−
−−−∞
=
−
−−−
−−
−=−−= ∑ e
ee
e
e
k
. (24)
With (24) formula (16) reduces to (23). The corollary is proved.
Corollary 4. If the conditions of corollary 3 are satisfied and ,τθ << then
( ) λθ
θ
−≈ AeKOR , (25)
where A is the LRU availability determined as [12]
30
1
MSMS
MS
A
−
= . (26)
Proof. When τθ << expression (24) simplifies and takes the following form:
( ) ( ) ( )
( )[ ] .
11
1
;)1(, 1
0 0
λθ−
λτ−
λτ−λθ−∞
=
θ−τ ∞
θ++τ
=
α−−λ
−≈ϑϑωϑττ−τ∑ ∫ ∫ eMS
e
ee
dxdkkP
k xk
PO (27)
Further, in equation (12) it is easy to observe that
( ) 1≈−θττ . (28)
Substituting (27) and (28) in (16), we obtain
( )
30
λθ
1θ
MSMS
eMS
KOR −
≈
−
. (29)
In view of (26), the expression (29) becomes (25). The corollary is proved.
It should be noted that formula (25) is widely used in reliability theory. However, as can
be seen from the comparison of expressions (16) and (25), the coefficient of operational readiness
representation, as a product of the availability and the probability of failure-free operation during
the operating time θ , is valid only for the exponential distribution of time to failure and τ<<θ .
Example 3. Calculate the coefficient of operational readiness, if 10000λ/1 ==MTBF h,
5,2τ = h, 5,0θ = h, 005,0βα == , 25,0== DM tt h, 8=PRt h, 4=FRt h and 5,05 =MS h.
Substituting the initial data in the following formulas obtained in [12]:
( )[ ];
11
1 λτ
1 λτeαλ
e
MS −
−
−−
−= (30)
( )
;
λ
1
β1
β1τ
α)1(1
1 λτλτ
λτ2
−−
−
−
−
−
−
−−
=
ee
e
MS ;
α)1(1
α
λτ
λτ
6 −−−
=
−
e
et
MS FR ,
α)1(1
)1(
λτ
λτ
7 −
−
−−
−=
e
et
MS PR
we calculate the values 2,4761 =MS h, 06,02 =MS h, 81,36 =MS h and 38,07 =MS h.
Next, using (25), we find ( ) 990,0=θORK . It can be seen from Example 3 that the mean
time 1MS is much smaller than MTBF . However, in case of ideal LRU checking, i.e. when
0=β=α , from (30) follows that λ== 11 MTBFMS .Thus, the false failures of operability
checks significantly reduce the mean time between unscheduled repairs (MTBUR). Airborne
ISSN 1028-9763. Математичні машини і системи, 2012, № 1 127
electronic systems are used in the interrupted time regime, which is caused by alternating areas of
aircraft operations and flight waiting in the parking lot. The finding of the LRU in states of 3E ,
4E , 6E , 7E and 8E is associated with the value costs because the relevant works can be carried
out during the stay of the aircraft in the parking lot or after removing the LRU from the aircraft
board. Therefore, in determining the probabilistic parameters of maintenance we can exclude the
intervals corresponding to these states from the regeneration cycle.
Let us to introduce a new time axis which is associated only with the use of LRU for per-
forming its intended functions. At any time on the new axis the LRU may be in one of the states
21, EE and 5E . The average regeneration cycle 0MS is now determined as follows
5210 MSMSMSMS ++= ,
and the operational readiness coefficient is given by
( )
0
1
MS
eMS
KOR
λθ−
≈θ . (31)
Example 4. Calculate the operational readiness coefficient, if 10000/1 =λ=MTBF h,
10=τ h, 1=θ h, 005,0=β=α and 05 =MS .
Substituting the initial data in (31) gives ( ) 999395,0≈θORK . The calculated value is
exactly equal to the result of Example 3, which indicates the correctness of the derived formulas.
4. Conclusions
The exploitation states have been determined in which a single block system of avionics may be
found when it is in use. The theorem that has been proved determines the non-stationary function
of the operational readiness for an arbitrary failure distribution with taking into account the
trustworthiness characteristics of multiple operability checks.
The proved corollary allows determining the non-stationary function and coefficient of
operational readiness for an exponential failure distribution.
The proved theorem determines the expression for the stationary operational readiness
coefficient for an arbitrary distribution of time to failure, as well as the corollaries, determining
the stationary coefficient for an exponential distribution of failures under different assumptions. It
has been shown that well-known expressions for the stationary coefficient of operational
readiness are special cases of the derived formulas, which give a more adequate value of this
coefficient. In a specific example of the exponential failure distribution it has been shown that the
mean time between unscheduled repairs is largely determined by the probability of the false fail-
ures registered by the built-in test equipment.
These results allow assessing the effectiveness of maintenance strategies of avionics sys-
tems till safe failure and justify the requirements for the built-in test equipment. The obtained re-
sults are advisable to use as in the design phase of avionics systems as well as in the phase of
avionics exploitation. Further development of these results can be conducted for optimizing the
maintenance of modern avionics.
REFERENCES
1. Надежность и эффективность в технике: справочник / Под ред. В.И. Кузнецова,
Е.Ю. Барзиловича. – М.: Машиностроение, 1990. – Т. 8. – 319 с.
2. Nakagava T. Maintenance theory of reliability/ Nakagava T. – N.Y.: Springer, 2005. – 258 p.
3. Pham H. Handbook of reliability engineering / Pham H. – London: Springer, 2003. – 298 p.
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4. Blocus A. Reliability and availability evaluation of large renewal systems / A. Blocus // International
Journal of Materials & Structural Reliability. – 2006. – Vol. 4, N 1. – P. 39 – 52.
5. 700 Series ARINC. Characteristics, Aeronautical Radio, Inc., USA [Електронний ресурс]. – Режим
доступу: http://www.arinc.com.
6. Барлоу Р. Статистическая теория надежности и испытания на безотказность / Р. Барлоу,
Ф. Прошан. – М.: Наука, 1984.– 329 с.
7. Cox D.R. Renewal theory / Cox D.R. – London: Methuen, 1960. – 247 p.
8. Уланский В.В. Оптимальные планы обслуживания радиоэлектронных систем на основе диагно-
стирования / В.В. Уланский // Вопросы технической диагностики: межвуз. сб. науч. тр. – Ростов-
на-Дону: РИСИ, 1987. – С. 137 – 143.
9. Уланский В.В. Математическая модель процесса эксплуатации легкозаменяемых блоков систем
авионики / В.В. Уланский, И.А. Мачалин // Авіаційно-космічна техніка і технологія. – 2006. –
№ 6 (32). – С. 74 – 80.
10. Уланский В.В. Стратегия обслуживания одноблочной системы авионики при наличии явных и
скрытых отказов / В.В. Уланский, И.А. Мачалин // Математичні машини і системи. – 2007. – № 3,4.
– С. 245 – 256.
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В.В. Уланский, И.А. Мачалин // Электронное моделирование. – 2008. – Т. 30, № 2. – С. 55 – 67.
12. Уланский В.В. Организация системы технического обслуживания и ремонта радиоэлектронно-
го комплекса Ту-204: уч. пособ. / Уланский В.В., Конахович Г.Ф., Мачалин И.А. – К.: КИИГА,
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мых радиоэлектронных систем / В.В. Уланский // Ресурсосберегающие технологии обслуживания и
ремонта авиационного и радиоэлектронного оборудования воздушных судов гражданской авиации:
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Стаття надійшла до редакції 29.11.2011
|
| id | nasplib_isofts_kiev_ua-123456789-83974 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1028-9763 |
| language | English |
| last_indexed | 2025-12-01T10:36:17Z |
| publishDate | 2012 |
| publisher | Інститут проблем математичних машин і систем НАН України |
| record_format | dspace |
| spelling | Ulansky, V.V. Machalin, I.A. 2015-07-01T13:17:39Z 2015-07-01T13:17:39Z 2012 Mathematical models for evaluation of operational readiness of periodically inspected electronic systems / V.V. Ulansky, I.A. Machalin // Мат. машини і системи. — 2012. — № 1. — С. 119-128. — Бібліогр.: 13 назв. — англ. 1028-9763 https://nasplib.isofts.kiev.ua/handle/123456789/83974 629.735.05:621.3(045) The mathematical models and theorems that identify stationary and non-stationary coefficients of operational readiness are developed for an arbitrary and exponential distribution of time to failure of avionics systems. Proven expressions, in contrast to all other known, take into account the trustworthiness characteristics of multiple inspections. It is shown that the mean time between unscheduled repairs of the system is largely determined by the probability of a false failure. Розроблено математичні моделі та доведено теореми, що дозволяють визначити стаціонарний і нестаціонарний коефіцієнти оперативної готовності при довільному і експоненційному законах розподілу напрацювання системи до відмови. Доведені вирази, які, на відміну від відомих, враховують характеристики достовірності багаторазового контролю працездатності. Показано, що середній час напрацювання на незапланований ремонт системи багато в чому визначається ймовірністю помилкової відмови. Разработаны математические модели и доказаны теоремы, позволяющие определить стационарный и нестационарный коэффициенты оперативной готовности при произвольном и экспоненциальном законах распределения наработки системы до отказа. Доказанные выражения, в отличие от известных, учитывают характеристики достоверности многократного контроля работоспособности. Показано, что среднее время наработки на незапланированный ремонт системы во многом определяется вероятностью ложного отказа. en Інститут проблем математичних машин і систем НАН України Математичні машини і системи Моделювання і управління Mathematical models for evaluation of operational readiness of periodically inspected electronic systems Математичні моделі для оцінки коефіцієнта оперативної готовності періодично контрольованих електронних систем Математические модели для оценки коэффициента оперативной готовности периодически контролируемых электронных систем Article published earlier |
| spellingShingle | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems Ulansky, V.V. Machalin, I.A. Моделювання і управління |
| title | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| title_alt | Математичні моделі для оцінки коефіцієнта оперативної готовності періодично контрольованих електронних систем Математические модели для оценки коэффициента оперативной готовности периодически контролируемых электронных систем |
| title_full | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| title_fullStr | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| title_full_unstemmed | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| title_short | Mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| title_sort | mathematical models for evaluation of operational readiness of periodically inspected electronic systems |
| topic | Моделювання і управління |
| topic_facet | Моделювання і управління |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/83974 |
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