- Email: [email protected]
Availability modeling for periodically inspection system with different lifetime and repair-time distribution
Chinese Journal of Aeronautics, (2019), 32(7): 1667–1672
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Availability modeling for periodically inspection system with different lifetime and repair-time distribution Junliang LI a,b, Yueliang CHEN a,*, Yong ZHANG a, Hailiang HUANG a a b
Qingdao Campus, Naval Aviation University, Qingdao 266041, China The 92635th Unit of Navy, Qingdao 266041, China
Received 26 June 2018; revised 27 August 2018; accepted 19 November 2018 Available online 26 April 2019
KEYWORDS Availability; Inspection period; Lifetime distribution; Reliability; Repair-time distribution
Abstract Availability is a main feature of design and operation of all engineering system. Recently, availability evaluation of periodical inspection systems with different structures is at the center of attention due to the wide application in engineering. In this paper, an analytical and probabilistic availability model for periodical inspection system is proposed by a new recursively algorithm, which can achieve limiting average availability and instantaneous availability of periodical inspection system under arbitrary lifetime and repair-time distributions. Then three application examples are presented, the systems lifetime and repair-time are respectively fellow exponential/exponential, Weibull/normal and Weibull/lognormal distribution. Finally, a Weibull/lognormal system is studied to analyze the dynamic relationship between inspection period and availability. The results indicate that the proposed approach can provide the technology support for improving system availability and determining reasonable inspection period. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Many deteriorating repairable systems, such as a vehicle, computer, or aircraft, though properly functioning, suffer from inevitable failures due to complex degradation processes and environmental conditions.1 These unexpected failures may * Corresponding author. E-mail address: [email protected] (Y. CHEN). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
result in severe consequences, including massive production losses, high corrective replacement (repair) costs and safety hazards to environment and personnel. For this reason, Preventive Maintenance (PM) is extremely important as it could effectively avoid occurrence of unexpected failures, and thereby save servicing costs. Periodic inspection is main PM policy of these systems, which is maintained through periodic inspection. Barlow and Proschan2 firstly put forward the availability model for this system. Hoyland and Rausand3 discussed the exact expressions of system availability model under some simple distributions. Sarkar and Chaudhuri4,5 analyzed the system availability model under gamma lifetime and exponential repair time or allowing a stand-by spare unit. J. Sarkar and S. Sarkar6 proposed an availability model of periodic inspec-
https://doi.org/10.1016/j.cja.2019.03.025 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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tion system, which is maintained under periodic inspection with a perfect repair policy and constant repair time. Moreover, the exact availability and the limiting average availability of a periodically inspected system were obtained, supported by a spare and maintained with perfect repairs or upgrades.7 Meng et al.8 analyzed dynamic relationship of system inspection period and availability based on the theory of renewal process. Ahmad and Soleimanmeigouni9 developed a reliability-based cost model for periodically inspected unit under Weibull lifetime distribution. Tiwary and Arya10 developed a technique for optimizing inspection and repair based on availability of distribution systems using Teaching Learning method. For an non-periodic inspection system, the availability model and lifetime/repair-time function distribution were studied in the broader areas. Ke et al.11 studied the statistical inferences of an availability system with imperfect coverage, in which the lifetime and repair-time of the components are assumed to follow an exponential and a general distribution respectively. Sun12 studied the availability assessment methods for complex system with various lifetime and repair-time distribution. Li and Teng13 proposed a general complex repairable system availability model when system fault and repair time obeys general distribution. The literature above constructed more availability model for some specific distribution system, but did not study the availability model for a system with competitive failures. Li Yang et al.14 proposed a preventive maintenance policy for a single-unit system whose failure had two competing and dependent causes, the objective was to determine the optimal preventive replacement interval. Li Yang et al.15,16 studied a system subject to two typical failure modes, degradation-
AðtÞ ¼
( FðtÞ
recursively method, which can achieve limiting average availability and instantaneous availability of system. The paper is organized as follows. Section 2 constructs the availability method of periodic inspection system through proposed method, including limiting average availability and instantaneous availability. Section 3 illustrates the proposed approach to compute several typical lifetime and repair-time distribution system availability. Section 4 illustrates a case study to demonstrate the dynamic relationship between instantaneous availability and inspection period under the same distribution. Section 5 concludes the whole paper. 2. Modeling of availability 2.1. Assumption We consider systems that are maintained through periodic inspection and perfect repair when they are failed. Suppose a system is placed on operation at time t ¼ 0 and is tested at regular intervals 2s,3 s,. . . which treats an un-failed system to be as good as new upon inspection, and makes perfect repair of a failed system with immediate restoration. 2.2. Availability model For a general system, if we assume FðtÞ denotes the lifetime function, GðtÞ denotes the repair-time function,
FðtÞ ¼ 1 FðtÞ denotes the system reliability function, AðtÞ denotes the availability function, t 0, and s denotes the inspection period, the availability function of the system can be expressed by
if 0 6 t 6 s
ð1Þ
AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ if ks < t 6 ðk þ 1Þs for k ¼ 1; 2; 3;
based failure and sudden failure. They developed a hybrid condition-based maintenance strategy to prevent competing failures and proposed a two-level CBM model for a production system subject to continuous degradation and random production waits. Qiu et al.17 proposed an analytical model on the instantaneous availability and the steady-state availability for a competing-risk system, which was subject to multiple modes. The model was then utilized to obtain the optimal inspection interval that maximized the system steady-state availability or minimized the average long-run cost rate‘. Qiu and Cui18 also derived a reliability functions by introducing a novel dependent two-stage failure process, which could be used to obtain the optimal inspection interval that maximizes the system steady-state availability or minimizes the average long-run cost rate. So far, there has not been a general availability model for periodic inspect system whose lifetime and repair-time obeys general distribution. In this paper, the availability of a periodically inspected system with a perfect repair policy under arbitrary lifetime and repair-time distributions is analyzed; an analytical availability model is developed for system with a
Proof (1) If 0 t s and no inspection happens during this time period, it is clear that
AðtÞ ¼ FðtÞ
ð2Þ
(2) If s t 2s and the system availability adopts full probability formula, AðtÞcan be deduced by AðtÞ ¼ PrðXðtÞ ¼ 1jXðsÞ ¼ 1Þ PrðXðsÞ ¼ 1Þ þ PrðXðtÞ ¼ 1jXðsÞ ¼ 0Þ PrðXðsÞ ¼ 0Þ
¼ FðsÞ Fðt sÞ þ FðsÞGðt sÞ
¼ FðsÞ Fðt sÞ þ ð1 FðsÞÞGðt sÞ
¼ AðsÞ Fðt sÞ þ ð1 AðsÞÞGðt sÞ
ð3Þ
(3) If ks t ðk þ 1Þs and the system availability adopts full probability formula, AðtÞ can be deduced by
Availability modeling for periodically inspection system
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AðtÞ ¼ PrðXðtÞ ¼ 1jXðksÞ ¼ 1Þ PrðXðksÞ ¼ 1Þ þ PrðXðtÞ
repair time is exponentially distribution. Without loss of generality, the model can be extended as follows: Z s Aav ½0; 1Þ ¼ s1 AðuÞdu ð9Þ
¼ 1jXðksÞ ¼ 0Þ PrðXðksÞ ¼ 0Þ
¼ AðksÞ Fðt ksÞ þ ð1 AðksÞÞGðt ksÞ
ð4Þ
0
From the above analysis, in order to obtain the expressions ofAðtÞ, it just needs to find the deduction relationship between AðksÞ and A½ðk þ 1Þs.
If AðtÞ ¼ AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ, then the system limiting availability shows as
Lemma 1. Suppose 0 < a; b < 1, the successive weighted averages of a and b can be defined as follows:
lim AðtÞ )
t¼ksþu
w0 ¼ 1; w1 ¼ a and k ¼ 1; 2; 3; and then,
wkþ1 ¼ wk a þ ð1 wk Þb,
t!1
ða bÞk ð1 aÞ þ b FðuÞ 1aþb
þ ½1
for
k!1
)
ða bÞk ð1 aÞ þ b wk ¼ 1aþb
ð5Þ
ða bÞk ð1 aÞ þ b GðuÞ 1aþb
b 1a FðuÞ þ ½ GðuÞ 1aþb 1aþb
ð10Þ
and the system limiting average availability shows as Z s Z s AðuÞdu ¼ s1 fAðksÞ Fðt ksÞ Aav ½0; 1Þ ¼ s1
and thus limk ! 1 wk ¼ b=ðb þ 1 aÞ
0
0 t¼ksþu
þ ½1 AðksÞGðt ksÞgdt ) Aav ½0; 1Þ Proof. Note that wkþ1 ¼ b þ ða bÞwk , which applied successively implies that
¼ s1
Z
k!1
s
f 0
b 1a FðuÞ þ ½ GðuÞgdu ð11Þ 1aþb 1aþb
wkþ1 wk ¼ ða bÞðwk wk1 Þ ¼ ¼ ða bÞk ðw1 w0 Þ 3. Availability model of the system with several typical lifetime and repair-time distribution
Hence, wk ¼ ðwk wk1 Þ þ ðwk1 wk2 Þ þ þ ðw1 w0 Þ þ w0 ¼ ½ða bÞk1 þ ða bÞk2 þ þ 1ðw1 w0 Þ þ w0 k
¼
k
1 ða bÞ ða bÞ ða 1Þ þ b ða 1Þ þ 1 ¼ 1aþb 1aþb
Proving Eq. (5). Finally, since ja bj < 1, ðb þ 1 aÞ.
limk ! 1
and
wk ¼ b=
a ¼ FðuÞ;b ¼ GðuÞ,
0us
and
t ¼ ksþ
Suppose u,u 2 ½0; s Then,
Aðks þ uÞ ¼ AðksÞ FðuÞ þ ð1 AðksÞÞGðuÞ
ð6Þ
Moreover, if we set wk ¼ A½ðk þ 1Þs, we get Að0Þ ¼ 1; AðsÞ ¼ a; A½ðk þ 1Þs ¼ AðksÞa þ ð1 AðksÞÞb
ð7Þ
k
t¼ksþu
)
AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ ðabÞk ð1aÞþb 1aþb
k
ð1aÞþb FðuÞ þ ½1 ðabÞ1aþb GðuÞ if ks t ðk þ 1Þs for k ¼ 1; 2; 3;
3.1. Availability model of the system lifetime and repair-time with exponential Suppose that system lifetime T has an exponential distribution, the reliability function is
ð1aÞþb Hence, by Lemma 1, A½ðk þ 1Þs ¼ ðabÞ1aþb . Then, the system availability at time ks t ðk þ 1Þs, for k ¼ 1; 2; 3; is given in Eq. (8)
AðtÞ ¼
For system lifetime and repair-time distribution are difficult to determine, many scholars summarize the basic distribution of certain system in practice. As depicted in Table 1, the lifetime distribution functions for some products are summarized by Zhao.19 Repair-time distribution functions for some products are studied by Xu20 and Li21 et al., as shown in Table 2.
FðtÞ ¼ expða tÞ
ð12Þ
where a is the scale parameter. And the repair-time function is also exponential distribution function, GðtÞ ¼ 1 expðl tÞ
ð13Þ
The system availability at time t can be rewritten as follows:
ð8Þ Table 1
Classic lifetime distribution.
If we get the express of the system lifetime functionFðtÞ, even the competing-risk system, and repair-time function GðtÞ, the system availability function can be obtained accordingly.
Distribution
Suitable set
Exponential
2.3. System limiting average availability
Lognormal
The limiting average availability of the system is obtained by Sarkar,4–6 when the lifetime has a density function and the
Normal
With constant failure rate of components, periodic maintenance unit. etc. Some capacitors, ball bearing, relay, motor, generator, cable, battery, material fatigue, etc. Motor winding insulation, semiconductor devices, silicon transistors, helicopter rotor blades and aircraft structure, metal fatigue. Aircraft tire wear and some mechanical products.
Weibull
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Table 2
Classic repair-time distribution.
Distribution
Suitable set
Exponential
For a short period of time to repair or instantaneous replacement product Suitable for simple fault, a single product maintenance or basic operations, such as simple removal and replacement of some parts of the maintenance time is fixed, general compound Gaussian distribution. Apply repair frequency and duration are not equal complex equipment maintenance tasks, composed of a number of activities suitable to describe the maintenance time of complex systems, such as mechanical and electrical, mechanical and electronic equipment.
Normal
Lognormal
AðtÞ ¼
expða tÞ; if 0 t s AðksÞ expða ðt ksÞÞ þ ½1 AðksÞ½1 expðu ðt ksÞÞ ; if ks < t ðk þ 1Þs for k ¼ 1; 2; 3;
where l is the scale parameter of GðtÞ. When a ¼ 0:01, l ¼ 0:5, s = 50, the system availability is shown in Fig. 1. Therefore, from Eq. (11) the limiting average availability of the system is Z 50 1 Aav ½0; 1Þ ¼ 501 expð0:01 uÞ 1 0:6050 þ 1 0 1 0:6050 ð1 expð0:5 uÞÞ du þ 1 0:6050 þ 1 ¼ 0:8358
ð15Þ
3.2. Availability model of the system lifetime and repair-time with Weibull/normal distribution Suppose that system lifetime T has a Weibull distribution, the reliability function is
FðtÞ ¼ expððt=gÞm Þ
where l is the mean, r is the variance. The system availability at time t, form Eq. (1) given in Eq. (18)
AðtÞ ¼
ð14Þ
Therefore, from Eq. (11) the limiting average availability of the system is Z 10 0:8413 Aav ½0; 1Þ ¼ 101 exp ðu=0:5Þ2 1 0:1353 þ 0:8413 0 1 0:1353 uðuÞ du ¼ 0:0471 þ 1 0:1353 þ 0:8413 ð19Þ According to Eq. (19), the limiting average availability of system is relatively low. Sarkar5 has studied the relationship between limiting average availability and inspection period of the system with exponential and constant distribution.5 The results illustrate that the varies inspection period can change the system limiting average availability. In order to describe our proposed methodology more exactly, the limiting average availability under differentswas calculated, as shown in Table 3.
ð16Þ
where g is the scale parameter, m is the shape parameter. And the repair-time function is normal distribution function, Z t 1 1 xl 2 GðtÞ ¼ pffiffiffiffiffiffi Þ dx exp½ ð ð17Þ 2 r 2pr 0
(
Fig. 1 Instantaneous availability of system under exponential and exponential distribution.
3.3. Availability model of the system with Weibull/lognormal distribution Suppose that system lifetime T has a Weibull distribution, the reliability function is
FðtÞ ¼ expððt=gÞm Þ
expððt=gÞm Þ; if 0 t s R tkt 1ffi expð 12 ðxl Þ2 Þdx ; AðksÞ expððt ks=gÞm Þ þ ½1 AðksÞ pffiffiffi r 2pr 0
When, g ¼ 0:5, m ¼ 2, u ¼ 0, r ¼ 1, s ¼ 10, the system availability is shown in Fig. 2
if ks < t ðk þ 1Þs for k ¼ 1; 2; 3;
ð20Þ
ð18Þ
And the repair-time function is lognormal distribution function,
Availability modeling for periodically inspection system Z GðtÞ ¼ 0
t
1671
"
2 #
1 1 lnx u lnx l pffiffiffiffiffiffi exp dx ¼ u 2 r r 2prx ð21Þ
( AðtÞ ¼
When, g ¼ 0:5, m ¼ 2, u ¼ 0, r ¼ 1, s ¼ 10, the system availability is shown in Fig. 3. Therefore, from Eq. (11) the limiting average availability of the system is
expððt=gÞm Þ; if 0 t s R tkt 1 2 ffi expð 12 ðlnxl Þ Þ dx; AðksÞ expððt ks=gÞm Þ þ ½1 AðksÞ 0 pffiffiffi r 2prx
if ks < t ðk þ 1Þs for k ¼ 1; 2; 3; ð22Þ
where l is the mean, r is the variance. The system availability at timet, form Eq. (1) given in Eq. (22)
Z 10 0:5 Aav ½0; 1Þ ¼ 101 exp ðu=0:5Þ2 1 0:1353 þ 0:5 0 1 0:1353 uðlnuÞ du ¼ 0:0248 ð23Þ þ 1 0:1353 þ 0:5 For different inspection period, the limiting average availability of system varies as shown in Table 4. 4. Analysis of instantaneous availability and inspection period under same distribution
Fig. 2 Instantaneous availability of system under Weibull/ normal distribution.
Table 3 Relation among Aav ½0; 1Þ and s, when the system lifetime distribution is Weibull and repair-time is normal. s
1
2
3
4
5
Aav ½0; 1Þ
0.4716
0.2302
0.1571
0.1151
0.0684
In Section 3, we calculate the instantaneous availability and the limiting average availability of the system, and analyze the relationship between limiting average availability and inspection period of two kinds of system with Weibull/normal and Weibull/lognormal. In Section 4 we will analyze the relationship between inspection period and instantaneous availability for complex system with the same distribution form. Weibull distribution is the common lifetime distribution for engineering and lognormal distribution is the common repairtime distribution for complex repairable system. So, assume the system with Weibull and lognormal distribution, and the distribution parameters for the system areg ¼ 0:5,m ¼ 2,u ¼ 0,r ¼ 1, then the instantaneous availability of the system varies with s ¼ 10,s ¼ 20 and s ¼ 40 as shown in Fig. 4. In Fig. 4, when t 2 ½10; 20 the highest system availability is s ¼ 10, when t 2 ð20; 40Þ the highest system availability is s ¼ 20, when t 2 ð40; 80Þ the highest system availability is s ¼ 40, when t 2 ð80; 100Þ the highest system availability is s ¼ 20. As show in Fig. 4, although the system under the same lifetime and repair-time function, different inspection period will cause availability fluctuation in different life stages. So it is important to make reasonable inspection period for improving the availability and reducing the cost of maintenance support system.
Table 4 Relation among Aav ½0; 1Þ and s, when the system lifetime distribution is Weibull and repair-time is lognormal. Fig. 3 Instantaneous availability of system under Weibull/ lognormal distribution.
s
1
2
3
4
5
Aav ½0; 1Þ
0.4723
0.2486
0.1494
0.1019
0.0749
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Fig. 4
Instantaneous availability of system with different s.
5. Conclusions This paper develops the periodic inspection system availability model with general probability distribution, and presents a recursive method to compute the availability. In addition, we propose the express of availability model in the case of exponential and exponential distributed system, Weibull and normal distributed system, Weibull and lognormal distributed system, and analyze the relationship between inspection period and availability. (1) The availability model can solve the instantaneous availability and limiting average availability of periodic inspection system with general probability distribution. (2) When the system lifetime and repair-time distribution function and distributed parameter are constant, different inspection period can effect on the system instantaneous availability and limiting average availability. It is reasonable to formulate reasonable inspection period in different stages for different system in order to improve the availability of system and reduce the maintenance cost of the system. For short life cycle system, we should determine the reasonable inspection period based on the instantaneous availability. For long life cycle system, we should determine the reasonable inspection period based on the limiting average availability. In the future, the optimization of inspection period and availability will be one important research directions. A new concept of sequential probability series system is proposed by Qiu et al.,22 in which three optimal allocation models are formulated, the analytical expressions for the optimal allocation solutions are derived when the lifetime of units obey exponential distributions, a genetic algorithm is given to search the optimal solutions when the lifetime of units obey general distributions, and a Monte Carlo method is provided to solve the optimal allocation problem when the lifetime distributions of units are non-identical. It might be worthwhile to characterize the nature of failure modes in a more realistic case for complex repairable system availability optimization.
1. Lin ZL, Huang YS, Fang CC. Non-periodic preventive maintenance with reliability thresholds for complex repairable systems. Reliab Eng Syst Saf 2015;136:145–56. 2. Barlow RE, Proschan F. Mathematical theory of reliability. California: Wiley; 1996. p. 119–36. 3. Rausand M, Hyland A. System reliability theory:Modeling, inference, statistical methods, and application. 2nd ed. Hoboken: Wiley; 2004. p. 361–83. 4. Sarkar J, Chaudhuri G. A note on the availability of a system with a stand-by spare under perfect repa ir[dissertation]. Indianapolis: University Purdue University; 1998. 5. Sarkar J, Chaudhuri G. Availability of a system with gamma life and exponential repair time under a perfect repair policy. Statist Probab Lett 1999;43(2):189–96. 6. Sarkar J, Sarkar S. Availability of a periodically inspected system under perfect repair. J Stat Plan Inference 2000;91(1):77–90. 7. Sarkar J, Sarkar S. Availability of a periodically inspected system supported by aspire unit, under perfect repair or perfect upgrade. Stat Probab Lett 2001;53(2):207–17. 8. Meng H, Liu J, Hei XH. Modeling and optimizing periodically inspected software rejuvenation policy based on geometric sequences. Reliab Eng Syst Saf 2015;133:184–91. 9. Ahmadi A, Soleimanmeigouni I. Optimum failure management strategy for periodically inspected units with imperfect maintenance. IFAC-Papers On Line 2016;49(12):799–804. 10. Tiwary A, Arya LD. Inspection–repair based availability optimization of distribution systems using teaching learning based optimization. Inst Eng India Ser(B) 2016;97(3):355–65. 11. Ke JC, Su ZL, Wang KH. Simulation inferences for an availability system with general repair distribution and imperfect fault coverage. Simul Model Pract Theory 2010;18(3):338–47. 12. Sun YQ. Availability assessment for passenger aircraft subsystem with different lifetime and repair-time distribution. J Mech Eng 2014;50(16):54–60 [Chinese]. 13. Li JL, Teng KN. An availability method for complex repairable system. Acta Aeronaut Astronaut Sinica 2017;38(12):150–8 [Chinese]. 14. Yang L, Ma XB, Peng R, Zhai QQ, Zhao Y. A preventive maintenance policy based on dependent two-stage deterioration and external shocks. Reliab Eng Syst Saf 2017;160:201–11. 15. Yang L, Zhao Y, Peng R, Ma XB. Hybrid preventive maintenance of competing failures under random environment. Reliab Eng Syst Saf 2018;64:699–712. 16. Yang L, Zhao Y, Peng R. Opportunistic maintenance of production systems subject to random wait time and multiple control limits. J Manuf Syst 2018;47:12–34. 17. Qiu Q, Cui LR, Gao HD. Availability and maintenance modelling for systems subject to multiple failure modes. Comput Ind Eng 2017;108:192–8. 18. Qiu Q, Cui LR. Reliability evaluation based on a dependent twostage failure process with competing failures. Appl Math Model 2018;64:699–712. 19. Zhao Y. Reliability maintainability supportability. Beijing: National Defense Industry Press; 2011. p. 33–45 [Chinese]. 20. Xu ZC. Supportability engineering. Beijing: Weapon Industry Press; 2002. p. 50–65 [Chinese]. 21. Li JL, Teng KN. Instantaneous availability of military aircraft during mission preparation period. J Beijing Univ Aeronaut Astronaut 2017;43(4):754–60 [Chinese]. 22. Qiu QG, Cui LR, Hongda GH, He Y. Optimal allocation of units in sequential probability series systems. Reliab Eng Syst Saf 2018;169:351–63.
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Availability modeling for periodically inspection system with different lifetime and repair-time distribution Junliang LI a,b, Yueliang CHEN a,*, Yong ZHANG a, Hailiang HUANG a a b
Qingdao Campus, Naval Aviation University, Qingdao 266041, China The 92635th Unit of Navy, Qingdao 266041, China
Received 26 June 2018; revised 27 August 2018; accepted 19 November 2018 Available online 26 April 2019
KEYWORDS Availability; Inspection period; Lifetime distribution; Reliability; Repair-time distribution
Abstract Availability is a main feature of design and operation of all engineering system. Recently, availability evaluation of periodical inspection systems with different structures is at the center of attention due to the wide application in engineering. In this paper, an analytical and probabilistic availability model for periodical inspection system is proposed by a new recursively algorithm, which can achieve limiting average availability and instantaneous availability of periodical inspection system under arbitrary lifetime and repair-time distributions. Then three application examples are presented, the systems lifetime and repair-time are respectively fellow exponential/exponential, Weibull/normal and Weibull/lognormal distribution. Finally, a Weibull/lognormal system is studied to analyze the dynamic relationship between inspection period and availability. The results indicate that the proposed approach can provide the technology support for improving system availability and determining reasonable inspection period. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Many deteriorating repairable systems, such as a vehicle, computer, or aircraft, though properly functioning, suffer from inevitable failures due to complex degradation processes and environmental conditions.1 These unexpected failures may * Corresponding author. E-mail address: [email protected] (Y. CHEN). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
result in severe consequences, including massive production losses, high corrective replacement (repair) costs and safety hazards to environment and personnel. For this reason, Preventive Maintenance (PM) is extremely important as it could effectively avoid occurrence of unexpected failures, and thereby save servicing costs. Periodic inspection is main PM policy of these systems, which is maintained through periodic inspection. Barlow and Proschan2 firstly put forward the availability model for this system. Hoyland and Rausand3 discussed the exact expressions of system availability model under some simple distributions. Sarkar and Chaudhuri4,5 analyzed the system availability model under gamma lifetime and exponential repair time or allowing a stand-by spare unit. J. Sarkar and S. Sarkar6 proposed an availability model of periodic inspec-
https://doi.org/10.1016/j.cja.2019.03.025 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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tion system, which is maintained under periodic inspection with a perfect repair policy and constant repair time. Moreover, the exact availability and the limiting average availability of a periodically inspected system were obtained, supported by a spare and maintained with perfect repairs or upgrades.7 Meng et al.8 analyzed dynamic relationship of system inspection period and availability based on the theory of renewal process. Ahmad and Soleimanmeigouni9 developed a reliability-based cost model for periodically inspected unit under Weibull lifetime distribution. Tiwary and Arya10 developed a technique for optimizing inspection and repair based on availability of distribution systems using Teaching Learning method. For an non-periodic inspection system, the availability model and lifetime/repair-time function distribution were studied in the broader areas. Ke et al.11 studied the statistical inferences of an availability system with imperfect coverage, in which the lifetime and repair-time of the components are assumed to follow an exponential and a general distribution respectively. Sun12 studied the availability assessment methods for complex system with various lifetime and repair-time distribution. Li and Teng13 proposed a general complex repairable system availability model when system fault and repair time obeys general distribution. The literature above constructed more availability model for some specific distribution system, but did not study the availability model for a system with competitive failures. Li Yang et al.14 proposed a preventive maintenance policy for a single-unit system whose failure had two competing and dependent causes, the objective was to determine the optimal preventive replacement interval. Li Yang et al.15,16 studied a system subject to two typical failure modes, degradation-
AðtÞ ¼
( FðtÞ
recursively method, which can achieve limiting average availability and instantaneous availability of system. The paper is organized as follows. Section 2 constructs the availability method of periodic inspection system through proposed method, including limiting average availability and instantaneous availability. Section 3 illustrates the proposed approach to compute several typical lifetime and repair-time distribution system availability. Section 4 illustrates a case study to demonstrate the dynamic relationship between instantaneous availability and inspection period under the same distribution. Section 5 concludes the whole paper. 2. Modeling of availability 2.1. Assumption We consider systems that are maintained through periodic inspection and perfect repair when they are failed. Suppose a system is placed on operation at time t ¼ 0 and is tested at regular intervals 2s,3 s,. . . which treats an un-failed system to be as good as new upon inspection, and makes perfect repair of a failed system with immediate restoration. 2.2. Availability model For a general system, if we assume FðtÞ denotes the lifetime function, GðtÞ denotes the repair-time function,
FðtÞ ¼ 1 FðtÞ denotes the system reliability function, AðtÞ denotes the availability function, t 0, and s denotes the inspection period, the availability function of the system can be expressed by
if 0 6 t 6 s
ð1Þ
AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ if ks < t 6 ðk þ 1Þs for k ¼ 1; 2; 3;
based failure and sudden failure. They developed a hybrid condition-based maintenance strategy to prevent competing failures and proposed a two-level CBM model for a production system subject to continuous degradation and random production waits. Qiu et al.17 proposed an analytical model on the instantaneous availability and the steady-state availability for a competing-risk system, which was subject to multiple modes. The model was then utilized to obtain the optimal inspection interval that maximized the system steady-state availability or minimized the average long-run cost rate‘. Qiu and Cui18 also derived a reliability functions by introducing a novel dependent two-stage failure process, which could be used to obtain the optimal inspection interval that maximizes the system steady-state availability or minimizes the average long-run cost rate. So far, there has not been a general availability model for periodic inspect system whose lifetime and repair-time obeys general distribution. In this paper, the availability of a periodically inspected system with a perfect repair policy under arbitrary lifetime and repair-time distributions is analyzed; an analytical availability model is developed for system with a
Proof (1) If 0 t s and no inspection happens during this time period, it is clear that
AðtÞ ¼ FðtÞ
ð2Þ
(2) If s t 2s and the system availability adopts full probability formula, AðtÞcan be deduced by AðtÞ ¼ PrðXðtÞ ¼ 1jXðsÞ ¼ 1Þ PrðXðsÞ ¼ 1Þ þ PrðXðtÞ ¼ 1jXðsÞ ¼ 0Þ PrðXðsÞ ¼ 0Þ
¼ FðsÞ Fðt sÞ þ FðsÞGðt sÞ
¼ FðsÞ Fðt sÞ þ ð1 FðsÞÞGðt sÞ
¼ AðsÞ Fðt sÞ þ ð1 AðsÞÞGðt sÞ
ð3Þ
(3) If ks t ðk þ 1Þs and the system availability adopts full probability formula, AðtÞ can be deduced by
Availability modeling for periodically inspection system
1669
AðtÞ ¼ PrðXðtÞ ¼ 1jXðksÞ ¼ 1Þ PrðXðksÞ ¼ 1Þ þ PrðXðtÞ
repair time is exponentially distribution. Without loss of generality, the model can be extended as follows: Z s Aav ½0; 1Þ ¼ s1 AðuÞdu ð9Þ
¼ 1jXðksÞ ¼ 0Þ PrðXðksÞ ¼ 0Þ
¼ AðksÞ Fðt ksÞ þ ð1 AðksÞÞGðt ksÞ
ð4Þ
0
From the above analysis, in order to obtain the expressions ofAðtÞ, it just needs to find the deduction relationship between AðksÞ and A½ðk þ 1Þs.
If AðtÞ ¼ AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ, then the system limiting availability shows as
Lemma 1. Suppose 0 < a; b < 1, the successive weighted averages of a and b can be defined as follows:
lim AðtÞ )
t¼ksþu
w0 ¼ 1; w1 ¼ a and k ¼ 1; 2; 3; and then,
wkþ1 ¼ wk a þ ð1 wk Þb,
t!1
ða bÞk ð1 aÞ þ b FðuÞ 1aþb
þ ½1
for
k!1
)
ða bÞk ð1 aÞ þ b wk ¼ 1aþb
ð5Þ
ða bÞk ð1 aÞ þ b GðuÞ 1aþb
b 1a FðuÞ þ ½ GðuÞ 1aþb 1aþb
ð10Þ
and the system limiting average availability shows as Z s Z s AðuÞdu ¼ s1 fAðksÞ Fðt ksÞ Aav ½0; 1Þ ¼ s1
and thus limk ! 1 wk ¼ b=ðb þ 1 aÞ
0
0 t¼ksþu
þ ½1 AðksÞGðt ksÞgdt ) Aav ½0; 1Þ Proof. Note that wkþ1 ¼ b þ ða bÞwk , which applied successively implies that
¼ s1
Z
k!1
s
f 0
b 1a FðuÞ þ ½ GðuÞgdu ð11Þ 1aþb 1aþb
wkþ1 wk ¼ ða bÞðwk wk1 Þ ¼ ¼ ða bÞk ðw1 w0 Þ 3. Availability model of the system with several typical lifetime and repair-time distribution
Hence, wk ¼ ðwk wk1 Þ þ ðwk1 wk2 Þ þ þ ðw1 w0 Þ þ w0 ¼ ½ða bÞk1 þ ða bÞk2 þ þ 1ðw1 w0 Þ þ w0 k
¼
k
1 ða bÞ ða bÞ ða 1Þ þ b ða 1Þ þ 1 ¼ 1aþb 1aþb
Proving Eq. (5). Finally, since ja bj < 1, ðb þ 1 aÞ.
limk ! 1
and
wk ¼ b=
a ¼ FðuÞ;b ¼ GðuÞ,
0us
and
t ¼ ksþ
Suppose u,u 2 ½0; s Then,
Aðks þ uÞ ¼ AðksÞ FðuÞ þ ð1 AðksÞÞGðuÞ
ð6Þ
Moreover, if we set wk ¼ A½ðk þ 1Þs, we get Að0Þ ¼ 1; AðsÞ ¼ a; A½ðk þ 1Þs ¼ AðksÞa þ ð1 AðksÞÞb
ð7Þ
k
t¼ksþu
)
AðksÞ Fðt ksÞ þ ½1 AðksÞGðt ksÞ ðabÞk ð1aÞþb 1aþb
k
ð1aÞþb FðuÞ þ ½1 ðabÞ1aþb GðuÞ if ks t ðk þ 1Þs for k ¼ 1; 2; 3;
3.1. Availability model of the system lifetime and repair-time with exponential Suppose that system lifetime T has an exponential distribution, the reliability function is
ð1aÞþb Hence, by Lemma 1, A½ðk þ 1Þs ¼ ðabÞ1aþb . Then, the system availability at time ks t ðk þ 1Þs, for k ¼ 1; 2; 3; is given in Eq. (8)
AðtÞ ¼
For system lifetime and repair-time distribution are difficult to determine, many scholars summarize the basic distribution of certain system in practice. As depicted in Table 1, the lifetime distribution functions for some products are summarized by Zhao.19 Repair-time distribution functions for some products are studied by Xu20 and Li21 et al., as shown in Table 2.
FðtÞ ¼ expða tÞ
ð12Þ
where a is the scale parameter. And the repair-time function is also exponential distribution function, GðtÞ ¼ 1 expðl tÞ
ð13Þ
The system availability at time t can be rewritten as follows:
ð8Þ Table 1
Classic lifetime distribution.
If we get the express of the system lifetime functionFðtÞ, even the competing-risk system, and repair-time function GðtÞ, the system availability function can be obtained accordingly.
Distribution
Suitable set
Exponential
2.3. System limiting average availability
Lognormal
The limiting average availability of the system is obtained by Sarkar,4–6 when the lifetime has a density function and the
Normal
With constant failure rate of components, periodic maintenance unit. etc. Some capacitors, ball bearing, relay, motor, generator, cable, battery, material fatigue, etc. Motor winding insulation, semiconductor devices, silicon transistors, helicopter rotor blades and aircraft structure, metal fatigue. Aircraft tire wear and some mechanical products.
Weibull
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J. LI et al.
Table 2
Classic repair-time distribution.
Distribution
Suitable set
Exponential
For a short period of time to repair or instantaneous replacement product Suitable for simple fault, a single product maintenance or basic operations, such as simple removal and replacement of some parts of the maintenance time is fixed, general compound Gaussian distribution. Apply repair frequency and duration are not equal complex equipment maintenance tasks, composed of a number of activities suitable to describe the maintenance time of complex systems, such as mechanical and electrical, mechanical and electronic equipment.
Normal
Lognormal
AðtÞ ¼
expða tÞ; if 0 t s AðksÞ expða ðt ksÞÞ þ ½1 AðksÞ½1 expðu ðt ksÞÞ ; if ks < t ðk þ 1Þs for k ¼ 1; 2; 3;
where l is the scale parameter of GðtÞ. When a ¼ 0:01, l ¼ 0:5, s = 50, the system availability is shown in Fig. 1. Therefore, from Eq. (11) the limiting average availability of the system is Z 50 1 Aav ½0; 1Þ ¼ 501 expð0:01 uÞ 1 0:6050 þ 1 0 1 0:6050 ð1 expð0:5 uÞÞ du þ 1 0:6050 þ 1 ¼ 0:8358
ð15Þ
3.2. Availability model of the system lifetime and repair-time with Weibull/normal distribution Suppose that system lifetime T has a Weibull distribution, the reliability function is
FðtÞ ¼ expððt=gÞm Þ
where l is the mean, r is the variance. The system availability at time t, form Eq. (1) given in Eq. (18)
AðtÞ ¼
ð14Þ
Therefore, from Eq. (11) the limiting average availability of the system is Z 10 0:8413 Aav ½0; 1Þ ¼ 101 exp ðu=0:5Þ2 1 0:1353 þ 0:8413 0 1 0:1353 uðuÞ du ¼ 0:0471 þ 1 0:1353 þ 0:8413 ð19Þ According to Eq. (19), the limiting average availability of system is relatively low. Sarkar5 has studied the relationship between limiting average availability and inspection period of the system with exponential and constant distribution.5 The results illustrate that the varies inspection period can change the system limiting average availability. In order to describe our proposed methodology more exactly, the limiting average availability under differentswas calculated, as shown in Table 3.
ð16Þ
where g is the scale parameter, m is the shape parameter. And the repair-time function is normal distribution function, Z t 1 1 xl 2 GðtÞ ¼ pffiffiffiffiffiffi Þ dx exp½ ð ð17Þ 2 r 2pr 0
(
Fig. 1 Instantaneous availability of system under exponential and exponential distribution.
3.3. Availability model of the system with Weibull/lognormal distribution Suppose that system lifetime T has a Weibull distribution, the reliability function is
FðtÞ ¼ expððt=gÞm Þ
expððt=gÞm Þ; if 0 t s R tkt 1ffi expð 12 ðxl Þ2 Þdx ; AðksÞ expððt ks=gÞm Þ þ ½1 AðksÞ pffiffiffi r 2pr 0
When, g ¼ 0:5, m ¼ 2, u ¼ 0, r ¼ 1, s ¼ 10, the system availability is shown in Fig. 2
if ks < t ðk þ 1Þs for k ¼ 1; 2; 3;
ð20Þ
ð18Þ
And the repair-time function is lognormal distribution function,
Availability modeling for periodically inspection system Z GðtÞ ¼ 0
t
1671
"
2 #
1 1 lnx u lnx l pffiffiffiffiffiffi exp dx ¼ u 2 r r 2prx ð21Þ
( AðtÞ ¼
When, g ¼ 0:5, m ¼ 2, u ¼ 0, r ¼ 1, s ¼ 10, the system availability is shown in Fig. 3. Therefore, from Eq. (11) the limiting average availability of the system is
expððt=gÞm Þ; if 0 t s R tkt 1 2 ffi expð 12 ðlnxl Þ Þ dx; AðksÞ expððt ks=gÞm Þ þ ½1 AðksÞ 0 pffiffiffi r 2prx
if ks < t ðk þ 1Þs for k ¼ 1; 2; 3; ð22Þ
where l is the mean, r is the variance. The system availability at timet, form Eq. (1) given in Eq. (22)
Z 10 0:5 Aav ½0; 1Þ ¼ 101 exp ðu=0:5Þ2 1 0:1353 þ 0:5 0 1 0:1353 uðlnuÞ du ¼ 0:0248 ð23Þ þ 1 0:1353 þ 0:5 For different inspection period, the limiting average availability of system varies as shown in Table 4. 4. Analysis of instantaneous availability and inspection period under same distribution
Fig. 2 Instantaneous availability of system under Weibull/ normal distribution.
Table 3 Relation among Aav ½0; 1Þ and s, when the system lifetime distribution is Weibull and repair-time is normal. s
1
2
3
4
5
Aav ½0; 1Þ
0.4716
0.2302
0.1571
0.1151
0.0684
In Section 3, we calculate the instantaneous availability and the limiting average availability of the system, and analyze the relationship between limiting average availability and inspection period of two kinds of system with Weibull/normal and Weibull/lognormal. In Section 4 we will analyze the relationship between inspection period and instantaneous availability for complex system with the same distribution form. Weibull distribution is the common lifetime distribution for engineering and lognormal distribution is the common repairtime distribution for complex repairable system. So, assume the system with Weibull and lognormal distribution, and the distribution parameters for the system areg ¼ 0:5,m ¼ 2,u ¼ 0,r ¼ 1, then the instantaneous availability of the system varies with s ¼ 10,s ¼ 20 and s ¼ 40 as shown in Fig. 4. In Fig. 4, when t 2 ½10; 20 the highest system availability is s ¼ 10, when t 2 ð20; 40Þ the highest system availability is s ¼ 20, when t 2 ð40; 80Þ the highest system availability is s ¼ 40, when t 2 ð80; 100Þ the highest system availability is s ¼ 20. As show in Fig. 4, although the system under the same lifetime and repair-time function, different inspection period will cause availability fluctuation in different life stages. So it is important to make reasonable inspection period for improving the availability and reducing the cost of maintenance support system.
Table 4 Relation among Aav ½0; 1Þ and s, when the system lifetime distribution is Weibull and repair-time is lognormal. Fig. 3 Instantaneous availability of system under Weibull/ lognormal distribution.
s
1
2
3
4
5
Aav ½0; 1Þ
0.4723
0.2486
0.1494
0.1019
0.0749
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J. LI et al. References
Fig. 4
Instantaneous availability of system with different s.
5. Conclusions This paper develops the periodic inspection system availability model with general probability distribution, and presents a recursive method to compute the availability. In addition, we propose the express of availability model in the case of exponential and exponential distributed system, Weibull and normal distributed system, Weibull and lognormal distributed system, and analyze the relationship between inspection period and availability. (1) The availability model can solve the instantaneous availability and limiting average availability of periodic inspection system with general probability distribution. (2) When the system lifetime and repair-time distribution function and distributed parameter are constant, different inspection period can effect on the system instantaneous availability and limiting average availability. It is reasonable to formulate reasonable inspection period in different stages for different system in order to improve the availability of system and reduce the maintenance cost of the system. For short life cycle system, we should determine the reasonable inspection period based on the instantaneous availability. For long life cycle system, we should determine the reasonable inspection period based on the limiting average availability. In the future, the optimization of inspection period and availability will be one important research directions. A new concept of sequential probability series system is proposed by Qiu et al.,22 in which three optimal allocation models are formulated, the analytical expressions for the optimal allocation solutions are derived when the lifetime of units obey exponential distributions, a genetic algorithm is given to search the optimal solutions when the lifetime of units obey general distributions, and a Monte Carlo method is provided to solve the optimal allocation problem when the lifetime distributions of units are non-identical. It might be worthwhile to characterize the nature of failure modes in a more realistic case for complex repairable system availability optimization.
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