LI Junliang, CHEN Yueliang, ZHANG Yong, ZHANG Zhuzhu, and FAN Weijie

Qingdao Campus, Naval Aeronautics University, Qingdao 266041, China

Abstract: The availability of a periodic inspection system under mixed maintenance policies is studied in this paper. To accommodate the characteristic of multiple failure modes for complex systems, the system failures can be divided into two failure modes: hard failure and soft failure. When hard failure occurs,the corresponding corrective maintenance will be performed,taking a random time under the perfect maintenance policy; in contrast, if the soft failure is found, the corresponding preventive maintenance will be performed, taking a random time under the imperfect maintenance policy. The dynamic age setback model is adopted for imperfect maintenance, which can accurately reflect the fault characteristics of the degraded system.Then an analytical model for system steady state availability and instantaneous availability are derived. Moreover, the optimal method to maximize the system steady-state availability through adjusting the inspection interval is researched. According to the above research, the optimization of system unit time cost, preventive maintenance intervals and availability is researched. Finally, the developed approach is demonstrated by a numerical example.

Keywords: availability, reliability, maintenance, repairable system, maintenance policy.

1. Introduction

Complex repairable systems such as aircraft, submarines and unmanned aerial vehicles consist of various interacting components that realize the specified functions of the systems. This leads to the multiple failure modes of the systems which require different maintenance methods to maintain their reliability. It is necessary to analyze the reliability and maintainability of the equipment, so the availability of the equipment becomes the focus of the research.

Most of the literature on equipment availability in the past only considered the single failure mode [1-3]. Sarkar et al. [4] studied the simple system availability for an inspection-based system using a recursive method. They analyzed the availability of the system which was maintained under periodic inspections and constant repair time. They also studied the periodically inspect system availability, with its lifetime and repair-time obeying discrete distributions [5]. Furthermore, Biswas et al. [6] developed a periodically inspected system which was maintained by a fixed number of imperfect-repairs before being replaced or perfectly repaired. Unlike the research in those papers [4-6], Cui et al. [7] assumed that the periodic inspection was adopted at fixed time points after each maintenance. Some results about instantaneous availability and steady-state availability of the repairable system under the assumption of random maintenance time were presented. Li et al. [8,9] analyzed the availability of a periodical inspection system with general lifetime and repair-time distributions under a perfect repair policy. It should be noted that all the above results were obtained under the assumption that down time caused by preventive maintenance is negligible. Tang et al. [10] investigated the availability of a periodically inspected system,considering non-negligible down time caused by inspection and repair/replacement.

However, in practice, with the system configuration and the failure modes of the units becoming more diverse, the conventional assumption in the system availability analysis that a system only has one failure mode is not adequate. Therefore, the modelling of systems with multiple failure modes is drawing increasing attention[11-15]. Specifically, Qiu et al. [16] studied a repairable system in a working state withMfailure modes. However, the study did not consider the duration of inspection time [17].

Although the availability analysis for periodic inspection systems has been extensively studied in literature,there are some drawbacks.

(i) In previous studies, mixed maintenance strategies were not considered simultaneously.

(ii) System downtime caused by inspections and delays has not yet been considered, only Tang [10] considered the repair/replacement time.

Our study has made three main contributions. Firstly,we propose the instantaneous availability and steady-state availability of a periodically inspected system with multiple failure modes. Secondly, the system is maintained under mixed maintenance policies, including imperfect and perfect maintenance. Thirdly, downtime caused by inspection and supply delay is investigated. Finally, the optimization model for system unit time cost, preventive maintenance intervals and availability is studied.

2. System description

2.1 Basic assumptions

The specific assumptions used for availability analysis and maintenance modelling are summarized as follows.

(i) The system components with multiple failure modes, the failure modes can be divided into hard failure or soft failure. The former requires corrective maintenance(CM) and the latter requires preventive maintenance (PM).

(ii) Each component has its own fault time distribution a nd the density functionfi(t). Additionally,Xi(i=1,2,···,M)is independent.Xi(i=1,2,···,M), withthe distributionfunctionFi(t),pendent ofeachother.

(iii)PMtimeY1,CMtimeY2and lifetimeXare inde-

2.2 System maintenance process

When the hard failure occurs between two successive PM activities, the CM will be taken under the perfect maintenance policy, which restores the system operating condition to as-good-as-new. In this paper we assume the CM restores the system operating condition to the initial state of each cycle.

In general, the CM time can be defined as the sum of the duration ofTcandTd[18,19], whereTcis the repair and/or replacement time,Tdis the support delay time. In general, complex repairable system support delays are mainly caused by spare parts delays. Therefore, the delay time can be defined as

whereTdsisthe sparedelaytime, andξ is thesupplementary coefficient ,0≤ξ＜1, which iscausedbyother delays instead of spare delays.

Thus, the total mean CM time can be expressed as

Without loss of generality, we define a random variableY1asthe CMtime,withthedistributionfunctionG1(y)andthe density functiong1(y).Asaspecial case,if the CM time is a constant, it is denoted byv.

PM is performed at the fixed intervals τ after the former PM activities, and after the inspection, necessary activities are taken to restore the system by imperfect maintenance. Imperfect maintenance activities can make the effective age of the system younger and improve the operating condition, but not as a new one. Specially, after each PM activity, the systems are available. PM time can berepresentedbytherandomvariableY2,whichincludes inspectionanddiagnosistime,aswellas therepair and/or replacement time. In general, the distribution function PM time isG2(y) and the density function isg2(y).

2.3 System operation process

According to the above analysis, a possible sample process of the system is illustrated in Fig. 1.

As shown in Fig. 1, the PM interval length is τ,systemoperation cycle,PMistaken attimeWhenhard failure occurs, the corresponding CM will be taken with a random timeY1. When the corrective maintenance is completed, the system is the same as the new one . If the soft failure is found, the system will be operated till the next PM, and after a preventive maintenance, the system completes one operation cycle. Therefore, each operating cycle includes a preventive maintenance time, the system operation time and the CM time after each hard failure.

3. System availability analysis

3.1 System reliability

According to the method of reliability logic, the complex system can be served as a multiple component series system, the system reliability can be expressed as

whereRi(t) is the reliability of theith component,Xiis its life,Mis the number of components in the series system.

We know that PM activities reduce the effective age of the system and hence improve its reliability. In other words, PM activities make the system younger. Martorell et al. [20] introduced the proportional age setback (PAS)model, which fits the effect of imperfect maintenance on deteriorating systems well [21,22]. Hence the system reliability after the (k-1)th PM and before thekth PM is

where

andTk=kT,t′∈(0,τ) , αirepresents the effective age of the system immediately after theith PM activity,0≤αi≤1 , and λ(t) is the system extrinsic failure rate.

3.2 System instantaneous availability

In this section, we study the instantaneous availability for the system under mixed maintenance policies.

Theorem 1The instantaneous availability of the system with age-based inspection policies is given by

ProofTo derive the instantaneous availability of the system, we define a stochastic process as follows:

Clearly, the instantaneous availability of the system can be expressed as

When 0＜t＜τ , the system is available at timet. According to the total probability decomposition method,A(t)can be expressed as

The system is available at time 0, thusP{X(0)=1}=1.The first part in the right of (9) isP{X＞t}=R(t), and the second part is 0. We know the maintenance time and the failure time are independent, therefore, by using convolution, the third part of (9) can be expressed as

whereQ(t)=P{X+Y≤t} . Then,A(t) in the first interval satisfy the renew equation, which can be expressed as

Furthermore, whenkT＜t＜kT+τ with the total probability decomposition, systemA(t) can be expressed as

We know that system is available after each PM, which meansA(kT)≡1. The soft fault can only be found in the PM time, thus the second part of (12) can be expressed as

Thus the system availability can be expressed as

Summing up, we can obtain

3.3 System steady-state availability

We assume thatXis the system life, and

When the system is available at time 0, the system instantaneous availability at time 0＜t＜τ can be gained from (11), (16) and (17).

Then Laplace transform is used for (18),

Since

so

and

WhenF(t),G1(t) andG2(t) are lattice distributions,is also a lattice distribution. According to the Tobel theorem and l’Hôpital’s rule, the steady state availability is denoted as

Therefore, the system steady state availability is given by

3.4 Cost function

We assume thatcfandcpare the cost of PM and CM,cdis the unit time loss cost in the system downtime. We expect to minimize the unit time loss costC(T) of the system operation cycle [23,24].

whereE(C) is the total cost in a operation cycle, including the PM cost, CM cost and penalty cost during system downtime, and E(T) is a system expect operation cycle.

Therefore, the system unit time loss cost is

Whencf,cp, λ(t) andG(y) are determined, the system unit time loss cost depends on the preventive maintenance interval τ.

4. Numerical example

4.1 Availability analysis

The helicopter vibration signal contains rich equipment status information, and its changing characteristics can reflect the abnormal state of the equipment. Therefore,the vibration signal analysis is very important for the status monitoring and fault diagnosis of the aircraft equipment, and is an important factor related to the flight safety. The overall structure of the vibration data acquisition system is shown in Fig. 2.

Fig. 2 Main structure of the vibration data acquisition system

As shown in Fig. 1, the system consists of four main components, they are power supply components, sensor components, integrated acquisition recorder and ground equipment. The power supply components, sensor components and integrated acquisition recorder are installed on the helicopter, and each time the data collection is completed, the ground staff will copy the collected data to the ground equipment. Therefore, during the flight, only the power supply components, sensor components and integrated acquisition recorder work actually. Under normal circumstances, the power supply is powered by the helicopter engine, so no separate analysis will be done here for the power supply components. The sensor components are installed in the engine compartment, the working environment is awful. When the system fails, the system will stop working, and its fault distribution function isF1(t)=1-e-t(t+1) , the CM distribution functionG(y)=1-e-y, the integrated acquisition recorder is installed in the cockpit, its failure does not cause system downtime,its distribution function isF2=1-e-2t, τ′=2 , andT=τ+τ′=12.

Theoretically, the system reliability is

The system extrinsic failure rate is λ (t)=(3t+2)/(t+1).We assume the dynamic age setback factor is αi=2i/(2i+1) [22]. Then the failure rate after the (k-1)th PM action is

Substituting (32) into (6), we have the instantaneous availability of the system as shown in Fig. 3.

Fig. 3 indicates that in each interval,A(t) decreases first and then increases because a CM may be performed,and between each PM time,A(t)=0.A(t) is different in PM intervals, but allA(t) is above 0.7. Since the system is a degraded one, the instantaneous availability of the system is gradually reduced in the order of the periods.This rule can be seen in the black dotted line linking the ordinates in Fig. 4, which can also be reflected in Table 1.

Fig. 3 Instantaneous availability over time

Fig. 4 System steady-state availability vs. inspection interval

Table 1 Steady-state availability in each time interval

Since in the operation stage the system instantaneous availability may be fluctuating, the relevant research could be seen in [24-27]. Through the analysis, it is shown that the model is correct and conforms to the engineering practice.

Using (25), we calculate the system steady-state availability for the first 5th life-cycle, as shown in Table 1.

From Table 1, we can see that the average availability is 0.757 4, the steady-state availability in the first PM interval is low and the average availability is also fluctuating.

4.2 Maximization of the system steady-state availability

The determination of an optimal PM strategy involves numerous uncertainties, such as reliability, availability and system operation/maintenance costs. Because the high frequency of preventive maintenance will lead to the increased system maintenance costs and downtime, and insufficient maintenance will lead to the system reliability reduction. Therefore, the preventive maintenance inspection and availability of the system should be optimized.

We consider the steady-state availability as a function of the inspection interval. Then we can calculate the steady state availability denoted byA(τ). Fig. 4 illustratesA(τ) as a function of τ. It can be seen from Fig. 4 thatA(τ) has an optimal value. The maximum steadystate availability is 36%, which is obtained at τ=2.2.Thus, the optimal inspection policy is to inspect the system every 2.2 unit time.

To verify the correctness of the model, we analyze the instantaneous availability and reliability of the system, as shown in Fig. 5. At 2 unit time, the system reliability is close to 0, and the availability of the maintenance behavior is about 0.72. The minimum instantaneous availability of the system occurs at 2 unit time, while the optimal steady-state availability PM time is also at 2 unit time.However, the system cost is not considered in the above research process.

Fig. 5 System instantaneous availability and reliability

4.3 Analysis of system steady-state availability and the cost function

Obviously, if the preventive maintenance interval is too small, it will not only reduce the availability of the system, but also increase the maintenance cost. If the preventive maintenance interval is longer, The cost of inspections will decrease, and failures rate and CR cost will increase. Therefore, an optimal interval should be chosen to meet the best requirements of inspections and the cost caused by the system downtime.defined inSection4.1. Webringthese parametersinto

We increase τ from 1 to 10 with each step size of 0.1,and the system unit time loss cost is shown in Fig. 6.

It can be seen from Fig. 6 that the system can get the minimum value at τ=1 . Since in (33), when λ(t) increases monotonically andcf＞cp, the system has an optimal solution, the proof process can refer to Cao et al. [23].

Fig. 6 Average long-run cost rate vs. inspection interval

In order to better analyze the relationship between availability, system unit time loss cost and PM maintenance interval, we can transform (24) to

Then with (29) and (35), we can get

It can be seen from (36) that the system unit time loss cost can be a function as availability, PM interval, mean PM time, mean CM time, PM cost, CM cost, system downtime and reliability function. Here, we only take the system unit time loss cost and the preventive maintenance interval as variables, and the remaining parameters are the fixed values, and are given in Subsection 4.1 and Subsection 4.2, τ′=2 ,cf=3 ,cp=2 ,cd=0.2 and E(Y1)=3.

Then we plot the system unit time loss cost vs. the steady availability and PM interval (as shown in Fig. 7),set the steady availability from 0 to 1 with step size 0.1,and set the PM interval from 0 to 10 with step size 1.

Fig. 7 System unit time loss cost

In order to describe the calculation results more clearly, the function values corresponding to different variables are enumerated, which can be seen in Table 2.The minimum unit time loss cost is 1.20, the system steady availability is 1 and the PM interval is 1. The inf represents a value that is infinite or does not exist.

Table 2 System unit time loss cost

Through the above analysis, we can see that the steady availability, the optimal PM maintenance interval as well asC(T) of the system can be obtained by (36). Furthermore, this research can provide the basis for parameter design and optimization of reliability, maintainability and support-ability of the system, which will be the focus of future research [24].

5. Conclusions

This paper mainly analyzes the modeling and optimization of the system availability by using hybrid maintenance policies. Perfect and imperfect maintenance policies are adopted simultaneously in the availability model.Then the instantaneous availability and steady-state availability model of repairable equipment is derived during the study. The system unavailable time is computed, both PM time and CM time including delay time are investigated,which obey the general distribution. An example is given to illustrate the application of the design model. The results show that the models and methods can meet the requirements of availability modelling for complex systems under a mixed maintenance strategy. Furthermore, the models can provide a basis for parameter design and optimization PM interval, loss cost and the steady availability of the system.

In this paper, the system is inspected periodically. It would be significant for the research of non-periodic preventive maintenance to find the optimal inspection interval for the complex system over the life-cycle [22,25].

Finally, we are also interested in the availability fluctuation research, which is a new topic in the system reliability analysis [26-30]. System reliability and maintainability parameters can be designed based on defining and analyzing fluctuation parameters, which will be a challenging task in both theoretical and engineering fields.