Availability modeling for periodically inspection system with different lifetime and repair-time distribution

Junling LI ,Yueling CHEN ,Yong ZHANG ,Hiling HUANG

a Qingdao Campus,Naval Aviation University,Qingdao 266041,China

b The 92635th Unit of Navy,Qingdao 266041,China

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.

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.1These unexpected failures may 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 Proschan2firstly put forward the availability model for this system.Hoyland and Rausand3discussed the exact expressions of system availability model under some simple distributions.Sarkar and Chaudhuri4,5analyzed the system availability model under gamma lifetime and exponential repair time or allowing a stand-by spare unit.J.Sarkar and S.Sarkar6proposed an availability model of periodic inspection 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.7Meng et al.8analyzed dynamic relationship of system inspection period and availability based on the theory of renewal process. Ahmad and Soleimanmeigouni9developed a reliability-based cost model for periodically inspected unit under Weibull lifetime distribution.Tiwary and Arya10developed 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.11studied 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.Sun12studied the availability assessment methods for complex system with various lifetime and repair-time distribution.Li and Teng13proposed 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.14proposed 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.15,16studied a system subject to two typical failure modes, degradationrecursively 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=0 and is tested at regular intervals 2τ,3 τ,...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 F(t)denotes the lifetime function, G(t) denotes the repair-time function,denotes the system reliability function,A(t)denotes the availability function,t ≥0,and τ denotes the inspection period,the availability function of the system can be expressed by 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.17proposed 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 Cui18also 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 ≤τ and no inspection happens during this time period,it is clear that

(2)If τ ≤t ≤2τ and the system availability adopts full probability formula,A(t)can be deduced by

(3)If kτ ≤t ≤(k+1)τ and the system availability adopts full probability formula,A(t)can be deduced by

From the above analysis,in order to obtain the expressions ofA(t),it just needs to find the deduction relationship between A(kτ)and A[(k+1)τ].

Lemma 1.Suppose 0 ＜a,b ＜1,the successive weighted averages of a and b can be defined as follows:

w0=1, w1=a and wk+1=wka+(1-wk)b, for k=1,2,3,···and then,

and thus

limk →∞wk=b/(b+1-a)

Proof.Note that wk+1=b+(a-b)wk,which applied successively implies that

Hence,

Proving Eq.(5).

Finally, since |a -b|＜1, limk →∞ and wk=b/(b+1-a).

Then,

Moreover,if we set wk=A[(k+1)τ],we get

Then,the system availability at time kτ ≤t ≤(k+1)τ,for k=1,2,3,···is given in Eq.(8)

If we get the express of the system lifetime functionF(t),even the competing-risk system, and repair-time function G(t), the system availability function can be obtained accordingly.

2.3.System limiting average availability

The limiting average availability of the system is obtained by Sarkar,4-6when the lifetime has a density function and the repair time is exponentially distribution.Without loss of generality,the model can be extended as follows:

and the system limiting average availability shows as

3.Availability model of the system with several typical lifetime and repair-time distribution

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.19Repair-time distribution functions for some products are studied by Xu20and Li21et al.,as shown in Table 2.

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

where α is the scale parameter.And the repair-time function is also exponential distribution function,

The system availability at time t can be rewritten as follows:

Table 1 Classic lifetime distribution.

Table 2 Classic repair-time distribution.

Fig.1 Instantaneous availability of system under exponential and exponential distribution.

where μ is the scale parameter of G(t).When α=0.01,μ=0.5,τ=50,the system availability is shown in Fig.1.

Therefore,from Eq.(11)the limiting average availability of the system is

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

where η is the scale parameter,m is the shape parameter.And the repair-time function is normal distribution function,

where μ is the mean,σ is the variance.The system availability at time t,form Eq.(1)given in Eq.(18)

Therefore,from Eq.(11)the limiting average availability of the system is

According to Eq.(19),the limiting average availability of system is relatively low.Sarkar5has 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 differentτwas calculated,as shown in Table 3.

3.3.Availability model of the system with Weibull/lognormal distribution

Suppose that system lifetime T has a Weibull distribution,the reliability function is

When,η=0.5,m=2,u=0,σ=1,τ=10,the system availability is shown in Fig.2

And the repair-time function is lognormal distribution function,

When,η=0.5,m=2,u=0,σ=1,τ=10,the system availability is shown in Fig.3.

Therefore,from Eq.(11)the limiting average availability of the system is

where μ is the mean,σ is the variance.The system availability at timet,form Eq.(1)given in Eq.(22)

Fig. 2 Instantaneous availability of system under Weibull/normal distribution.

Table 3 Relation among Aav[0,∞)and τ,when the system lifetime distribution is Weibull and repair-time is normal.

Fig. 3 Instantaneous availability of system under Weibull/lognormal distribution.

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

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 areη=0.5,m=2,u=0,σ=1,then the instantaneous availability of the system varies with τ=10,τ=20 and τ=40 as shown in Fig.4.

In Fig.4,when t ∈[10,20]the highest system availability is τ=10,when t ∈(20,40)the highest system availability is τ=20,when t ∈(40,80)the highest system availability is τ=40,when t ∈(80,100)the highest system availability is τ=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[0,∞)and τ,when the system lifetime distribution is Weibull and repair-time is lognormal.

Fig.4 Instantaneous availability of system with different τ.

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.,22in 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.