SOGOBO_2012_v35n3_45.pdf678.2KB

1. Introduction

In engineering field, we can find such kinds of systems that need to be kept on standby for operation until needed. For examples, a standby diesel generator of the cooling system in the nuclear power plant should always be ready only for when a priority diesel generator is failed. A fire protection system also should be kept on standby for operation whenever fire occurs. Missiles and spare parts of aircraft is also examples of such systems in that they should remain in storage state for operation whenever required. Such systems are usually called standby units and frequently hired for high mission reliability. 
 

 Comparing with the time on standby, the operation time of the standby units are very short, even nobody knows when the units are called. But failing to operate their functions when they are called may lead to catastrophic consequences, therefore standbys unit have been intensively investigated in reliability engineering fields [1, 4, 5, 7~12, 15~17].

In particular, regular inspection and maintenance policies for the standby units to avoid the occurrence of catastrophic consequences have attracted much attention to many researchers [4, 5, 10~12, 15]. More frequent tests can increase the likelihood of disclosing a failure, however, they may also deteriorate the unit faster [7]. Therefore, unit deterioration by test should be considered when test scheduling is determined.
 

This paper considers a standby unit which should be ready for operation for a long time whenever needed. Assuming the standby unit can fail during the standby period, the period test is considered for the standby unit [2, 3, 5, 10~14]. Although the periodic test is capable of detecting failures during the standby period, it can cause test-induced failures for the standby unit at the start of the test [6, 10, 12~15]. Also, the aging process of the standby unit during the test period needs to be taken into account, which is not treated in the existing studies.
 

This study adopts an availability of the standby unit as a meaningful measure of unit performance. The limiting avail-ability is derived for the standby unit by incorporating three failure types : (i) type I failure; failure during availability for the standby period, (ii) type Ⅱ failure; test-induced failure at the start of the test and (iii) type Ⅲ failure; operating failures during the test period due to the aging effect.  

2. Limiting Availability of the Standby Unit

The following assumptions and notations are used throughout this paper.
<Assumptions>
(1) Three failure types of a standby unit are independent of each other
(2) Type I failure rate is constant, i.e., type I failure occurs according to the Homogeneous Poison Process.
(3) Probability of test-induced(type Ⅱ) failure is constant.
(4) Type Ⅲ failure rate is increasing over cumulative test time until the standby unit is called for operation in place of the priority unit, i.e.,  

<Notations> 
T : test interval for the standby unit
τ : test period for the standby unit
F₀(t) : operating time distribution of the standby unit
C₀(t) : repair time distribution of the standby unit
H₀(t) : standby failure time distribution of the standby unit
 α : probability of type Ⅱ failure
 : probability of type I failure in each test interval  

To derive availability of the standby unit, mean cycle time (MCT) and mean up time (MUT) must be calculated respec-tively. The standby unit undergoes standby and test states repeatedly until a failure for the standby unit occurs. The failure rate of the standby unit is constant during standby period. The failure rate function of the standby unit during test period is increasing function of the cumulative test duration. If no failure is found during a specific test period, the state of the standby unit at the beginning of the next test period is equal to that at the end of the previous test period. If any failure is found through the test, the standby unit undergoes repair action and is restored to the state of as-good-as-new. Therefore, from the initial standby status to the completion of the repair of the standby unit constitutes a cycle. <Figure 1> illustrates such aspect of the failure rate function of the standby unit. 
 

<Figure 1> Failure Rate Function of the Standby Unit

<Figure 2>~<Figure 4> represent the mean cycle time and the mean up time for the following three failure type of the standby unit:
(ⅰ) type I failure; failure during standby period,
(ⅱ) type Ⅱ failure; test-induced failure at the start of the test,
(ⅲ)type Ⅲ failure; operating failure during the test period.  

2.1 Mean Cycle Time (MCT)

To derive MCT of the standby unit, three failure types of the standby unit should be considered. Firstly, if the standby unit fails according to the type I, i.e., the standby unit has failed during the standby period and this failure is found at the beginning of the test as depicted in <Figure 2>, the mean cycle time due to type I failure can be written as 

Eq. (2) implies that there is no failure from the first standby period to(j - 1)^th   test, and the failure occurring during j^th  standby period is found at . 

<Figure 2> Mean Cycle Time of Type I Failure Case

Secondly, if the standby unit fails according to the type II failure (test-induced failure case), i.e., the standby unit fails and is found at the start of the test as depicted in <Figure 3>, the mean time due to type II failure can be written as 

Eq. (3) can be obtained by similar way to obtain eq. (2), and note that the time to find the failure is same to the type I failure case. 

<Figure 3> Mean Cycle Time of Type II Failure Case

Thirdly, if the standby unit fails according to the type Ⅲ failure, i.e., the standby unit fails during the test operation and the failure is found immediately as depicted in <Figure 4>, the mean cycle time due to type Ⅲ failure can be written as 

Eq. (4) means that there is no failure from first standby period to j^th  standby period, and  j^th test starts successfully but the failure occurs during  j^th test, i.e., between  and . 

<Figure 4> Mean Cycle Time of Type III Failure Case

Lastly, we should consider mean repair time (MRT) by each failure types to compute the mean cycle time. the MRTs due to three failure types are all the same, and easily ob`tained by the distribution of repair time. Therefore, the mean cycle time can be obtained by  

2.2 Mean Up Time (MUT)

 In a similar way, the mean up time (MUT) of the standby unit during cycle is derived by considering three failure types as depicted in <Figure 2>~<Figure 4>. Note that only standby period is considered in MUT. 

 Firstly, if the standby unit fails according to the type I, the mean up time can be written as

If the standby unit has failed during the j^th  standby period, the up time of the standby unit is the sum ( j-1 ) · T and working time during  j^th  standby period.

 secondly, if the standby unit fails according to the type Ⅱ, the failure occurs at the beginning of test, i.e., at , and the uptime of the standby is  j· T , therefore  the means up time due to type Ⅱ failure can be written as

Lastly, if type III failure occurs at between  and , the up time of the standby unit is , therefore, the means up time due to type III failure can be written as  

By eq(6)~(8), the mean up time can be obtained by  

In conclusion, limiting availability of the standby unit can be derived as  

3. Numerical analysis

To identify relationships between the periodic test interval and various failure characteristics of the standby unit, several experiments are performed using the availability of the standby unit as measure of system performances.

 In the experiments, life distribution of the standby unit in test operation is assumed to be Weibull with scale and shape parameters of 100 and 2 respectively. Repair time distribution of the standby unit is assumed to be the exponential with parameter of 0.2, i.e., MRT = 5. The type I failure rate is assumed to be constant, i.e., type I failure occurs according to the Homogeneous Poison Process, therefore, the failure time during each standby period follows exponential distribution with parameter of the type I failure (λ).

3.1 Sensitivity Analysis for Availability

Variations of the test interval with respect to the changes of test duration (aging effect during test period) are observed for fixed α = 0.01and  type I failure rate is 0.0001 (H_s (t) = 1 - e^-t/1000 , i.e.,f_s (t) = 0.0001 -e^-t/1000  ). Result is depicted in <Figure 5> against performance measure of availability. For more comprehension, detailed values are provided in <Table 1> of appendix.  

<Figure 5> Availability Versus T and t

The availability is shown to be decreased as the test duration time  increase (as the aging effect increases), especially, it is noticeable that availability is decreasing more rapidly when the test duration time t is small. However, Availability with respect to increasing test interval T has turning point from increasing to decreasing as shown <Figure 5>. which means that the test interval has optimal point for maximizing availability of the standby unit.

 <Figure 6> shows the changes of availability with respect to the test interval and the probability of type II failure  α(test-induced failure) for fixed t=2 and type I failure rate = 0.0001. The availability is shown to be decreased as the probability of type II failure α (test-induced failure) increases, which is commonly predictable result (Detailed values are provided in <Table 2> of appendix).

<Figure 6> Availability Versus T and α

The changes of availability with respect to the test interval and the type I failure rate for fixed t=2 and α=0.01 are shown in <Figure 7> (Detailed values are provided in <Table 3> of appendix). The type I failure rate is assumed to be constant but we should keep in mind that the probability  of type I failure occurrence is obtained by  where H_s (x) is cumulative density function of exponential distribution with parameter of the type I failure rate.

<Figure 7> is commonly predictable such that the availability tends to decrease as increasing type I failure rate. However, It is somewhat interesting that the changes of the availability versus type I failure rate are more rapid with respect to increasing test interval.  

<Figure 7>Availability Versus t and Type I Failure Rate

3.2 Sensitivity analysis for optimal test interval

Numerical analyses are performed to observe the change of optimal test interval by changing the test duration (t), probability of type II failure (α) and type I failure rate.

 <Table 1> shows the optimal test interval with respect to the test duration and the probability of type II failure. The optimal test interval tends to increase as test duration increase, which implies that the aging process of the standby unit during the test period affects strongly to the optimal test interval.

We can observe somewhat interesting point in the change of optimal test interval with respect to the probability of type II failure of <Table 1>. the optimal test interval is increasing with respect to the increasing α when t ≤ 4 , however, the optimal test interval is decreasing with respect to the increasing  α when t>4.
 

<Table 1>The Optimal Test Interval with Respect to αand t

 The optimal test interval with respect to the test duration and the type II failure rate is computed in <Table 2>. As expected, the optimal test interval decrease with respect to increasing type II failure rate.

<Table 21>The Optimal Test Interval with Respect to t and  α

4. Concluding Remarks

This paper analyzes the availability of the standby unit which should be ready for operation for a long time whenever needed. The periodic test is adopted to disclose a failure during standby period, however, it may also deteriorate the unit faster. Therefore, operating failure due to the aging effect during the test period, which is not treated in the previous works.
 

 By considering three failure types of the standby unit, we can obtain limiting availability of the standby unit. Results of the experiments show that the availability decreases with respect to increasing test duration time and, probability of the test-induced failure and type I failure rate. Also, the availability increases rapidly to a certain value of the test interval and decreases slowly after this value. These results agree with what we have expected. However, an appropriate periodic test interval of the standby unit for maximizing the limiting availability can reasonably be identified under three types of failures of the standby unit. This paper illustrates the practical implementation of the optimal test interval by numerical examples.

The further study can be possible to determine optimal periodic test interval, and extend to the maintenance model considering cost variables. In addition, although three failure types of the standby unit are treated as independent in this study, the degree of dependency between three failures types need to be investigated further, which is left as future research works.  

<Appendix>

Detailed values of numerical analysis are given below. 

<Table A1> Availability Versus T and t

<Table A2>Availability Versus T and α

<Table A3>Availability Versus T and α