A Start-Up Demonstration Test Involving Distant Failures

Abstract

A new procedure for determining the acceptance or rejection of a system that undergoes a start-up demonstration set of tests is presented. It is a generalization of the recently introduced CSDF model (consecutive successes distant failures). According to the new total successes consecutive successes total failures distant failures (TSCSTFDF) procedure, a unit is accepted when either a total number of successful tests or a specified number of consecutive successes are observed before a total number of failures or the occurrence of near failures which are too close to each other. The practical advantage of this new procedure is the significant reduction in the expected number of required tests together with improved second-order statistics (standard deviation).

1. Introduction

Start-up demonstration tests are performed in order to prove the reliability of power generating equipment like lawn mowers, water pumps, car batteries, and many sorts of electronic equipment. An example for an application has been mentioned by Antzoulakos et al. [1] regarding a conference center that wishes to replace its old slide projectors with modern ones. The outcomes of a set of demonstration tests that are carried out on the new projectors are recorded. The requirement may be in form of observing consecutive successes before the occurrence of a certain number of failures or before meeting a consecutive set of failures. As a result, the reliability of the equipment is determined.

Various procedures exist for deciding whether the set of tests is successful which imply the decision on accepting or rejecting the unit in question.

They are based on the commonly known theory of consecutive𝑘-out-of-𝑛systems [2]. Among others, there exist the consecutive successes (CS), total successes consecutive successes (TSCS), consecutive failures total failures (CSTF), consecutive successes distant failures (CSDF), and a general TSCSTFCF procedure. According to the simplest CS procedure, the equipment is accepted if there exists a certain run of successes of prespecified length (𝑘_𝑐𝑠) when performing the set of tests [3, 4]. The combination of the requirement for consecutive successes and/or the total number of successes (TSCS) (𝑘_𝑠,  𝑘_𝑐𝑠) as a basis for accepting the unit has been presented by Gera [5]. A more advanced procedure (CSTF) has been given by Balakrishnan and Chan [6], Smith and Griffith [7, 8], and Martin [9, 10]. Like before, the unit is accepted if there exists a run of successes (𝑘_𝑐𝑠). However, it is rejected if a certain number of failures is reached before that run (𝑘_𝑓). More general models are the TSCSTF and TSCSTFCF procedures which have been presented by Gera [11, 12]. According to the TSCSTFCF model, a unit is accepted if either a run of successes or a total number of successes are observed before either a total number of failure or a run of failures (𝑘_𝑠, 𝑘_𝑐𝑠, 𝑘_𝑓, and 𝑘_𝑐𝑓), and vice versa for rejection. The CSDF model has been recently introduced by Antzoulakos et al. [1] according to which a unit is accepted if a run of𝑘_𝑐𝑠consecutive successes is obtained before the occurrence of two failures that are not sufficiently distant (𝑟) from each other. It is rejected if two failures that are not sufficiently distant from each other are met before a certain run of successes (𝑘_𝑐𝑠, 𝑟).

A generalization of the previously mentioned CSDF procedure is presented here: total successes consecutive successes total failures distant failures (TSCSTFDF). It involves the addition of two parameters: total number of successes (𝑘_𝑠) and total number of failures (𝑘_𝑓). Thus, the tested equipment is accepted if there exists a run of successes of a fixed length (𝑘_𝑐𝑠) or a total number of successes (𝑘_𝑠) before either a total number of𝑘_𝑓failures or the occurrence of two failures that have less than𝑟−1successes between them (so that they are too close to each other). The following questions will be answered. What is the probability of accepting a unit undergoing such a process? What are the chances of observing a set of𝑘_𝑐𝑠consecutive successes before the occurrence of a set of consecutive𝑘_𝑐𝑓failures? Given the various parameters like𝑘_𝑐𝑠,𝑘_𝑠,𝑘_𝑓, and𝑟, what is the expected number of tests till arriving a decision? From analysis to design, a method of determining the values of those parameters due to some optimization criterion will be presented.

The statistical properties of the new model are examined.𝑠-independence between the tests is assumed and the probability of success of each trial (𝑝) is assumed to be identical (i.i.d.). A practical problem is the number of required tests which surely should be as low as possible subject to some limits on the confidence level of the tests. This problem of minimizing the expected number of tests that are required is set up in form of a constrained optimization problem. It will be observed that a significant reduction in that number is achieved by using the new procedure.

The problem of estimating the underlying probability of success of each test (𝑝) is also tackled. Previous work in this direction has been carried out by Viveros and Balakrishnan [4], Smith and Griffith [7], Gera [12], and Antzoulakos et al. [1]. Maximum likelihood (MLE) is applied to obtain interval estimation. An illustrative example is presented.

2. The TSCSTFDF Procedure

Let𝑘_𝑠be the number of successes required for acceptance, let𝑘_𝑐𝑠be the number of consecutive successes required for acceptance, let𝑘_𝑓be the number of failures yielding rejection, and let𝑟−2be the maximal number of successes between failures causing rejection. Also, denote by𝑇_𝑛𝑠the number of successful startups throughout𝑛tests, by𝐿_𝑛𝑠the length of the longest run of successes throughout𝑛tests, and by𝐷_𝑛𝑓the minimal spacing between any two adjacent failures throughout𝑛tests.𝑋_𝑖is the outcome of the𝑖th start-up test (=1 for success, =0 for failure). The previously known CSDF procedure made use only of the𝑘_𝑐𝑠and𝑟parameters. Here we increase the number of parameters (TSCSTFDF), so that we will obtain significantly shortening of the number of tests required to reach a decision with respect to the tested equipment.

Following the notation of Gera [12], let 𝑓₀(𝑖,𝑛)=𝑃𝑇_𝑛𝑠=𝑖,𝐿_𝑛𝑠<𝑘_𝑐𝑠,𝐷_𝑛𝑓>𝑟−2,𝑋_𝑛,𝑓=0₁(𝑖,𝑛)=𝑃𝑇_𝑛𝑠=𝑖,𝐿_𝑛𝑠<𝑘_𝑐𝑠,𝐷_𝑛𝑓>𝑟−2,𝑋_𝑛=1. (1)

Example  1. Using the values of𝑘_𝑠=𝑘_𝑓=3,  𝑘_𝑐𝑠=2, and  𝑟=3, 𝑓₀()(),𝑓1,2=𝑞𝑝caseof“10”₁()(),𝑓1,2=𝑝𝑞caseof“01”₀()𝑓2,3=0,₁(2,3)=𝑝²𝑞(),𝑓caseof“101”₀()𝑓2,4=0,₁(2,4)=0. (2) Summing up on the various possibilities, for𝑛>𝑖>𝑟−2: 𝑓₀()𝑋𝑖,𝑛=𝑃_𝑛⋅=0_{min(𝑖,𝑘𝑐𝑠−1,𝑛−1)𝑗=𝑟−1}𝑃𝑋_𝑛−1=1,…,𝑋_{𝑛−(𝑟−2)}=1,…,𝑋_𝑛−𝑗=1⋅𝑓₀,𝑓𝑖−𝑗,𝑛−𝑗−1₁()=𝑖,𝑛_{min(𝑘𝑐𝑠−1,𝑖)𝑗=1}𝑃𝑋_𝑛=1,𝑋_𝑛−1=1,…,𝑋_{𝑛−(𝑗−1)}=1⋅𝑓₀𝑖−𝑗,𝑛−𝑗 (3) so that 𝑓₀(𝑖,𝑛)=𝑞⋅^{min(𝑖,𝑘𝑐𝑠−1,𝑛−1)}_{𝑗=𝑟−1}𝑝^𝑗⋅𝑓₀,𝑓𝑖−𝑗,𝑛−𝑗−1₁(𝑖,𝑛)=^{min(𝑘𝑐𝑠−1,𝑖)}_𝑗=1𝑝^𝑗⋅𝑓₀𝑖−𝑗,𝑛−𝑗. (4) The boundary value relation is 𝑖<𝑛<𝑟:𝑓₀(𝑖,𝑛)=𝑞𝑝^𝑖⋅𝑢𝑘_𝑐𝑠)].−1−𝑖⋅𝛿[(𝑛−1−𝑖 (5) Let𝑁be the random variable representing the total number of startups (trials) until termination of the experiment.

Due to the limit on the total number of failures𝑘_𝑓, the total number of successes is limited by 𝑇_𝑛𝑠>𝑛−𝑘_𝑓. (6) The tail distribution of𝑁may then be derived with aid of(1,2): 𝑃{𝑁>𝑛}=^𝑘𝑠−1_{𝑖=max𝑛−𝑘𝑓+1,0}𝑓₀(𝑖,𝑛)+𝑓₁(𝑖,𝑛), (7) and the point distribution function is given by 𝑃{𝑁=𝑛}=𝑃{𝑁>𝑛−1}−𝑃{𝑁>𝑛}. (8) The expected number of trials is as follows: 𝐸{𝑁}=^∞_𝑛=1𝑛⋅𝑃{𝑁=𝑛}. (9) Antzoulakos et al. [1] gave a closed-form expression for𝐸{𝑁}for the CSDF case: 𝐸{𝑁}=1−𝑝^𝑘𝑐𝑠2−𝑝^𝑟−1𝑞1+𝑝^𝑘𝑐𝑠−𝑝^𝑟−1. (10) The probability of acceptance of the tested unit is also of interest. The acceptance may be due to two possibilities: either the required total number of successes (𝑘_𝑠) is encountered or a specified run of𝑘_𝑐𝑠consecutive successes is achieved before meeting either a certain total number of failures (𝑘_𝑓) or before two failures that occur too close to each other (less than𝑟−1successes between them). Denote by𝑇_𝑛𝑓the number of failed startups throughout𝑛tests. For the first possibility, 𝑃_𝑎,1=^∞_{𝑛=𝑘𝑠}𝑃𝑇_𝑛,𝑠=𝑘_𝑠,𝐿_𝑛,𝑠<𝑘_𝑐𝑠,𝑇_𝑛,𝑓<𝑘_𝑓,𝐷_𝑛𝑓>𝑟−2,𝑋_𝑛=1. (11) Since𝑇_𝑛,𝑠is limited (6), 𝑃_𝑎,1=^{𝑘𝑠+𝑘𝑓−1}_{𝑛=𝑘𝑠}𝑃𝑇_𝑛,𝑠=𝑘_𝑠,𝐿_𝑛,𝑠<𝑘_𝑐𝑠,𝐿_𝑛,𝑓<𝑘_𝑐𝑓,𝑋_𝑛=1 (12) or 𝑃_𝑎,1=^{𝑘𝑠+𝑘𝑓−1}_{𝑛=𝑘𝑠}𝑓₁𝑘_𝑠,𝑛. (13) The second term contributing to𝑃_𝑎due to the outcome of the𝑛th test will be 𝑃_{𝑎,𝑖,𝑛}=𝑃𝑇_𝑛,𝑠=𝑖,𝐿_𝑛,𝑠=𝑘_𝑐𝑠,𝐿_{𝑛−1,𝑠}<𝑘_𝑐𝑠,𝑇_𝑛,𝑓<𝑘_𝑓,𝐷_𝑛𝑓>𝑟−2. (14) Using conditional probabilities, for𝑖>𝑛−𝑘_𝑓, 𝑃_{𝑎,𝑖,𝑛}=𝑃𝑋_𝑛=1∣𝑇_{𝑛−1,𝑠}𝑋=𝑖−1⋅𝑃_𝑛−1=1∣𝑇_{𝑛−2,𝑠}𝑋=𝑖−2⋅⋯⋅𝑃_{𝑛−𝑘𝑐𝑠+1}=1∣𝑇_{𝑛−𝑘𝑐𝑠,𝑠}=𝑖−𝑘_𝑐𝑠⋅𝑃𝑇_{𝑛−𝑘𝑐𝑠,𝑠}=𝑖−𝑘_𝑐𝑠,𝐿_{𝑛−𝑘𝑐𝑠,𝑠}<𝑘_𝑐𝑠,𝑇_{𝑛−𝑘𝑐𝑠,𝑓}<𝑘_𝑓,𝐷_{𝑛−𝑘𝑐𝑠,𝑓}>𝑟−2,𝑋_{𝑛−𝑘𝑐𝑠}=0. (15) Therefore, 𝑃_{𝑎,𝑖,𝑛}=𝑝^𝑘𝑐𝑠⋅𝑓₀𝑖−𝑘_𝑐𝑠,𝑛−𝑘_𝑐𝑠⋅𝑢(𝑖−1)−𝑛−𝑘_𝑓,𝑃_𝑎,2=^{𝑘𝑠+𝑘𝑓−1}_{𝑛=𝑘𝑐𝑠𝑖}𝑃_{𝑎,𝑖,𝑛}=𝑝^𝑘𝑐𝑠⋅𝛿𝑛−𝑘_𝑐𝑠+𝑞𝑝^𝑘𝑐𝑠⋅𝛿𝑛−𝑘_𝑐𝑠+−1_{𝑘𝑠+𝑘𝑓−1𝑛=𝑘𝑐𝑠+2min(𝑛,𝑘𝑠)𝑖=max0,𝑛−𝑘𝑓+1,𝑘𝑐𝑠}𝑝^𝑘𝑐𝑠⋅𝑓₀𝑖−𝑘_𝑐𝑠,𝑛−𝑘_𝑐𝑠. (16) The probability of accepting the unit will be given by 𝑃_𝑎=𝑃_𝑎,1+𝑃_𝑎,2. (17) Again, Antzoulakos et al. [1] presented a closed-form formula for the CSDF procedure: 𝑃_𝑎=𝑝^𝑘𝑐𝑠2−𝑝^𝑟−11−𝑝^𝑟−1+𝑝^𝑘𝑐𝑠. (18)

3. Constrained Optimization

The present procedure is a generalization of the previous CSDF introduced by Antzoulakos et al. and it is advantageous compared to the simpler model in reducing the expected number of tests required for the demonstration. Smith and Griffith [7, 8] have suggested a procedure for minimizing the expected number of tests through a correct choice of the𝑘_𝑐𝑠,𝑘_𝑓parameters for the CSTF model. The optimization is carried out with respect to some constraints. Antzoulakos et al. [1] pointed out such an optimization regarding the CSDF model. Gera [12] applied this procedure to the more general TSCSTFCF. A similar procedure will be applied here using the TSCSTFDF model.

The tested unit should be accepted if the value of the probability of success (𝑝) is higher than some specific value𝑝_𝑈, and it is rejected if that value is lower than some initially set value𝑝_𝐿. Explicitly, it is required that 𝑃𝑎𝑐𝑐𝑒𝑝tan𝑐𝑒/𝑝=𝑝_𝑈𝑃>1−𝛽,𝑎𝑐𝑐𝑒𝑝tan𝑐𝑒/𝑝=𝑝_𝐿<𝛼. (19) The constrained optimization problem is then to find the value of𝑘_𝑠,𝑘_𝑐𝑠,𝑘_𝑓, and𝑟that will minimize the expected number of required tests subject to the above constraints (19), on the confidence level.

4. Interval Estimation of “𝑝”

It is assumed here that the probability of success (𝑝) of each unit is the same, and it is independent of the other trials (i.i.d.). Performing𝑛tests on several units with prespecified values of𝑘_𝑠,𝑘_𝑓,𝑘_𝑐𝑠, and𝑟, the number of successes (𝑠_𝑖) and of failures (𝑓_𝑖) are observed for each unit. The number of tests performed on the𝑖^′th unit is given by𝑛_𝑖=𝑠_𝑖+𝑓_𝑖. Then, the total number of tests for all units will be given by summing on all of the𝑛_𝑖.

The lines of argumentation presented by Gera [12] for the TSCSTFCF model are observed to apply also in our case with the conclusion that the estimate of𝑝is given by ∑𝑝=^𝑛𝑖=1𝑠_𝑖∑^𝑛_𝑖=1𝑠_𝑖+∑^𝑛_𝑖=1𝑓_𝑖 (20) together with the observed Fisher information 𝐼𝑝=∑^𝑛_𝑖=1𝑠_𝑖+𝑓_𝑖𝑝,𝑝⋅1− (21) and the interval estimate of𝑝will be given by 𝑧𝑝±^𝛼/2𝐼𝑝. (22)

5. Numerical Results

As an example, the effect of introducing additional parameters like𝑘_𝑓on the expected number of tests and on probability of acceptance may be seen in Figures 1, 2, 3, and 4.

Figure 1 

The expected number of tests according to the CSDF plan and when considering also𝑘𝑓(CSTFDF):𝑟=2,𝑘𝑓=3.

Figure 2 

The probability of acceptance of the unit according to the CSDF plan and when considering also𝑘𝑓(CSTFDF):𝑟=2,𝑘𝑓=3.

Figure 3 

The expected number of tests according to the CSDF plan and when considering also𝑘𝑓(CSTFDF):𝑟=3,𝑘𝑓=5.

Figure 4 

The probability of acceptance of the unit according to the CSDF plan and when considering also𝑘𝑓(CSTFDF):𝑟=3,𝑘𝑓=5.

As to the design problem of choosing the optimal values of the parameters, they obviously depend on the values of the upper and lower probabilities and the confidence limits. The optimization procedure presented above has been applied to the test case given by Antzoulakos et al. [1]. They derived optimal values of the parameters using the simpler CSDF model. Our numerical results fully correlate theirs for that case. Using the values of𝑝_𝑈=0.9,𝑝_𝐿=0.7,𝛼=0.25, and𝛽=0.2and applying their relation (10) for finding the expected number of tests, their results are presented in Table 1.

Table 1 

The expected number of tests and probability of acceptance using CSDF.

It is observed that the optimal values are obtained when choosing𝑘_𝑐𝑠=6,𝑟=5which fulfill the constraints.

We suggest an improved optimum for the same problem by using instead the more general TSCSTFDF procedure, as shown in Table 2.

Table 2 

The expected number of tests, standard deviation and probability of acceptance using TSCSTFDF.

The optimal value of𝐸{𝑁/𝑝_𝑈}owing to using the TSCSTFDF procedure is thus reduced significantly, 7.5% improvement, with respect to the previously derived optimum value using the CSDF model. We also gain simultaneously an enormous improvement in the value of the standard deviation. A systematic approach to locate the optimum will surely yield a further reduction in the optimal value.

The statistical inference on the value of “𝑝” may be illustrated for instance by using the simulated data presented by Antzoulakos et al. [1] for𝑛=30units by adding a value of𝑘_𝑠=12to the input values of𝑝=0.9,𝑘_𝑐𝑠=10, and𝑟=3. Referring to their data, the following results are shown in Table 3. The reason for terminating the set of tests is indicated in the last row (𝑘_𝑠: reaching the limit number of successes for acceptance,𝑘_𝑐𝑠: encountering a run of successes for acceptance, and𝐶_0,1and𝐶_0,2: close failures <𝑟−1).

Table 3 

Simulation results: 𝑛=30 units, 𝑝=0.9, TSCSTFDF, 𝑘_𝑠=12,  𝑘_𝑐𝑠=10,  𝑘_𝑓=100, and  𝑟=3.

It is counted that ³⁰_𝑖=1𝑠_𝑖=284,³⁰_𝑖=1𝑓_𝑖=36,_30𝑖=1𝑛_𝑖=320. (23) Using (20),𝑝≈0.8875and𝐼(𝑝)≈3205.

Therefore, a 95% confidence interval for𝑝will be 0.8875−1.96√3205,0.8875+1.96√=().32050.8529,0.9221 (24)

6. Summary and Conclusions

A new model (TSCSTFDF) has been presented for performing a set of start-up demonstration tests. It is a generalization of the previously known CSDF procedure by adding the parameters representing the total number of successes (𝑘_𝑠) and of failures (𝑘_𝑓) into the analysis and the design of the test procedure. The main advantage in using the more general model is in the reduction of the expected number of required tests together with improved second-order statistics. Although some more measurements need to be carried out, this advantage still seems to be significant and the procedure seems to be practical. A constrained optimization problem has been set up together with an illustrated example. Future work should involve a systematic algorithm for yielding the optimum.

Abbreviation

Acronym

CS:	Total successes
TSCS:	Total successes consecutive successes
CSTF:	Consecutive successes total failures
CSDF:	Consecutive successes distant failures
TSCSTF:	Total successes consecutive successes total failures
TSCSTFCF:	Total successes consecutive successes total failures consecutive failures
TSCSTFDF:	Total successes consecutive successes total failures distant failures.

Notation

𝑝,𝑞:	Probability of success, failure of each trial
𝑘𝑠:	The number of successes required for acceptance
𝑘𝑐𝑠:	The number of consecutive successes required for acceptance
𝑘𝑓:	The number of failures yielding rejection
𝐿𝑛𝑠:	The length of the longest run of successes throughout𝑛tests
𝐷𝑛𝑓:	The minimal spacing between adjacent failures throughout𝑛tests
𝑟−2:	The maximal number of successes between failures causing rejection
𝑇𝑛𝑠:	The number of successful startups throughout𝑛tests
𝑇𝑛𝑓:	The number of failed startups throughout𝑛tests
𝑁:	The total number of startups (trials) until termination of the experiment
𝐸{𝑁}:	Expected number of𝑁
SD:	The standard deviation for𝑁
𝑃𝑎:	The probability of acceptance of the unit
𝑢[⋅]:	The unit step function
𝑋𝑖:	Outcome of𝑖th start-up test (=1 for success, =0 for failure).