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 (kcs) when performing the set of tests [3, 4]. The combination of the requirement for consecutive successes and/or the total number of successes (TSCS) (ks, kcs) 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 (kcs). However, it is rejected if a certain number of failures is reached before that run (kf). 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 (ks, kcs, kf, and kcf), 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 ofkcsconsecutive 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 (kcs, 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 (ks) and total number of failures (kf). Thus, the tested equipment is accepted if there exists a run of successes of a fixed length (kcs) or a total number of successes (ks) before either a total number ofkffailures 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 ofkcsconsecutive successes before the occurrence of a set of consecutivekcffailures? Given the various parameters likekcs,ks,kf, 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
Letksbe the number of successes required for acceptance, letkcsbe the number of consecutive successes required for acceptance, letkfbe the number of failures yielding rejection, and letr-2be the maximal number of successes between failures causing rejection. Also, denote byTnsthe number of successful startups throughoutntests, byLnsthe length of the longest run of successes throughoutntests, and byDnfthe minimal spacing between any two adjacent failures throughoutntests.Xiis the outcome of theith start-up test (=1 for success, =0 for failure). The previously known CSDF procedure made use only of thekcsandrparameters. 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
(1)f0(i,n)=P{Tns=i,Lns<kcs,Dnf>r-2,Xn=0},f1(i,n)=P{Tns=i,Lns<kcs,Dnf>r-2,Xn=1}.
Example 1. Using the values ofks=kf=3, kcs=2, and r=3,
(2)f0(1,2)=qp(caseof“10”),f1(1,2)=pq(caseof“01”),f0(2,3)=0,f1(2,3)=p2q(caseof“101”),f0(2,4)=0,f1(2,4)=0.
Summing up on the various possibilities, forn>i>r-2:
(3)f0(i,n)=P{Xn=0}·∑j=r-1min(i,kcs-1,n-1)P{Xn-1=1,…,Xn-(r-2)=1,…,Xn-j=1}·f0(i-j,n-j-1),f1(i,n)=∑j=1min(kcs-1,i)P{Xn=1,Xn-1=1,…,Xn-(j-1)=1}·f0(i-j,n-j)
so that
(4)f0(i,n)=q·∑j=r-1min(i,kcs-1,n-1)pj·f0(i-j,n-j-1),f1(i,n)=∑j=1min(kcs-1,i)pj·f0(i-j,n-j).
The boundary value relation is
(5)i<n<r:f0(i,n)=qpi·u[(kcs-1)-i]·δ[(n-1)-i].
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 failureskf, the total number of successes is limited by
(6)Tns>n-kf.
The tail distribution ofNmay then be derived with aid of(1,2):
(7)P{N>n}=∑i=max(n-kf+1,0)ks-1[f0(i,n)+f1(i,n)],
and the point distribution function is given by
(8)P{N=n}=P{N>n-1}-P{N>n}.
The expected number of trials is as follows:
(9)E{N}=∑n=1∞n·P{N=n}.
Antzoulakos et al. [1] gave a closed-form expression forE{N}for the CSDF case:
(10)E{N}=(1-pkcs)(2-pr-1)q(1+pkcs-pr-1).
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 (ks) is encountered or a specified run ofkcsconsecutive successes is achieved before meeting either a certain total number of failures (kf) or before two failures that occur too close to each other (less thanr-1successes between them). Denote byTnfthe number of failed startups throughoutntests. For the first possibility,
(11)Pa,1=∑n=ks∞P{Tn,s=ks,Ln,s<kcs,TfKTn,f<kf,Dnf>r-2,Xn=1}.
SinceTn,sis limited (6),
(12)Pa,1=∑n=ksks+kf-1P{Tn,s=ks,Ln,s<kcs,Ln,f<kcf,Xn=1}
or
(13)Pa,1=∑n=ksks+kf-1f1(ks,n).
The second term contributing toPadue to the outcome of thenth test will be
(14)Pa,i,n=P{Tn,s=i,Ln,s=kcs,Tn,fLn-1,s<kcs,Tn,f<kf,Dnf>r-2}.
Using conditional probabilities, fori>n-kf,
(15)Pa,i,n=P{Xn=1∣Tn-1,s=i-1}·P{Xn-1=1∣Tn-2,s=i-2}·⋯·P{Xn-kcs+1=1∣Tn-kcs,s=i-kcs}·P{Tn-kcs,s=i-kcs,Ln-kcs,s<kcs,Tn-kcs,f<kf,Dn-kcs,f>r-2,Xn-kcs=0}.
Therefore,
(16)Pa,i,n=pkcs·f0(i-kcs,n-kcs)·u[(i-1)-(n-kf)],Pa,2=∑n=kcsks+kf-1∑iPa,i,n=pkcs·δ[n-kcs]+qpkcs·δ[n-kcs-1]+∑n=kcs+2ks+kf-1∑i=max(0,n-kf+1,kcs)min(n,ks)pkcs·f0(i-kcs,n-kcs).
The probability of accepting the unit will be given by
(17)Pa=Pa,1+Pa,2.
Again, Antzoulakos et al. [1] presented a closed-form formula for the CSDF procedure:
(18)Pa=pkcs(2-pr-1)1-pr-1+pkcs.
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 thekcs,kfparameters 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 valuepU, and it is rejected if that value is lower than some initially set valuepL. Explicitly, it is required that
(19)P{acceptance/p=pU}>1-β,P{acceptance/p=pL}<α.
The constrained optimization problem is then to find the value ofks,kcs,kf, andrthat will minimize the expected number of required tests subject to the above constraints (19), on the confidence level.
4. Interval Estimation of “<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M92"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>”
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 ofks,kf,kcs, andr, the number of successes (si) and of failures (fi) are observed for each unit. The number of tests performed on thei′th unit is given byni=si+fi. Then, the total number of tests for all units will be given by summing on all of theni.
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
(20)p^=∑i=1nsi∑i=1nsi+∑i=1nfi
together with the observed Fisher information
(21)I(p^)=∑i=1n(si+fi)p^·(1-p^),
and the interval estimate ofpwill be given by
(22)p^±zα/2I(p^).
5. Numerical Results
As an example, the effect of introducing additional parameters likekfon the expected number of tests and on probability of acceptance may be seen in Figures 1, 2, 3, and 4.
The expected number of tests according to the CSDF plan and when considering alsokf(CSTFDF):r=2,kf=3.
The probability of acceptance of the unit according to the CSDF plan and when considering alsokf(CSTFDF):r=2,kf=3.
The expected number of tests according to the CSDF plan and when considering alsokf(CSTFDF):r=3,kf=5.
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 ofpU=0.9,pL=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.
The expected number of tests and probability of acceptance using CSDF.
kcs
r
E {N/pU}
E {N/pL}
P {a/pU}
P {a/pL}
6
5
7.1937
5.8984
0.8159
0.2359
7
3
9.2897
7.7974
0.8517
0.2099
12
2
20.6397
13.6163
0.8124
0.0573
pU=0.9; pL=0.7; α=0.25; β=0.2.
It is observed that the optimal values are obtained when choosingkcs=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.
The expected number of tests, standard deviation and probability of acceptance using TSCSTFDF.
ks
kcs
kf
r
E {N/pU}
SD
P {a/pU}
P {a/pL}
200
6
200
5
7.19
2.64
0.8159
0.2359
11
6
200
5
7.09
2.27
0.8210
0.2478
11
6
3
5
7.09
2.26
0.8185
0.2215
8
6
2
5
6.64
1.47
0.8046
0.2468
The SD value is the standard deviation around E {N/pU}.
pU=0.9; pL=0.7; α=0.25; β=0.2.
The optimal value ofE{N/pU}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 ofks=12to the input values ofp=0.9,kcs=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 (ks: reaching the limit number of successes for acceptance,kcs: encountering a run of successes for acceptance, andC0,1andC0,2: close failures <r-1).
It is counted that
(23)∑i=130si=284,∑i=130fi=36,∑i=130ni=320.
Using (20),p^≈0.8875andI(p^)≈3205.
Therefore, a 95% confidence interval forpwill be
(24)(0.8875-1.963205,0.8875+1.963205)=(0.8529,0.9221).
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 (ks) and of failures (kf) 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.
AbbreviationAcronymCS:
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.
Notationp,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).
AntzoulakosD. L.KoutrasM. V.RakitzisA. C.Start-up demonstration tests based on run and scan statisticsEryilmazS.Review of recent advances in reliability of consecutive k-out-of-n and related systemsHahnG. J.GageJ. B.Evaluation of a start-up demonstration testViveroR.BalakrishnanN.Statistical inference from start-up demonstration tests dataGeraA. E.Combined k-out-of-n: G, and consecutive k_{c}-out-of-n: G systemsBalakrishnanN.ChanP. S.Start-up demonstration tests with rejection of units upon observing d failuresSmithM. L. D.GriffithW. S.Start-up demonstration tests based on consecutive successes and total failuresSmithM. L.GriffithW. S.The analysis and comparison of start-up demonstration testsMartinD. E. K.Markovian start-up demonstration tests with rejection of units upon observing d failuresMartinD. E. K.Application of auxiliary Markov chains to start-up demonstration testsGeraA. E.A new start-up demonstration testGeraA. E.A general model for start-up demonstration tests