Annals of Operations Research

, Volume 212, Issue 1, pp 225–239

On generalized start-up demonstration tests

Authors

    • School of Management & EconomicsBeijing Institute of Technology
Article

DOI: 10.1007/s10479-012-1279-y

Cite this article as:
Zhao, X. Ann Oper Res (2014) 212: 225. doi:10.1007/s10479-012-1279-y

Abstract

Start-up demonstration tests and various extensions have been discussed, in which a unit under the test is accepted or rejected according to some criteria. CSTF, CSCF, TSCF and TSTF are most well known start-up demonstration tests. In this paper, two kinds of more general start-up demonstration tests are introduced. CSTF, TSTF, TSCF and CSCF are all special situations of the new tests. For the new generalized start-up demonstration tests, under the assumption of independent and identically distributed trials for each test, the analytic expressions for the expectation, the probability mass function and the distribution of the test length, as well as the probability of acceptance or rejection of the unit are given. All the analyses are based on the finite Markov chain imbedding approach which avoids the complexities of the probability generating function approach and makes the results readily understood and easily extended to the non-i.i.d. cases. Furthermore, an optimal model for generalized start-up demonstration tests is proposed. Finally, a numerical example is presented to make our results more transparent, and it can demonstrate the advantages of the new tests.

Keywords

Start-up reliabilityOptimal modelFinite Markov chain imbedding approach

Abbreviations

CS

consecutive successes;

CSTF

consecutive successes total failures;

TSTF

total successes total failures;

CSCF

consecutive successes consecutive failures;

TSCF

total successes consecutive failures;

R1-CS/TS/R2-CF/TF

R1 runs of consecutive successes, total successes, R2 runs of consecutive failures, total failures;

R1

the number of non-overlapping successful runs required for acceptance;

kC

the number of consecutive successes in a successful run;

kT

the number of total successes required for acceptance;

R2

the number of non-overlapping failed runs required for rejection;

dC

the number of consecutive failures in a failed run;

dT

the number of total failures required for rejection;

Tl

the test length (e.g. the number of tests) until termination of the test;

α

the producer’ risk;

β

the consumer’ risk.

1 Introduction

For some special products, such as outboard motors, water pumps, engines, lawn mowers and car batteries etc., sometimes failed start-ups may happen. If these products can’t start up successfully, then their normal functions can not be performed, and as a result, the discussion about reliability is meaningless. Therefore the start-up reliability is a very important issue, and the research on start-up demonstration tests is necessary. By using start-up demonstration tests, on one hand, products with high start-up reliability can be selected; on the other hand, the start-up reliability of products can be estimated. A start-up demonstration test consists of two steps: the first step is attempting to start up a unit several times, and recording the outcomes (successful or failed start-up); the second step is making a decision on accepting or rejecting the unit according to some criteria. A start-up demonstration test is often named in accordance with the acceptance and rejection criteria.

The first start-up demonstration test was proposed by Hahn and Gage (1983), in which a unit is accepted if a pre-specified number of consecutive successful start-ups are observed in a series of attempted start-ups. This kind of start-up demonstration test was named CS (consecutive successes) start-up demonstration test. One obvious drawback of CS start-up demonstration test is that the test may not be terminated if the start-up reliability of the unit is very low, since there are not any rejection criteria. As a modification of CS start-up demonstration test, CSTF (consecutive successes total failures) start-up demonstration test was presented by Balakrishnan and Chan (2000), in which the test is terminated and the unit is rejected if total d failed start-ups are observed prior to consecutive k successful start-ups, or the test is terminated and the unit is accepted if consecutive k successful start-ups are observed prior to total d failed start-ups. After that, TSTF (total successes total failures), CSCF (consecutive successes consecutive failures) and TSCF (total successes consecutive failures) start-up demonstration tests were suggested by Smith and Griffith (2008). Their definitions are almost the same as CSTF start-up demonstration test except that the words “consecutive” or “total” need to be replaced appropriately. Smith and Griffith (2008) analyzed and compared CSTF, CSCF, TSCF, and TSTF start-up demonstration tests. The results show that none of them is always the best test, and it is impossible for any test to be preferred among all the others in any specified situations. In this paper, in order to assemble the advantages and dismiss the disadvantages of the above four tests, two generalized start-up demonstration tests are introduced. They are R1-CS/TS/R2-CF/TF start-up demonstration test and R1-CS/R2-CF start-up demonstration test. CSTF, TSCF and TSTF start-up demonstration tests are all special situations of R1-CS/TS/R2-CF/TF start-up demonstration test, and CSCF start-up demonstration test is a special situation of R1-CS/R2-CF start-up demonstration test. The motivation for introducing the new start-up demonstration tests lies in the following two points. (1) The new start-up demonstration tests are more flexible and precise for modeling the actual conditions than traditional tests. (2) The number of start-ups needed for terminating the test can be reduced by the new start-up demonstration tests, which can effectively decrease the test costs (the numerical example in Sect. 6 indicate that the test length expectation of R1-CS/TS/R2-CF/TF start-up demonstration test is obviously less than CSTF, TSCF and TSTF start-up demonstration tests subjecting to the same constraints on the two kinds of risks).

A literature review shows that all researches about CS, CSTF, CSCF, TSCF and TSTF start-up tests can be divided into two categories. One assumes the tests are independent and identically distributed. The other supposes the tests are first-order or higher-order Markovian. Otherwise, for probabilistic analysis, three kinds of methodologies are used, which are the probability generating function approach, recursive formulas approach and the finite Markov chain imbedding approach. By using the probability generating function approach, Viveros and Balakrishnan (1993) derived the mean and variance of CS test with i.i.d. start-ups; Balakrishnan et al. (1997) analyzed CS test with Markov dependence start-ups; Balakrishnan and Chan (2000) obtained the probability mass function, the mean and the conditional distribution of the test length of CSTF test with i.i.d. start-ups. By using the recursive formulas, Martin (2004) analyzed CSTF test with Markov dependent start-ups. By using the finite Markov chain imbedding approach, Smith and Griffith (2005, 2008), Martin (2008) derived the probabilistic results of CSTF, CSCF, TSCF and TSTF tests with i.i.d. and Markov dependent start-ups separately. All researches indicate that the finite Markov chain imbedding approach can not only avoid the complexity of the probability generating function, but also be extended to study the probabilistic analysis of Markov dependent start-up tests. The finite Markov chain imbedding approach is an efficient method for the studying of run related problems. It was first formally named by Fu and Koutras (1994). After that some investigations about the improving and application of this approach have been published, such as Balakrishnan and Koutras (2002), Fu and Lou (2003), Aston and Martin (2005), Zhao et al. (2007), Zhao and Cui (2009), and Cui et al. (2010) etc. In this paper, the finite Markov chain imbedding approach will be used to discuss some problems about the new start-up demonstration tests, including the acceptance or rejection probability, and other indexes which can describe the performance of start-up tests.

The rest of the paper is organized as follows. In Sect. 2, the definition of the generalized start-up demonstration tests is given, and the relationship between the traditional start-up demonstration tests and the generalized start-up demonstration tests is discussed. In Sect. 3, the generalized start-up demonstration tests are analyzed by using the finite Markov chain imbedding approach. In Sect. 4, a procedure of determining the parameters is proposed. In Sect. 5, all researches above are generalized to the non-i.i.d. cases. In Sect. 6, a numerical example is presented to illustrate the studies and demonstrate the advantages of the new tests. Finally, conclusions and future work are summarized in Sect. 7.

2 The generalized start-up demonstration tests

Let event I represents that R1 non-overlapping runs of kC consecutive successful start-ups are recorded; event II means that kT in total successful start-ups are recorded; event III stand for that R2 non-overlapping runs of dC consecutive failed start-ups are observed; event IV refers to that dT in total failed start-ups are observed. Definitions of two new start-up demonstration tests are as follows.

Definition 1

A start-up demonstration test will be terminated and the unit will be accepted if either event I or II is the first to occur and rejected if either event III or IV is the first to occur. This test is named R1-CS/TS/R2-CF/TF start-up demonstration test.

Definition 2

For a start-up demonstration test, if event I occurs before event III, the test will be terminated and the unit under the test will be accepted; if event III occurs before event I, the test will be terminated and the unit under the test will be rejected. This test is called R1-CS/R2-CF start-up demonstration test.

For example, for a R1-CS/TS/R2-CF/TF start-up demonstration test with R1=2, kC=4, kT=14, R2=1, dC=2, dT=5, successful and failed start-up is represented by ‘0’ and ‘1’ separately. If the test result sequence is ‘0000100010000’, the unit will be accepted, since event I occurs before event III and event IV; if the test result sequence is ‘1000110000101’, the unit will be rejected, as event IV occurs before event I and event II.

R1-CS/TS/R2-CF/TF and R1-CS /R2-CF start-up demonstration tests defined in Definition 1 and Definition 2 are both called the generalized start-up tests which will be studied later in this paper.

When the parameters of R1-CS/TS/R2-CF/TF test meet some rules, R1-CS/TS/R2-CF/TF test will degenerate to CSTF, TSTF or TSCF etc. For instance, if kT>(kC−1)(dT−1)+R1kC, R2dC>dT and R1=1, then R1-CS/TS/R2-CF/TF test will degenerate to CSTF test. All possible degenerating transitions from R1-CS/TS/R2-CF/TF test and the corresponding rules that the parameters should satisfy are as shown in Fig. 1. So CSTF test, TSTF test and TSCF test are all the special situations of R1-CS/TS/R2-CF/TF test. Otherwise, some demonstration tests in Fig. 1 just were published in 2010 (such as CSTSCF test Gera 2010), or have never been proposed (such as CSCFTF, TSTFCF, CSTSTF and CSTSCFTF tests). But all of them are special situations of R1-CS/TS/R2-CF/TF test. It makes R1-CS/TS/R2-CF/TF start-up demonstration test more flexible and suitable that R1-CS/TS/R2-CF/TF start-up demonstration test includes much more criteria for terminating a test than CSTF, TSTF and TSCF tests for the actual conditions.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Fig1_HTML.gif
Fig. 1

All possible degenerating transitions of the R1-CS/TS/R2-CF/TF and R1-CS/R2-CF tests

In Fig. 1, the rules from (1) to (10) and some relationships between them are as follows.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equa_HTML.gif
It must be noted that CSCF and R1-CS/R2-CF tests are different from others, which can be found from Fig. 1. They can not be obtained by the degeneration of R1-CS/TS/R2-CF/TF test. But R1-CS/R2-CF test can degenerate to CSCF test if R1=R2=1. So CSCF test is a special case of R1-CS/R2-CF test. In the next section, a detailed study on R1-CS/R2-CF test and R1-CS/TS/R2-CF/TF test will be conducted separately. Otherwise, it is worth mentioning that the test length may be unbounded for R1-CS/R2-CF and CSCF tests, but it is always finite for all the other tests.

3 Analysis of the generalized start-up demonstration tests

3.1 Markov chain presentation for the generalized start-up demonstration tests

In this section, we will assume that the start-ups are i.i.d. Bernoulli start-ups with constant probability of successful start-up p (p=1−q). In order to analyze R1-CS/TS/R2-CF/TF test by using finite Markov chain imbedding approach, a Markov chain {Yt} with the following state space Ω is defined. Yt represents the total situations of the first t start-up tests when the tth start-up test is just finished.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equb_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equc_HTML.gif
corresponds to the initial state that none of start-up tests has been done; (r1,sC,sT,r2,fC,fT)T represents the states that the start-up demonstration test has not been terminated; Ea represents the state that the start-up demonstration test has been terminated and the unit has been accepted; Er means the state that the start-up demonstration test has been terminated and the unit has been rejected. The detailed meaning of Ω is as follows.
$$Y_{t} = \left \{ \begin{array}{@{}l@{\quad }l} (0,0,0,0,0,0)^{T}, &\mbox{if the start-up test is in initial time and none of tests has}\\ &\mbox{been done;} \\ (r_{1},s_{C},s_{T},r_{2},f_{C},f_{T})^{T}, &\mbox{if the start-up test has just been done }t\mbox{ times, but do not}\\ &\mbox{meet the terminating criteria, and there are just }\\ &r_{1}\mbox{ non-overlapping runs of }k_{C} \mbox{ consecutive successful}\\ &\mbox{start-ups, the last }s_{C}\mbox{ start-ups are all successful,} \\ &\mbox{there are just }s_{T}\mbox{ in total successful start-ups, there are}\\ &\mbox{just }r_{1}\mbox{ non-overlapping runs of }d_{C}\mbox{ consecutive failed}\\ &\mbox{start-ups, the last }f_{C} \mbox{ start-ups are all failed, and there are}\\ &\mbox{just }f_{T}\mbox{ in total failed start-ups;} \\ E_{a}, &\mbox{if the start-up test has just been done }t\mbox{ times and one}\\ &\mbox{of the criteria for terminating the test and accepting}\\ &\mbox{the unit has been satisfied;} \\ E_{r}, &\mbox{if the start-up test has just been done }t\mbox{ times and one of}\\ &\mbox{the criteria for terminating the test and rejecting the unit}\\ &\mbox{has been satisfied}, \\ \end{array} \right . $$
where (0,0,0,0,0,0)T and (r1,sC,sT,r2,fC,fT)T are transient states, Ea and Er are absorbing states. The transient state set can be expressed as A={(0,0,0,0,0,0)T}∪{(r1,sC,sT,r2,fC,fT)T}. The setup of state space consists of two steps: (1) constructing all possible states, (2) constructing the state space by finding the states which are satisfied with the relationship among the parameters. So the computational complexity of setup of state space is O(R1kCkTR2dCdT). By using computer, this process can be finished in short time for the fixed parameter values, which can be found in Sect. 6.
For the Markov chain {Yt}, if the current state Yt−1 is a transient state, that is to say Yt−1=(r1(t−1),sC(t−1),sT(t−1),r2(t−1),fC(t−1),fT(t−1))T and (r1(t−1),sC(t−1),sT(t−1),r2(t−1),fC(t−1),fT(t−1))TA, then Yt=(r1(t),sC(t),sT(t),r2(t),fC(t),fT(t))T after the tth test, where (r1(t),sC(t),sT(t),r2(t),fC(t),fT(t))T may belong to A or {Ea} with probability p, and it may also belong to A or {Er} with probability q. According to the flow as shown in Fig. 2, the transition rules from Yt−1 to Yt can be set down. For example, for a R1-CS/TS/R2-CF/TF test with R1=3, kC=3, kT=10, R2=2, dC=2, dT=6, if Yt−1=(2,0,9,1,1,5)T, then Yt=Ea with probability p, and Yt=Er with probability q. If Yt−1=(2,0,8,1,0,3)T, then Yt=(2,1,9,1,0,3)T with probability p, and Yt=(2,0,8,1,1,4)T with probability q. According to the transition rules, the transition probability matrix of Markov chain {Yt} can be obtained easily.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Fig2_HTML.gif
Fig. 2

A flow for determining the transition rules of R1-CS/TS/R2-CF/TF test

All the above analyses about the construction of Markov chain and the definitions of the state transition rules are for R1-CS/TS/R2-CF/TF test. For all special cases of R1-CS/TS/R2-CF/TF test, the construction of Markov chain and the definitions of state transition rules are almost the same as R1-CS/TS/R2-CF/TF test except that some elements of the Markov chain state and related conditions need to be deleted. For example, in order to analyze a R1-CS/R2-CF/TF test by using finite Markov chain imbedding approach, a Markov chain {Yt} with the following state space Ω is defined.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Eque_HTML.gif
where the meanings of all notations in the above state space are the same as R1-CS/TS/R2-CF/TF test. It is obvious that the state space of Markov chain for R1-CS/R2-CF/TF test can be obtained only by deleting the third element of (0,0,0,0,0,0)T and (r1,sC,sT,r2,fC,fT)T which belong to the state space of Markov chain for R1-CS/TS/R2-CF/TF test and all conditions related to the third element. Otherwise, the state transition rules can also be gained according to the flow as shown in Fig. 2 by deleting all equations related to the third element.
It is worth mentioning that R1-CS/R2-CF test is not a special case of R1-CS/TS/R2-CF/TF test, but the construction of Markov chain and the definitions of the state transition rules are similar to the special cases of R1-CS/TS/R2-CF/TF test. For the sake of analyzing R1-CS/R2-CF test by using finite Markov chain imbedding approach, a Markov chain {Xt} with the following state space Φ is defined.
$$\varPhi = \left \{ \begin{pmatrix}0 \cr 0 \cr 0 \cr 0\end{pmatrix} \right \} \cup \left \{ \begin{array}{@{}l@{\quad }l} (r_{1},s_{C},r_{2},f_{C})^{T}{:}&0 \le r_{1} \le R_{1} - 1,0 \le s_{C} \le k_{C} - 1; \\ &0 \le r_{2} \le R_{2} - 1,0 \le f_{C} \le d_{C} - 1;\\ &s_{C}f_{C} = 0; \\ &s_{C} + f_{C} > 0,\mbox{when }r_{1} + r_{2} = 0; \\ &s_{C} + f_{C} \ge 0,\mbox{when }r_{1} + r_{2} > 0; \end{array} \right \} \cup \{ A_{1} \} \cup \{ A_{2} \} $$
where the meanings of all notations in the above state space are the same as R1-CS/TS/R2-CF/TF test. The state transition rules can be obtained according to a flow, which is same as the Fig. 2 except that all conditions related to the elements sT and fT are deleted.

3.2 Probabilistic analysis for the generalized start-up demonstration tests

In this section, by using the Markov chain theory, two kinds of general approaches for obtaining the acceptance/rejection probability, the probability mass function, the distribution function and the mean of the test length will be given. In Sect. 3.1, R1-CS/TS/R2-CF/TF test and its special cases, and R1-CS/R2-CF test are described by the Markov chain. Their state transition probability matrices can be obtained according to the flow for determining the transition rules. For the Markov chain which is used to describe the generalized start-up demonstration tests, the one-step transition probability matrix is
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equg_HTML.gif
where T(|Ω|−2)×(|Ω|−2) is a (|Ω|−2)×(|Ω|−2) matrix containing the one-step transition probabilities among the transient states, Q(|Ω|−2)×2 is a (|Ω|−2)×2 matrix including the one-step transition probabilities from the transient states to the absorbing states, I2×2 is a 2×2 identity matrix, and 02×(|Ω|−2) is a 2×(|Ω|−2) zero matrix. The first row of Λ contains the one-step transition probabilities from state (0,0,0,0,0,0)T to any state in state space Ω.
For a R1-CS/TS/R2-CF/TF test with i.i.d. Bernoulli start-ups which has constant probability of a successful start-up p and the first transient state (0,0,0,0,0,0)T, according to the finite Markov chain imbedding approach, π0=(1,0,…,0)1×|Ω| is the initial state distribution, U(Ea) and U(Er) are |Ω|×1 vectors, in which the elements corresponding to the absorbing states Ea and Er are 1s respectively and other elements are all 0s. U(Ea)+U(Er)=(0,0,…,0,1,1)T. The distribution function of test length Tl can be obtained by Eq. (1).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ1_HTML.gif
(1)
The exact probability mass function and the expected value of the test length Tl are given by Eqs. (2) and (3).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ3_HTML.gif
(3)
Otherwise, because the test length of R1-CS/TS/R2-CF/TF test must be finite if the parameters are fixed, so the expected value of test length can also be obtained by Eq. (3′).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figa_HTML.gif
where nmax=min{kT+dT−1,kC(dT+R1−1),dC(kT+R2−1)} is the maximum value of the test length of R1-CS/TS/R2-CF/TF test.
Equation (4) is the expression of the probability of acceptance and rejection.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ4_HTML.gif
(4)

Derivation processes of formulas (1)–(4) are described in Fu and Lou (2003). For all special cases of R1-CS/TS/R2-CF/TF test, the acceptance/rejection probability and other indexes related to the test length can be obtained by Eqs. (1)–(4) and (3′) once they are described by the corresponding Markov chains. Because R1-CS/R2-CF test is not a special case of R1-CS/TS/R2-CF/TF test and its test length may be unbounded (that is to say nmax=∞), the acceptance/rejection probability and the expected value of the test length can not be obtained by Eqs. (4) and (3′) respectively. But the probability mass function, the distribution function and the mean of the test length can be obtained by Eqs. (1)–(3) respectively, for those equations do not depend on the maximum value of the test length nmax.

For any of R1-CS/TS/R2-CF/TF test and its special cases, and R1-CS/R2-CF test with i.i.d. Bernoulli start-ups, which has a constant probability of successful start-up p and the first transient state (0,0,…,0)T, by using formulas in Bhat (1984), the acceptance/rejection probability and some indexes related to test length can be obtained by the following Eqs. (1″)–(4″), which have been proved by Smith and Griffith (2005). In the following Eqs. (1″)–(4″), 1a is a column vector of dimension a×1, where all elements are 1s.

The distribution function, the probability mass function and the expected value of the test length Tl can be obtained by Eqs. (1″)–(3″).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figb_HTML.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figc_HTML.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figd_HTML.gif
The probability of the rejection and acceptance can be obtained by Eq. (4″).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Fige_HTML.gif

4 Optimization for generalized start-up demonstration tests

One of the main goals of start-up demonstration tests is checking if the start-up reliability of the unit meets the pre-specified level. In order to achieve the above target efficiently, a start-up demonstration test must satisfy the following requirements.
  1. (a)

    The unit should be accepted with a high probability value, if the start-up reliability of the unit is greater than the lowest pre-specified acceptable level.

     
  2. (b)

    The unit should be accepted with a low probability value, if the start-up reliability of the unit is less than the greatest pre-specified unacceptable level.

     
The above two requirements can be expressed by Eqs. (5) and (6) respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ6_HTML.gif
(6)
where p is the real level of the start-up reliability, pA represents the lowest pre-specified acceptable level of the start-up reliability, pr is the greatest pre-specified unacceptable level (pA>pr), α and β are two kinds of risks. We hope that the unit is accepted if the start-up reliability ppA. But it is possible that the unit is rejected according to a start-up demonstration test plan. This probability is just the producer’s risk α. We hope that the unit is rejected if the start-up reliability p<pr. But it is possible that the unit is accepted according to a start-up demonstration test plan. This probability is just the consumer’s risk β. Applying Eqs. (4) and (4″), it is obvious to see that conditions (a) and (b) are only related to the parameters (R1, kC, kT, R2, dC and dT) of start-up demonstration tests besides the start-up reliability p. So in order to design a test to satisfy the conditions (a) and (b), choosing the suitable parameters is the only thing needed to do. Otherwise, for R1-CS/TS/R2-CF/TF test and its special cases, and R1-CS/R2-CF test, P(Accepted) is an increasing function in p and P(Rejected) is a decreasing function in p, so Eqs. (5) and (6) can be rewritten as Eqs. (5′) and (6′).
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figf_HTML.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Figg_HTML.gif
For specified values of α and β, maybe there are a lot of different combinations of parameters which meet both conditions (a) and (b). In this situation, the test length of a good test design should be as short as possible. In this paper, for pre-specified α and β, a combination of the parameters, which satisfies the conditions (a) and (b) and results in the shortest expected test length E(Tl), should be chosen as the best test design. So Eq. (7) is the optimization model for generalized start-up demonstration tests.
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ7_HTML.gif
(7)
In order to obtain the best test design for the pre-specified α and β, a procedure consisting of two steps is proposed as follows.
Step 1:

We should find all combinations of parameters which meet both conditions (a) and (b). This step is similar to the one used in Smith and Griffith (2005). The initial values of the six parameters are set to 1. We fix the values of the five parameters, and increase the rest parameter as long as equations (a) and (b) is not met during the process of finding parameter combinations.

Step 2:

We should compute the expected values of the test length in all possible combinations of the parameters by using formula (3″), and choose the test with the shortest test length as the best design.

5 Generalization to the non i.i.d. situations

All the above studies in this paper focus on the generalized start-up demonstration tests with i.i.d. Bernoulli start-ups. However, all analyses based on the finite Markov chain imbedding approach can be readily extended to the generalized start-up demonstration tests with independent and non-identically distributed, or higher-order (including first-order) Markov dependent start-ups.

For independent and non-identically distributed start-up tests, all analyses based on the finite Markov chain imbedding approach are the same as the i.i.d. case except the construction of transition matrix Λt of the Markov chain {Yt}. The construction of transition matrix Λt needs to apply the transition rules as shown in Fig. 2, but in which transition probabilities p and q need to be changed to pt and qt.

For higher-order (including first-order) Markov dependent start-up tests all analyses based on the finite Markov chain imbedding approach are also the same as the i.i.d. case except three points which are shown as follows.
  1. (a)

    The definition of the state space Ω of the Markov chain {Yt} for higher-order Markov dependent case is different from i.i.d. case. For m-order Markov dependent start-up tests, each transient state in the state space Ω needs to add a new element which is a m-dimension vector to record the last m test results. That is to say, (r1,sC,sT,r2,fC,fT)T is changed to (r1,sC,sT,r2,fC,fT,yt)T, where yt=(ytm+1,ytm+2,…,yt) is the added element, and any element yi in yt represents the result of the ith test from the bottom.

     
  2. (b)

    The Markov chain {Yt} is initialized at time m, when the first test results xm=(x1,x2,…,xm) is formed. There are 2m possible results for Xm=(X1,X2,…,Xm). The probability P(Xm=xm) of each possible results xm=(x1,x2,…,xm) can be obtained according to historical data. For t=1,2,…,m, r1t,sCt,sTt,r2t,fCt,fTt is used to describe the current state, which is determined by (x1,x2,…,xt). To obtained the initial state, we set r11=0, sC1=1−x1, sT1=1−x1, r21=0, fC1=x1, fT1=x1. For t=2,3,…,m, sCt=(sC(t−1)+1)(1−xt), sTt=sT(t−1)+1−xt, fCt=(fC(t−1)+1)xt, fTt=fT(t−1)+1−(1−xt). r1t=r1(t−1)+1 and sCt=0, if sCt=kC. r2t=r2(t−1)+1 and fCt=0, if fCt=dC. Now, for each xm=(x1,x2,…,xm) such that r1t<R1, sTt<kT, r2t<R2 and fTt<dT, t=1,2,…,m, a probability of an initial transient state π0(r1m,sCm,sTm,r2m,fCm,fTm,xm) can be obtained. Probabilities for initial absorbing states Ea and Er can be computed by summing over xm=(x1,x2,…,xm) corresponding to r1tR1 or sTtkT, and r2tR2 or fTt<dT (t=min(R1kC,R2dC,kT,dT),…,m). Based on above computing, initial probability π0 for Markov chain {Yt} are presented.

     
  3. (c)
    The state transition rules for the higher-order Markov dependent case are different from the i.i.d. case. For m-order Markov dependent start-up tests, in order to obtain the transition matrix, the last m test results must be considered. To illustrate the difference between the higher-order Markov dependent case and the i.i.d. case, R1-CS/R2-CF test with m-order Markov dependent Bernoulli start-ups is considered. Its state transition rules can be obtained according to the flow as shown in Fig. 3. In Fig. 3, yt−1=(ytm,ytm+2,…,yt−1) is the added element according to the point (a) which records the last m test results, vi=(v1i,v2i,…,vmi) (1≤i≤2m) is one of the possible results of consecutive m tests. The first step is to judge which possibility the test results yt−1 of the last m tests belong to. If yt−1=vi, that is to say, the results of the last m tests are just the ith possibility, then the next test result is successful and yt=(v2,v3,…,vm,1) with probability pi (as shown in Eq. (7)), the next test result is failed and yt=(v2,v3,…,vm,0) with probability qi (as shown in Eq. (8)).
    https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ8_HTML.gif
    (8)
    https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equ9_HTML.gif
    (9)
    https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Fig3_HTML.gif
    Fig. 3

    A flow for determining the transition rules of R1-CS/R2-CF test with m-order Markov dependent start-ups

     

Except for the above three points, the formulas for the expectation, probability mass function and the distribution of test length, the acceptance or rejection probability, and the procedures for the determination of parameters are all the same as the formulas and procedures given in the previous section.

6 Example

In this section, the design procedure and the related index computing of a R1-CS/TS/R2-CF/TF test based on the formulas and flows studied in the previous sections will be given in an example. It is worth mentioning that this example partly shows that R1-CS/TS/R2-CF/TF test is more flexible than other existing start-up demonstration tests.

A buyer wishes to establish a R1-CS/TS/R2-CF/TF test to provide an aid of the decision-making in purchasing for equipment, and then this test should satisfy the following two requirements.
  1. (a)

    If the start-up reliability of an equipment is greater or equal to 0.9, it should be accepted with probability at least 0.9, that is to say P(Accepted|p≥0.9)>0.9;

     
  2. (b)

    If the start-up reliability of an equipment is less than or equal to 0.7, it should be rejected with probability at least 0.7, that is to say P(Accepted|p≤0.7)<0.3.

     

By using formulas (4) or (4″), it can be found that there are many different combinations of the parameters R1,kC,kT,R2,dC,dT satisfying the above requirements (a) and (b) for a R1-CS/TS/R2-CF/TF test. By using the design procedure which is introduced in Sect. 4, it can be easily found that when R1=2, kC=4, kT=16, R2=1, dC=2, dT=5, R1-CS/TS/R2-CF/TF test satisfies requirements (a) and (b), and the expect value of test length is the smallest (E(Tl)=9.7384 is calculated with p=pA). So it is a good choice of the parameter values. For the fixed parameter values, the state space including 262 states is constructed by using the finite Markov chain imbedding approach introduced in Sect. 3. This construction process is carried out on a computer (2.93 GHz 2 Duo CPU, 2.00 GB memory, Window 2003) in Matlab 7.9.0., and the computing time is less than 0.0001 s.

For the same requirements, if the buyer wishes to establish a CSTF test, or a TSTF test, or a TSCF test to provide aid of the decision-making in purchasing for equipments, it can be seen that kC=9, dT=5 must be the best choice for CSTF test (E(Tl)=14.4478), kT=12, dT=4 must be the best choice for TSTF test (E(Tl)=13.1014), kT=44, dC=3 must be the best choice for TSCF test (E(Tl)=47.8045). Through the design results of the above four start-up demonstration tests with the same requirements, it can be known that the smallest expect value of the test length of R1-CS/TS/R2-CF/TF test is less than any of CSTF test, TSTF test and TSCF test (as show in Table 1). A decrease of 26% in the value of the expected test length is observed compared to the value of the expected test length using the best one among CSTF, TSTF and TSCF. So we can say that R1-CS/TS/R2-CF/TF test is more flexible and economical.
Table 1

The parameters satisfying P(Accepted|p≥0.9)>0.9 and P(Accepted|p≤0.7)<0.3 and the corresponding index values for R1-CS/TS/R2-CF/TF, TSTF, CSTF and TSCF tests

Test name

R1

kC

kT

R2

dC

dT

P(Accepted|p=pA)

P(Accepted|p=p0)

E(Tl)

R1-CS/TS/R2-CF/TF

2

4

16

1

2

5

0.9060

0.2995

9.7384

TSTF

  

12

  

4

0.9444

0.2969

13.1014

CSTF

 

9

   

5

0.9137

0.1861

14.4478

TSCF

  

44

 

3

 

0.9569

0.2999

47.8045

Otherwise, by using the formulas which is introduced in Sect. 3, the distribution of the test length for R1-CS/TS/R2-CF/TF test with R1=2, kC=4, kT=16, R2=1, dC=2, dT=5 can be obtained (as show in Fig. 4). We know that the maximum value of the test length of this test is
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Equh_HTML.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs10479-012-1279-y/MediaObjects/10479_2012_1279_Fig4_HTML.gif
Fig. 4

The probability distribution of the test length for R1-CS/TS/R2-CF/TF test with R1=2, kC=4, kT=16, R2=1, dC=2, dT=5

This result can also be found from the Fig. 4, because it is obvious that the probability that the test length is greater than 20 is zero.

7 Summary

In this paper, two generalized start-up demonstration tests are proposed, which are the extensions of the existing CSTF, TSCF, TSTF and CSCF tests. Any known test is a special case of the new start-up demonstration tests, so the new start-up demonstration tests are more generalized. By using the finite Markov chain imbedding approach, the acceptance and rejection probabilities, the probability mass function, the distribution function, the mean of the test length of the generalized start-up demonstration tests have been derived. This method can be generalized to the special cases of R1-CS/TS/R2-CF/TF test and the non-i.i.d. case easily. An example is presented to illustrate the design procedure of a start-up demonstration test and the usage of the formulas in the previous sections. Otherwise, the example partly shows that R1-CS/TS/R2-CF/TF test is more flexible than known start-up demonstration tests.

All studies in this paper are based on the assumption that the probability of a successful start-up is a known constant p. Making an inference on the probability of a successful start-up p based on the start-up demonstration test results will be considered in the future.

Copyright information

© Springer Science+Business Media New York 2012