On generalized start-up demonstration tests
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DOI: 10.1007/s10479-012-1279-y
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- Zhao, X. Ann Oper Res (2014) 212: 225. doi:10.1007/s10479-012-1279-y
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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 approachAbbreviations
- CS
consecutive successes;
- CSTF
consecutive successes total failures;
- TSTF
total successes total failures;
- CSCF
consecutive successes consecutive failures;
- TSCF
total successes consecutive failures;
- R_{1}-CS/TS/R_{2}-CF/TF
R_{1} runs of consecutive successes, total successes, R_{2} runs of consecutive failures, total failures;
- R_{1}
the number of non-overlapping successful runs required for acceptance;
- k_{C}
the number of consecutive successes in a successful run;
- k_{T}
the number of total successes required for acceptance;
- R_{2}
the number of non-overlapping failed runs required for rejection;
- d_{C}
the number of consecutive failures in a failed run;
- d_{T}
the number of total failures required for rejection;
- T_{l}
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 R_{1}-CS/TS/R_{2}-CF/TF start-up demonstration test and R_{1}-CS/R_{2}-CF start-up demonstration test. CSTF, TSCF and TSTF start-up demonstration tests are all special situations of R_{1}-CS/TS/R_{2}-CF/TF start-up demonstration test, and CSCF start-up demonstration test is a special situation of R_{1}-CS/R_{2}-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 R_{1}-CS/TS/R_{2}-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 R_{1} non-overlapping runs of k_{C} consecutive successful start-ups are recorded; event II means that k_{T} in total successful start-ups are recorded; event III stand for that R_{2} non-overlapping runs of d_{C} consecutive failed start-ups are observed; event IV refers to that d_{T} 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 R_{1}-CS/TS/R_{2}-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 R_{1}-CS/R_{2}-CF start-up demonstration test.
For example, for a R_{1}-CS/TS/R_{2}-CF/TF start-up demonstration test with R_{1}=2, k_{C}=4, k_{T}=14, R_{2}=1, d_{C}=2, d_{T}=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.
R_{1}-CS/TS/R_{2}-CF/TF and R_{1}-CS /R_{2}-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.
3 Analysis of the generalized start-up demonstration tests
3.1 Markov chain presentation for the generalized start-up demonstration tests
3.2 Probabilistic analysis for the generalized start-up demonstration tests
Derivation processes of formulas (1)–(4) are described in Fu and Lou (2003). For all special cases of R_{1}-CS/TS/R_{2}-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 R_{1}-CS/R_{2}-CF test is not a special case of R_{1}-CS/TS/R_{2}-CF/TF test and its test length may be unbounded (that is to say n_{max}=∞), 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 n_{max}.
For any of R_{1}-CS/TS/R_{2}-CF/TF test and its special cases, and R_{1}-CS/R_{2}-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″), 1_{a} is a column vector of dimension a×1, where all elements are 1s.
4 Optimization for generalized start-up demonstration tests
- (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.
- (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.
- 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 Λ_{t} of the Markov chain {Y_{t}}. The construction of transition matrix Λ_{t} needs to apply the transition rules as shown in Fig. 2, but in which transition probabilities p and q need to be changed to p_{t} and q_{t}.
- (a)
The definition of the state space Ω of the Markov chain {Y_{t}} 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, (r_{1},s_{C},s_{T},r_{2},f_{C},f_{T})^{T} is changed to (r_{1},s_{C},s_{T},r_{2},f_{C},f_{T},y_{t})^{T}, where y_{t}=(y_{t−m+1},y_{t−m+2},…,y_{t}) is the added element, and any element y_{i} in y_{t} represents the result of the ith test from the bottom.
- (b)
The Markov chain {Y_{t}} is initialized at time m, when the first test results x_{m}=(x_{1},x_{2},…,x_{m}) is formed. There are 2^{m} possible results for X_{m}=(X_{1},X_{2},…,X_{m}). The probability P(X_{m}=x_{m}) of each possible results x_{m}=(x_{1},x_{2},…,x_{m}) can be obtained according to historical data. For t=1,2,…,m, r_{1t},s_{Ct},s_{Tt},r_{2t},f_{Ct},f_{Tt} is used to describe the current state, which is determined by (x_{1},x_{2},…,x_{t}). To obtained the initial state, we set r_{11}=0, s_{C1}=1−x_{1}, s_{T1}=1−x_{1}, r_{21}=0, f_{C1}=x_{1}, f_{T1}=x_{1}. For t=2,3,…,m, s_{Ct}=(s_{C(t−1)}+1)(1−x_{t}), s_{Tt}=s_{T(t−1)}+1−x_{t}, f_{Ct}=(f_{C(t−1)}+1)x_{t}, f_{Tt}=f_{T(t−1)}+1−(1−x_{t}). r_{1t}=r_{1(t−1)}+1 and s_{Ct}=0, if s_{Ct}=k_{C}. r_{2t}=r_{2(t−1)}+1 and f_{Ct}=0, if f_{Ct}=d_{C}. Now, for each x_{m}=(x_{1},x_{2},…,x_{m}) such that r_{1t}<R_{1}, s_{Tt}<k_{T}, r_{2t}<R_{2} and f_{Tt}<d_{T}, t=1,2,…,m, a probability of an initial transient state π_{0}(r_{1m},s_{Cm},s_{Tm},r_{2m},f_{Cm},f_{Tm},x_{m}) can be obtained. Probabilities for initial absorbing states E_{a} and E_{r} can be computed by summing over x_{m}=(x_{1},x_{2},…,x_{m}) corresponding to r_{1t}≥R_{1} or s_{Tt}≥k_{T}, and r_{2t}≥R_{2} or f_{Tt}<d_{T} (t=min(R_{1}k_{C},R_{2}d_{C},k_{T},d_{T}),…,m). Based on above computing, initial probability π_{0} for Markov chain {Y_{t}} are presented.
- (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, R_{1}-CS/R_{2}-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, y_{t−1}=(y_{t−m},y_{t−m+2},…,y_{t−1}) is the added element according to the point (a) which records the last m test results, v_{i}=(v_{1i},v_{2i},…,v_{mi}) (1≤i≤2^{m}) is one of the possible results of consecutive m tests. The first step is to judge which possibility the test results y_{t−1} of the last m tests belong to. If y_{t−1}=v_{i}, that is to say, the results of the last m tests are just the ith possibility, then the next test result is successful and y_{t}=(v_{2},v_{3},…,v_{m},1) with probability p_{i} (as shown in Eq. (7)), the next test result is failed and y_{t}=(v_{2},v_{3},…,v_{m},0) with probability q_{i} (as shown in Eq. (8)).
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 R_{1}-CS/TS/R_{2}-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 R_{1}-CS/TS/R_{2}-CF/TF test is more flexible than other existing start-up demonstration tests.
- (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;
- (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 R_{1},k_{C},k_{T},R_{2},d_{C},d_{T} satisfying the above requirements (a) and (b) for a R_{1}-CS/TS/R_{2}-CF/TF test. By using the design procedure which is introduced in Sect. 4, it can be easily found that when R_{1}=2, k_{C}=4, k_{T}=16, R_{2}=1, d_{C}=2, d_{T}=5, R_{1}-CS/TS/R_{2}-CF/TF test satisfies requirements (a) and (b), and the expect value of test length is the smallest (E(T_{l})=9.7384 is calculated with p=p_{A}). 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.
The parameters satisfying P(Accepted|p≥0.9)>0.9 and P(Accepted|p≤0.7)<0.3 and the corresponding index values for R_{1}-CS/TS/R_{2}-CF/TF, TSTF, CSTF and TSCF tests
Test name | R_{1} | k_{C} | k_{T} | R_{2} | d_{C} | d_{T} | P(Accepted|p=p_{A}) | P(Accepted|p=p_{0}) | E(T_{l}) |
---|---|---|---|---|---|---|---|---|---|
R_{1}-CS/TS/R_{2}-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 |
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 R_{1}-CS/TS/R_{2}-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 R_{1}-CS/TS/R_{2}-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.