Reliability Engineering and System Safety 39 (1993) 109-111

Technical note

Estimating expected waiting-time in Weibull SPRT

K. K. Sharma, R. S. Rana & R. K. Bhutani Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut-250004, India

(Received 5 December 1991; accepted 29 February 1992)

The present paper deals with the development of a sequential test plan when lifetime has a Weibull distribution with a known shape parameter. Formulae for the OC function, the ASN function and the EWT function are also given and analysed.

1 INTRODUCTION

Many life-testing studies 1-3 have assumed that the distribution of lifetime is exponential. Epstein and Sobel I were concerned with fixed-length test plans for an exponential distribution. Sequential life-test plans in the corresponding situation are given in Ref. 3. The study in Ref. 3 developed sequential probability ratio test (SPRT) for testing mean time between failures (MTBF) and gave the formulae for the operating characteristic (OC), the expected number of failures (ASN) and the expected waiting time (EWT) before reaching a decision. A very useful expression giving the relationship between ASN and EWT was also obtained. However, in real situations, an exponential distribution seems unfair in view of its assumption of a constant failure rate (CFR) pattern as increasing and decreasing failure rates (IFR and DFR) are generally encountered in practice, e.g. in the lifetimes of electronic and mechanical equipment. One such commonly-known distribution is the Weibull distribu- tion which can define IFR(DFR) with its shape parameter other than one. Later, Harter and Moore 4

Reliability Engineering and System Safety 0951-8320/92/$05.00 (~) 1992 Elsevier Science Publishers Ltd, England.

109

developed SPRT for the Weibull distribution and studied some of its properties in respect to MIL-STD-781B plans. Using a simulation study, they obtained the ASN and EWT without considering necessary expressions for these characteristics.

In the light of the above developments, the present article deals with SPRT procedures for testing the scale parameter of a Weibull distribution with a known shape parameter. The present test, with a simple formulation, can be used for testing reliability of the device for a mission time. Formulae giving the OC function, the ASN function, the EWT function and their inter-relationship for such SPRT procedures have been developed and analysed.

2 DEVELOPMENT OF WEIBULL SPLIT

Let us assume that the lifetime of a system or device say X, follows Weibull density with probability density function (p.d.f.)

k exp[ x ~

where 0 > 0 and k > 0 are the scale and shape parameters respectively. For values of k > 1, k < 1

110 K. K. Sharma, R. S. Rana, R. K. Bhutani

and k = 1, the model in eqn (1) respectively defines IFR, D F R and CFR. Also, the rth raw moment for this distribution is

r r E ( X O = O F(-~+ I) (2)

With k assumed known, Ref. 5 can be used for developing sequential testing of the hypothesis H o : 0 = 00 versus 1-11:0= 01(<0o) with pre-assigned producer's risk, cr, the probability of rejection given that Ho is true, and consumer's risk, fl, the probability of acceptance given that/-/1 is true.

To initiate our test plan, it is assumed that n items are put on life-test and Xl < x z < ' " <Xr represent r observed times to failure up to time t. Here, n is assumed to be sufficiently large so that the probability of reaching no decision by time x , is negligible. Now, if we use the transformation Y = X k, the p.d.f, in eqn (1) becomes

1

which is an exponential distribution with mean O h and variance 02k. Since k is known, y l < y 2 < . . . < y r become the observed lifetimes of the r failed items before time t '(=t k) in the transformed form. With this information and following the developments in Ref. 3, testing continues until

B < (0o /01 ) kr exp[ - (01 k - Ook)T(t')] < A (4)

where B = fl/(1 - tr), A -- (1 - fl)lo: and T(t') can be interpreted as total life observed up to time t'. If the failed items are not replaced, then

3 OC AND ASN FUNCTIONS FOR THE TEST

The approximate formula for the OC function, L(O), the probability of accepting Ho when 0 is true, of the test can be obtained by considering the expression [ f (y ; 01, k ) / f ( y ; 0o, k)] h where h = h(O):/:0 satisfies

E[f(y; 01, k ) / f (y ; 0o, k)] h = 1 (8)

Then, approximations to the OC curve is given by the following pair of equations:

[(0o/01) kh - 1] 1/k A h - 1 O=lh-~Z-Oor) ] , L ( O ) = A h _ B h (9)

A plot of L(O) for variations in 0 gives the OC curve. Moreover, for k = 1, eqn (9) resembles eqn (7) of Ref. 3.

Similarly, an approximation to the ASN function, Eo(r), the expected number of failures required to reach a decision, is obtained as

Eo(r) = t ( 0 ) l o g B + [1 - t ( 0 ) ] l og A, Eo(Z)

Here,

Eo(Z), o

(10)

[ l o g f ( y ; 01, k) eo(z)=e°L Oo,

= k log(00/01) - Ok(O1 * - Oo k) (11)

A plot of Eo(r) values for 0 gives the ASN curve. Again, for k = 1, eqn (10) reduces to eqn (10) of Ref. 3.

T(t') = i

(n - i + 1)(yi - Yi-0 + (n - r)(t' - Yr) (5) 4 RELATIONSHIP BETWEEN Eo(r) AND Eo(t')

However, if the failed items are replaced immediately, then

T(t') = nt' (6)

The test is terminated in favour of Ho(H1) with the violation of the left(right) inequality in eqn (4). For k = 1, the inequality in eqn (4) reduces to expression (2) in Ref. 3.

Remark: In the case of Weibull life model in eqn (1), the probability of a system or device surviving for at least time to (mission time), i.e. R(to) is given by

Ro(to) = exp[ - ( ~ ) k] (7)

Then the problem of testing H o : 0 = 0o versus Hi:0 = 01(<00) is equivalent to testing Ho: 0 = 0o = to/[log(1/Roo(to))] Ilk against Hi : 0 = 01 = to/[ log(1/ Ro,( to) ) ]l/* ( < 0o).

Since our transformed variable y is exponential with p.d.f, in eqn (3), obviously, if we put V(t) -- T(t ' ) and 0 = 0 k in eqns (9), (12) and (20) of Ref. 3 (pp. 84-86), we get the following results

Eo[ T(t')] = OkEo(r) (12)

This gives a relation between T(t') and r. To obtain the relationship between Eo(r) and Eo(t') (the expected waiting time required to reach a decision on the transformed scale) in the replacement case, we use eqn (6) in eqn (12) to get

OkEo(r) Eo(t') = ~ (13)

n

Similarly, following eqn (20) of Ref. 3, in the non-replacement case, we have

Eo(t') = ~ Eo(ys)P(r = s/O) (14) s

Estimating expected waiting-time in Weibull S P R T 111

where

r

= Ok ~, 1 Eo(yr) • n - i + l

and the probability distribution of r, P(r =s /O) has been developed in Ref. 2.

5 BOUNDS FOR E o ( t )

Clearly, in replacement and non-replacement cases, Eo(t') or Eo(t k) can be obtained from eqns (13) and (14) respectively. However, we are interested in getting Eo(t) for which a relation between Eo(t') and Eo(t) is needed. While no such exact relationship exists, Jenson's inequality can still be used to get bounds for Eo(t). Taking E o ( t ' ) = # and using Jenson's inequality for the convex function g( t )= t k, let us consider the following two cases

(i) when k > 1

Eo(t) <- [Eo(tk)] ~/* = [Eo(t')] l'k =/~l/k (15)

(ii) when k < 1

In this case, let us consider a = k - 1 > 1 . Therefore

Eo(t') <--- [Eo(t'~)] TM = [Eo((tk)l/*)] k

= [Eo(t)]* ::> [l,l 1/k ~ Eo(t) (16)

Thus for a Weibull distribution with IFR, eqn (15) gives an upper bound for Eo(t) in terms of Eo(t'). Similarly, for a Weibull distribution with DFR, a lower bound for Eo(t) in terms of Eo(t') is given by eqn (16).

REFERENCES

1. Epstein, B. & Sobel, M., Life testing. J. Amer. Stat. Assoc., 48 (1953) 486-502.

2. Epstein, B., Truncated life-tests in the exponential case. Ann. Math. Stat., 2,5 (1954) 555-64.

3. Epstein, B. & Sobel, M., Sequential life-tests in the exponential case. Ann. Math. Star., 26 (1955) 82-93.

4. Harter, L. & Moore, A. H., An evaluation of exponential and Weibull test plans. IEEE Trans. on Reliab., 25 (1976) 100-4.

5. Wald, A., Sequential Analysis, J. Wiley and Sons, New York, 1947.