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Confidence limits for steady state availability of systems with lognormal operating time and inverse Gaussian repair time
Microelectron. Reliab., Vol. 37, No. 6, pp. 969-971, 1997
~)
Pergamon PII:S0026-2714(96)00116-3
© 1997 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0026-2714/97 $17.00+.00
RESEARCH N O T E C O N F I D E N C E LIMITS FOR STEADY STATE AVAILABILITY OF SYSTEMS WITH L O G N O R M A L O P E R A T I N G TIME A N D INVERSE G A U S S I A N REPAIR TIME P. C H A N D R A S E K H A R and R. N A T A R A J A N t Department of Statistics, Loyola College, Madras 600 034, India t Department of Mathematics, Presidency College, Madras 600 005, India (Received Jbr publication 28 May 1996)
Abstract A 100 p% confidence interval for the steady state availability of a system is derived, when the operating time distribution is lognormal and the repair time distribution is Inverse Gaussian (IG). It is assumed that one of the parameters of Iognormal distribution and also the ratio of parameters of IG distribution are known. © 1997 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION It is well known that the steady state availability is a satisfactory measure for systems which are to be operated continuously (e.g. a detection radar system). A point estimate of steady state availability is usually the only statistic calculated, although decisions about the true steady state availability of the system should take uncertainty into account. Since A~ = M T B F / ( M T B F + MTTR), the uncertainties in the values of M T B F and M T T R reflect an uncertainty in the value of the point steady state availability. By treating these uncertain parameters as random variables, we can obtain the distribution of point steady state availability by using the distribution of operation and repair times. Hence, we can construct estimators and confidence intervals for A:~. Table 1 indicates the details of earlier work in this direction on systems with several operating time and repair time distributions. The main purpose of this paper is to construct a 100 p°/o confidence interval for A~, when the
operating time distribution is lognormal and repair time distribution is IG. It is assumed that one of the parameters of lognormal distribution and also the ratio of parameters of I G distribution are known. The assumptions and notation are discussed in detail in the following section.
2. ASSUMPTIONS AND NOTATION (i)
The operating time X has a lognormal distribution with probability density function (pdf) given by 1 x/~/~x
_ f(x)=
exp 0
0,
1 log x - ~ 2 ~\~ ,
)]
< oo, c t e ~ , f l > 0
(1)
otherwise
and is denoted by A (~, f12). Note that,
Table 1 Author(s)
System
Operating time distribution
Repair time distribution
Chandrasekhar et al. [8]
One unit system One unit system One unit system One unit system Parallel system One unit system Standby system One unit system
Cbandrasekhar and Natarajan [9]
Parallel system
exponential exponential exponential Gamma exponential Weibull exponential lognormal Inverse Gaussian exponential
exponential lognormal general lognormal Erlang lognormal Erlang lognormal lognormal Erlang
Thompson [-1] Gray and Lewis [2] Butterworth and Nikolaisen [3] Masters and Lewis [4] Mohammad Abu-Salih et al. [5] Masters et al. [6] Chandrasekhar and Natarajan [7]
969
970
Research Note
(ii) There is only one repair facility and the repair time Y has an IG distribution with a pdfgiven by
where
~mn2 2nflp
C--
exp[
2(Y - #)2],
L -:z#:~ j 0
~,2,/~ > 0
(2)
and
otherwise
\0,
Pr(a < W < b) = c
and is denoted by IG(2, p). It may be noted that e(r3 = ~. (iii) The parameter fl of the lognormal distribution and only the ratio 2/p of the IG distribution are known.
3. CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM Let X1, X2,..., X= be a random sample of times to operate with the pdf as in eqn (1). It can be shown that
wz~z l~m(logwz) 2
x exp - ~ (.
f12
(7) Since the distribution of W is completely specified, given p(0 < p < 1), we can determine a and b such that Pr[a < W < b] = p. To obtain the exact confidence interval for A ~, let G 2 " W = e ~ #2jE.=, YJ
E(y) exp{l°g(Gp)+fl2} e•
e•
E(X)
where G is the sample geometric mean of time to operate. Let ?'I, Y2. . . . . Y, be another random sample of times to repair with a pdfas in eqn (2), It can be shown that
~ j~I Yj~ IG \ p: '
Let the random variables U and V be distributed according to and
'
where Y is the sample mean of time to repair. Hence, we have, Pr[a< W
(4)
•
exp{log(2n ~}
< E(X)
~Pr
I1
exp{log(2nY)}
-----
1 +
l] B2
=P,
< b
= p
(8)
where
//n2~.2 IG[, p2 ,
, b exp{log(2n ~} n I
respectively. If U and V are independent, then the joint density of (U, V) is given by
f(u, v) -
~mn2
[
and
l~m(logu) z
(2nflFOuvx/v exp - } [
exp{log(G/~) + ~ }
a exp{log(2n ~}
f12 n 2
+
V--
,
0
~.
(5)
By making transformations W = U/V and Z = V, it can be shown that the pdf of W is
h(w) = c f o
1
wz,/z +
z-
exp{iog(G#) + ~ t " Clearly, eqn (8) is an exact confidence interval for A~, if the parameter fl for lognormal distribution and only the ratio 2/p of IG distribution are known.
expV_ 1 ~m(logwz) 2
L
REFERENCES
t dz,
0
(6)
1. Thompson, M., Lower confidence limits and a test of hypotheses for system availability. IEEE Trans. Reliab., 1966, R-15, 32-36.
Research Note 2. Gray, H. L. and Lewis. T. O., A confidence interval for the availability ratio. Tech., 1967, 9, 465. 3. Butterworth, R. W. and Nikolaisen, T., Bounds on the availability function. Naval Research Logistics Quarterly, 1973, 20, 289-296. 4. Masters, B. N. and Lewis, T. O., A note on the confidence interval for the availability ratio. Microelectron. Reliab., 1987, 27(3), 487-492. 5. Abu-Salih, M., Anakerh, N. N. and Salahuddin Ahmed, M., Confidence limits for steady state availability. Pakistan J. Star., 1990, 6(2)A, 189-196. 6. Masters, B. N., Lewis, T. O. and Kolarik, W. J., A
971
confidence interval for the availability ratio for systems with Weibull operating time and lognormal repair time. Microelectron. Reliab., 1992, 32, 89-99. 7. Chandrasekhar, P and Natarajan, R., Confidence limits for steady state availability of a two unit standby system. Microelectron. Reliab., 1994, 34(7), 1249-1251. 8. Chandrasekhar, P., Natarajan, R., and Sheryl Sujatha, H., Confidence limits for steady state availability of systems. Microelectron. Reliab., 1994, 34(8), 1365-1367. 9. Chandrasekhar, P. and Natarajan, R., Confidence limits for steady state availability of a parallel system. Microelectron. Reliab., 1994, 34(11), 1847-1851.
~)
Pergamon PII:S0026-2714(96)00116-3
© 1997 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0026-2714/97 $17.00+.00
RESEARCH N O T E C O N F I D E N C E LIMITS FOR STEADY STATE AVAILABILITY OF SYSTEMS WITH L O G N O R M A L O P E R A T I N G TIME A N D INVERSE G A U S S I A N REPAIR TIME P. C H A N D R A S E K H A R and R. N A T A R A J A N t Department of Statistics, Loyola College, Madras 600 034, India t Department of Mathematics, Presidency College, Madras 600 005, India (Received Jbr publication 28 May 1996)
Abstract A 100 p% confidence interval for the steady state availability of a system is derived, when the operating time distribution is lognormal and the repair time distribution is Inverse Gaussian (IG). It is assumed that one of the parameters of Iognormal distribution and also the ratio of parameters of IG distribution are known. © 1997 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION It is well known that the steady state availability is a satisfactory measure for systems which are to be operated continuously (e.g. a detection radar system). A point estimate of steady state availability is usually the only statistic calculated, although decisions about the true steady state availability of the system should take uncertainty into account. Since A~ = M T B F / ( M T B F + MTTR), the uncertainties in the values of M T B F and M T T R reflect an uncertainty in the value of the point steady state availability. By treating these uncertain parameters as random variables, we can obtain the distribution of point steady state availability by using the distribution of operation and repair times. Hence, we can construct estimators and confidence intervals for A:~. Table 1 indicates the details of earlier work in this direction on systems with several operating time and repair time distributions. The main purpose of this paper is to construct a 100 p°/o confidence interval for A~, when the
operating time distribution is lognormal and repair time distribution is IG. It is assumed that one of the parameters of lognormal distribution and also the ratio of parameters of I G distribution are known. The assumptions and notation are discussed in detail in the following section.
2. ASSUMPTIONS AND NOTATION (i)
The operating time X has a lognormal distribution with probability density function (pdf) given by 1 x/~/~x
_ f(x)=
exp 0
0,
1 log x - ~ 2 ~\~ ,
)]
< oo, c t e ~ , f l > 0
(1)
otherwise
and is denoted by A (~, f12). Note that,
Table 1 Author(s)
System
Operating time distribution
Repair time distribution
Chandrasekhar et al. [8]
One unit system One unit system One unit system One unit system Parallel system One unit system Standby system One unit system
Cbandrasekhar and Natarajan [9]
Parallel system
exponential exponential exponential Gamma exponential Weibull exponential lognormal Inverse Gaussian exponential
exponential lognormal general lognormal Erlang lognormal Erlang lognormal lognormal Erlang
Thompson [-1] Gray and Lewis [2] Butterworth and Nikolaisen [3] Masters and Lewis [4] Mohammad Abu-Salih et al. [5] Masters et al. [6] Chandrasekhar and Natarajan [7]
969
970
Research Note
(ii) There is only one repair facility and the repair time Y has an IG distribution with a pdfgiven by
where
~mn2 2nflp
C--
exp[
2(Y - #)2],
L -:z#:~ j 0
~,2,/~ > 0
(2)
and
otherwise
\0,
Pr(a < W < b) = c
and is denoted by IG(2, p). It may be noted that e(r3 = ~. (iii) The parameter fl of the lognormal distribution and only the ratio 2/p of the IG distribution are known.
3. CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM Let X1, X2,..., X= be a random sample of times to operate with the pdf as in eqn (1). It can be shown that
wz~z l~m(logwz) 2
x exp - ~ (.
f12
(7) Since the distribution of W is completely specified, given p(0 < p < 1), we can determine a and b such that Pr[a < W < b] = p. To obtain the exact confidence interval for A ~, let G 2 " W = e ~ #2jE.=, YJ
E(y) exp{l°g(Gp)+fl2} e•
e•
E(X)
where G is the sample geometric mean of time to operate. Let ?'I, Y2. . . . . Y, be another random sample of times to repair with a pdfas in eqn (2), It can be shown that
~ j~I Yj~ IG \ p: '
Let the random variables U and V be distributed according to and
'
where Y is the sample mean of time to repair. Hence, we have, Pr[a< W
(4)
•
exp{log(2n ~}
< E(X)
~Pr
I1
exp{log(2nY)}
-----
1 +
l] B2
=P,
< b
= p
(8)
where
//n2~.2 IG[, p2 ,
, b exp{log(2n ~} n I
respectively. If U and V are independent, then the joint density of (U, V) is given by
f(u, v) -
~mn2
[
and
l~m(logu) z
(2nflFOuvx/v exp - } [
exp{log(G/~) + ~ }
a exp{log(2n ~}
f12 n 2
+
V--
,
0
~.
(5)
By making transformations W = U/V and Z = V, it can be shown that the pdf of W is
h(w) = c f o
1
wz,/z +
z-
exp{iog(G#) + ~ t " Clearly, eqn (8) is an exact confidence interval for A~, if the parameter fl for lognormal distribution and only the ratio 2/p of IG distribution are known.
expV_ 1 ~m(logwz) 2
L
REFERENCES
t dz,
0
(6)
1. Thompson, M., Lower confidence limits and a test of hypotheses for system availability. IEEE Trans. Reliab., 1966, R-15, 32-36.
Research Note 2. Gray, H. L. and Lewis. T. O., A confidence interval for the availability ratio. Tech., 1967, 9, 465. 3. Butterworth, R. W. and Nikolaisen, T., Bounds on the availability function. Naval Research Logistics Quarterly, 1973, 20, 289-296. 4. Masters, B. N. and Lewis, T. O., A note on the confidence interval for the availability ratio. Microelectron. Reliab., 1987, 27(3), 487-492. 5. Abu-Salih, M., Anakerh, N. N. and Salahuddin Ahmed, M., Confidence limits for steady state availability. Pakistan J. Star., 1990, 6(2)A, 189-196. 6. Masters, B. N., Lewis, T. O. and Kolarik, W. J., A
971
confidence interval for the availability ratio for systems with Weibull operating time and lognormal repair time. Microelectron. Reliab., 1992, 32, 89-99. 7. Chandrasekhar, P and Natarajan, R., Confidence limits for steady state availability of a two unit standby system. Microelectron. Reliab., 1994, 34(7), 1249-1251. 8. Chandrasekhar, P., Natarajan, R., and Sheryl Sujatha, H., Confidence limits for steady state availability of systems. Microelectron. Reliab., 1994, 34(8), 1365-1367. 9. Chandrasekhar, P. and Natarajan, R., Confidence limits for steady state availability of a parallel system. Microelectron. Reliab., 1994, 34(11), 1847-1851.