ON THE COMPUTATION OF LAPLACE-STIELJES TRANSFORM FOR LOGNORMAL AND WEIBULL

In this paper, we perform the computation of the Laplace-Stieljes transform of Lognormal and Weibull distribution, in which the upper limit of the definite integral from infinite to 1 by using proper transformation. Some advantages are found in this study. Key WordsLaplace-Stieljes transform; Lognormal; Weibull

criterion with three different repair time distributions which are exponential, 3-Erlang, and deterministic.Nevertheless, the Lognormal and Weibull distributions are not considered in this paper due to the difficulty in integral computation of their LST.
The aim of this paper is that an efficient reciprocal transformation is tried to improve the computation time spent for the integral computation of their LST.Firstly, we derive a new Laplace-Stieljes transform as definite integral from zero to one.Secondly, we discuss the availability of these three models by Wang and Chiu [3] when the repair time is Lognormal or Weibull distribution.Finally, we make some comparisons on the computation time spent between the integral formula from zero to infinity and reciprocal transformation developed in this paper (by MAPLE software).

TRANSFORMATION COMPUTATION OF LST
The LST of X is as The computation listed above, the upper limit of the definite integral is infinite which it is cumbersome if one doesn't adopt by mathematical software.Making the reciprocal transformation of 1 / (1 ) y x   , the LST can be rewritten as (i) If the random variable X exponential with mean 1 /  , the probability density function is ( ) We have which is accordance with that of Trivedi [1].
(ii) If the random variable X k -stage Erlang, each with mean 1 / k , the probability density function is We have which is consistent with that of Trivedi [1].
Then we have which is consistent with that of Trivedi [1].
(iv) If the random variable X LogN( 2 ,   ), the probability density function is which leads to B s e e dy y y which implies It should be noted that when no mathematical software is used, the computation of

EXAMPLE ILLUSTRATION
To demonstrate the validness of this reciprocal transformation, the four cases in Wang and Chiu [3] are considered to perform a comparative analysis of the Av for the availability models 1, 2, and 3 when the repair time distribution is exponential ( M ), 3stage Erlang ( 3 E ), deterministic ( D ), Lognormal ( LN ), and Weibull (W ), respectively.In order to make the comparisons, the parameters of Lognormal (Weibull) distribution are setting

Weibull repair time (W )
0.001    0.00152 Table 9.The CPU mean time of 100 computation times spent for calculating availabilities when  changes from 0.001 to 0.01.

THE EFFICIENCY OF RECIPROCAL TRANSFORMATION METHOD
To investigate the efficiency of computation of the reciprocal transformation method, the CPU computation time spent for calculating availabilities of three models for five different repair time distributions is recorded.Numerical results of the mean CPU time of 100 computation times using original method and the reciprocal transformation method are shown in Tables 9-12.Noting that the computation of original method indicates the MAPLE integral code used with lower bound zero and upper bound infinity.The percentages in the last column denote the relative improvement (worsening) of the computation time using reciprocal transformation.
From Tables 9-12, it can be observed that the reciprocal transformation significantly reduces the CPU computation time when the repair time distribution is Lognormal.Alternatively, the reciprocal transformation to the M and 3 E distributions doesn't bring out the decreasing of computation time due to the explicitly closed-form with simple computation.

2 b
 ) such that the expected repair time equals 1 /  (as Wang and Chiu[3]).All numerical results are obtained by the mathematical program MAPLE 9, which are based on the equipment of the following setting: CPU-Pentium 4 3.0 GHz, RAM 1.50 GB.Numerical results of the ( ) each availability model i ( i =1, 2, 3) are shown in Tables1-4for Cases 1-4, respectively.The comparative analysis of the cost/benefit ratios is made based on assumed the cost models of the models in Wang and Chiu's model[3, pp.1251-1252].The rank (cost/benefit) are shown in

Table 12 .
The CPU mean time of 100 computation times spent for calculating availabilities when c changes from 0.5 to 1.0.