663
5. Is it possible to evaluate the renewal function numerically?
The difficulty of evaluating the renewal function numerically is
tuofold: both the evaluation of the Laplace transform (F,) and the
inversion of the Laplace transform to evaluate m(t) are numerically
sensitive operations.
The numerical evaluation of the Laplace transform given in (15) is
very difficult. Since the Laplace transform parameter (1) is a complex
number, its sine and cosine components result in very large fluctuations
when t is close to zero. Figure 3 presents the L(t) function whose integral
is the Laplace transform of the Erlangian density with shape parameter 2.
It can be seen that both the real and the imaginary parts of L(t) fluctuate
when t is small. However, nearly all numerical integration algorithms
assume reasonable smoothness of the function to be integrated. When this
assumption does not hold, the F values become very sensitive to the choice
of the parameters of the numerical algorithm.
The second source of difficulty lies in the numerical inversion of the
Laplace transform of the renewal function. The numerical inversion of the
Laplace transform is a difficult problem known to many branches of the
engineering sciences (for an overview of such problems see Bellman et al.,
1966). This is mainly because of the instability of the Laplace
transformation: a small perturbation in the original function may have a
large impact on its Laplace transform. For a reasonably accurate numerical
inversion, the Laplace transform itself has to be evaluated to a
considerable precision. This is not possible unless one uses extra¬
ordinarily accurate numerical algorithms of integration.