Page 472 - A First Course In Stochastic Models

F. NUMERICAL LAPLACE INVERSION 467

recommended. The parameter M is taken as a power of 2 (say, M = 32 or M = 64)
while the parameter m is chosen equal to 4M. The choices of M and are not
particularly relevant when f is smooth enough (theoretically, the accuracy increases
when gets smaller). In practice it is advisable to apply the algorithm for and
1 to see whether or not the results are affected by the choice of .
2

Non-smooth functions
The Den Iseger algorithm may also perform unsatisfactorily when f or its derivative
has discontinuities. In such cases the numerical difficulties may be circumvented
by using a simple modification of the algorithm. To do this, assume that f (s) can
∗
be represented as
∗
f (s) = v(s, e x 0 s ) (F.5)
for some real scalar x 0 and some function v(s, u) with the property that for any
fixed u the function v(s, u) is the Laplace transform of a smooth function. As an
example, consider the complementary waiting-time distribution f (t) = P {W q > t}
in the M/D/1 queue with deterministic service times D and service in order of
arrival; see Chapter 9. This function f (t) is continuous but is not differentiable at
the points t = D, 2D, . . . . The Laplace transform f (s) of f (t) is given by
∗

ρs − λ + λe −sD
∗
f (s) = −sD , (F.6)
s[s − λ + λe ]
where λ is the average arrival rate and ρ = λD < 1. Then (F.5) applies with

ρs − λ + λu
x 0 = −D and v(s, u) = .
s(s − λ + λu)
In this example we have indeed that for any fixed u the function v(s, u) is the
Laplace transform of an analytic (and hence smooth) function.
In the modified Den Iseger algorithm the basic relation (F.2) should be modi-
fied as n
e bℓ 1 j
f (ℓ ) ≈ α j v (t) cos(πℓ(t + 1)) dt (F.7)
−1
j=1

with
b + iλ j + iπt iπx 0 b + iπt

j
v (t) = Re v , exp − .

It is essential that in (F.7) the constant > 0 is chosen such that |x 0 | is a multiple of
, where x 0 comes from (F.5). As before, the integral in (F.7) can be approximated