Page 474 - A First Course In Stochastic Models

F. NUMERICAL LAPLACE INVERSION 469

(t) is a probability
for appropriately chosen constants a 0 and a 1 such that f a 0 ,a 1
density with mean t 0 . The choice of the parameters a 0 and a 1 is intended to make
(t 0 ) is computed by numerically
f a 0 ,a 1 (t) not too small. The unknown value f a 0 ,a 1
inverting its Laplace transform f ∗ (s), which is given by
a 0 ,a 1
∗
f ∗ (s) = a 0 f (s + a 1 ).
a 0 ,a 1
(t 0 ) is computed the desired value f (t 0 ) is easily obtained. The com-
Once f a 0 ,a 1
putation of the constants a 0 and a 1 is as follows:
∞ −sx
1. Determine the smallest real number s such that e f (x) dx is convergent
∗
0
for all s with Re(s) > s (possibly s = −∞).
∗
∗
2. Try to find the real root a 1 of the equation
∗
df (s)/ds
+ t 0 = 0
∗
f (s)
∗
on the interval (s , ∞). Since the function −[1/f (s)] df (s)/ds can be shown
∗
∗
∗
to be decreasing on the interval (s , ∞), this equation has at most one root.
3. Determine a 0 = 1/f (a 1 ).
∗
In many applications this procedure works surprisingly well. We used the mod-
ified Den Iseger algorithm in combination with the scaling procedure to compute
P {W q > t} for t = 75, 100 and 125 in the M/D/1 example discussed above.
The respective values 8.022E − 15, 1.685E − 19 and 3.537E − 24 were calculated.
Those values were exactly the same as the values obtained from the asymptotic
expansion for P {W q > t} for t large.
Analytically intractable Laplace transforms
Sometimes the Laplace transform f (s) of the unknown function f (t) is not given
∗
in an explicit form, but contains an analytically intractable expression. To illustrate
∗
this, consider the Laplace transform M (s) of the renewal function M(t) for a
renewal process. As shown by formula (E.12) in Appendix E, the Laplace transform
M (s) is given by
∗
b (s)
∗
∗
M (s) = ,
s[1 − b (s)]
∗
where b (s) is the Laplace transform of the interoccurrence-time density b(t).
∗
Suppose now that this density is given by a lognormal density. In this particular
case it is not possible to give an explicit expression for b (s) and one has to handle
∗
an analytically intractable integral. How do we handle this? Suppose we wish to
compute M(t) for a number of t-values in the interval [0, t 0 ]. The key observation
is that, by the representation (E.11), the renewal function M(t) for 0 ≤ t ≤ t 0
uses the interoccurrence-time density b(t) only for 0 ≤ t ≤ t 0 . The same is true