Page 467 - A First Course In Stochastic Models

462 APPENDICES

−1 n
∗ ∞ ∗
Thus we find M (s) = s [b (s)] , which yields the general formula
n=1
∗
b (s)
∗
M (s) = . (E.13)
s[1 − b (s)]
∗
This expression can be inverted for the Erlang density. As an illustration, consider
2
2
∗
the case of b(x) = λ xe −λx . Then b (s) = [λ/(λ + s)] and so
λ 2
∗
M (s) = .
2
s (s + 2λ)
Partial-fraction expansion gives
1 1 1
− s + λ
∗ 4 2 4
M (s) = 2 + .
s s + 2λ
By applying (E.3), (E.9) and (E.10), we obtain
1 1 1 −2λt
M(t) = λt − + e , t ≥ 0.
2 4 4
APPENDIX F. NUMERICAL LAPLACE INVERSION
For a long time numerical Laplace inversion had the reputation of being difficult and
numerically unreliable. However, contrary to previous impressions, it is nowadays
not difficult to compute probabilities and other quantities of interest in probability
models by using reliable Laplace inversion methods. This appendix briefly discusses
two effective Laplace inversion algorithms. These algorithms involve complex cal-
culations. There is nothing magic about doing calculations with complex numbers.
These calculations can be reduced to operations with real numbers by dealing sep-
arately with the real part and the imaginary part of the complex numbers. Simple
facts such as the relation e ix = cos(x) + i sin(x) for any real x and the represen-
tation z = re iθ for any complex number z are typically used in the calculations in
addition to the basic rules for adding and multiplying two complex numbers. Here
2
i denotes the complex number with i = −1. Certain computer languages such as
the language C++ have automatic provision for doing complex calculations. In
many applied probability problems it is possible to derive an expression for the
Laplace transform of some unknown function. Let the real-valued function f (t) be
an unknown function in the variable t ≥ 0. Suppose its Laplace transform
∞

∗ −st
f (s) = e f (t) dt
0
in the complex variable s is known. Assume that the function f (t) satisfies the
following conditions:

1. f (t) is of bounded variation on any finite interval.