Page 45 - A First Course In Stochastic Models

36 RENEWAL-REWARD PROCESSES

In Exercise 2.4 the reader is asked to prove that M(t) < ∞ for all t ≥ 0. In
Chapter 8 we will discuss how to compute the renewal function M(t) in general.
The infinite series (2.1.4) is in general not useful for computational purposes. An
exception is the case in which the interoccurrence times X 1 , X 2 , . . . have a gamma
distribution with shape parameter α > 0 and scale parameter λ > 0. Then the sum
X 1 +· · ·+X n has a gamma distribution with shape parameter nα and scale parameter
λ. In this case F n (t) is the so-called incomplete gamma integral for which efficient
numerical procedures are available; see Appendix B. Let us explain this in more
detail for the case that α is a positive integer r so that the interoccurrence times
X 1 , X 2 , . . . have an Erlang (r, λ) distribution with scale parameter λ. Then F n (t)
becomes the Erlang (nr, λ) distribution function
nr−1 k
−λt (λt)
F n (t) = 1 − e , t ≥ 0
k!
k=0
and thus
nr−1
∞ k
−λt (λt)
M(t) = 1 − e , t ≥ 0. (2.1.5)
k!
n=1 k=0
In this particular case M(t) can be efficiently computed from a rapidly converg-
ing series. For the special case that the interoccurrence times are exponentially
distributed (r = 1), the expression (2.1.5) reduces to the explicit formula

M(t) = λt, t ≥ 0.
This finding is in agreement with earlier results for the Poisson process.

Remark 2.1.1 The phase method
A very useful interpretation of the renewal process {N(t)} can be given when the
interoccurrence times X 1 , X 2 , . . . have an Erlang distribution. Imagine that tokens
arrive according to a Poisson process with rate λ and that the arrival of each rth
token triggers the occurrence of an event. Then the events occur according to a
renewal process in which the interoccurrence times have an Erlang (r, λ) distri-
bution with scale parameter λ. The explanation is that the sum of r independent,
exponentially distributed random variables with the same scale parameter λ has an
Erlang (r, λ) distribution. The phase method enables us to give a tractable expres-
sion of the probability distribution of N(t) when the interoccurrence times have an
Erlang (r, λ) distribution. In this case P {N(t) ≥ n} is equal to the probability that
nr or more arrivals occur in a Poisson arrival process with rate λ. You are asked
to work out the equivalence in Exercise 2.5.

Asymptotic expansion
A very useful asymptotic expansion for the renewal function M(t) can be given
under a weak regularity condition on the interoccurrence times. This condition