Page 323 - A First Course In Stochastic Models

318 ADVANCED RENEWAL THEORY

The function a(t) = 1 − F(t + u), t ≥ 0 is monotone and integrable. By applying
the key renewal theorem it now follows that

1 ∞ 1 ∞
lim Z(t) = {1 − F(y + u)} dy = {1 − F(y)} dy,
t→∞ µ 1 0 µ 1 u
∞
yielding the desired result by using the fact that {1 − F(y)} dy = µ 1 .
0
In many practical applications the asymptotic expansion (8.2.8) gives a useful
approximation to the distribution of γ t already for moderate values of t. The limiting
distribution of the excess life is called the equilibrium excess distribution and has
applications in a wide variety of contexts. The equilibrium excess distribution can
be given the following interpretation. Suppose that an outside person observes the
state of the process at an arbitrarily chosen point in time when the process has
been in operation for a very long time. Assuming that the outside person has no
information about the past history of the process, the best prediction the person
can give about the residual life of the item in use is according to the equilibrium
excess distribution.
The asymptotic expansions in Theorem 8.2.3 will be illustrated by the next
example.

Example 8.2.1 The D-policy for controlling the workload

Batches of fluid material arrive at a processing plant according to a Poisson process
with rate λ. The batch amounts are independent random variables having a contin-
uous probability distribution with finite first two moments µ 1 and µ 2 . It is assumed
that λµ 1 < 1. The unprocessed material is temporarily stored in an infinite-capacity
buffer. If the processing plant is open, the material is processed at a unity rate.
The plant is controlled by the so-called D-policy. If the inventory of unprocessed
material becomes zero, the plant is temporarily closed down. The plant is reopened
as soon as the buffer content exceeds the threshold value D. The set-up time to
restart the processing is zero. The following costs are incurred. A holding cost at
rate hx is incurred when the buffer content is x. A fixed set-up cost of K > 0 is
incurred each time the plant is reopened. What value of the control parameter D
minimizes the long-run average cost per time unit?

Preliminary analysis
To answer the above question, we first derive some preliminary results for the
M/G/1 queue. Note that the control problem can be seen as an M/G/1 queue in
which the workload is controlled. The workload is defined as the remaining amount
of work for the server. Define the basic functions

t(x) = the expected amount of time until the workload is zero
when the current workload is x and the server is working