= | |||
= | (5.37) |
Our problem is to find an admissible control that minimizes the cost function
|
(5.38) |
In place of Assumptions 3.4, 3.5 and 3.6 in Section 3.3 on the cost function and and the machine capacity process, we impose the following assumptions on the random process throughout this section.
Assumption 5.5 Let , and for, that is, pn is the average capacity of the machine n, and n(i,j) is the number of the machine located on the arc (i,j). Furthermore, we assume that there exist such that
|
(5.39) | ||
|
(5.40) |
|
(5.41) |
|
(5.42) |
In the case of the long-run average cost criterion used here, we know, by Theorem 2.4 in Presman, Sethi, and Zhang (1999b), that under Assumption 5.5, is independent of the initial condition . Thus we will use instead of . We use to denote our control problem, i.e.,
|
(5.43) |
As in Section 5.1, the positive attrition rate implies a uniform bound for .
Next we examine elementary properties of the potential function and obtain the limiting control problem as .
The Hamilton-Jacobi-Bellman equation in the directional derivative sense with the average-cost optimal control problem in , as shown in Sethi, Zhang, and Zhang (1999b), takes the form
= | |||
(5.44) |
First we can get the boundedness of .
Theorem 5.9 The minimum average expected cost of is bounded in, i.e., there exists a constant M1>0 such that
For the proof, see Sethi, Zhang, and Zhang (1999b).
Now we derive the limiting control problem as . As in Sethi and Zhou (1994), we give the following definition.
Definition 5.6 For , let denote the set of the following measurable controls
such that for all , j=1,...,p, and n=1,...,nc, and the corresponding solutions of the following system
We use to denote the above problem, and will regard this as our limiting problem. Then we define the limiting control problem as follows:
|
(5.45) |
where is a potential function for and is the directional derivative of along the direction with. From Presman, Sethi, and Zhang (1999b), we know that there exist and such that (5.45) holds. Moreover, is the limit of as .
Now we are ready to discuss the convergence of the minimum average expected cost as goes to zero, and establish the corresponding convergence rate. First we give two lemmas which are used in proving the convergence.
Lemma 5.7 For and any sufficiently small, there exist,, and
|
(5.46) |
|
(5.47) |
where is the trajectory under
Lemma 5.8 For , there exist, and, and
|
(5.48) |
|
(5.49) |
where
is the state trajectory under the control .For
the proof of Lemmas 5.7 and 5.8,
see Sethi, Zhang, and Zhang (1999b).
Using these two lemmas, Sethi, Zhang, and Zhang (1999b) derive the following theorem.
Theorem 5.10 For any there exists a constant such that for all sufficiently small
|
(5.50) |
This implies in particular that
Remark 5.4 The theorem says that the problem is indeed a limiting problem in the sense that the of converges to of . Moreover, it gives the corresponding convergence rate.