In this paper we study the fluid optimal control model which corresponds to
the optimal server control in a multiclass queueing network. In the last decade one
of the most effective ways for analysing multiclass queueing networks(qn) have
been
fluid approximations. The convergence of qn under fluid scaling to fluid networks(fn) have been studied in many papers. The analysis of the optimal control of qn through the corresponding fluid network has attracted less attention yet. The structure of the optimal fluid control for obtaining good policies in the qn has been used for specific models and it is well-known that fluid approximations can be used to obtain asymptotically optimal policies for the qn. The procedure consists of the following steps:
1. formulation of a corresponding deterministic fluid control model;
2. construction of a optimal solution of the fn;
3. translation of the optimal solution of the fn to obtain an asymptotically optimal solution of the qn.
The first step for the qn with optimal server control is straightforward. This paper is devoted to the analysis of the step 2. For the construction of an optimal control for a given initial state in a specified qn Pontryagin's maximum principle can been used. This approach has been used in several papers and for the special models analysed in these papers it was shown that the optimal policy for the fluid network, when properly translated is asymptotically optimal for the qn.
However, in general the constructed policy depends on the initial state, and for another starting state a different policy might be optimal. In most optimal control problems of qn it is well-known that a stationary optimal policy exists. For obtaining an asymptotically optimal policy which is stationary, we have to construct a stationary optimal control for the corresponding fn. This is the main goal of this paper, and it is precisely what in the classical book on optimal control has been called the synthesis problem. In the optimal control theory as developed by Pontryagin the synthesis of the solution of the optimal control problem is a vector field defined on the state space such that the solutions of this vector field give the optimal trajectories of the optimal control problem for all initial states. In our analysis for the fn we determine the value of the optimal control as function of the state. Hence, we determine optimal controls which only depend on the state, hence are stationary. So the goal of this paper is to derive the structure of this vector field for fluid networks with service control. One of the main results of this paper shows that there is a covering of the state space with a finite number of polyhedron convex cones such that the control and hence the vector field is constant on each polyhedral convex cone.
The structure of the vector field for the fluid model can be applied to get asymptotically optimal stationary policy for the queueing network. But, this needs an extended analysis which we do not include in this paper.
We discuss some assumptions which allow us to introduce generalized Klimov indices. They extend the well-known Klimov index for the single server network, and in the more general setting of a multi server network they also play a central role.
We derive a necessary and sufficient condition for a myopic optimal policy to minimize also the total cost over any time-interval. The generalized Klimov policy is in fact a myopic optimal policy, and another main result proves that it is time-uniformly optimal under an extra condition. This extra condition requires that the support of the control vector never changes in more than one component.