Network Control

The source model makes it possible to investigate several different kinds of controls. As in Sections 4 and 9 of [10], we can consider rejecting new sources or removing existing sources. We can also consider changing the levels of existing sources. For example, on-off sources in the on state might be turned off. We also can consider rate controls corresponding to changing the bandwidths assigned to the levels. With each of these controls, we can calculate the resulting conditional mean aggregate bandwidth to evaluate the performance of the control.

To illustrate, suppose that we wish to consider which of n candidate sources to serve over the time interval [0,T]. Suppose that source i earns a fixed revenue R_i plus a revenue rate r_i per unit of bandwidth per time. Also suppose that we want to keep the demand below a capacity c at all times. Then we can solve the following integer program. Let y_i be the decision variable, with y_i = 1 if source i is served and y_i = 0 if not. The integer program can be formulated as:  

$$\max \sum_{i=1}^n y_i (R_i + r_i \int_0^T m_{j_i}^i (t \vert x_i ) dt )$$ (36)

subject to  

$$\sum_{i=1}^n y_i m_{j_i}^i (t_k \vert x_i ) \le c ~, \quad 0 \le k \le K~,$$ (37)

for a set of time points $0 = t_0 < t_1 < \ldots < t_K = T$.In (39) and (40) the level j_i and age x_i of source i are included because these are presumed to be known.