Two-buffer fluid models with multiple ON-OFF inputs and threshold assistance

We consider a two-buffer fluid model with N ON-OFF inputs and threshold assistance, which is an extension of the same model with N = 1 in [18]. While the rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates slight modifications in the original rules of output-capacity sharing from [18], and considerably complicates both the theoretical analysis and numerical computation of various performance measures.

Here, we give a short explanation on how to derive the marginal probability distribution of Buffer 1, and bounds for that of Buffer 2. In an upcoming paper, we describe the procedures in more details. Furthermore, restricting Buffer 1 to a finite size, we determine its marginal probability distribution in the specific case of N = 1, thus providing numerical comparisons to the corresponding results in [18] where Buffer 1 is assumed to be infinite. We also demonstrate how this imposed restriction effects the bounds of marginal probabilities for Buffer 2.