The one-level functional equation of multi-rate loss systems

Motivated by the discrete multirate Kaufmann-Roberts recurrence relations we derive a functional equation (FE), which covers nonintegral states. This FE implies a unique effective step parameter d, which defines an equivalent one-level recurrence depth, or bitrate, at each state under progress. This state-dependent depth results from the equality requirement of the multirate and the one-level model in the moment generating function transform domain. By this method it is possible to model d by a few moments of the original multirate statistic. In this case we obtain an explicit FE solution covering the entire (global) state space. Next we verify that the resulting state probability density incorporates iteratively enumerated discrete state probabilities including the state-dependent depth. With a system capacity C the iterations then need time complexities between O(C) and O(C). In contrast to this each FE state is performed at a time complexity O(1). By the efficient coverage of the whole state space fast optimizations of multirate networks and multiresource systems can be improved.