Near-explicit state probabilities of multiservices loss systems

The Kaufman-Roberts (1981) multirate recurrence equation is converted to a functional equation subject to a single step parameter d, which defines the state-dependent recurrence depth of the multirate model. This parameter, d, is verified to be the joint solution of two moment-generating functions. In contrast to the multirate transform domain solution, the characteristic function of the functional model may be inverted explicitly. But this inverse is scaled by d and yields a new state-dependent probability density based on two types of scaled gamma functions. They uniformly cover wide areas of continuous states, capacities C, and demands. A two-moments series expansion of d in the transform domain signifies that it models the effective bitrate of the multirate connections under progress within a complete sharing mode of C. Selected case studies show very good agreements to the exact Kaufman-Roberts' state probabilities and to the high accurate Mitra-Morrison's (1994) tail probabilities. Further partial sharing mode investigations include an efficient approximation for connection admission controls by trunk reservation covering partial and full fairness. The resulting state and tail probabilities deviate only slightly from Roberts' approximations of the past. All solutions achieve time complexities of O(1) per state, which amount to O(SC) with iterative multirate recurrence solutions for S service classes.