Analysis of Finite Buffer Queue: Maximum Entropy Probability Distribution With Shifted Fractional Geometric and Arithmetic Means

Abstract

A theoretical method based on maximum Shannon entropy framework (MSEF) in the presence of the geometric/or shifted fractional geometric mean of the queue size is applied to study the finite buffer system. Analytical expression of the loss probability for large buffer size is found to depict power law behavior. The maximum entropy framework is extended to incorporate additional shifted fractional arithmetic mean constraint to yield the expression of loss probability which is similar to the one derived by Kim and Shroff who have employed an entirely different approach based on maximum variance asymptotic (MVA) approximation. An important finding is the relationship between fractional order and the Hurst parameter. The advantage of MSEF is that it has enabled one to derive the analytical closed form generalized expression of the probability distribution of queue size in finite buffer system. Further, MSEF with shifted fractional geometric mean constraint gives the same distribution of queue size as the one obtained by maximizing Tsallis entropy subject to the fractional arithmetic mean of queue size.