Power Law Behavior of Queue Size: Maximum Entropy Principle with Shifted Geometric Mean Constraint

A theoretical framework based on the maximum Shannon entropy principle with the specification of geometric mean or shifted geometric mean is proposed to generate the power law behavior of the system size in broadband communication networks. It is shown that the equilibrium distribution of an M/M/1 queue is obtained as a limiting case when the shifted geometric mean is specified. The various quality-of-service parameters, such as the probability of exceeding buffer size, exhibit the power law behavior. It is noted that the results based on the Shannon entropy with shifted geometric mean are found to be similar to the results when the Tsallis entropy subject to expected number of jobs in the system is prescribed. This brings out a deeper issue of the relevance of shifted geometric mean within Shannon entropy framework when the input traffic has broadband characteristics. The proposed approach gives closed-form expressions for a queuing distribution in a much simpler way and thus establishes the efficacy of Shannon entropy in communication networks exhibiting power law.