Modulated Branching Processes, Origins of Power Laws and Queueing Duality

TL;DR

This paper introduces reflected modulated branching processes as a versatile model for power law distributions observed across various fields, establishing their mathematical properties and duality with queueing processes.

Contribution

It provides a general framework for modeling power laws using reflected modulated branching processes and explores their asymptotic behavior and duality with queueing systems.

Findings

Power law distributions arise under broad polynomial conditions.

Reflected branching processes are asymptotically equivalent to multiplicative processes.

Duality between branching and queueing processes is established.

Abstract

Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991), is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes.