# Modulated branching processes, origins of power laws, and queueing duality

## 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 because of the replication of their many independent components, e.g., births and deaths of individuals and replications of cells. Furthermore, the rates of replications 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 below 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 [Goldie, C. M. 1991. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1) 126-166], is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes. © 2010 INFORMS.