Modulated Branching Processes, Origins of Power Laws and Queueing Duality

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.Comment: 36 pages, 2 figures; added references; a new theorem in Subsection 4.

Characterizing Heavy-Tailed Distributions Induced by Retransmissions

Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamics is modeled as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During each period of time that the channel becomes available, say A_i, we attempt to transmit the data unit. If L<A_i, the transmission was considered successful; otherwise, we wait for the next available period and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was shown that this model results in power law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional dependence between P[L>x] and P[A>x] impacts the distributions of N and T. In particular, we discover several functional criticality points that separate classes of different functional behavior of the distribution of N. We also discuss the engineering implications of our results on communication networks since retransmission strategy is a fundamental component of the existing network protocols on all communication layers, from the physical to the application one.Comment: 39 pages, 2 figure

The vortex dynamics of a Ginzburg-Landau system under pinning effect

It is proved that the vortices are attracted by impurities or inhomogeities in the superconducting materials. The strong H^1-convergence for the corresponding Ginzburg-Landau system is also proved.Comment: 23page

Learning Loosely Connected Markov Random Fields

We consider the structure learning problem for graphical models that we call loosely connected Markov random fields, in which the number of short paths between any pair of nodes is small, and present a new conditional independence test based algorithm for learning the underlying graph structure. The novel maximization step in our algorithm ensures that the true edges are detected correctly even when there are short cycles in the graph. The number of samples required by our algorithm is C*log p, where p is the size of the graph and the constant C depends on the parameters of the model. We show that several previously studied models are examples of loosely connected Markov random fields, and our algorithm achieves the same or lower computational complexity than the previously designed algorithms for individual cases. We also get new results for more general graphical models, in particular, our algorithm learns general Ising models on the Erdos-Renyi random graph G(p, c/p) correctly with running time O(np^5).Comment: 45 pages, minor revisio

Q-CSMA: Queue-Length Based CSMA/CA Algorithms for Achieving Maximum Throughput and Low Delay in Wireless Networks

Recently, it has been shown that CSMA-type random access algorithms can achieve the maximum possible throughput in ad hoc wireless networks. However, these algorithms assume an idealized continuous-time CSMA protocol where collisions can never occur. In addition, simulation results indicate that the delay performance of these algorithms can be quite bad. On the other hand, although some simple heuristics (such as distributed approximations of greedy maximal scheduling) can yield much better delay performance for a large set of arrival rates, they may only achieve a fraction of the capacity region in general. In this paper, we propose a discrete-time version of the CSMA algorithm. Central to our results is a discrete-time distributed randomized algorithm which is based on a generalization of the so-called Glauber dynamics from statistical physics, where multiple links are allowed to update their states in a single time slot. The algorithm generates collision-free transmission schedules while explicitly taking collisions into account during the control phase of the protocol, thus relaxing the perfect CSMA assumption. More importantly, the algorithm allows us to incorporate mechanisms which lead to very good delay performance while retaining the throughput-optimality property. It also resolves the hidden and exposed terminal problems associated with wireless networks.Comment: 12 page

Variational wave functions of a vortex in cyclotron motion

In two dimensions the microscopic theory, which provides a basis for the naive analogy between a quantized vortex in a superfluid and an electron in an uniform magnetic field, is presented. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with the energy, and scales with the size of the vortex.Comment: LaTeX, 4 pages, 2 EPS figures, To appear in ``Series on Advances in Quantum Many-Body Theory'' ed. by R.F. Bishop, C.E. Campbell, J.W. Clark and S. Fantoni (World Scientific, 2000

Direct Detection of Planets Orbiting Large Angular Diameter Stars: Sensitivity of an Internally Occulting Space-based Coronagraph

High-contrast imaging observations of large angular diameter stars enable complementary science questions to be addressed compared to the baseline goals of proposed missions like the Terrestrial Planet Finder-Coronagraph, New World's Observer, and others. Such targets, however, present a practical problem in that finite stellar size results in unwanted starlight reaching the detector, which degrades contrast. In this paper, we quantify the sensitivity, in terms of contrast, of an internally occulting, space-based coronagraph as a function of stellar angular diameter, from unresolved dwarfs to the largest evolved stars. Our calculations show that an assortment of band-limited image masks can accommodate a diverse set of observations to help maximize mission scientific return. We discuss two applications based on the results: the spectro-photometric study of planets already discovered with the radial velocity technique to orbit evolved stars, which we elucidate with the example of Pollux b, and the direct detection of planets orbiting our closest neighbor, α Centauri, whose primary component is on the main sequence but subtends an appreciable angle on the sky. It is recommended that similar trade studies be performed with other promising internal, external, and hybrid occulter designs for comparison, as there is relevance to a host of interesting topics in planetary science and related fields