Lingering Issues in Distributed Scheduling

Recent advances have resulted in queue-based algorithms for medium access control which operate in a distributed fashion, and yet achieve the optimal throughput performance of centralized scheduling algorithms. However, fundamental performance bounds reveal that the "cautious" activation rules involved in establishing throughput optimality tend to produce extremely large delays, typically growing exponentially in 1/(1-r), with r the load of the system, in contrast to the usual linear growth. Motivated by that issue, we explore to what extent more "aggressive" schemes can improve the delay performance. Our main finding is that aggressive activation rules induce a lingering effect, where individual nodes retain possession of a shared resource for excessive lengths of time even while a majority of other nodes idle. Using central limit theorem type arguments, we prove that the idleness induced by the lingering effect may cause the delays to grow with 1/(1-r) at a quadratic rate. To the best of our knowledge, these are the first mathematical results illuminating the lingering effect and quantifying the performance impact. In addition extensive simulation experiments are conducted to illustrate and validate the various analytical results

How non-Gibbsianness helps a metastable Morita minimizer to provide a stable free energy

We analyze a simple approximation scheme based on the Morita-approach for the example of the mean field random field Ising model where it is claimed to be exact in some of the physics literature. We show that the approximation scheme is flawed, but it provides a set of equations whose metastable solutions surprisingly yield the correct solution of the model. We explain how the same equations appear in a different way as rigorous consistency equations. We clarify the relation between the validity of their solutions and the almost surely discontinuous behavior of the single-site conditional probabilities.Comment: 15 page

The Law of Large Numbers in a Metric Space with a Convex Combination Operation

We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with subsets of the basic space in order to develop a systematic way of proving laws of large numbers for such operations with random set