Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps

We study the positive recurrence of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. We first provide a generic method to construct a Lyapunov function when the drift can be extended to a smooth function on $\mathbb R^N$, using an associated deterministic dynamical system. This approach gives an elementary proof of ergodicity without needing to establish the convergence of the scaled version of the process towards a fluid limit and then proving that the stability of the fluid limit implies the stability of the process. We also provide a counterpart result proving instability conditions. We then show how discontinuous drifts change the nature of the stability conditions and we provide generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piece-wise constant drifts in dimension 2.Comment: 18 pages, 4 figure

Counting cliques and cycles in scale-free inhomogeneous random graphs

Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis. We obtain various asymptotic descriptions for how the average number of cliques and cycles, of any size, grow with the network size. For the cycle asymptotics we invoke the theory of circulant matrices

Surfing, sweeping, and assembly of particles by a moving liquid crystal phase boundary

Non-equilibrium transport of particles embedded in a liquid crystal host can, by cooling through a phase transition, be exploited to create a remarkable variety of structures including shells, foams, and gels. Due to the complexity of the multicomponent system and protocol-dependent experimental results, the physical mechanisms behind structure selection remain only partially understood. Here we formulate a new model coupling LC physics to a Fokker-Planck equation as is commonly used in studies of transport. The resulting model allows us to draw an analogy between the LC-nanocomposite system and chemotaxis, enriching the space of possible target structures that could be produced. We study the model in one dimension both analytically and numerically to identify different parameter regimes where soliton-like pulses of particles ``surf'' the phase boundary or where the interface ``sweeps'' particles from one domain to another. We also consider an extended model that includes agglomeration of the particles and observe formation of periodic structures as a prototypical example of hierarchical self assembly. Results are compared with experimental observations of transport by isolated phase boundaries.Comment: 11 pages, 6 figure

A unified approach to the heavy-traffic analysis of the maximum of random walks

Wachtel V, Shneer S. A general approach to the analysis of the maximum of a random walk in heavy traffic. Teor. Veroyatn. Primen. 2010;55(2):335-344

Mean-field limits for multi-hop random-access networks

Recent years have seen wireless networks increasing in scale, interconnecting a vast number of devices over large areas. Due to their size these networks rely on distributed algorithms for control, allowing each node to regulate its own activity. A popular such algorithm is Carrier-Sense Multi-Access (CSMA), which is at the core of the well-known 802.11 protocol. Performance analysis of CSMA-based networks has received significant attention in the research literature in recent years, but focused almost exclusively on saturated networks where nodes always have packets available. However, one of the key features of emerging large-scale networks is their ability to transmit packets across large distances via multiple intermediate nodes (multi-hop). This gives rise to vastly more complex dynamics, and to phenomena not captured by saturated models. Consequently, performance analysis of multi-hop random-access networks remains elusive. Based on the observation that emerging multi-hop networks are typically dense and contain a large number of nodes, we consider the mean-field limit of multihop CSMA networks. We show that the equilibrium point of the resulting initial value problem provides a remarkably accurate approximation for the pre-limit stochastic network in stationarity, even for sparse networks with few nodes. Using these equilibrium points we investigate the performance of linear networks under di erent back-o rates, which govern how fast each node transmits. We find the back-o rates which provide the best end-to-end throughput and network robustness, and use these insights to determine the optimal back-o rates for general networks. We confirm numerically the resulting performance gains compared to the current practice of assigning all nodes the same back-o rate

Comparison of stability regions for a line distribution network with stochastic load demands

We compare stability regions for different power flow models in the process of charging electric vehicles (EVs) by considering their random arrivals, their stochastic demand for energy at charging stations, and the characteristics of the electricity distribution network. We assume the distribution network is a line with charging stations located on it. We consider the Distflow and the Linearized Distflow models, and we assume that EVs have an exponential charging requirement, that voltage drops on the distribution network stay under control, and that the number of charging stations N goes to infinity. We investigate the stability of utility-optimizing power allocations in large distribution networks for both power flow models by controlling the arrival rate of EVs to charging stations. For both power flow models, we show that, to obtain stability, the maximum feasible arrival rate, i.e., stability region of vehicles, is decaying as 1 / N2, and the difference between those arrival rates is up to constants, which we compare explicitly