Timestep Stochastic Simulation of Computer Networks using Diffusion Approximation

Performance evaluation of modern computer networks is challenging because of their large sizes, high speeds of communication links, and complex state-dependent control mechanisms. In particular, TCP congestion control reacts in a nonlinear fashion to the state of the network at the time scale of round-trip times, making analysis intractable. Thus packet-level simulation is the only widely used method of performance evaluation. Although it can be accurate, it is computationally expensive and thus can be applied only to small networks and low link speeds. Timestep Stochastic Simulation (TSS) is a novel method for generating sample paths of computer networks, in increments of time steps rather than packet transmissions. TSS has a low computation cost independent of packet rates and provides adequate accuracy for evaluating general state-dependent control mechanisms. TSS generates the evolution of the system state S(t) on a sample path in time steps of size delta. At each step, S(t + delta) is randomly chosen according to S(t) and the probability distribution Pr[S(t+delta)|S(t)], obtained using the diffusion approximation. Because packet transmission and reception events are replaced by time steps, TSS generates sample paths at a fraction of the cost of packet-level simulation. Because TSS generates sample paths, it can accurately model state-dependent control mechanisms, including TCP congestion control, adaptive dynamic routing, and so on. We have a TSS implementation for general computer networks with state-dependent control. We have applied this to numerous networks with TCP and state-dependent UDP flows, and confirmed its accuracy against packet-level simulation

Learning Dynamic Boltzmann Distributions as Reduced Models of Spatial Chemical Kinetics

Finding reduced models of spatially-distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the interaction parameters of a graphical model, we derive Boltzmann-machine like learning algorithms to estimate directly the functionals dictating the time evolution of these parameters. By incorporating analytic solutions from simple reaction motifs, an efficient simulation method is demonstrated for systems ranging from toy problems to basic biologically relevant networks. The broadly applicable nature of our approach to learning spatial dynamics suggests promising applications to multiscale methods for spatial networks, as well as to further problems in machine learning

GridCell: a stochastic particle-based biological system simulator

<p>Abstract</p> <p>Background</p> <p>Realistic biochemical simulators aim to improve our understanding of many biological processes that would be otherwise very difficult to monitor in experimental studies. Increasingly accurate simulators may provide insights into the regulation of biological processes due to stochastic or spatial effects.</p> <p>Results</p> <p>We have developed GridCell as a three-dimensional simulation environment for investigating the behaviour of biochemical networks under a variety of spatial influences including crowding, recruitment and localization. GridCell enables the tracking and characterization of individual particles, leading to insights on the behaviour of low copy number molecules participating in signaling networks. The simulation space is divided into a discrete 3D grid that provides ideal support for particle collisions without distance calculation and particle search. SBML support enables existing networks to be simulated and visualized. The user interface provides intuitive navigation that facilitates insights into species behaviour across spatial and temporal dimensions. We demonstrate the effect of crowing on a Michaelis-Menten system.</p> <p>Conclusion</p> <p>GridCell is an effective stochastic particle simulator designed to track the progress of individual particles in a three-dimensional space in which spatial influences such as crowding, co-localization and recruitment may be investigated.</p

Simulating Brownian suspensions with fluctuating hydrodynamics

Fluctuating hydrodynamics has been successfully combined with several computational methods to rapidly compute the correlated random velocities of Brownian particles. In the overdamped limit where both particle and fluid inertia are ignored, one must also account for a Brownian drift term in order to successfully update the particle positions. In this paper, we present an efficient computational method for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics that handles both computations and provides a similar approximation as Stokesian Dynamics for dilute and semidilute suspensions. This advancement relies on combining the fluctuating force-coupling method (FCM) with a new midpoint time-integration scheme we refer to as the drifter-corrector (DC). The DC resolves the drift term for fluctuating hydrodynamics-based methods at a minimal computational cost when constraints are imposed on the fluid flow to obtain the stresslet corrections to the particle hydrodynamic interactions. With the DC, this constraint need only be imposed once per time step, reducing the simulation cost to nearly that of a completely deterministic simulation. By performing a series of simulations, we show that the DC with fluctuating FCM is an effective and versatile approach as it reproduces both the equilibrium distribution and the evolution of particulate suspensions in periodic as well as bounded domains. In addition, we demonstrate that fluctuating FCM coupled with the DC provides an efficient and accurate method for large-scale dynamic simulation of colloidal dispersions and the study of processes such as colloidal gelation

Discrete stochastic models for traffic flow

We investigate a probabilistic cellular automaton model which has been introduced recently. This model describes single-lane traffic flow on a ring and generalizes the asymmetric exclusion process models. We study the equilibrium properties and calculate the so-called fundamental diagrams (flow vs.\ density) for parallel dynamics. This is done numerically by computer simulations of the model and by means of an improved mean-field approximation which takes into account short-range correlations. For cars with maximum velocity 1 the simplest non-trivial approximation gives the exact result. For higher velocities the analytical results, obtained by iterated application of the approximation scheme, are in excellent agreement with the numerical simulations.Comment: Revtex, 30 pages, full postscript version (including figures) available by anonymous ftp from "fileserv1.mi.uni-koeln.de" in the directory "pub/incoming/" paper accepted for publication in Phys.Rev.

Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion

We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and simulation results to appear in J Phys