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997 results in Applied probability and stochastic networks

Page 16 of 50

Processing Networks
  • Fluid Models and Stability
  • J. G. Dai, J. Michael Harrison
  • Published online:
    17 September 2020
    Print publication:
    15 October 2020
  • This state-of-the-art account unifies material developed in journal articles over the last 35 years, with two central thrusts: It describes a broad class of system models that the authors call 'stochastic processing networks' (SPNs), which include queueing networks and bandwidth sharing networks as prominent special cases; and in that context it explains and illustrates a method for stability analysis based on fluid models. The central mathematical result is a theorem that can be paraphrased as follows: If the fluid model derived from an SPN is stable, then the SPN itself is stable. Two topics discussed in detail are (a) the derivation of fluid models by means of fluid limit analysis, and (b) stability analysis for fluid models using Lyapunov functions. With regard to applications, there are chapters devoted to max-weight and back-pressure control, proportionally fair resource allocation, data center operations, and flow management in packet networks. Geared toward researchers and graduate students in engineering and applied mathematics, especially in electrical engineering and computer science, this compact text gives readers full command of the methods.

  • 1 - Stochastic Simulation of Chemical Reactions

  • Radek Erban, University of Oxford, S. Jonathan Chapman, University of Oxford
  • Book:
    Stochastic Modelling of Reaction–Diffusion Processes
    Published online:
    04 November 2019
    Print publication:
    30 January 2020, pp 1-32

  • Summary

    This chapter provides an introduction to stochastic methods for modelling spatially homogeneous systems of chemical reactions. The Gillespie stochastic simulation algorithm and the chemical master equation are presented using simple examples of chemical systems. The chemical master equation is analysed for chemical systems containing zeroth-order, first-order and second-order chemical reactions. For zeroth-order and first-order chemical reactions, the average behaviour of the stochastic chemical system is described by the ordinary differential equations (ODEs) given by the standard deterministic model. However, when we consider higher-order chemical reactions, for which the deterministic description is nonlinear, the deterministic ODE model does not provide an exact description of the average behaviour of the stochastic system.

  • 8 - Microscopic Models of Brownian Motion

  • Radek Erban, University of Oxford, S. Jonathan Chapman, University of Oxford
  • Book:
    Stochastic Modelling of Reaction–Diffusion Processes
    Published online:
    04 November 2019
    Print publication:
    30 January 2020, pp 226-267

  • Summary

    This chapter presents microscopic models of diffusion (Brownian motion). The discussed diffusion models explicitly describe the dynamics of solvent molecules. Such molecular dynamics models provide many more details than the models discussed in Chapter 4 (which simply postulate that the diffusing molecule is subject to a random force) and can be used to assess the accuracy of the stochastic diffusion models from Chapter 4. The analysis starts with theoretical solvent models, including a simple “one-particle” description of the solvent (heat bath), which is used to introduce the generalized Langevin equation and the generalized fluctuation–dissipation theorem. Analytical insights are provided by theoretical models with short- and long-range interactions. The chapter concludes with less analytically tractable, but more realistic, computational models, introducing molecular dynamics (molecular mechanics) and applying it to the Lennard-Jones fluid and to simulations of ions in aquatic solutions.

  • 7 - SSAs for Reaction–Diffusion–Advection Processes

  • Radek Erban, University of Oxford, S. Jonathan Chapman, University of Oxford
  • Book:
    Stochastic Modelling of Reaction–Diffusion Processes
    Published online:
    04 November 2019
    Print publication:
    30 January 2020, pp 192-225

  • Summary

    This chapter shows how active transport (for example, by an electrical field, molecular and cellular motors, running, swimming or flying, all in response to external cues) can be incorporated into the stochastic diffusion and reaction–diffusion algorithms we have introduced in Chapters 4 and 6. The resulting stochastic diffusion–advection and reaction–diffusion–advection models are analysed. Applications include systems consisting of many interacting “particles”, where individual particles can range in size from small ions and molecules to individual cells and animals. Three examples illustrate this: mathematical modelling of ions and ion channels, modelling bacterial chemotaxis, and studying collective behaviour of social insects. The chapter concludes with the discussion of the Metropolis–Hastings algorithm, which can be used to compute stationary (equilibrium) properties of complicated diffusion–advection problems.

Page 16 of 50