Modeling information flow in a computer processor with a multi-stage queuing model

In this paper, we introduce a nonlinear stochastic model to describe the propagation of information inside a computer processor. In this model, a computational task is divided into stages, and information can flow from one stage to another. The model is formulated as a spatially-extended, continuous-time Markov chain where space represents different stages. This model is equivalent to a spatially-extended version of the M/M/s queue. The main modeling feature is the throttling function which describes the processor slowdown when the amount of information falls below a certain threshold. We derive the stationary distribution for this stochastic model and develop a closure for a deterministic ODE system that approximates the evolution of the mean and variance of the stochastic model. We demonstrate the validity of the closure with numerical simulations.

在本文中，我们引入一个非线性随机模型来描述计算机处理器内部信息的传播。在该模型中，一个计算任务被划分为多个阶段，信息可以从一个阶段流向另一个阶段。该模型被表述为一个空间扩展的连续时间马尔可夫链，其中空间代表不同的阶段。这个模型等同于M/M/s队列的空间扩展版本。主要的建模特征是节流函数，它描述了当信息量低于某一阈值时处理器的减速情况。我们推导了这个随机模型的稳态分布，并为一个确定性常微分方程系统开发了一种闭合形式，该系统近似于随机模型的均值和方差的演化。我们通过数值模拟证明了这种闭合形式的有效性。