Modeling Information Flow with a Multi-Stage Queuing Mode

TL;DR

The stationary distribution is derived and a closure for a deterministic ODE system that approximates the evolution of the mean and variance of the stochastic model is developed, demonstrating the validity of the closure with numerical simulations.

Abstract

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.

Figures (7)

• Figure 1: Throttling function $v$, defined in \ref{['v']}, with $\sigma_*=3$. A slowdown in the processing rate occurs when $x < \sigma_*$.
• Figure 2: Numerical comparison of the mean $\mathbb{E}[\sigma_k(t)]$ in deterministic system \ref{['eq:naive']} and Monte-Carlo simulations of the stochastic system $\boldsymbol{\sigma}(t)$ for $c=10$, $c_0=6$, $\sigma_* = 3$, $t=30$, zero initial condition. Stochastic results are averaged over 10,000 Monte-Carlo trajectories. C.f. with results in Section \ref{['num1']} and Figure \ref{['fig:p1']} in particular, where numerical results for our improved closure model are discussed.
• Figure 3: $\mathbb{E}_k$ and $\rho_k$ vs $k$ with $c=10$, $\sigma_*=3$ and piece-wise $c_0(t)$, for $300$ stages and times $t=10,20,50,100$
• Figure 4: $\mathbb{V}_k$ and $\eta_k$ vs $k$ with $c=10$, $\sigma_*=3$ and piece-wise $c_0(t)$, for $300$ stages and times $t=10,20,50,100$
• Figure 5: $\mathbb{E}_k$ and $\rho_k$ with $c=10$, $\sigma_*=5$ and piece-wise $c_0(t)$, for $300$ stages and times $t=10,20,50,80$

Theorems & Definitions (35)

• Definition 1
• Theorem 1
• Definition 2
• Remark 1
• Definition 3
• Theorem 2
• Definition 4
• Definition 5
• Theorem 3
• Theorem 4