[Paper Review] Gaussian Fluid Queue with Autocorrelated Input

Kerry Fendick|arXiv (Cornell University)|Apr 25, 2011

Complex Systems and Time Series Analysis24 references3 citations

TL;DR

This paper introduces a generalized Gaussian fluid queue model with autocorrelated input by extending Brownian motion to include stationary, autocorrelated increments. It characterizes such processes via three parameters—drift, volatility, and autocorrelation—enabling derivation of the transient queue distribution conditional on history, offering a tractable framework for financial and business modeling with memory effects.

ABSTRACT

This paper develops a generalization of Brownian motion with stationary, autocorrelated increments as a tractable model for problems in business and finance. We show that any real continuous Gaussian Markov process with stationary increments and smooth covariance function is characterized by three parameters quantifying drift, volatility, and autocorrelations. We model a queue as a functional of a process defined by those characteristics and derive its transient distribution conditional on its history.

Motivation & Objective

• To develop a tractable stochastic model for fluid queues with memory effects in business and finance.
• To generalize Brownian motion by incorporating stationary, autocorrelated increments.
• To characterize continuous Gaussian Markov processes with smooth covariance via three key parameters: drift, volatility, and autocorrelation.
• To derive the transient distribution of the queue process conditional on its history.
• To provide a computationally feasible framework for modeling systems with persistent dependence in input processes.

Proposed method

• Model the input process as a continuous Gaussian Markov process with stationary increments and smooth covariance function.
• Parameterize the process using three characteristics: drift (μ), volatility (σ), and autocorrelation (ρ).
• Define the fluid queue as a functional of this process, representing cumulative workload or content.
• Use stochastic calculus to derive the transient distribution of the queue state given its history.
• Apply results from Gaussian process theory and conditional expectation to express the distribution in closed form.
• Ensure the model remains analytically tractable by maintaining Markovian and Gaussian properties.

Experimental results

Research questions

• RQ1 How can Brownian motion be generalized to include autocorrelated increments while preserving tractability?
• RQ2 What minimal set of parameters fully characterizes a continuous Gaussian Markov process with stationary increments and smooth covariance?
• RQ3 How does the inclusion of autocorrelation affect the transient behavior of a fluid queue?
• RQ4 Can the conditional distribution of a fluid queue be derived explicitly when input increments are autocorrelated?
• RQ5 What is the mathematical structure of the queue process under a Gaussian input with memory?

Key findings

• Any real continuous Gaussian Markov process with stationary increments and smooth covariance is fully characterized by three parameters: drift, volatility, and autocorrelation.
• The fluid queue is modeled as a functional of this three-parameter Gaussian process, enabling analytical treatment.
• The transient distribution of the queue is derived conditionally on its history, providing a path-dependent solution.
• The model maintains analytical tractability despite the inclusion of autocorrelation, a significant extension over standard Brownian fluid queues.
• The framework supports applications in business and finance where input processes exhibit long-memory or persistence.
• The derived distribution allows for exact inference on queue content at finite time points, given initial conditions and history.