Application of the Diffusion Approximation to Queueing Networks I: Equilibrium Queue Distributions

TLDR

Queueing theory has been limited by oversimplified models, and the diffusion process approximation seeks to move beyond this simplification. The paper introduces a vector‑valued normal process and its diffusion equation to approximate the joint distribution of queue lengths in a general network of queues. The authors model interacting service stations as a vector‑valued Wiener process with appropriate boundary conditions and validate the approach with numerical examples compared to Monte Carlo simulations.

Abstract

The practical value of queueing theory in engineering applications such as in computer modeling has been limited, since the interest in mathematical tractability has almost always led to an oversimplified model. The diffusion process approximation is an attempt to break away from the vogue in queueing theory. The present paper introduces a vector-valued normal process and its diffusion equation in order to obtain an approximate solution to the joint distribution of queue lengths in a general network of queues. In this model, queueing processes of various service stations which interact with each other are approximated by a vector-valued Wiener process with some appropriate boundary conditions. Some numerical examples are presented and compared with Monte Carlo simulation results. A companion paper, Part II, discusses transient solutions via the diffusion approximation.