Point queue models: a unified approach

TL;DR

This paper introduces a unified framework for deriving and analyzing point queue models from link-based queueing models, enabling systematic study of queueing processes at point facilities and their networks.

Contribution

It presents a unified approach to derive, analyze, and extend point queue models from link-based models, clarifying their relationships and properties.

Findings

All existing point and fluid queue models are special cases of the derived models.

The paper provides analytical solutions for Vickrey's point queue model.

Demonstrates properties like equivalence, well-definedness, and spillback analytically and numerically.

Abstract

In transportation and other types of facilities, various queues arise when the demands of service are higher than the supplies, and many point and fluid queue models have been proposed to study such queueing systems. However, there has been no unified approach to deriving such models, analyzing their relationships and properties, and extending them for networks. In this paper, we derive point queue models as limits of two link-based queueing model: the link transmission model and a link queue model. With two definitions for demand and supply of a point queue, we present four point queue models, four approximate models, and their discrete versions. We discuss the properties of these models, including equivalence, well-definedness, smoothness, and queue spillback, both analytically and with numerical examples. We then analytically solve Vickrey's point queue model and stationary states in various models. We demonstrate that all existing point and fluid queue models in the literature are special cases of those derived from the link-based queueing models. Such a unified approach leads to systematic methods for studying the queueing process at a point facility and will also be helpful for studies on stochastic queues as well as networks of queues.