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.
Highlights
When the demands of service are higher than the supplies at such facilities as road networks, security check points in airports, supply chains, water reservoirs, document processors, task managers, and computer servers, there arise queues of vehicles, customers, commodities, water, documents, tasks, and programs, respectively
Link Transmission Model (LTM) and Link Queue Model (LQM) differ in the definitions of demand and supply functions and the corresponding state variables related to vehicle accumulations
In this study we presented a unified approach for point queue models, which can be derived as limits of two link-based queueing model: the link transmission and link queue models
Summary
When the demands of service are higher than the supplies at such facilities as road networks, security check points in airports, supply chains, water reservoirs, document processors, task managers, and computer servers, there arise queues of vehicles, customers, commodities, water, documents, tasks, and programs, respectively. Many queueing models have been proposed to understand the characteristics of such systems, including waiting times, queue lengths, and stationary states and their stability (Kleinrock, 1975; Newell, 1982) These models can be represented by arrival and service of individual customers or accumulation and dissipation of continuum fluid flows; they can be continuous or discrete in time; and the arrival and service patterns of queueing contents can be random or deterministic. Point queue models inherit two critical components from both LTM and LQM: we first define demand and supply of a point queue, and apply macroscopic junction models, which were originally proposed in CTM and later adopted for both LTM and LQM, to calculate boundary fluxes from upstream demands and downstream supplies This approach enables us to derive all existing point and fluid queue models and their generalizations, discuss their relationships, and analyze their properties.
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