In this paper, an overview of different analytic queueing models for traffic on road networks is presented. In the literature, it has been shown that queueing models can be used to adequately model uninterrupted traffic flows. This paper gives a broad review on this literature. Moreover, it is shown that the developed published methodologies (which are mainly single node oriented) can be extended towards queueing networks. First, an extension towards queueing networks with infinite buffer sizes is evaluated. Secondly, the assumption of infinite buffer sizes is dropped leading to queueing networks with finite buffer sizes. The impact of the buffer size when comparing the different queueing network methodologies is studied in detail. The paper ends with an analytical application tool to facilitate the optimal positioning of the counting points on a highway.

1. Introduction and Motivation

Congestion is a function of the number of vehicles on the road, showing the need for well-performing traffic models that capture this specific relationship. Traffic Hows are usually modelled empirically: speed and How data are collected for a Sj)(1CiHc road and econometrically fitted into curves, i.e.. the spoed-How-donsity diagrams (Daganzo, 1997). Alternatively, (mainly supported by the increasing computer performance), one may use simulation to model traffic Hows [e.g., leading to the wellknown car-following models, see Transportation Research Board (1996); Zhang and Kim (2005)]. Th(1Se approaches are however limited in terms of predictive power and sensitivity analysis. Moreover, these techniques are highly data-dependent and (computer) time-dependent and as such, not directly applicable in the decision process of various policy makers (Jain and MacGregor Smith, 1997).

As an alternative to these methodologies, analytical models based on queueing theory could be used to model traffic flows. This review paper intends to give an overview of the different efforts in the relevant traffic flow literature where queueing models are used. The contributions of this paper are twofold:

First, the methodology to model road networks using analytical queueing models is reviewed in detail. The current methodology is however mostly limited to single node analysis, i.e., single stage queueing models. As in practice traffic passes through a multitude of nodes, the extension towards network models is necessary. In this paper, it is proposed that a road network can be represented as a queueing network where vehicles spend time. This time spent is dependent upon the occupation of the road network, i.e., a high occupation or traffic intensity will lead to more time en route. Consequently, the performance indicators of the queueing networks will be used to determine the time on the road. Note that the term congestion will be used here in a strictly queueing theory sense meaning more than one customer in the system leading to traffic intensity strictly larger than zero. When considering getting stuck in traffic (stand still), the term traffic jam will be used.

Secondly, in the late 1990s, more and more vehicle detectors have been installed throughout the world to record the passing of vehicles (Newell, 2002; Ehlert et ai, 2005). Mostly, the decision concerning the location of the detector is arbitrarily (e.g., near an off-ramp or on-ramp). Based on the insights obtained from the literature on finite versus infinite queueing networks applied to traffic environments, a policy tool is developed to determine the optimal positions of the vehicle detectors on highways. The tool i)roi)osed in this paper determines, based on the expected traffic intensity, the optimal number and the best, locations for the different detectors to adequately monitor traffic.

This paper is organized as follows. First, in Sec. 2, a broad literature review on queueing models applied to traffic flows is presented. Based on the latter, an extension in the direction of queueing network analysis for traffic networks is presented. It is split up into two major paths depending upon the buffer size: nodes having an infiniti! buffer size (Sec. 3.1) or nodes having a finite buffer size (Sec. 3.2). The developed models (networks with infinite and finite buffer size) are compared with each other and differences are evaluated (Sec. 4). In Sec. 5. a tool (based on the elaborated queueing analysis) to determine the optimal places of the different counting points on the road is presented. Then, future research opportunities are discussed (Sec. 6). The last section concludes this review.

2. Literature Overview

In this paper, traffic How models based on queueing theory are considered. The following subsections explain the basic concepts in detail and give relevant references. The interested reader on the history of traffic flow theory in general, is referred to, e.g., Newell (2002) or Daganzo (1997).