## 1 Introduction  
There is now a growing interest in the development of smart cities [1]. These cities will provide ubiquitous communication, intelligent transport systems (ITS), advanced digital health platforms and urban and infrastructure management. ITS will be brought about by the fusion of communication and transport infrastructures to create city-wide vehicular networks that form digital ring roads around the city and will be realised through the deployment of satellite navigation systems, vehicular networking and connected and autonomous vehicles. This should result in less congestion, shorter journey times and a significantly reduced number of road deaths. The emergence of vehicular ad hoc networks (VANETs) that are deployed using roadside units (RSUs) along the road infrastructure and onboard units (OBUs) in vehicles represents another significant change. Information about vehicles, including their location, speed and state, as well as the source and destination of their journeys, is provided in real-time, allowing new traffic management algorithms to be developed [2]. Such detailed information combined with analytical models will allow more accurate estimates of journey times [3; 4]. These systems will support current networking technologies including 5G and ITS-G5 and future technologies such as IEEE 802.11 bd and 6G. A VANET network is shown in Figure 1.  
Though we now have several experimental vehicular testbeds [6], in order to build an efficient ITS, it is necessary to develop better traffic management algorithms that will lead to reduced journey times and less traffic congestion. Thus, new techniques are required to analyse journey times in detail. This is the focus of this paper. In the solution approach, a journey is defined as the traversal of junctions and links. A technique called the zero-server Markov chain (ZSMC) is used to analyse delays at a junction. This is then combined with the Jackson network technique to analyse the delays at multiple junctions. The delay along road links is analysed using an M/M/K/K model. The results while traversing three junctions were compared with SUMO and VISSIM for the same scenario and showed an excellent comparison. Using this approach, [PERSON]'s algorithm was then used to analyse the different routes between a source and a destination to determine the best route [7]. In this paper, a junction is an intersection with four entrances and exits, as shown in Figure 2. Traffic lights are stationed at each entrance lane to the junction. This paper concentrates on a three-junction scenario. Figure 1: A VANET network [5]  
Figure 2: A standard junction. The scenario being considered is on a weekday in the Thurmaston region of the city of Leicester in the U.K. The setup is shown in Figure 3. There are three junctions and each junction has four exits. The distance between Junction 1 and Junction 2 (Link 1) is 2000 m, while the distance between Junction 2 and Junction 3 (Link 2) is 2500 m.  
The contributions of this paper can be summarised below:  
* A technique called the zero-server Markov chain (ZSMC) is used to analyse delays at the junction. * A new concept called lost service is introduced. * An M/M/K/K model is used to analyse delays along the links. * An analysis showing how the delay for a route containing three junctions and two links is demonstrated. * The results are validated using two simulators: SUMO and VISSIM. * An analysis of a real scenario is performed using [PERSON]'s algorithm and the new techniques. The rest of the paper is organized as follows: In Section 2, an overview of related work is detailed, while Section 3 explains the zero-server Markov chain technique. Section 4 examines the solution approach and highlights the analysis of delays at junctions. Section 5 develops the Markov chain to analyse the delay along road links, while in Section 6, an analysis of three junctions is presented. In Section 7, SUMO and VISSIM are used to evaluate the analysis, and Section 8 examines a real-life journey to determine the shortest route.