# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 2 Literature Review  
This section provides an overview of different capacity calculation methods, where first a summary of the state-of-the-art methods for determining the capacity of individual elements is given. Further, it is described which approaches have been pursued to calculate a network-wide capacity. The focus is on analytical methods and optimization approaches relying on methods from flow theory, while a complete overview of methods for capacity analysis in railways can be found in [7]. One widely used method to determine capacity consumption on lines is the UIC Code 406 compression method [1], which was developed with the aim of standardizing capacity assessment across different countries. It is applied to a specific timetable and based on the occupancy rate, which is obtained from the total occupancy derived via the compression of the single train rides' blocking time stairways in relation to the investigation period. To ensure a certain quality, the UIC defined permissible standard values for the occupancy rate depending on the type and usage of the line. For the assessment of node capacity, however, the method is not unrestrictedly suitable [2]. Analytical methods are often applied to determine capacities independently of an actual timetable. These methods mostly rely on the division of the infrastructure into nodes and lines and are based on queueing theory. In the case of nodes, the corresponding queueing system is solved for the scheduled waiting times occurring in the investigated infrastructure. For lines, in Germany, unscheduled waiting times are determined using the STRELE-formula [5; 6]. In both cases, the resulting waiting times serve as quality criteria for which a defined quality standard can be applied to relate to the optimal number of trains, i.e., the infrastructure element's capacity. Since the respective capacities for nodes and lines are not comparable and do not sufficiently consider interactions between the single elements, the question arises whether capacities can also be determined network-wide to be used as a basis for long-term infrastructure planning. There are already a few approaches for deriving such a capacity, selected ones of which are presented below. In [8], queueing theory is applied to calculate unscheduled waiting times in a network, for example. However, the calculations assume simplified station layouts and there is no possibility for trains to pass each other. In [9], the methods for lines and nodes described above are modified such that the resulting capacities are approximately comparable. Subsequently, the train rides in the network are iteratively optimized based on the calculated capacities for the single elements. Another approach that explicitly includes interactions between the individual infrastructure elements on a corridor is presented in [10]. Here, individual infrastructure sections are modeled by queueing systems, the throughput for a corridor is calculated, and the influence of travel time variability on it is examined. Due to the quickly increasing complexity of the problem, the approach is only suited for corridors, though, and cannot be applied to branched networks. Also, in [11], interactions are considered by investigating the influence of junctions on the capacity of lines by means of Markov models. Within the scope of the analysis, different operating strategies are mapped and compared with each other. This approach is extended in [12] to include the state of the infrastructure. In [13], the queueing approach is also extended to calculate capacity indicators for networks. Also, for the model developed in this approach, drawbacks consist in limited network sizes and a substantial generalization of the underlying infrastructure models, for example. In addition to analytical calculation methods, there are approaches that rely on the formulation of an optimization model to calculate a network-wide capacity. Here, the idea is that calculating a network capacity can be traced back to calculating a maximum flow in a restricted flow network.