The Merlin-Randell problem of train journeys

Abstract

The following synchronization problem is studied: “There is a finite set of trains, a finite set of stations, and the engine house. Each station can hold only one train. Each train has a route to follow starting and ending at the engine house. The route may be repeated one or more times. A train may leave a station (or the engine house) only when its immediate destination is an empty station or the engine house. At the beginning all trains are placed at the engine house. The problem is to find a synchronization among train movements which allows parallel movements of trains where possible and enables the completion of each train journey”. A formal model of the train system is proposed assuming that each train route is deterministic and contains no repeated stations. The model is based on the notion of a movement graph being a labeled directed multigraph, the nodes of which represent stations, the arcs of which represent possible train movements. The solution proposed is based on a characterization of inadmissible situations by means of some subgraphs of the movement graph. The properties of these subgraphs, called minimal critical patterns, are investigated. Optimal and suboptimal synchronization strategies are defined. A specification of the optimal strategy in the basic COSY notation is included, and its correctness is demonstrated.