Modelling the Operation of Urban Passenger Transport with the Help of Spatial Matrices

This article addresses the problem of mathematical modelling of the operation of urban passenger transport, a problem that has great practical importance for both the design of urban transport networks and the organisation of transport services for the urban population. It demonstrates that a model of urban passenger trips will only be adequate for this task if it reflects the distribution of those trips over the twenty-four-hour time period, as well as their distribution between different parts of the city: in other words, that it will only be adequate if it is dynamic. It is thus impossible to apply here classical (plane) matrices accommodating only two dimensions of the system—in this case zones of departure (origins) and arrival (destinations). For modelling urban passenger movements we have therefore to move to spatial (three-dimensional) matrices, and the article describes such a model which has been built. A method is proposed for processing the spatial matrices by carrying out partitions of them. A new operation of the superposition (short multiplication) of plane and spatial matrices is introduced, which makes it possible to solve the problem of mathematical modelling of complex systems. A method is described whereby the dynamic model of urban passenger journeys has been used to optimise design solutions of urban transport networks and to improve the operation of urban passenger transport. Other possibilities for the use of spatial matrices in the modelling of complex urban systems are also indicated.