Abstract
A mathematical model of the queuing system for the passenger flow of urban public transport is proposed. The resulting model differs from canonical models of queuing theory by taking into account the fundamental features of real systems. Firstly, the service process is divided into different successive service sessions. Secondly, arrival and departures are batch. Thirdly, the arrival rates vary in different service sessions. Fourthly, the laws of distribution of the number of jobs in batch arrivals for different sessions are different. Fifth, the laws of distribution of the number of batch arrivals and departures are also different.A criterion of efficiency of the service system is developed. The criterion is based on the calculation of the probability distribution of the service system states at the input and similar distribution at the output. These distributions are determined independently for each service session, into which the entire service cycle is divided. The numerical value of the criterion is set by the ratio of the average number of service rejections to the average number of jobs in the batch arrival for the entire service cycle. It can be used to assess the efficiency of the service system at any selected time interval during the day, because the value of the proposed criterion depends on the length of the interval between sessions, determined by the number of vehicles on the route.The resulting models adequately reflect the functioning of the system, which makes it possible to predict many different situations and evaluate the consequences of proposed solutions. Thus, it becomes possible to predict the provision of the population with public transport and determine quantitative values of efficiency of the urban public transport system
Highlights
Urban public transport is the industry that provides the city with services of delivering people on established routes, guaranteeing accessible, regular and mass traffic
We introduce the necessary notations: – ζ – random number of jobs in the batch arrival; – η – random number of jobs received by the system on free channels; – ν – random number of jobs rejected due to the business of all channels; – ξ – random number of jobs departing from the system upon completion of service; – j – number of channels engaged in service
A method for calculating the number of busy channels in case of batch arrival is developed. It is based on the description of the distribution of the number of busy channels after the batch arrival, taking into account the fact that the probability of business of e channels is equal to the sum of the products of the probabilities of j ≤ e channels business multiplied by the conditional probability of arrival of the number of jobs transferring the system in the state with e busy channels
Summary
Urban public transport is the industry that provides the city with services of delivering people on established routes, guaranteeing accessible, regular and mass traffic. Due to the fact that many citizens do not have personal vehicles, timely and adequate satisfaction of transport demands develops from a purely transport problem into a social one This problem determines the attitude of the population to the quality of transport services, and to the processes that occur in the regions and the country as a whole. Distinguishing between machine simulation and analytical models is sometimes difficult They allow describing the urban public transport system as an interconnected set of numerical models. These models adequately reflect the functioning of the system, which makes it possible to predict many different situations and evaluate the consequences of proposed solutions. The analysis of the organization of the urban public transport system is an urgent and important task
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