RUDN University mathematicians proposed a model of a queue at passport control at the airport or in other similar systems. The new stochastic model was studied theoretically and its performance was calculated under different parameters. The results were published in Mathematics.

Newswise — From a mathematical point of view, a queue in a store and computational tasks in a computer are the same thing. Both, as well as dozens of similar examples, can be described using models of queuing theory. They contain one or more service devices (store cashier, processor, or other server) and applications (customers, requests, tasks, and so on), and also set laws for new applications receiving and processing. In classical models, it is assumed that these actions are independent of each other. However, in real problems, this is not always the case. For example, in busy airports or computer systems, it is necessary to manage the flow of new requests, directing them to the right service devices. RUDN University mathematicians created and studied a model in which the emergence of new requests and the process of servicing them are interconnected.

“Models with two queues are especially important from a practical point of view. They reflect those application scenarios in which several servers process incoming requests. Traditional queuing models assume that new tasks arrive and are processed independently. However in many practical situations, these processes are interrelated, and this significantly affects the performance and behavior of the system. The study of systems with interdependent arrival and service is necessary to understand how this mutual influence shapes the dynamics of queues, waiting time, and the overall efficiency of the system,” Dmitry Kozyrev, Ph.D., Associate Professor of the Probability Theory and Cyber Security Department of the RUDN University said.

Mathematicians considered a model with two different servers processing incoming requests. The emergence of new applications is interconnected with the processing of old ones. Both of these streams are based on what is called a Markov process. This is the name of a random process in which the future state does not depend on the past with a fixed present. RUDN University mathematicians studied the system theoretically - using the matrix-geometric method - and constructed examples of numerical solutions for given parameters.

RUDN University mathematicians have described several practical application scenarios, which, however, do not exhaust the entire range of possibilities. For example, large airports where, after the arrival of an international flight, passengers are sent to passport control. The constructed model can perform the functions of an airport employee, who usually divides the queue into branches depending on the number of open counters. The second example relates to the field of distributed computing and communication systems. In situations where data packets arrive at a server node with an intensity depending on the current load, the model can distribute requests across servers.

“We used a matrix-geometric method to analyze the model calculated performance metrics and presented numerical solutions. In the future, we can consider the case where service processes are executed in stages and can make several transitions back and forth. In addition, it would be interesting to study how the system develops with a larger number of servicing devices,”  Dmitry Kozyrev, Ph.D., Associate Professor of the Probability Theory and Cyber Security Department of the RUDN University said.

Journal Link: Mathematics 2023, 11(22), 4692