# Mathematicians Found Optimal Parameters for Self-Service System

RUDN mathematician in collaboration with scientists from Belarus created a mathematical model of a self-service system. The results of the study can help calculating the optimal parameters to ensure maximum profit and users’ satisfaction — for example, customers in stores. The results are published in Mathematics.

Mathematical modeling techniques are used to improve the efficiency of customer service systems for different types of users. For example, customers in stores or requests in telecommunications systems.  To do this, they replace real concepts by abstract quantities. Processes are translated to mathematical terms and the resulting problem is investigated using mathematical tools. So for example, self-service counters in a store or airport turn into a system that can be evaluated and optimized using probability theory. How to make the system to work most efficiently and ensure customers’ satisfaction? RUDN University mathematician with colleagues from Belarus answered this question by building a model of a self-service system.

«Queuing theory is useful for modeling real-world systems, such as airports, banks, telecommunications and retail networks. Our queuing model corresponds to many real-world systems, such as call centers, catering services, and retail chains. It can make the system work as efficiently as possible. That is, to ensure maximum profit and a high degree of customer satisfaction,” said Alexander Dudin, Doctor of Physical and Mathematical Sciences, Head of the Research Center for Applied Probability Analysis of RUDN University

The model includes a group of servers, a potentially infinite queue, and assistant servers that help the main servers in case of a problem. The work of the system is described by the classical Markov arrival process. However RUDN University mathematicians made two novel features. First, they introduced the rating of a system. That is the value that determines the arrival of a new client. Secondly, service is provided via self-service devices. In real life, these can be, for example, self-checkout machines at the store.

Mathematicians tested their model on a numerical example. The researchers examined a system with a maximum of 50 main servers and 10 assistants. Mathematicians took the probability that the client will need an assistant as 25%, while the probability of raising and lowering the rating was 0.1% and 0.5%. Using the obtained theoretical calculations, mathematicians plotted the dependence of rating, profit, and other parameters on the number of servers. In the same way, one can perform calculations for any specific system.

«Our results can be extended to models with different mechanisms for calculating the rating value, more complex distributions of service time and assistants, unreliable servers and assistants, experienced and inexperienced customers, etc,” said Alexander Dudin, Doctor of Physical and Mathematical Sciences, Head of the Research Center for Applied Probability Analysis of RUDN University