RUDN Mathematicians on a New Approach to Mass Service Theory

Mass service theory or queuing theory is a branch of applied probability theory. It solves the problem of evaluation and optimization of service quality for users of real systems, due to optimal allocation of limited resource between them. An important contribution to the development of this theory was made by RUDN scientists G.P. Basharin (1927-2018), P.P. Bocharov (1943-2006), their colleagues and students.

Scientists analyzing unilinear mass-service systems often assume that requests received from users are served one at a time or in groups of fixed size. Serving requests one at a time is often inadequate to the reality, since modern systems can technically handle an entire group. For example, to conduct sessions for broadcast users’ service requests of information and entertainment networks, received with a small-time gap. If we need to accumulate a group of users, we may face a long wait: until the system accumulates the necessary number of requests to perform further actions.

RUDN mathematicians Alexander Dudin and Sergey Dudin, together with scientists from the University of Salerno (Italy), studied a mathematical model of a single-line system:

• with repeated calls and simultaneous service of groups of variable size requests,
• with service users by means of preliminary, remotely performed, reservation of the required resource.

Analysis of the model was carried out with the help of previously developed in RUDN matrix-analytical methods, the study of multidimensional Markov chains with a special generator structure and their generalization in the case of spatially inhomogeneous chain behavior. Thereby scientists managed to avoid the imposition of assumptions about the Poisson stationary of the incoming flow of requests and the exponential distribution of service times, typical in the study of complex systems, and significantly simplify the mathematical analysis of systems.

The obtained results allow to determine the ability or inability of the system to successfully perform its functions, as well as to calculate the stationary probability distribution of the system states and the main characteristics of its performance. This helps to solve the problems of determining the lower and upper threshold values of the possible group size, providing the minimum possible time of stay of an arbitrary request in the system. The achievement of such minimum is possible. On the one hand, by maximizing the advantage of group, rather than individual, servicing of requests through a shorter service time (in terms of one request). On the other hand, by minimizing the negative effect of group service due to the possible long waiting time for an arbitrary arriving request to accumulate a large group.

Let us consider an example. If the service time of one request has an exponential distribution with the average value equal to 1, then the time of successive servicing a group of N requests will be equal to N, and the time of their group servicing (with offsetting by the last) will be 1+1/2+1/3+...+1/N. If N = 5, the service time for groups will be equal to 5 and 2.28. Accordingly, the group service in this example gives a gain in service time of more than twice. As the number of simultaneously served requests N increases, the gain continuously grows. Also, the gain is significantly higher when the dispersion of individual service times is smaller than for exponential distribution, e.g., degenerate distribution, Erlang distribution and uniform distribution on the small segment. In particular, with deterministic service times, the gain will be N times.

The results of the study allow us to solve the problem of efficient use of resources in various areas of human life. Based on them, it is possible to develop recommendations for working with transportation systems. For example, providing a vehicle with a smaller capacity to serve a group of users waiting for their bus to depart. Or we can talk about using a smaller system resource for remote service for training and video conferencing. The results may also be useful for the health care system, for example, in using fewer tests for the group testing procedure, but only if the size of the group served was greater than the minimum threshold but significantly less than the maximum threshold.

For more on the results of the study, see the article D’Arienzo M.P., Dudin A.N., Dudin S.A., Manzo R. Analysis of Retrial Queue With Group Service of Impatient Customers // Journal of Ambient Intelligence and Humanized Computing. 2020. V. 11. P. 2591-2599.