The Numerical Analysis of the Time-Scale Shortest Queue Model Under the Dobrushin Mean-Field Approach

5G/6G networks are a next technological step in the field of telecommunications. 5G/6G networks provide the implementation of the required quality of communication with the growth of subscriber devices and lack of frequency bands. The application of queuing theory methods to analyze network performance is very important at the design, implementation and operation stages, as it is necessary to ensure a high return on investment that will be directed to the introduction of this new technology. Consequently, the attention of 5G/6G researchers is particularly focused on the analysis of the shortest queue problem which is widely used as balancing mechanisms in time-scale queueing system (TSQS). In this paper we employ simulation analysis of the TSQS evolution dynamics under the supposition that there are the large number of identical single-service devices and it is suppose this number increases indefinitely. It is assumed that all single-service devices have identical exponentially distributed service time with a finite mean value and a finite service intensity. It is supposed that there is a Poisson incoming stream of arriving requests with a finite intensity and TSQS fulfills a service discipline so that for each incoming request is provided a random selection a server device from random selected m-set server devices that has the s-th shortest queue size. The evolution of TSQS states can be represent by solutions of a system of differential equations of infinite degree. We investigate the Cauchy problem for this singularly perturbed system with a small parameter. We use the mean-field approach and formulate the Cauchy problem for the truncated singularly perturbed finite order system of differential equations and the initial condition problem for the singularly perturbed nonlinear first order partial differential equation with a small parameter. We construct an analytical solution of the initial condition problem for the singularly perturbed nonlinear first order partial differential equation and apply a high-order non-uniform grid scheme for numerical analysis of the solutions of the truncated singularly perturbed Cauchy problem. We use the numerical scheme with different sets which gives to evaluate the impact of a small parameter in time-scaling processes for TSQS. This grid scheme demonstrates good convergence of the solutions of the truncated singularly perturbed Cauchy problem when a small parameter tend to zero. The final outcome of our numerical simulations shows that this TSQS can support execution of the services with a high incoming flow of requests. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.