Numerical Analysis of Shortest Queue Problem for Time-Scale Queueing System with a Small Parameter

5G networks are a new technological step in the field of telecommunications. 5G 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 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 formulate the singularly perturbed Cauchy problem for this system of differential equations with a small parameter. We apply the truncation procedure for this singularly perturbed Cauchy problem and formulate the finite order system of differential equations. We use a high-order non-uniform grid scheme for numerical solving of the truncated Cauchy problem. We implement the numerical scheme with different sets which allows to evaluate the impact of a small parameter in time-scaling processes for TSQS. The grid scheme demonstrates good convergence of solutions of the singularly perturbed Cauchy problem when a small parameter tend to zero. The results of the numerical analysis demonstrate that this TSQS keep services with a high incoming flow of requests. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.