Numerical analysis of large-scale queueing system with a small parameter

In this work we study large-scale queueing systems (LSQS) with a small parameter using numerical analysis. We assume that there is a Poisson input flow of requests to LSQS with a limited intensity and there is a service discipline for any request which provides a randomly selection from any m-set servers such server that has the s-th shortest queue size. We consider Tikhonov problem for a system of differential equations with a small parameter. Solutions of Tikhonov problem are shares of the servers that have the queues lengths with not less than 1. We describe the processes of rapid changes of LSQS and time scaling in this LSQS using a small parameter. We apply the adaptive numerical methods for this LSQS analysis using a piecewise-uniform grid. The results of the numerical analysis demonstrate the high efficiency of this numerical method.