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

In this work we study large-scale queueing systems with with a small parameter using numerical analysis. We assume that there is a Poisson input flow of requests to large-scale queueing systems 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 k. We describe the processes of rapid changes of large-scale queueing systems and time scaling in this large-scale system using a small parameter. We apply the adaptive numerical methods for this large-scale queueing systems analysis using a piecewise-uniform grid. The results of the numerical analysis demonstrate the high efficiency of this numerical method.