Dobrushin mean-field approach for time-scaling queueing systems with a small parameter

In this paper we consider the dynamics of time-scaling queueing systems (TSQS) with an infinite number of servers. We assume that TSQS has a Poisson input flow of requests with constant intensity. Each request selects two any servers randomly. A share of the servers that have the queues lengths with not less than k can be described using an system of differential equations with a small parameter. This small parameter allows us to describe the processes of rapid changes of TSQS with time-scaling property. We use the Dobrushin mean-field approach and the simulation methods for this time-scaling queueing system analysis.