A closed network consists of two multi-servers with n customers. Service requirements of customers at a multi-server have a common cdf. State parameters of the network: for each multi-server empirical measure of the age of customers being serviced and for the queue the number of customers in it, all multiplied by 1/n. Our objective: asymptotics of dynamics as n→∞. The asymptotics of dynamics of a single multi-server and its queue with an arrival process as the number of servers n→∞is currently studied by famous scientists K. Ramanan, W. Whitt et al. In the last publications the arrival process is generalized to time-dependent. We develop our previous asymptotics results for a network also in this direction: instead of a simple time dependence a markov swithching behavior of one multi-server is introduced. For the asymptotic process we in a rough way find equilibrium and prove convergence as n→∞. Motivation for studying such models: they represent call/contact centers, and switching expresses the changes of the system environment.