Fluid Limit for Switching Closed Queueing Network with Two Multi-servers

Abstract

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 $n^{-1}$.

Our objective: asymptotics of dynamics as $n\rightarrow \infty$. The asymptotics of dynamics of a single multi-server and its queue with an arrival process as the number of servers $n\rightarrow \infty$ 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 $n\rightarrow \infty$.

Motivation for studying such models: they represent call/contact centers, and switching expresses the changes of the system environment.

S. Anulova—This work was partially supported by RFBR grants No. 16-08-01285 and No. 17-01- 00633.